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https://raweb.inria.fr/rapportsactivite/RA2014/regal/uid29.html
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Research Program
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
PDF e-Pub
## Section: New Results
### Management of distributed data
Participants : Pierpaolo Cincilla, Raluca Diaconu, Jonathan Lejeune, Mesaac Makpangou, Olivier Marin, Sébastien Monnet, Karine Pires, Dastagiri Reddy Malikireddy, Masoud Saeida Ardekani, Pierre Sens, Marc Shapiro, Véronique Simon, Julien Sopena, Vinh Tao Thanh, Serdar Tasiran, Marek Zawirski.
Storing and sharing information is one of the major reasons for the use of large-scale distributed computer systems. Replicating data at multiple locations ensures that the information persists despite the occurrence of faults, and improves application performance by bringing data close to its point of use, enabling parallel reads, and balancing load. This raises numerous issues:
• Where to store or replicate the data, in order to ensure that it is available quickly and remains persistent despite failures and disconnections.
• How many copies, located where, are needed to face dynamically-changing demand (load) and offer (elasticity).
• How to parallelize writes and hence how to ensure consistency between replicas.
• Tradeoffs between synchronised, consistent but slow updates, and fast but weakly-consistent ones.
• When and how to move data to computation, or computation to data, in order to improve response time while minimizing storage or energy usage.
• How to apply our approaches towards addressing the above issues onto a challenging use case: achieving true scalability for online games.
#### Long term durability
To tolerate failures, distributed storage systems replicate data. However, despite the replication, pieces of data may be lost (i.e. all the copies are lost). We have previously proposed a mechanism, RelaxDHT, to make distributed hash tables (DHT) resilient to high churn rates.
We have observed that a given system with a given replication mechanism can store a certain amount of data above which the loss rate would be greater than an “acceptable”/fixed threshold. This amount of data can be used as a metric to compare replication strategies. We have studied the impact of the data distribution layout upon the loss rate. The way the replication mechanism distribute the data copies among the nodes has a great impact. If node contents are very correlated, the number of available sources to heal a failure is low. On the opposite, if the data copies are shuffled/scattered among the nodes, many source nodes may be available to heal the system, and thus, the system losses less pieces of data. In order to study data durability on a long term, we have designed a model, and implemented a discrete event based simulator that can simulate a 100 node system over years within several hours. Our model, SPLAD [49] (for scattering and placing data replicas to enhance long-term durability), allows us to vary the data scattering degree by tuning a selection range width. We are also studying the impact of the policy used while choosing a storing node within the selection range (e.g., randomly, the least loaded, or smarter policies like the power of two choices). This policy has an important impact on both the storage load distribution among nodes and the number of lost pieces of data.
#### Achieving scalability for online games
Massively Multiplayer Online Games (MMOGs) such as World of Warcraft constitute a great use case for the management of distributed data on a large scale. Commercial support systems for MMOGs rely almost exclusively on traditional client/server architectures that are centralized. These architectures do not scale properly, both in terms of the number of players and of the number objects used to model virtual universes that grow ever more complex. Most MMOGs avoid this problem by limiting the scale of the universe: the virtual environment is partitioned into several parallel and totally disconnected worlds, such as the Realms in World of Warcraft. Each partition, handled in a centralized way, limits the number of players it can host; avatars created on different partitions will never meet in the game.
From a systems point of view, achieving true scalability raises many challenging issues for MMOGs. For instance the system must be very reactive: if the update latency on a player node is too high, the game becomes unplayable. Since these games are meant to operate on a large scale, they induce a trade-off between availability and consistency of data. The consistency aspect is critical because MMOGs incur a high degree of cheating.
Designing and implementing a scalable service for Multiplayer Online Games requires an extensive knowledge of the habits, behaviors and expectations of the players. The first part of our work on MMOGs aimed at gathering and analyzing traces of real games offers to gain insight on these matters. We collected public data from a League of Legends server (information over more than 56 million game sessions): the resulting database is freely available online, and an ensuing publication [34] details the analysis and conclusions we draw from this data regarding the expected requirements for a scalable MMOG service.
We steered a second part of our work on MMOGs in 2014 towards designing a peer to peer refereeing system that remains highly efficient, even on a large scale, both in terms of performance and in terms of cheat prevention. Simulations show that such a system scales easily to more than 30,000 nodes while leaving less than 0.013% occurrences of cheating undetected on a mean total of 24,819,649 refereeing queries. This work got published in the Multimedia Systems Journal [21] .
Finally, we also worked on the design of a scalable architecture for online games. The goal is to balance the load among nodes to allow the simulation of a whole, contiguous, virtual space.
#### Management of dynamic big data
Managing and processing Dynamic Big Data, where multiple sources produce new data continuously, is very complex. Static cluster- or grid-based solutions are prone to induce bottleneck problems, and are therefore ill-suited in this context. Our objective in this domain is to design and implement a Reliable Large Scale Distributed Framework for the Management and Processing of Dynamic Big Data. In 2014, we focused our research on data placement and on gathering traces from target applications in order to assess our future solutions.
With respect to application traces, we targeted sport tracker applications. Designing and implementing a big data service for sport tracker applications requires an extensive knowledge of both data distribution and input load. Gathering and analysing traces from a real world sports tracker service provides insight on these matters, but such services are very protective of their data due to competition as well as privacy issues. We avoided these issues by gathering public data from a popular sports tracker server called EndoMondo. The resulting database is freely available online, and allowed an in-depth analysis from a dynamic big data perspective. This study has lead to the publication of an Inria research report (RR- 8636) [47] .
#### Keyword-based Indexing and Search Substract for Structured P2P Information System
Number of large scale information systems rely on a DHT-based storage infrastructure. To help users to find suitable information, one attractive solution is to maintain an index that maps keywords to suitable data. Maintaining and exploiting an index distributed towards a DHT is confronted to the performance issue. Mainly, the computation of the intersection of postings related to provided keywords could generate too large traffic over the network; also one is confronted to some unbalanced on peers' load due to the fact that certain world are too popular!
In 2014, we propose FreeCore, a DHT-based distributed indexing substract that can be used to build efficient keyword-based search facilities for large scale information systems. A FreeCore index, considers keyword sets, then summarizes each set with a Bloom Filter. To limit the probability of false positive, we anticipate that one will use large size filters together enough hash functions. Thanks to this representation, we transform the searching problem, to the one of bitmaps matching as each query is also coded by a Bloom Filter. To distribute resulting summaries towards peers, FreeCore considers each summary as a sequence of binary keywords. Each binary keyword is assigned a peer and all summaries containing this binary keyword are stored at its assigned peer. Finally, to reduce the traffic overhead as well as the the size of local indices, FreeCore fragments each filter such as to factorize sequence of bits that occur more than once. In [40] , we report the performances of the initial implementation of FreeCore. Thought a number of improvements were not included within this initial evaluation, FreeCore offers better performances than existing state of the art. Current work focusses on developping applications that exploit FreeCore.
#### Large-Scale File Systems
Storage architectures for large enterprises are evolving towards a hybrid cloud model, mixing private storage (pure SSD solutions, virtualization-on-premise) with cloud-based service provider infrastructures. Users will be able to both share data through the common cloud space, and to retain replicas in local storage. In this context we need to design data structures suitable for storage, access, update and consistency of massive amounts of data at the object, block or file system level.
Current designs consider only data structures (e.g., trees or B+-Trees) that are strongly consistent and partition-tolerant (CP). However, this means that they are not available when there is a network problem, and that replicating a CP index across sites is painful. The traditional approaches include locking, journaling and replaying of logs, snapshots and Merkle trees. All of these are difficult to scale using generic approaches, although it is possible to scale them in some specific instances. For instance, synchronization in a single direction (the Active/Passive model) is relatively simple but very limited. A multi-master (Active/Active) model, where updates are allowed at multiple replicas and synchronization occurs in both directions, is difficult to achieve with the above techniques.
This work is part of a CIFRE agreeement with Scality (see Section 7.2.1 ).
#### Strong consistency
When data is updated somewhere on the network, it may become inconsistent with data elsewhere, especially in the presence of concurrent updates, network failures, and hardware or software crashes. A primitive such as consensus (or equivalently, total-order broadcast) synchronises all the network nodes, ensuring that they all observe the same updates in the same order, thus ensuring strong consistency. However the latency of consensus is very large in wide-area networks, directly impacting the response time of every update. Our contributions consist mainly of leveraging application-specific knowledge to decrease the amount of synchronisation.
When a database is very large, it pays off to replicate only a subset at any given node; this is known as partial replication. This allows non-overlapping transactions to proceed in parallel at different locations and decreases the overall network traffic. However, this makes it much harder to maintain consistency. We designed and implemented two genuine consensus protocols for partial replication, i.e., ones in which only relevant replicas participate in the commit of a transaction.
Another research direction leverages isolation levels, particularly Snapshot Isolation (SI), in order to parallelize non-conflicting transactions on databases. We prove a novel impossibility result: under standard assumptions (data store accesses are not known in advance, and transactions may access arbitrary objects in the data store), it is impossible to have both SI and GPR. Our impossibility result is based on a novel decomposition of SI which proves that, like serializability, SI is expressible on plain histories.
We designed an efficient protocol that maintains side-steps this impossibility but maintains the most important features of SI:
1. (Genuine Partial Replication) only replicas updated by a transaction $T$ make steps to execute $T$;
2. (Wait-Free Queries) a read-only transaction never waits for concurrent transactions and always commits;
3. (Minimal Commit Synchronization) two transactions synchronize with each other only if their writes conflict.
The protocol also ensures Forward Freshness, i.e., that a transaction may read object versions committed after it started.
Non-Monotonic Snapshot Isolation (NMSI) is the first strong consistency criterion to allow implementations with all four properties. We also present a practical implementation of NMSI called Jessy, which we compare experimentally against a number of well-known criteria. Our measurements show that the latency and throughput of NMSI are comparable to the weakest criterion, read-committed, and between two to fourteen times faster than well-known strong consistencies.
An interesting side-effect of this research is an apples-to-apples comparison of many strong-consistency protocols. This work was published at LADIS 2014 [41] and at Middleware 2014 [33] .
This research is supported in part by ConcoRDanT ANR project (Section 8.1.7 ) and by the FP7 grant SyncFree (Section 8.2.1.1 ).
#### Distributed Transaction Scheduling
Parallel transactions in distributed DBs incur high overhead for concurrency control and aborts. Our Gargamel system proposes an alternative approach by pre-serializing possibly conflicting transactions, and parallelizing non-conflicting update transactions to different replicas. This system provides strong transactional guarantees. In effect, Gargamel partitions the database dynamically according to the update workload. Each database replica runs sequentially, at full bandwidth; mutual synchronisation between replicas remains minimal. Both our simulations and the experimental results obtained with our prototype show that Gargamel improves both response time and load by an order of magnitude when contention is high (highly loaded system with bounded resources), and that otherwise slow-down is negligible.
We have studied Gargamel's behavior while running over multiple geographically distant sites. One instance of Gargamel runs on each site, synchronizations among the different sites occur off the critical path [39] . Our experiments with the Amazon platform show that or solution can be used to support failures of whole sites.
#### Eventual consistency
Eventual Consistency (EC) aims to minimize synchronisation, by weakening the consistency model. The idea is to allow updates at different nodes to proceed without any synchronisation, and to propagate the updates asynchronously, in the hope that replicas converge once all nodes have received all updates. EC was invented for mobile/disconnected computing, where communication is impossible (or prohibitively costly). EC also appears very appealing in large-scale computing environments such as P2P and cloud computing. However, its apparent simplicity is deceptive; in particular, the general EC model exposes tentative values, conflict resolution, and rollback to applications and users. Our research aims to better understand EC and to make it more accessible to developers.
We propose a new model, called Strong Eventual Consistency (SEC), which adds the guarantee that every update is durable and the application never observes a roll-back. SEC is ensured if all concurrent updates have a deterministic outcome. As a realization of SEC, we have also proposed the concept of a Conflict-free Replicated Data Type (CRDT). CRDTs represent a sweet spot in consistency design: they support concurrent updates, they ensure availability and fault tolerance, and they are scalable; yet they provide simple and understandable consistency guarantees.
This new model is suited to large-scale systems, such as P2P or cloud computing. For instance, we propose a “sequence” CRDT type called Treedoc that supports concurrent text editing at a large scale, e.g., for a wikipedia-style concurrent editing application. We designed a number of CRDTs such as counters (supporting concurrent increments and decrements), sets (adding and removing elements), graphs (adding and removing vertices and edges), and maps (adding, removing, and setting key-value pairs).
CRDTs are the main topic of the ConcoRDanT ANR project (Section 8.1.7 ) and the FP7 grant SyncFree (Section 8.2.1.1 ). After developing the SwiftCloud extreme-scale CRDT platform (see Section 5.3 ), we are currently developing a flexible cloud database called Antidote (see Section 5.4 ).
#### Lower bounds and optimality of CRDTs
CRDTs raise challenging research issues: What is the power of CRDTs? Are the sufficient conditions necessary? How to engineer interesting data types to be CRDTs? How to garbage collect obsolete state without synchronisation, and without violating the monotonic semi-lattice requirement? What are the upper and lower bounds of CRDTs?
We co-authored an innovative approach to these questions, published at Principles of Programming Languages (POPL) 2014 [25] . Geographically distributed systems often rely on replicated eventually consistent data stores to achieve availability and performance. To resolve conflicting updates at different replicas, researchers and practitioners have proposed specialized consistency protocols, called replicated data types, that implement objects such as registers, counters, sets or lists. Reasoning about replicated data types has however not been on par with comparable work on abstract data types and concurrent data types, lacking specifications, correctness proofs, and optimality results. To fill in this gap, we propose a framework for specifying replicated data types using relations over events and verifying their implementations using replication-aware simulations. We apply it to seven existing implementations of 4 data types with nontrivial conflictresolution strategies and optimizations (last-writer-wins register, counter, multi-value register and observed-remove set). We also present a novel technique for obtaining lower bounds on the worst-case space overhead of data type implementations and use it to prove optimality of four implementations. Finally, we show how to specify consistency of replicated stores with multiple objects axiomatically, in analogy to prior work on weak memory models. Overall, our work provides foundational reasoning tools to support research on replicated eventually consistent stores.
#### Explicit Consistency: Strengthening Eventual Consistency to support application invariants
The designers of the replication protocols for geo-replicated storage systems have to choose between either supporting low latency, eventually consistent operations, or supporting strong consistency for ensuring application correctness. We propose an alternative consistency model, explicit consistency, that strengthens eventual consistency with a guarantee to preserve specific invariants defined by the applications. Given these application-specific invariants, a system that supports explicit consistency must identify which operations are unsafe under concurrent execution, and help programmers to select either violation-avoidance or invariant-repair techniques. We show how to achieve the former while allowing most of operations to complete locally, by relying on a reservation system that moves replica coordination off the critical path of operation execution. The latter, in turn, allow operations to execute without restriction, and restore invariants by applying a repair operation to the database state. We designed and evaluated Indigo, a middleware that provides Explicit Consistency on top of a causally-consistent data store. Indigo guarantees strong application invariants while providing latency similar to an eventually consistent system.
This work was presented at W-PSDS 2014 [24] and LADIS 2014 [38] . It was selected for presentation at EuroSys 2015 [23] . This research is supported in part by the FP7 grant SyncFree (Section 8.2.1.1 ).
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2021-05-16 09:58:00
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https://www.e-olymp.com/en/problems/924
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favorite We need a little bit of your help to keep things running, click on this banner to learn more
Problems
# Ring
The area of a ring and radius of an external circle are given. Find the radius of an internal circle.
#### Input
Two real numbers are given: the area of a ring and the radius of an external circle, which does not exceed 100.
#### Output
Print the radius of an internal circle with 2 digits after the decimal point.
Time limit 1 seconds
Memory limit 122.17 MiB
Input example #1
50.2655 5
Output example #1
3.00
Source SFE-2010 Variant 24
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2020-11-27 18:02:07
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https://gitlab.cba.mit.edu/erik/lufa/-/blame/ed9d77aeee3a192852ca2419b7cfa58d39073036/LUFA/Drivers/USB/Class/Host/MIDI.h
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MIDI.h 10.1 KB
Dean Camera committed May 08, 2010 1 2 3 /* LUFA Library Copyright (C) Dean Camera, 2010. 4 Dean Camera committed May 08, 2010 5 dean [at] fourwalledcubicle [dot] com Dean Camera committed Oct 28, 2010 6 www.lufa-lib.org Dean Camera committed May 08, 2010 7 8 9 10 11 */ /* Copyright 2010 Dean Camera (dean [at] fourwalledcubicle [dot] com) 12 Permission to use, copy, modify, distribute, and sell this Dean Camera committed May 08, 2010 13 software and its documentation for any purpose is hereby granted 14 without fee, provided that the above copyright notice appear in Dean Camera committed May 08, 2010 15 all copies and that both that the copyright notice and this 16 17 18 permission notice and warranty disclaimer appear in supporting documentation, and that the name of the author not be used in advertising or publicity pertaining to distribution of the Dean Camera committed May 08, 2010 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 software without specific, written prior permission. The author disclaim all warranties with regard to this software, including all implied warranties of merchantability and fitness. In no event shall the author be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of this software. */ /** \file * \brief Host mode driver for the library USB MIDI Class driver. * * Host mode driver for the library USB MIDI Class driver. * Dean Camera committed Oct 24, 2010 36 37 * \note This file should not be included directly. It is automatically included as needed by the USB module driver * dispatch header located in LUFA/Drivers/USB.h. Dean Camera committed May 08, 2010 38 39 40 41 42 43 44 */ /** \ingroup Group_USBClassMIDI * @defgroup Group_USBClassMIDIHost MIDI Class Host Mode Driver * * \section Sec_Dependencies Module Source Dependencies * The following files must be built with any user project that uses this module: Dean Camera committed Jul 19, 2010 45 * - LUFA/Drivers/USB/Class/Host/MIDI.c (Makefile source module name: LUFA_SRC_USBCLASS) Dean Camera committed May 08, 2010 46 47 48 49 50 51 52 53 54 55 56 57 58 * * \section Module Description * Host Mode USB Class driver framework interface, for the MIDI USB Class driver. * * @{ */ #ifndef __MIDI_CLASS_HOST_H__ #define __MIDI_CLASS_HOST_H__ /* Includes: */ #include "../../USB.h" #include "../Common/MIDI.h" 59 Dean Camera committed May 08, 2010 60 61 62 63 64 65 66 /* Enable C linkage for C++ Compilers: */ #if defined(__cplusplus) extern "C" { #endif /* Preprocessor Checks: */ #if !defined(__INCLUDE_FROM_MIDI_DRIVER) Dean Camera committed Oct 24, 2010 67 68 69 70 71 #error Do not include this file directly. Include LUFA/Drivers/USB.h instead. #endif #if defined(__INCLUDE_FROM_MIDI_HOST_C) && defined(NO_STREAM_CALLBACKS) #error The NO_STREAM_CALLBACKS compile time option cannot be used in projects using the library Class drivers. Dean Camera committed May 08, 2010 72 #endif 73 Dean Camera committed May 08, 2010 74 75 76 77 78 79 80 81 82 83 84 85 /* Public Interface - May be used in end-application: */ /* Type Defines: */ /** \brief MIDI Class Host Mode Configuration and State Structure. * * Class state structure. An instance of this structure should be made within the user application, * and passed to each of the MIDI class driver functions as the MIDIInterfaceInfo parameter. This * stores each MIDI interface's configuration and state information. */ typedef struct { const struct { Dean Camera committed Jun 17, 2010 86 87 uint8_t DataINPipeNumber; /**< Pipe number of the MIDI interface's streaming IN data pipe. */ bool DataINPipeDoubleBank; /**< Indicates if the MIDI interface's IN data pipe should use double banking. */ 88 Dean Camera committed Jun 17, 2010 89 90 uint8_t DataOUTPipeNumber; /**< Pipe number of the MIDI interface's streaming OUT data pipe. */ bool DataOUTPipeDoubleBank; /**< Indicates if the MIDI interface's OUT data pipe should use double banking. */ Dean Camera committed May 08, 2010 91 92 93 94 95 } Config; /**< Config data for the USB class interface within the device. All elements in this section * must be set or the interface will fail to enumerate and operate correctly. */ struct { Dean Camera committed Sep 30, 2010 96 97 98 99 100 bool IsActive; /**< Indicates if the current interface instance is connected to an attached device, valid * after \ref MIDI_Host_ConfigurePipes() is called and the Host state machine is in the * Configured state. */ uint8_t InterfaceNumber; /**< Interface index of the MIDI interface within the attached device. */ Dean Camera committed May 08, 2010 101 Dean Camera committed Jun 17, 2010 102 103 uint16_t DataINPipeSize; /**< Size in bytes of the MIDI Streaming Data interface's IN data pipe. */ uint16_t DataOUTPipeSize; /**< Size in bytes of the MIDI Streaming Data interface's OUT data pipe. */ Dean Camera committed May 08, 2010 104 105 106 107 108 } State; /**< State data for the USB class interface within the device. All elements in this section * may be set to initial values, but may also be ignored to default to sane values when * the interface is enumerated. */ } USB_ClassInfo_MIDI_Host_t; 109 Dean Camera committed May 08, 2010 110 111 /* Enums: */ /** Enum for the possible error codes returned by the \ref MIDI_Host_ConfigurePipes() function. */ Dean Camera committed Sep 28, 2010 112 enum MIDI_Host_EnumerationFailure_ErrorCodes_t Dean Camera committed May 08, 2010 113 { Dean Camera committed Jun 17, 2010 114 115 MIDI_ENUMERROR_NoError = 0, /**< Configuration Descriptor was processed successfully. */ MIDI_ENUMERROR_InvalidConfigDescriptor = 1, /**< The device returned an invalid Configuration Descriptor. */ Dean Camera committed Sep 30, 2010 116 MIDI_ENUMERROR_NoCompatibleInterfaceFound = 2, /**< A compatible MIDI interface was not found in the device's Configuration Descriptor. */ Dean Camera committed Dec 02, 2010 117 MIDI_ENUMERROR_PipeConfigurationFailed = 3, /**< One or more pipes for the specified interface could not be configured correctly. */ Dean Camera committed May 08, 2010 118 }; 119 Dean Camera committed May 08, 2010 120 121 122 123 124 125 126 /* Function Prototypes: */ /** Host interface configuration routine, to configure a given MIDI host interface instance using the Configuration * Descriptor read from an attached USB device. This function automatically updates the given MIDI Host instance's * state values and configures the pipes required to communicate with the interface if it is found within the device. * This should be called once after the stack has enumerated the attached device, while the host state machine is in * the Addressed state. * Dean Camera committed Sep 30, 2010 127 128 129 130 * \note The pipe index numbers as given in the interface's configuration structure must not overlap with any other * interface, or pipe bank corruption will occur. Gaps in the allocated pipe numbers or non-sequential indexes * within a single interface is allowed, but no two interfaces of any type have have interleaved pipe indexes. * Dean Camera committed Jun 17, 2010 131 132 133 * \param[in,out] MIDIInterfaceInfo Pointer to a structure containing an MIDI Class host configuration and state. * \param[in] ConfigDescriptorSize Length of the attached device's Configuration Descriptor. * \param[in] DeviceConfigDescriptor Pointer to a buffer containing the attached device's Configuration Descriptor. Dean Camera committed May 08, 2010 134 * Dean Camera committed Sep 28, 2010 135 * \return A value from the \ref MIDI_Host_EnumerationFailure_ErrorCodes_t enum. Dean Camera committed May 08, 2010 136 */ Dean Camera committed Jul 21, 2010 137 138 uint8_t MIDI_Host_ConfigurePipes(USB_ClassInfo_MIDI_Host_t* const MIDIInterfaceInfo, uint16_t ConfigDescriptorSize, Dean Camera committed May 08, 2010 139 140 void* DeviceConfigDescriptor) ATTR_NON_NULL_PTR_ARG(1) ATTR_NON_NULL_PTR_ARG(3); Dean Camera committed Oct 27, 2010 141 142 143 144 145 146 147 /** General management task for a given MIDI host class interface, required for the correct operation of the interface. This should * be called frequently in the main program loop, before the master USB management task \ref USB_USBTask(). * * \param[in,out] MIDIInterfaceInfo Pointer to a structure containing an MIDI Class host configuration and state. */ void MIDI_Host_USBTask(USB_ClassInfo_MIDI_Host_t* const MIDIInterfaceInfo) ATTR_NON_NULL_PTR_ARG(1); Dean Camera committed May 08, 2010 148 149 /** Sends a MIDI event packet to the device. If no device is connected, the event packet is discarded. * Dean Camera committed Oct 12, 2010 150 * \pre This function must only be called when the Host state machine is in the \ref HOST_STATE_Configured state or the Dean Camera committed Jun 15, 2010 151 * call will fail. Dean Camera committed May 08, 2010 152 * Dean Camera committed Jun 17, 2010 153 154 * \param[in,out] MIDIInterfaceInfo Pointer to a structure containing a MIDI Class configuration and state. * \param[in] Event Pointer to a populated USB_MIDI_EventPacket_t structure containing the MIDI event to send. Dean Camera committed May 08, 2010 155 * Dean Camera committed Jun 17, 2010 156 * \return A value from the \ref Pipe_Stream_RW_ErrorCodes_t enum. Dean Camera committed May 08, 2010 157 158 159 160 161 */ uint8_t MIDI_Host_SendEventPacket(USB_ClassInfo_MIDI_Host_t* const MIDIInterfaceInfo, MIDI_EventPacket_t* const Event) ATTR_NON_NULL_PTR_ARG(1) ATTR_NON_NULL_PTR_ARG(2); /** Flushes the MIDI send buffer, sending any queued MIDI events to the device. This should be called to override the Dean Camera committed Jul 30, 2010 162 * \ref MIDI_Host_SendEventPacket() function's packing behaviour, to flush queued events. Events are queued into the Dean Camera committed May 08, 2010 163 164 165 * pipe bank until either the pipe bank is full, or \ref MIDI_Host_Flush() is called. This allows for multiple MIDI * events to be packed into a single pipe packet, increasing data throughput. * Dean Camera committed Jun 17, 2010 166 * \param[in,out] MIDIInterfaceInfo Pointer to a structure containing a MIDI Class configuration and state. Dean Camera committed May 08, 2010 167 * Dean Camera committed Jun 17, 2010 168 * \return A value from the \ref Pipe_WaitUntilReady_ErrorCodes_t enum. Dean Camera committed May 08, 2010 169 */ 170 uint8_t MIDI_Host_Flush(USB_ClassInfo_MIDI_Host_t* const MIDIInterfaceInfo) ATTR_NON_NULL_PTR_ARG(1); 171 Dean Camera committed May 08, 2010 172 173 /** Receives a MIDI event packet from the device. * Dean Camera committed Oct 12, 2010 174 * \pre This function must only be called when the Host state machine is in the \ref HOST_STATE_Configured state or the Dean Camera committed Jun 15, 2010 175 * call will fail. Dean Camera committed May 08, 2010 176 * Dean Camera committed Jun 17, 2010 177 178 * \param[in,out] MIDIInterfaceInfo Pointer to a structure containing a MIDI Class configuration and state. * \param[out] Event Pointer to a USB_MIDI_EventPacket_t structure where the received MIDI event is to be placed. Dean Camera committed May 08, 2010 179 * Dean Camera committed Jun 17, 2010 180 * \return Boolean true if a MIDI event packet was received, false otherwise. Dean Camera committed May 08, 2010 181 182 183 184 185 186 187 */ bool MIDI_Host_ReceiveEventPacket(USB_ClassInfo_MIDI_Host_t* const MIDIInterfaceInfo, MIDI_EventPacket_t* const Event) ATTR_NON_NULL_PTR_ARG(1) ATTR_NON_NULL_PTR_ARG(2); /* Private Interface - For use in library only: */ #if !defined(__DOXYGEN__) /* Function Prototypes: */ Dean Camera committed Oct 24, 2010 188 #if defined(__INCLUDE_FROM_MIDI_HOST_C) Dean Camera committed May 09, 2010 189 190 static uint8_t DCOMP_MIDI_Host_NextMIDIStreamingInterface(void* const CurrentDescriptor) ATTR_NON_NULL_PTR_ARG(1); static uint8_t DCOMP_MIDI_Host_NextMIDIStreamingDataEndpoint(void* const CurrentDescriptor) ATTR_NON_NULL_PTR_ARG(1); 191 #endif Dean Camera committed May 08, 2010 192 #endif 193 Dean Camera committed May 08, 2010 194 195 196 197 198 199 200 201 /* Disable C linkage for C++ Compilers: */ #if defined(__cplusplus) } #endif #endif /** @} */ 202
|
2023-02-08 13:54:14
|
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|
http://phxmarker.blogspot.com/2017/11/the-oracle-precision-physics-constant.html
|
## Wednesday, November 8, 2017
### The Oracle Precision Physics Constant Generator
$$\pi={{\alpha^2 m_e}\over{r_pR_Hm_p}}$$
$\alpha=fine\;structure\;constant$
$m_e=mass\;of\;electron$
$r_p=2010\;and\;2013\;muonic\;hydrogen\;proton\;radius\;(Haramein's\;Equation)$
$R_H=Rydberg\;constant$
$m_p=mass\;of\;proton$
The precision of pi and the equation above for $\pi$ may(?) be used to increase the digits of resolution of these 5 fundamental physics constants.
Will be posting results as this investigation proceeds...
It'll be like a table with before and after "The Oracle says:" comparison.
The Surfer, OM-IV
|
2018-09-21 00:54:03
|
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|
https://www.mathlearnit.com/what-is-73-149-as-a-decimal
|
# What is 73/149 as a decimal?
## Solution and how to convert 73 / 149 into a decimal
73 / 149 = 0.49
Converting 73/149 to 0.49 starts with defining whether or not the number should be represented by a fraction, decimal, or even a percentage. Fractions and decimals represent parts of a whole, sometimes representing numbers less than 1. Choosing which to use starts with the real life scenario. Fractions are clearer representation of objects (half of a cake, 1/3 of our time) while decimals represent comparison numbers a better (.333 batting average, pricing: \$1.50 USD). So let’s dive into how and why you can convert 73/149 into a decimal.
## 73/149 is 73 divided by 149
The first step of teaching our students how to convert to and from decimals and fractions is understanding what the fraction is telling is. 73 is being divided into 149. Think of this as our directions and now we just need to be able to assemble the project! The numerator is the top number in a fraction. The denominator is the bottom number. This is our equation! We use this as our equation: numerator(73) / denominator (149) to determine how many whole numbers we have. Then we will continue this process until the number is fully represented as a decimal. Here's how our equation is set up:
### Numerator: 73
• Numerators are the top number of the fraction which represent the parts of the equation. Overall, 73 is a big number which means you'll have a significant number of parts to your equation. 73 is an odd number so it might be harder to convert without a calculator. Large two-digit conversions are tough. Especially without a calculator. Let's look at the fraction's denominator 149.
### Denominator: 149
• Denominators are located at the bottom of the fraction, representing the total number of parts. 149 is a large number which means you should probably use a calculator. But 149 is an odd number. Having an odd denominator like 149 could sometimes be more difficult. Ultimately, don't be afraid of double-digit denominators. Next, let's go over how to convert a 73/149 to 0.49.
## Converting 73/149 to 0.49
### Step 1: Set your long division bracket: denominator / numerator
$$\require{enclose} 149 \enclose{longdiv}{ 73 }$$
To solve, we will use left-to-right long division. This is the same method we all learned in school when dividing any number against itself and we will use the same process for number conversion as well.
### Step 2: Extend your division problem
$$\require{enclose} 00. \\ 149 \enclose{longdiv}{ 73.0 }$$
We've hit our first challenge. 73 cannot be divided into 149! So we will have to extend our division problem. Add a decimal point to 73, your numerator, and add an additional zero. This doesn't add any issues to our denominator but now we can divide 149 into 730.
### Step 3: Solve for how many whole groups you can divide 149 into 730
$$\require{enclose} 00.4 \\ 149 \enclose{longdiv}{ 73.0 }$$
How many whole groups of 149 can you pull from 730? 596 Multiply this number by 149, the denominator to get the first part of your answer!
### Step 4: Subtract the remainder
$$\require{enclose} 00.4 \\ 149 \enclose{longdiv}{ 73.0 } \\ \underline{ 596 \phantom{00} } \\ 134 \phantom{0}$$
If there is no remainder, you’re done! If you still have numbers left over, continue to the next step.
### Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.
Sometimes you won't reach a remainder of zero. Rounding to the nearest digit is perfectly acceptable.
### Why should you convert between fractions, decimals, and percentages?
Converting between fractions and decimals is a necessity. Remember, fractions and decimals are both representations of whole numbers to determine more specific parts of a number. And the same is true for percentages. It’s common for students to hate learning about decimals and fractions because it is tedious. But they all represent how numbers show us value in the real world. Here are examples of when we should use each.
### When you should convert 73/149 into a decimal
Contracts - Almost all contracts leverage decimal format. If a worker is logging hours, they will log 1.48 hours, not 1 and 73/149 hours. Percentage format is also used in contracts as well.
### When to convert 0.49 to 73/149 as a fraction
Distance - Any type of travel, running, walking will leverage fractions. Distance is usually measured by the quarter mile and car travel is usually spoken the same.
### Practice Decimal Conversion with your Classroom
• If 73/149 = 0.49 what would it be as a percentage?
• What is 1 + 73/149 in decimal form?
• What is 1 - 73/149 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.49 + 1/2?
|
2023-03-30 14:02:25
|
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|
http://www.numdam.org/item/ALCO_2021__4_1_175_0/
|
Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 175-188.
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill–Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner–Tenner–Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.144
Classification: 05E05, 05E14
Keywords: Schur functions, Grothendieck polynomials, insertion algorithms, set-valued tableaux, Brill–Noether theory.
Chan, Melody 1; Pflueger, Nathan 2
1 Brown University Department of Mathematics Box 1917 Providence RI 02912, USA
2 Amherst College Department of Mathematics and Statistics Amherst MA 01002, USA
@article{ALCO_2021__4_1_175_0,
author = {Chan, Melody and Pflueger, Nathan},
title = {Combinatorial relations on skew {Schur} and skew stable {Grothendieck} polynomials},
journal = {Algebraic Combinatorics},
pages = {175--188},
publisher = {MathOA foundation},
volume = {4},
number = {1},
year = {2021},
doi = {10.5802/alco.144},
language = {en},
url = {http://www.numdam.org/articles/10.5802/alco.144/}
}
TY - JOUR
AU - Chan, Melody
AU - Pflueger, Nathan
TI - Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
JO - Algebraic Combinatorics
PY - 2021
SP - 175
EP - 188
VL - 4
IS - 1
PB - MathOA foundation
UR - http://www.numdam.org/articles/10.5802/alco.144/
UR - https://doi.org/10.5802/alco.144
DO - 10.5802/alco.144
LA - en
ID - ALCO_2021__4_1_175_0
ER -
%0 Journal Article
%A Chan, Melody
%A Pflueger, Nathan
%T Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
%J Algebraic Combinatorics
%D 2021
%P 175-188
%V 4
%N 1
%I MathOA foundation
%U https://doi.org/10.5802/alco.144
%R 10.5802/alco.144
%G en
%F ALCO_2021__4_1_175_0
Chan, Melody; Pflueger, Nathan. Combinatorial relations on skew Schur and skew stable Grothendieck polynomials. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 175-188. doi : 10.5802/alco.144. http://www.numdam.org/articles/10.5802/alco.144/
[1] Anderson, Dave; Chen, Linda; Tarasca, Nicola K-classes of Brill–Noether loci and a determinantal formula (2017) (https://arxiv.org/abs/1705.02992)
[2] Assaf, Sami H.; McNamara, Peter R. W. A Pieri rule for skew shapes, J. Combin. Theory Ser. A, Volume 118 (2011) no. 1, pp. 277-290 | DOI | MR | Zbl
[3] Bandlow, Jason; Morse, Jennifer Combinatorial expansions in $K$-theoretic bases, Electron. J. Combin., Volume 19 (2012) no. 4, 39, 27 pages | MR | Zbl
[4] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | DOI | MR | Zbl
[5] Buch, Anders Skovsted A Littlewood–Richardson rule for the $K$-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | DOI | MR | Zbl
[6] Buch, Anders Skovsted; Kresch, Andrew; Shimozono, Mark; Tamvakis, Harry; Yong, Alexander Stable Grothendieck polynomials and $K$-theoretic factor sequences, Math. Ann., Volume 340 (2008) no. 2, pp. 359-382 | DOI | MR | Zbl
[7] Chan, Melody; López Martín, Alberto; Pflueger, Nathan; Teixidor i Bigas, Montserrat Genera of Brill–Noether curves and staircase paths in Young tableaux, Trans. Amer. Math. Soc., Volume 370 (2018) no. 5, pp. 3405-3439 | DOI | MR | Zbl
[8] Chan, Melody; Pflueger, Nathan Euler characteristics of Brill–Noether varieties (to appear in Transactions of the AMS) | DOI
[9] Chan, Melody; Pflueger, Nathan Relative Richardson varieties (https://arxiv.org/abs/1909.12414)
[10] Fomin, Sergey; Greene, Curtis Noncommutative Schur functions and their applications, Discrete Math., Volume 193 (1998) no. 1-3, pp. 179-200 Selected papers in honor of Adriano Garsia (Taormina, 1994) | DOI | MR | Zbl
[11] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang-Baxter equation, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, sd, pp. 183-189 | MR
[12] Galashin, Pavel; Grinberg, Darij; Liu, Gaku Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions, Electron. J. Combin., Volume 23 (2016) no. 3, 3.14, 28 pages | DOI | MR | Zbl
[13] Lascoux, Alain; Schützenberger, Marcel-Paul Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 | MR | Zbl
[14] Lenart, Cristian Combinatorial aspects of the $K$-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82 | DOI | MR | Zbl
[15] Reiner, Victor; Tenner, Bridget Eileen; Yong, Alexander Poset edge densities, nearly reduced words, and barely set-valued tableaux, J. Combin. Theory Ser. A, Volume 158 (2018), pp. 66-125 | DOI | MR | Zbl
[16] Sagan, Bruce E.; Stanley, Richard P. Robinson–Schensted algorithms for skew tableaux, J. Combin. Theory Ser. A, Volume 55 (1990) no. 2, pp. 161-193 | DOI | MR | Zbl
[17] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | DOI | MR | Zbl
Cited by Sources:
|
2023-03-20 10:41:52
|
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|
https://www.lmfdb.org/knowledge/show/rcs.cande.ec.q
|
show · rcs.cande.ec.q all knowls · up · search:
The database of elliptic curves over $\Q$ consists of the following sets of curves:
• all curves of conductor less than $500,000$;
• all curves whose conductor is $7$-smooth (that is, only divisible by primes $p\le 7$;
• all curves of prime conducto r$p \le 200,000,000$.
Currently, the database includes $3,824,372$ elliptic curves over $\Q$ in $2,917,287$ isogeny classes, with conductor at most $299,996,953$.
Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Sutherland on 2021-04-17 09:04:22
Referred to by:
History:
|
2021-05-17 22:25:25
|
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|
http://www.oofem.org/resources/doc/oofemrefman/stressstrainprincmode_8h.html
|
OOFEM 2.3
stressstrainprincmode.h File Reference
Go to the source code of this file.
## Namespaces
oofem
This class contains a Neumann type boundary condition given as
where is a prescribed constant (eg pressure), is the gradient (pressure gradient), is the coordinate, is the centre of the structure and $n$ is the outward pointing normal.
## Enumerations
enum oofem::stressStrainPrincMode { oofem::principal_strain, oofem::principal_stress, oofem::principal_deviatoricstress }
We have only one algorithm for computing eigenvalues and vectors in order to be able to distinguish between some different modes we define this new type. More...
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2016-05-01 15:28:26
|
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https://tex.stackexchange.com/questions/125620/autocite-mcite-functionality-in-biblatex
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# autocite + mcite functionality in Biblatex
When using Biblatex's autocite command, I sometimes come across instances when I need to format the citations as a reference set. There is the option of using mcite, but then I lose the high level markup that autocite offers, .e.g. switching between plain, inline and superscript. If I use autocite or autocites, each reference would get it's own citation in the bibliography. Is there a solution that offers something like what I call \mautocite in this example?
\usepackage[backend=biber,mcite,subentry,autocite=plain]{biblatex}
...
\begin{document}
Some text here\autocite{key1}. More text here\mautocite{set1,*key2,*key3,*key4}.
Cite a subentry\autocite{key3}.
\end{document
In the text, it would appear as
Some text here [1]. More text here [2].
Cite a subentry [2b].
And this would give two entries in the bibliography, where the second reference, [2], would be a set. Is there a way to get biblatex to do this?
The mcite-like commands are defined in the biblatex module blx-mcite.def. There you will find a citation command modifier \mcitelike, used as follows.
\newrobustcmd*{<new mcite command>}{\mcitelike<existing citation command>}
An mcite-like \autocite is somewhat odd because \autocite is intended to be style-independent, but mcite-like commands are relevant only to numeric citation styles. That said you can define an mcite-variant of \autocite with:
\newrobustcmd*{\mautocite}{\mcitelike\autocite}
Here's an example.
\documentclass{article}
\usepackage[backend=biber,style=numeric,mcite,subentry,autocite=superscript]{biblatex}
% just for demo
\ExecuteBibliographyOptions{firstinits,sorting=none}
% define mcite-like variant of \autocite
\newrobustcmd*{\mautocite}{\mcitelike\autocite}
\begin{document}
\mtextcite{companion,bertram,knuth,*knuth:ct:a,*knuth:ct:b} showed that...
Filler text \autocite{knuth:ct:a}.
Filler text \mautocite{model,*weinberg,*glashow}.
Filler text \autocite{weinberg}.
\printbibliography
\end{document}
Note that here \mtextcite{companion,bertram,knuth,*knuth:ct:a,*knuth:ct:b} is the equivalent of:
\defbibentryset{knuth}{knuth:ct:a,knuth:ct:b}
\textcite{companion,bertram,knuth}
Similarly \mautocite{model,*weinberg,*glashow} becomes:
\defbibentryset{model}{weinberg,glashow}
\autocite{model}
Futher details can be found in the biblatex manual sections "mcite-like Citation Commands" and "Dynamic Entry Sets". Various label number and entry set bugs were introduced/fixed between versions 1.1 and 2.7, so you should upgrade to the latest biblatex and biber releases to get all relevant features working properly.
• Cheers! This is the solution I was looking for! Jul 29, 2013 at 1:01
• @SamuelTan My initial post uncovered a bug, which I documented here. The fix will be in the next release, but it is pretty easy to apply yourself. Jul 29, 2013 at 1:47
For this kind biblatex provides the set entry:
@SET{a-set,
entryset={key2,key3,key4},
}
In your example this would then follow like this:
\usepackage[backend=biber,mcite,subentry,autocite=plain}
...
\begin{document}
Some text here\autocite{key1}. More text here\autocite{a-set}.
Cite a subentry\autocite{key3}.
\end{document}
• mcite-like commands are for defining dynamic (on-the-fly) entry sets. Sets defined with @set are static. This is still relevant information, though, so +1. Jul 27, 2013 at 18:41
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2022-07-01 17:42:08
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https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.120.117402
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# Synopsis: Two-Face Dipole
A proposed dipole source of electromagnetic waves can selectively couple its emission into either of two neighboring waveguides.
Combinations of magnetic and electric dipoles are often used to generate directional electromagnetic emission in devices like nanoantennas or on-chip light emitters. Current approaches control directionality by separately manipulating the electric and magnetic fields generated by the dipoles. These approaches, however, ignore effects due to relative phase and amplitude differences between the electric and magnetic fields. Michela Picardi and colleagues at King’s College London have proposed a new type of dipole source that can control these differences. The scheme could lead to a broad array of photonic devices such as nanorouters and polarimeters.
By taking the phase and amplitude of both electric and magnetic fields into account, the researchers developed a general theoretical framework that describes the emission of an ensemble of dipoles as well as all possible ways to couple the dipoles’ emission into a waveguide. Such a framework allowed them to introduce the Janus-dipole source, consisting of two perpendicularly oriented electric and magnetic dipoles oscillating with a $9{0}^{\circ }$ phase difference. The source, like its namesake Roman god Janus, has two faces. One face can excite electromagnetic waves in a nearby waveguide, while the second cannot. The authors suggest that a Janus dipole could be realized using a single nanoparticle that can be simultaneously polarized magnetically and electrically.
Using numerical simulations, the team analyzed a scheme in which the Janus dipole sits between two parallel waveguides. They showed that one could select which of the waveguides carries the emitted signal by switching the orientation of the dipole’s two faces. This switching could be achieved by changing the polarization or wavelength of a laser beam shined on the nanoparticle.
This research is published in Physical Review Letters.
–Christopher Crockett
Christopher Crockett is a freelance writer based in Montgomery, Alabama.
More Features »
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2018-12-14 20:40:55
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http://cpl.iphy.ac.cn/EN/abstract/abstract69688.shtml
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Chin.Phys.Lett. 2017, Vol. 34 Issue (03): 037101 DOI: 10.1088/0256-307X/34/3/037101
CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES |
Electronic Structure and Thermoelectric Power Factor of Na$_{x}$CoO$_{2}$ from First-Principles Calculation
Peng-Xian Lu1**, Rui-Xia Zhao2
1College of Materials Science and Engineering, Henan University of Technology, Zhengzhou 450001
2Department of Civil Engineering and Architecture, Henan Technical College of Construction, Zhengzhou 450001
Download: Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract To investigate the relationship between the electronic structure and the power factor of Na$_{x}$CoO$_{2}$ ($x=0.3$, 0.5 and 1.0), the first-principles calculation is conducted by using density functional theory and the semi-classical Boltzmann theory. Our results suggest that with the decreasing Na content, a transition from semiconductor to semimetal is observed. Na$_{0.3}$CoO$_{2}$ possesses a higher electrical conductivity at 1000 K due to its increased density of states near the Fermi energy level. However, an optimal Seebeck coefficient at 1000 K is obtained in Na$_{0.5}$CoO$_{2}$ because of its broadened band gap near the Fermi energy level. Consequently, a maximum power factor is realized in Na$_{0.5}$CoO$_{2}$. Thus our work provides a complete understanding of the relationship between the electronic structure and the thermoelectric power factor of Na$_{x}$CoO$_{2}$.
Received: 04 November 2016 Published: 14 March 2017
PACS: 71.20.Nr (Semiconductor compounds) 72.20.Pa (Thermoelectric and thermomagnetic effects)
Fund: Supported by the Science Foundation of Henan University of Technology under Grant No 2015XTCX10.
Issue Date: 14 March 2017
Cite this article: Peng-Xian Lu,Rui-Xia Zhao 2017 Chin.Phys.Lett. 34 037101 URL: http://cpl.iphy.ac.cn/newweb/10.1088/0256-307X/34/3/037101 OR http://cpl.iphy.ac.cn/newweb/Y2017/V34/I03/037101
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2020-01-25 22:41:18
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https://mathoverflow.net/questions/366765/issue-update-in-graph-theory-different-definitions-of-edge-crossing-numbers/366876
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# Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications?
QUICK FINAL UPDATE: Just wanted to thank you MO users for all your support. Special thanks for the fast answers, I've accepted first one, appreciated the clarity it gave me. I've updated my torus algorithm with $${\rm cr}(G)$$. Works fine on my full test set, i.e. evidence for $${\rm cr}(G)={\rm pcr}(G)$$ on torus. More on this later, will test sharper bound from last answer as well. I'm going to submit in time! Thanks again MO users for all your help!
Original post:
I apologize if „crisis“ is too strong a word, but I am in a mode of panic, if that's the right word: In two weeks, I should be submitting my Ph.D. Thesis, but I have just received bad news, or I should say information that makes me very concerned. It is really an emergency situation:
My thesis is in computer science, algorithms related to graph drawings on the sphere and the torus. One of the cornerstone mathematical results I am relying on is the graph edge crossing lemma (or edge crossing inequality). It gives a lower bound for the minimum number of edge crossings $${\rm cr}(G)$$ for any drawing of the graph $$G$$ with $$n$$ vertices and $$e$$ edges $${\rm cr}(G)\geq \frac{e^3}{64n^2}$$ for $$e>4n$$.
PROBLEM: I am reading in the article of Pach and Tóth that there is a possibility that mathematics papers on crossing numbers operate with different definitions. There is the crossing number $${\rm cr}(G)$$ (minimum of edge crossings in a drawing of $$G$$), but also the pair crossing number $${\rm pcr}(G)$$, the minimum number of edge pairs crossing in a drawing of $$G$$. I double-checked my algorithms and, based on this definition, I clearly apply the pair crossing number $${\rm pcr}(G)$$
CRITICAL QUESTION: Can you confirm to me that the edge crossing lemma remains valid on the sphere and the torus also for the pair crossing number $${\rm pcr}(G)$$?
Reference: János Pach and Géza Tóth. Which crossing number is it anyway? J. Combin. Theory Ser. B, 80(2): 225–246, 2000.
And Wikipedia article as a starting point https://en.wikipedia.org/wiki/Crossing_number_inequality
• I don't really know anything about crossing numbers, but I can appreciate how stressful this must be for you. I hope that you are able to patch things up in time! Jul 28, 2020 at 8:24
• @PerAlexandersson --- as I understand it two edges may intersect multiple times; this multiplicity is counted in cr but not in pcr, hence pcr $\leq$ cr. Jul 28, 2020 at 15:53
• I suppose that the downvoter never felt any stress while doing his/her Ph.D Thesis...
– efs
Jul 28, 2020 at 17:15
• Frege once had to write "A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press. "... Jul 29, 2020 at 8:49
• Maybe you could accept the answer that is here, and then add your own answer after you successfully defend your dissertation. The vast majority of people who read this question are rooting for you. Jul 29, 2020 at 13:34
$$\DeclareMathOperator\cr{cr}\DeclareMathOperator\pcr{pcr}$$For the pair crossing number $$\pcr(G)$$, the short answer is yes the crossing lemma holds for drawings on the sphere, but it is not known whether it also holds on the torus.
The best and most current reference for you could be the survey article from Schaefer, updated in February 2020: “The Graph Crossing Number and its Variants: A Survey” from the Electronic Journal of Combinatorics (https://doi.org/10.37236/2713).
The relevant pages for you are pages 5 and 6 with the following quote from Schaefer:
“Since the Hanani–Tutte theorem is not known to be true for the torus, this means that we do not currently have a proof of the crossing lemma for $$\pcr$$ or $$\pcr_−$$ on the torus.”
Generally, $$\pcr(G)\leq \cr(G)$$. It is still an open problem whether they are equal or not. The first proofs of the crossing lemma did not make the distinction. The first one to raise the ambiguity was Mohar (1995) in a conference talk.
The Pach and Tóth (2000) paper that you mention does make the distinction between $$\pcr(G)$$ and $$\cr(G)$$, and applies Hanani–Tutte in the proof of the crossing lemma, which ensures that it also holds for $$\pcr(G)$$.
The issue is that you can apply Hanani–Tutte for the sphere (and the projective plane), but you cannot apply it for the torus. For surfaces of genus $$\geq4$$ it is known to be false, see Fulek and Kynčl (2019). This means the torus is really “in-between”.
Bojan Mohar (1995): Problem mentioned at the special session on Topological Graph Theory, Mathfest, Burlington, Vermont. (cited from: L.A. Székely (2016): Turán’s Brick Factory Problem: The Status of the Conjectures of Zarankiewicz and Hill. In: R. Gera et al. (eds.)(2016): Graph Theory—favorite conjectures and open problems. 1.)
Hanani–Tutte Theorem https://en.wikipedia.org/wiki/Hanani%E2%80%93Tutte_theorem
Radoslav Fulek and Jan Kynčl (2019): Counterexample to an Extension of the Hanani–Tutte Theorem on the Surface of Genus 4. Combinatorica, 39(6):1267–1279
• From OP's point of view this could be viewed as glass half-full rather than glass half-empty. Their dissertation results hold unequivocally on the sphere and might hold on the torus, though it is an open problem if they do. It is certainly legitimate to study what follows from a given conjecture being true. It could even be spun as a feature rather than a bug of the dissertation. If the results in fact fail on the torus then you know that the conjecture must be false. Potentially, it could open up a fruitful avenue of attack. Jul 29, 2020 at 0:25
• I think in algebraic topology literature, Hanani-Tutte is known as a version of the Flores-Van Kampen theorem, if that's a helpful link
– Matt
Jul 29, 2020 at 8:53
• Adding to my comment, here is a 2019 reference "Invariants of graph drawings in the plane" from A. Skopenkov arxiv.org/pdf/1805.10237.pdf
– Matt
Jul 29, 2020 at 9:05
• @ClausDollinger . The clarity is helpful. Thanks for your fast help. Much appreciated. I'm glad I can keep my sphere algorithm. I'm checking whether I can adapt my torus algorithm to ${\rm cr}(G)$ within the next 5 days Jul 29, 2020 at 17:22
Assuming an unpublished Ramsey-type result by Robertson and Seymour about Kuratowski minors [FK18, Claim 5], which is now "folklore" in the graph-minor community, an asymptotic variant of the crossing lemma, $$\operatorname{cr}(G)\ge \Omega(e^3/n^2)$$, is true even for the pair crossing number on a fixed surface, such as a torus.
With Radoslav Fulek [FK18, Corollary 9] we have shown that [FK18, Claim 5] implies an approximate version of the Hanani–Tutte theorem on orientable surfaces. In particular, [FK18, Claim 5] implies that there is a constant $$g$$ such that for every graph $$G$$ that can be drawn on the torus with every pair of independent edges crossing an even number of times, $$G$$ can be drawn on the orientable surface of genus $$g$$ without crossings. This gives an upper bound $$3n + O(g)$$ on the number of edges of every such graph $$G$$, and this can be used in the probabilistic proof of the crossing lemma, as described on p. 5-6 of Marcus Schaefer's survey [S20], mentioned in Claus Dollinger's answer. See also [SSSV96, Theorem 4.1].
References:
[FK18] https://dx.doi.org/10.4230/LIPIcs.SoCG.2018.40, https://arxiv.org/abs/1803.05085 - R. Fulek and J Kynčl, The $$\mathbb Z_2$$-genus of Kuratowski minors
[SSSV96] https://doi.org/10.1007/BF02086611 - F. Shahrokhi, L. A. Székely, O. Sýkora and I. Vrt'o, Drawings of graphs on surfaces with few crossings, Algorithmica 16, 118-131 (1996)
[S20] https://doi.org/10.37236/2713 - M. Schaefer, The Graph Crossing Number and its Variants: A Survey, The Electronic Journal of Combinatorics, DS21: Feb 14, 2020.
Edit: "Strong Hanani-Tutte for the Torus" by Radoslav Fulek, Michael J. Pelsmajer and Marcus Schaefer has just appeared on arxiv: https://arxiv.org/abs/2009.01683
• Is Schaefer's survey Hanani–Tutte and related results (MSN)? Jul 29, 2020 at 16:30
• I meant the survey about crossing numbers, mentioned in Claus Dollinger's answer. I will make an edit and add the reference to make it clear. Jul 29, 2020 at 16:47
• By the way, you know that diacrits like ˇ are allowed in your username if you want, right? Jul 29, 2020 at 17:03
• @JanKyncl Thank you for your help as well, much appreciated. My advisor says the faculty will not accept asymptotics. I'm checking whether I can put ${\rm cr}(G)$ into my torus algorithm Jul 29, 2020 at 17:26
• "the faculty will not accept asymptotics"? Jul 29, 2020 at 20:39
@user161819 I wanted to make a comment but it got too long, so putting it as an answer. But please take it just as a comment for later, once everything is finished:
If I understand your comment to my answer correctly, you are aiming to change your algorithm for the torus so it works with $${\rm cr}(G)$$. I think the whole MO community is keeping their fingers crossed, wishing you that you can successfully complete everything in time!
Looking at the far horizon, I wanted to make a suggestion to you. Once you have changed your torus algorithm and completed your thesis, you will have effectively two algorithms in your hands for the torus: The old one based on $${\rm pcr}(G)$$ and the new one based on $${\rm cr}(G)$$. I am saying the obvious here, keep both of them, they can really be fruitful for future research.
(1) Obviously, your two algorithms could support research on the big open question whether $${\rm pcr}(G)\stackrel{\rm ?}{=}{\rm cr}(G)$$ or not. They could produce experimental evidence, ideas, and insights for a future proof of equality, or an actual counterexample. (Again, I am saying the obvious here.)
(2) To really pressure-test $${\rm pcr}(G)\stackrel{\rm ?}{=}{\rm cr}(G)$$ on the torus, it would be interesting to also try the best known to date lower bound for $${\rm cr}(G)$$ $$\frac{1}{29}\frac{e^3}{n^2}$$ for graphs with $$e>7n$$. This lower bound is from Eyal Ackerman (2019): "On topological graphs with at most four crossings per edge", Computational Geometry, 85: 101574, 31, doi:10.1016/j.comgeo.2019.101574 (probably you are aware of it from the Wikipedia article that you quoted).
I think your question and this whole topic are really important. László Székely calls it one of the "foundational problems" and devotes a whole section to it in his article Turán’s Brick Factory Problem: The Status of the Conjectures of Zarankiewicz and Hill. In: R. Gera et al. (eds.)(2016): Graph Theory—favorite conjectures and open problems. 1.)
For now, fingers crossed that you can complete your thesis in time!
• Thanks for your comment!! Much appreciated. And thanks again for your help. I'm very interested in this Ackermann bound, will take a look at it Jul 30, 2020 at 16:28
• One good thing, my prototype for updated torus algorithm: tested on on first 2 graphs from my test set, and went ok Jul 30, 2020 at 16:33
• The term "Ackermann bound" generally refers to a bound of a type far, far worse than the one given here: en.wikipedia.org/wiki/Ackermann_function Aug 3, 2020 at 17:37
• @TerryTao Terry you are very right. Huge difference between the Wilhelm Ackermann bound that you mention and the Eyal Ackerman bound I am quoting here! Good to make the distinction. Aug 3, 2020 at 18:25
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2023-04-01 23:37:28
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https://www.dml.cz/handle/10338.dmlcz/119309?show=full
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Article
Title: Disasters in metric topology without choice (English) Author: Tachtsis, Eleftherios Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 43 Issue: 1 Year: 2002 Pages: 165-174 . Category: math . Summary: We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X,d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply {\it the disjoint union of metrizable spaces is normal\/}. (English) Keyword: Axiom of Choice Keyword: Axiom of Multiple Choice Keyword: Principle of Dependent Choice Keyword: Ordering Principle Keyword: metric spaces Keyword: separable metric spaces Keyword: second countable metric spaces Keyword: paracompact spaces Keyword: compact T$_2$ spaces Keyword: ccc spaces. MSC: 03E25 MSC: 54A35 MSC: 54D20 MSC: 54E35 MSC: 54E45 MSC: 54F05 idZBL: Zbl 1072.03030 idMR: MR1903316 . Date available: 2009-01-08T19:20:40Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119309 . Reference: [1] Cohen P.J.: Set Theory and the Continuum Hypothesis.Benjamin, 1966. Zbl 0182.01401, MR 0232676 Reference: [2] van Douwen E.K.: Horrors of topology without AC: a non normal orderable space.Proc. Amer. Math. Soc. 95 (1985), 101-105. MR 0796455 Reference: [3] Good C., Tree I.J.: Continuing horrors of topology without choice.Topology Appl. 63 (1995), 79-90. Zbl 0822.54001, MR 1328621 Reference: [4] Good C., Tree I.J., Watson W.S.: On Stone's theorem and the axiom of choice.Proc. Amer. Math. Soc. 126 (1998), 1211-1218. Zbl 0893.54016, MR 1425122 Reference: [5] Herrlich H., Steprāns J.: Maximal filters, continuity and choice principles.Quaestiones Math. 20 (1997), 697-705. MR 1625478 Reference: [6] Herrlich H., Strecker G.E.: When is $\Bbb N$ Lindelöf?.Comment. Math. Univ. Carolinae 38.3 (1997), 553-556. Zbl 0938.54008, MR 1485075 Reference: [7] Howard P., Keremedis K., Rubin H., Rubin J.E.: Disjoint unions of topological spaces and choice.Math. Logic Quart. 44 (1998), 493-508. Zbl 0922.03069, MR 1654348 Reference: [8] Howard P., Keremedis K., Rubin J.E., Stanley A.: Paracompactness of metric spaces and the axiom of multiple choice.Math. Logic Quart. 46 (2000). Zbl 0993.03059, MR 1755811 Reference: [9] Howard P., Keremedis K., Rubin J.E., Stanley A., Tachtsis E.: Non-constructive properties of the real numbers.Math. Logic Quart. 47 (2001), 423-431. MR 1847458 Reference: [10] Howard P., Rubin J.E.: Consequences of the Axiom of Choice.Math. Surveys and Monographs 59, Amer. Math. Soc., Providence R.I., 1998. Zbl 0947.03001, MR 1637107 Reference: [11] Jech T.: The Axiom of Choice.North-Holland, Amsterdam, 1973. Zbl 0259.02052, MR 0396271 Reference: [12] Keremedis K.: Disasters in topology without the axiom of choice.Arch. Math. Logic, to appear. Zbl 1027.03040, MR 1867681 Reference: [13] Keremedis K.: Countable disjoint unions in topology and some weak forms of the axiom of choice.Arch. Math. Logic, submitted. Reference: [14] Keremedis K., Tachtsis E.: Compact metric spaces and weak forms of the axiom of choice.Math. Logic Quart. 47 (2001), 117-128. Zbl 0968.03057, MR 1808950 Reference: [15] Keremedis K., Tachtsis E.: On Lindelöf metric spaces and weak forms of the axiom of choice.Math. Logic Quart. 46 (2000), 35-44. Zbl 0952.03060, MR 1736648 Reference: [16] Kunen K.: Set Theory, An Introduction to Independence Proofs.North-Holland, Amsterdam, 1983. Zbl 0534.03026, MR 0756630 Reference: [17] Willard S.: General Topology.Addison-Wesley Publ. Co., Reading, MA, 1968. Zbl 1052.54001, MR 2048350 .
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CommentatMathUnivCarolRetro_43-2002-1_15.pdf 230.2Kb application/pdf View/Open
Partner of
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2021-04-13 04:17:33
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https://www.biostars.org/p/9479769/
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usearch11 cluster_fast option
1
1
Entering edit mode
11 weeks ago
g.papp-co ▴ 10
I have a little example fasta file with protein fragments to figure out how usearch works:
fragments
>seq1
TKEHALSKERAA
>seq2
KKEHALSKERAR
>seq3
AAHAASAERAAE
>seq4
AAHAASAERAAS
I used usearch with the following options:
usearch -cluster_fast ex.fasta -id 0.5 -uc cluster.uc
I expect at most 2 clusters (1,2) and (3,4) but the result contains only singletons.I decreased the identity and also the gap penalties but the result is the same.
Any idea?
Thank You
usearch proteins clustering • 179 views
2
Entering edit mode
10 weeks ago
Mensur Dlakic ★ 13k
I think your sequences are too short. Try reducing the -minhsp value, but it simply may not work with these sequence lengths. If you duplicate each sequence and make the length 24 instead of 12, you will get the desired clusters.
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2021-09-25 10:22:35
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https://cs.stackexchange.com/questions/44249/the-running-time-of-the-knapsack-problem-is-on-cdot-minb-v-and-is-not-pol
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# The running time of the knapsack problem is $O(n\cdot \min(B,V))$ and is not polynomial, why?
My question is why the dynamic programming of the knapsack problem does run in polynomial time? The question is answered here Why is the O(nW) algorithm for the Knapsack problem not a polynomial one?
I understand the argument of the answer above. Also the answer is given in this book ''The Desing of Approximation Algorithms'' that I am reading. In [pp. 67, §2], the authors said:
Algorithm 3.1 takes $O(n\cdot \min(B, V))$ time. This is not a polynomial-time algorithm, since we assume that all input numbers are encoded in binary; thus, the size of the input number $B$ is essentially $\lg B$, and so the running time $O(n\cdot B)$ is exponential in the size of the input number $B$, not polynomial. $\cdots$
I understand that $O(n\cdot B)$ is exponential in the size of the input number $B$, but also, in my point of view (which is wrong), $O(n\cdot B)$ is exponetial in the size of the input number $n$, i.e., the size of the input $n$ is $\lg n$. Why not? (If I were to represent $n$ in binary I would take $\lg n$ size, no?)
$n$ is not part of the input, $n$ denotes the number of objects in the input.
The input consists of the capacity of the knapsack, a list of objects, each with a value and weight. If there are $n$ objects in the instance then the size of the instance is at least $n$ since each object needs to be represented (with at least one bit). Therefore $n$ is polynomial in the size of the input.
In general, the size of the input (if all weights and values are at most $B$) will be $O(n \log B)$, since there are $n$ objects to represent and representing an object (which is two numbers at most $B$) will take around $2\log B$ bits.
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2019-06-24 22:01:18
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https://math.eretrandre.org/tetrationforum/showthread.php?tid=882
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Real-analytic tetration uniqueness criterion? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/09/2014, 07:35 AM (This post was last modified: 06/09/2014, 10:25 AM by mike3.) Hi. I was wondering about this. it appears that the Kneser tetrational satisfies the following real-analytic property on $(-2, \infty)$: $\mathrm{tet}^{(2n)}(x)$ is strict-monotone increasing for $n \ge 0$, $n \in \mathbb{Z}$ $\mathrm{tet}^{(2n+1)}(x)$ is positive for $n \ge 0$, $n \in \mathbb{Z}$ and $\mathrm{tet}^{(2n+1)}(x)$ is convex for $n \ge 0$, $n \in \mathbb{Z}$. (the notation denotes differentiation) These are equivalent. Easy theorems from calculus concerning monotonicity, convexity, and derivatives and integrals shows that 1 and 2 imply each other and that 3 implies 1. But here's the thing: could this be a uniqueness criterion for tetration? (here, I'm thinking of base $e$) I tried some numerical tests of Kneser's tetration solution, warping it with small $\theta(z)$ 1-cyclic warping mappings (specifically $\theta(z) = \frac{\sin(2\pi z)}{K}$ and $\theta(z) = \frac{\sin(2\pi z - \pi) + 1}{K}$) with amplitudes of down to $10^{-5}$ (which is $K = 10^5$) and the criterion seems to fail if the derivative is high enough (for amplitudes of the given magnitude, at around the 32nd derivative). The derivative at which it fails seems to increase rapidly as $K$ shrinks, so I'm not sure if it is singular, meaning there is a range of $K$ for which the condition is satisfied and hence uniqueness is not obtained, or whether or not it will eventually fail no matter how small the 1-cyclic wobble is, which would mean this should provide a uniqueness condition when combined with, perhaps, analyticity or maybe even just smoothness and the usual functional equations. What do you think? tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 06/09/2014, 09:51 AM Im very convinced its not unique. Many reasons. Asymptotics , interpolation , " fake function theory " ,... In fact I posted similar conjectures years ago and came to reconsider them. Hence I posted " TPID 16 " and a related uniqueness claim for exp^[1/2]. See : http://math.eretrandre.org/tetrationforu...hp?tid=881 http://math.eretrandre.org/tetrationforu...hp?tid=879 ( were the suggestion in post 2 is considered solid now ) Somewhat different : http://math.eretrandre.org/tetrationforu...hp?tid=842 --- Here I was a bit upset because I posted these things first http://math.eretrandre.org/tetrationforu...hp?tid=503 post 13. --- http://math.eretrandre.org/tetrationforu...hp?tid=474 http://math.eretrandre.org/tetrationforu...hp?tid=484 and probably more posts. ( I currently posted more than 13% of this forum so I dont remember them all so well ) So the idea is far from completely new. But its intresting. I hope you get more responses than I did. As a remark : Notice this type of uniqueness was not sufficient for a unique real-analytic gamma function in the past. Gamma also grows fast and satisfies a functional equation ! See the Bohr-Möllerup and Wielandt theorems. If you want to replace convex with log convex , probably better to replace with arc2sinh or so ... which then again resembles my conjectures. Imho the lenght idea is also underrated although I owe some credit to gottfried. A few members here have already proved analogues of the Wielandt for tetration. regards tommy1729 mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/09/2014, 10:26 AM I am not convinced by the "2sinh"-stuff. A composition with inverse of 2sinh superfunction may not even be analytic (I think sheldonison did some work on related stuff involving tetrational base changes -- and maybe even 2sinh, I don't remember), and converges to a linear function plus a small 1-cyclic wobble. A small, 1-cyclic oscillation doesn't seem too good for "convexity"-related purposes -- something that wiggles a little about a line is not quite convex. Also, with more testing it looks like that this method of repeated differentiation eventually teases out even tiny 1-cyclic wobbles applied to the gamma function, i.e. $\Gamma(x) \theta(x)$ where $\theta(x)$ is a small-amplitude 1-cyclic wobble with $\theta(0) = 1$. Just that it seems with tetration, you need to differentiate it a whole lot more (probably because the growth rate is soo much faster than the gamma function's). 32 derivatives was more than enough to tease out a wobble of amplitude $10^{-12}$. tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 06/09/2014, 10:44 AM (This post was last modified: 06/09/2014, 12:34 PM by tommy1729.) I assume mike3 meant : $\Gamma(\theta(x))$ As for the numerical testing , did mike test for x > 0 or x >-2 or a Taylor series at some x ? Or x > some C ? If his conjecture means for x > -2 then it is quite a strong idea , but Im not sure about existance since going from f(-1.9) to f(-0.9) is not likely convex. SO I assume for x > 0. Right ? regards tommy1729 mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/09/2014, 10:50 AM (This post was last modified: 06/09/2014, 10:51 AM by mike3.) (06/09/2014, 10:44 AM)tommy1729 Wrote: I assume mike3 meant : $\Gamma(\theta(x))$ As for the numerical testing , did mike test for x > 0 or x >-2 or a Taylor series at some x ? Or x > some C ? If his conjecture means for x > -2 then it is quite a strong idea , but Im not sure about existance since going from 0 to 1 is not convex. SO I assume for x > 0. Right ? regards tommy1729 Um, it should be $\Gamma(x) \theta(x)$. $\Gamma(\theta(x))$ is a 1-periodic function itself and not at all a solution of the Gamma function equations! Convexity is over odd derivatives (for tetration), and even derivatives (for gamma function). mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/09/2014, 12:20 PM (This post was last modified: 06/09/2014, 12:22 PM by mike3.) IMPORTANT! I just was running some more tests and I've discovered that these results may not be trustworthy. Apparently, increasing the precision on sheldonison's Kneser PARI/GP program does not seem to make it necessarily generate more than 64 digits of the tetrational. So results requiring high precision seem to be suspect. I'll need to figure out how to get more digits out of the code before trying again. I had done a "\p 128" and it only would display 64 digits -- upon "\p 128"ing again to force it to cough up more digits, I found all the succeeding digits after the initial 64 were different from those I got for "\p 256", suggesting it is not getting beyond 64 digits. I'm not sure how to get the program to give a correct result with more than 64 digits -- sheldonison? tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 06/09/2014, 12:30 PM (This post was last modified: 06/09/2014, 12:31 PM by tommy1729.) (06/09/2014, 10:50 AM)mike3 Wrote: (06/09/2014, 10:44 AM)tommy1729 Wrote: I assume mike3 meant : $\Gamma(\theta(x))$ As for the numerical testing , did mike test for x > 0 or x >-2 or a Taylor series at some x ? Or x > some C ? If his conjecture means for x > -2 then it is quite a strong idea , but Im not sure about existance since going from 0 to 1 is not convex. SO I assume for x > 0. Right ? regards tommy1729 Um, it should be $\Gamma(x) \theta(x)$. $\Gamma(\theta(x))$ is a 1-periodic function itself and not at all a solution of the Gamma function equations! Convexity is over odd derivatives (for tetration), and even derivatives (for gamma function). Sorry I meant to say $\Gamma(x+\theta(x))$. Afterall you consider $sexp(x+\theta(x))$ right ? Or are you talking about $sexp(x)\theta(x)$ ?? ( that would not make sense ) Anyway for all clarity , I like the idea and wish it was true. But I fear not. You tested theta function that have $|\theta'(x)| > 0$ for x=0,1. I think that is the mistake. When I tried periodic functions I took functions that satisfied $|D^m \theta(x)| = 0$ for all 24 >= m >= 0 and all x = n/6 for any integer n. I did however take x > 1 I think. regards tommy1729 mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/09/2014, 12:49 PM (This post was last modified: 06/09/2014, 12:51 PM by mike3.) @Tommy1729: The Gamma function is different from tetration. To make an alternative tetration, use $\mathrm{tet}^{*}(z) = \mathrm{tet}(z + \theta(z))$ where $\theta(z)$ is a 1-cyclic function with $\theta(0) = 0$. To make an alternative "Gamma function", use $\Gamma^{*}(z) = \Gamma(z) \theta(z)$. where $\theta(z)$ is a 1-cyclic function with $\theta(0) = 1$. This is because the functional equation for tetration is $\mathrm{tet}(z + 1) = \exp(\mathrm{tet}(z))$ whereas that for the Gamma function is $\Gamma(z + 1) = z \Gamma(z)$. Take the ratio of two solutions of this equation, and you will see it is 1-periodic. Thus, a 1-periodic multiplication factor (unlike for tetration, where you need composition!) will convert one "Gamma-like function" into another. Also, $\Gamma((z + 1) + \theta(z + 1)) = \Gamma(z + \theta(z) + 1) = (z + \theta(z)) \Gamma(z + \theta(z)) \ne z \Gamma(z + \theta(z))$. tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 06/09/2014, 12:54 PM (06/09/2014, 12:49 PM)mike3 Wrote: @Tommy1729: The Gamma function is different from tetration. To make an alternative tetration, use $\mathrm{tet}^{*}(z) = \mathrm{tet}(z + \theta(z))$ where $\theta(z)$ is a 1-cyclic function with $\theta(0) = 0$. To make an alternative "Gamma function", use $\Gamma^{*}(z) = \Gamma(z) \theta(z)$. where $\theta(z)$ is a 1-cyclic function with $\theta(0) = 1$. This is because the functional equation for tetration is $\mathrm{tet}(z + 1) = \exp(\mathrm{tet}(z))$ whereas that for the Gamma function is $\Gamma(z + 1) = z \Gamma(z)$. Take the ratio of two solutions of this equation, and you will see it is 1-periodic. Thus, a 1-periodic multiplication factor (unlike for tetration, where you need composition!) will convert one "Gamma-like function" into another. Also, $\Gamma((z + 1) + \theta(z + 1)) = \Gamma(z + \theta(z) + 1) = (z + \theta(z)) \Gamma(z + \theta(z)) \ne z \Gamma(z + \theta(z))$. Yes of course you are right. Sorry silly of me. But the argument still remains ... What is your opinion about the periodic functions that I tried ? Did you try those ? regards tommy1729 sheldonison Long Time Fellow Posts: 631 Threads: 22 Joined: Oct 2008 06/09/2014, 01:16 PM (This post was last modified: 06/09/2014, 02:05 PM by sheldonison.) (06/09/2014, 12:20 PM)mike3 Wrote: IMPORTANT! I just was running some more tests and I've discovered that these results may not be trustworthy. Apparently, increasing the precision on sheldonison's Kneser PARI/GP program does not seem to make it necessarily generate more than 64 digits of the tetrational. So results requiring high precision seem to be suspect. I'll need to figure out how to get more digits out of the code before trying again. I had done a "\p 128" and it only would display 64 digits -- upon "\p 128"ing again to force it to cough up more digits, I found all the succeeding digits after the initial 64 were different from those I got for "\p 256", suggesting it is not getting beyond 64 digits. I'm not sure how to get the program to give a correct result with more than 64 digits -- sheldonison? Hey Mike, After changing the precision with "\p 134" or any large number, type in "init(exp(1));" You will get approximately n/2 decimal digits for the results, limited by the Schroeder functions algorithm. Code:\p 134 init(exp(1)); This regenerates the tetration system with arbitrarily high precision numbers. The "init(exp(1));" after the default of "\p 67" takes 3 or 4 seconds. After "\p 134" takes about 30 seconds, and after "\p 221" takes 3-4 minutes. There s an out of memory (>4meg) error around "\p 230" or so. But I have used "\p 512" I think with 30 minutes computer time. The out of memory error is due to initializing my base eta approximation function, in the initcheta code. Here is a patch to remove that memory error. Code:initcheta() = { /* automatically initialize cheta during program initialization */ local(z,local); z=1.0; precis=precision(z); /* initialization for xcheta and xsexpeta required for cheta, sexpeta, invcheta, invsexpeta functions */ /* initalizes to match precis, 50 digits/67 digits, this routine aims for 75% precision */ /* called from init(initbase) when program is loaded, and when init detects the precision has changed */ if (precis<=180, chterms = 2*(floor(precis/2)-8)+1; , chterms = 2*(floor(180/2)-8)+1; ); chdelta = (chterms-1)*2; xcheta = genpoly(chterms,chdelta,0); invchetr = imag(cheta(-chdelta+(chterms-1)*I/2)); invprecis = 10000.*(10^-chterms); /* invprecis = 100*abs(chetaerr(I+0.5,chdelta)); */ /* invprecis = 1E-47; */ chetadlt=cheta(-chdelta); xsexpeta = genpoly(chterms,chdelta,1); sxpetadlt=sexpeta(chdelta); return(0); } - Sheldon « Next Oldest | Next Newest »
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2019-12-16 07:20:24
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https://mathoverflow.net/questions/184920/why-is-it-so-hard-to-compute-pi-nsn
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# Why is it so hard to compute $\pi_n(S^n)$?
Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how to do the calculation within the homotopy category of topological spaces. Approaches that I'm aware of include:
-Homology theory (the Hurewicz theorem)
-Degree theory
-The divergence theorem
and each of these reduces to a calculation within some other category (PL or Diff). My question is: is there something "wrong" with hTop that precludes a computation of $\pi_n(S^n)$ within that category?
Certainly if my assumption that such a proof does not exist is wrong, I would be very interested to know it.
I have been thinking and reading further about this question for the past couple days, and I wanted to summarize some of the main points in the answers and questions:
1. Some techniques - e.g. the Freudenthal Suspension Theorem via the James construction or the Hurewicz Theorem with singular homology - might actually lead to proofs without any approximation arguments. But so far I'm not sure we quite have it: the proof of the Freudenthal Suspension Theorem uses the fact that $J(X)$ is homotopy equivalent to the loop space of the suspension of $X$, but the only proofs I can find of this fact use a CW structure on $X$, and similarly for the proofs of the Hurewicz theorem. Can these results be proved for the sphere without PL or smooth approximation?
2. Perhaps this question is entirely wrong-headed: the techniques of PL and smooth approximation are very well adapted to homotopy theory, so why try to replace them with a language which may end up adding more complications with little additional insight? Fair enough. But the goal of this question is not to disparage or seek alternatives to existing techniques, it is to understand exactly what role they play in the theory. The statement "The identity map $S^n \to S^n$ is homotopically nontrivial and freely generates $\pi_n(S^n)$" makes no mention of CW complexes or smooth structures, yet apparently the statement is difficult or impossible to prove without that sort of language (except in the case $n=1$!) To seek an understanding of this observation is different from lamenting it.
• One option which is perhaps overkill is to use the Hopf fibration to show that pi_2(S^2) is Z and then to use Freudenthal suspension theorem (which can be proved with homotopy theoretic ideas e.g. the James construction). – Callan McGill Oct 20 '14 at 23:44
• Maybe I'm missing something, but how is homology outside of the homotopy category of topological spaces? – Denis Nardin Oct 21 '14 at 2:49
• @PaulSiegel To compute $H_n(S^n)$ I would prove the suspension isomorphism in homology (that uses only homotopy invariance and Mayer-Vietoris) and then proceed to compute $H_0(S^0)$, which is elementary. I think that this proof does not leave the homotopy category, unless you regard the subdivision isomorphism to prove excision as "PL approximation" – Denis Nardin Oct 21 '14 at 13:33
• @PaulSiegel I must admit to being slightly confused by the question. Most early calculations of $\pi_n(S^n)$ do rely heavily on hard manipulations with homotopies to reduce to continuous maps of nice forms, but the definition of hTop is in terms of such homotopies. Without some of these basic tools under our control, we have almost no way to distinguish hTop from an arbitrary category. – Tyler Lawson Oct 21 '14 at 14:47
• If you like, though, one defect in hTop is that has no categorical niceties. You can't really build new objects out of old ones or use that many elementary "building block" calculations to get larger calculations. Several decades of experience with this have resulted in a working methodology where, even if a homotopy category is our goal, it almost always comes accompanied with a "chart" (like the category of spaces, or CW-complexes, or simplicial sets, or...) where all the important constructions can be carried out; the homotopy category is just a place to examine the results. – Tyler Lawson Oct 21 '14 at 14:53
I suppose that the proof that $\pi_1(S^1) \cong \mathbb{Z}$ using covering spaces is homotopy-theoretic.
The Freudenthal Suspension Theorem (via the James construction) tells us that $\Sigma: \pi_n(S^n)\to \pi_{n+1}(S^{n+1})$ is an isomorphism for $n \geq 2$ and surjective for $n =1$.
Since $S^1$ is an H-space, the suspension map $\sigma: S^1\to \Omega\Sigma S^1$ has a retraction $r: \Omega\Sigma S^1\to S^1$. Therefore $\Sigma = \sigma_*: \pi_1(S^1)\to \pi_{2}(S^{2})$ is injective (in addition to being surjection).
Now Freudenthal completes the calculation.
Note that we don't just get an abstract isomorphism, we get that these groups are generated by $[\mathrm{id}_{S^n}]$.
EDIT: Regarding getting the James Construction homotopically: the paper
Fantham, Peter; James, Ioan(4-OX); Mather, Michael On the reduced product construction. (English summary) Canad. Math. Bull. 39 (1996), no. 4, 385–389.
derives the relevant properties using the Cube Theorems (which are about the mixing of homotopy pushouts and homotopy pullbakcs) of an earlier paper of Mather's.
Even when computing $\pi_k(S^n), k < n$ all of the hard work, as far as I can tell, comes from showing that continuous maps behave reasonably up to homotopy; there is no difficulty once you show, whatever way you like (simplicial approximation, smooth approximation, etc.), that you can ignore space-filling curves. Until you know that it's necessary to consider the possibility that continuous maps behave in totally ungeometric ways that make them unsuitable for modeling homotopy theory.
Topological spaces and continuous maps are both absurdly general objects, and in fact too general to model homotopy theory: instead of working with the homotopy category of topological spaces we should of course be working with the homotopy category of CW complexes, and of course there are other ways to describe the homotopy category that don't involve topological spaces at all. In some of these descriptions (starting from simplicial sets, I would guess) it may be quite easy to compute $\pi_k(S^n), k \le n$.
Morally this computation should be easy starting from a heuristic picture of $\infty$-groupoids: $S^n$ is the free $\infty$-groupoid on an $n$-morphism, and in particular has no interesting $k$-morphisms for $k < n$. For $k = n$, again heuristically, all you can do starting from an $n$-morphism is compose it with itself a lot; the $\mathbb{Z}$ appearing here is the free group on one generator. Possibly this is the sort of reasoning that homotopy type theory is supposed to make precise.
Edit: That heuristic reasoning above may not seem too convincing because it doesn't seem to say anything about the higher homotopy groups, so let me spell out what it suggests about $\pi_3(S^2)$.
To compute this it suffices to ask what the free $3$-groupoid on a $2$-morphism is. A $3$-category with one object and one $1$-morphism is precisely a braided monoidal category, so the question now is what the free grouplike braided monoidal groupoid on an object $X$ is like. Well, it has a dual $X^{\ast}$ (which must in fact be its inverse) and there are unit and counit maps $1 \to X \otimes X^{\ast}, X^{\ast} \otimes X \to 1$. Finally there is a braiding $X \otimes X^{\ast} \to X^{\ast} \otimes X$. These three maps can be composed, and we get a map $1 \to 1$ which in no way is required to be the identity; in fact it can be identified with the generator of $\pi_3(S^2)$.
Incidentally, the corresponding argument for $\pi_2(S^1)$ fails because we don't have a braiding; the corresponding question is what the free grouplike monoidal groupoid on an object is like. The key is to be extremely careful what the unit and counit look like in a monoidal category which is not assumed to be symmetric, and once we are, the argument correctly suggests that $\pi_2(S^1)$ is trivial.
Edit #2: And the Freudenthal suspension theorem appears here in the fact that for $\pi_{n+1}(S^n), n \ge 3$ the question stabilizes to looking at the free grouplike symmetric monoidal groupoid on an object.
• My question (which I may not be able to pose very well) is: what demands does your heuristic picture place on the underlying category of spaces? Where would the heuristic run aground if nobody had ever invented CW complexes or simplicial sets and we were forced to contend with space filling curves? – Paul Siegel Oct 21 '14 at 9:05
• @Paul: the specific heuristic argument I wrote down above requires more or less that there be a nerve / geometric realization functor $X \mapsto |X|$ from $n$-groupoids to spaces and that this functor have the property that $\pi_k(|X|, x)$ is the automorphism group of the identity automorphism of the identity... of $x$ in $X$. The simplest version of this demand is that we should be able to create Eilenberg-MacLane spaces, although that isn't enough to run the heuristic argument. – Qiaochu Yuan Oct 21 '14 at 17:53
• In other words I'm taking for granted that the homotopy hypothesis provides the correct model-independent description of homotopy theory and that everything else is the quest for usable models, which may or may not involve topological spaces. – Qiaochu Yuan Oct 21 '14 at 17:56
This example and many other illustrate that geometric arguments cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra does not seem to have a simple purely algebraic proof. (I'm out on a limb with this statement.)
It looks to me that a large part of the fundamental functors of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory. I cannot imagine formal arguments, devoid of geometric intuition leading to such concepts. Obviously geometric arguments alone cannot get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact or spectral sequences.
Being a mathematical "mutt" myself, I always favor impure arguments. They give me the comforting feeling of not being isolated. Also, they broaden my sources of inspiration.
One can show that $\Omega^n S^n$ is the group completion of the configuration space of distinct unordered particles in $R^n$. Now $\pi_0$ of this configuration space is the natural numbers. This shows that $\pi_0(\Omega^n S^n)$ is the Grothendieck group of the natural number. Thus, $Z=\pi_0(\Omega^n S^n)=\pi_n(S^n)$.
• The arguments I've seen for this would fail the "internal to the homotopy category" criterion. – Ryan Budney Oct 21 '14 at 4:31
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2019-11-21 22:10:50
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http://ipmu2006.lip6.fr/soumissions.php
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## IPMU 2006 - Instructions for authors, special sessions and track on real applications
### IPMU 2006 - Instructions for presentations
All presentations will be done with a video projector. Computers will be available in all rooms.
### General Information
All submitted papers should present research advances. Contributions will be selected based upon their quality as evaluated by at least two reviewers.
A number of special sessions will be included in the program. Authors will be invited to contribute to these sessions. The special sessions will be chaired by recognized experts in the topics. Proposals for special sessions are welcome to be considered by the program committee.
One full registration will cover acceptance of up to two papers; each additional accepted paper associated with the same registration will be subject to an additional fee.
• Papers Submission Deadline: December 20, 2005
• Notification of acceptance/rejection: March 1, 2006
Each paper accepted for the conference (Oral presentation, track on real applications, poster) must be uploaded on the website (after March 1 and no later than March 31, 2006).
Guidelines
The authors should electronically submit their papers, written either in English or in French, not later than December 20, 2005, through the web upload procedure.
The only allowed formats for the submission are PDF and Postcript.
### Paper length
• Full papers are to be 6 to 8 pages length.
• Papers for the track on real applications are to be 2 to 4 pages length.
• Poster papers are to be 4 pages length.
### Paper Format
All the camera ready papers should be written on two columns with an overall width of 16 cm (7.6 cm each column and 0.8 cm of space between columns). Left and upper margins should be of 2.5 cm, and the length of the text of 24 cm. The normal text should be in Times 11 points. The pages must not be numbered. Paper title is centered. Authors' names are centered too.
Footnotes must be numbered and placed at the bottom of the column where they appear.
Citations must include the reference number between brackets. References must be sorted by the author name.
All figures must be centered. The number and caption of the figure must always appear below the figure.
All tables must be centered and clear. The number and title always appear above the table.
The word acknowledgements must be aligned to the left, not numbered and bolded.
Note for Latex users: You can use your own bibliography file with the ipmu2006.tex file:
1. replace in the ipmu2006.tex the following lines
\begin{thebibliography}{99}
\bibitem{bibli} P. Ipmu (2006). Instructions for Authors. In {\em Proceedings of the
conference IPMU'2006}, volume 2, pages 1-42, Paris, France, July 2006.
\end{thebibliography}
by the lines:
\bibliographystyle{abbrv}
\bibliography{bibliofile}
where bibliofile is replaced by the name of the file with your bibtex entries.
2. compile your file and use bibtex as usual to obtain you pdf file.
### Track on real applications
A track is organized on real applications of the methodologies dealt with by the IPMU 2006 conference. Industrial products and softwares developed by companies or research laboratories are concerned.
The contributors to this special track will not be required to submit a paper for publication in the proceedings. Any material related to the track will be gathered in a separate booklet. Oral presentations by the contributors will be included in the program of the conference. A regular registration allows participants to submit a communication in this track and no additional fee is required.
The schedule is the following:
• Submission of proposals (title, authors, one or two page abstract describing the real application) : February 12, 2006
• Notification of acceptance/rejection: March 1, 2006
• Camera-ready papers (2 to 4 pages) : March 31, 2006
### Poster
A paper accepted in the poster session will be published in a separate booklet that will be given to all participants.
Thus, a paper accepted as a poster should be uploaded on the website. This camera ready paper should be no longuer than 4 pages length, and should respect the general guidelines and paper format given in this page.
During the conference, two poster sessions will be organized. During these sessions, each poster will be displayed on a specific board. The size of the poster should be set to paper size A1 (height: 84 cm x width: 59.4cm).
There will be no oral presentation of a poster paper but the author of the poster will stand near the poster during the poster session in order to answer questions from participants.
### Special Sessions
A special session should be composed by at least 4 papers. All communications must present research advances and no state of the art.
The session organizer can take care of the reviewing process or can ask the communications to follow the standard reviewing process. He must inform the secretariat of his choice. In any case, all communications must be reviewed by at least two reviewers.
### Secretariat IPMU 2006
Secretariat IPMU 2006
Pôle IA, LIP6,
Université Pierre et Marie Curie
8 rue du Capitaine Scott, 75015 PARIS, FRANCE
Fax: +33 1 45 75 08 90
email:
Webmaster: webmaster
Last updated: July 24th, 2009.
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2018-06-22 20:21:52
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https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-12-vectors-and-the-geometry-of-space-12-1-three-dimensional-coordinate-systems-12-1-exercises-page-837/31
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## Calculus 8th Edition
Published by Cengage
# Chapter 12 - Vectors and the Geometry of Space - 12.1 Three-Dimensional Coordinate Systems - 12.1 Exercises - Page 837: 31
#### Answer
Circle of radius 2 centered at $(0,0,-1)$ and parallel to xy-plane.
#### Work Step by Step
The first constraint is the equation that constrains x and y to form a circle pattern, or a cylinder in 3 dimensions. With the z-constraint added, we end with a circle.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2019-12-16 05:41:49
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https://www.wikihow.com/Be-Successful-in-Seventh-Grade
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# How to Be Successful in Seventh Grade
Seventh grade can be a big change for many. It may be your first year of middle school, and you're another step closer to high school. While being successful in seventh grade may seem like a difficult task at first, with some preparation, achieving success is possible.
Method 1
Method 1 of 3:
1. 1
Review and learn seventh grade math concepts.[1] [2] In seventh grade, you'll learn many math concepts that are related to things you learned in sixth grade. Seventh grade math usually encompasses fractions and integers, some geometry, and some pre-algebra. If you're in an advanced math class, you'll usually take algebra 1. (it's common for private schools and kids in advanced math courses to take algebra 1 in seventh grade).
• You will learn many number-related math skills in seventh grade. This includes solving proportions, integer arithmetic (such as ${\displaystyle -5-(+41)}$), and fraction arithmetic (such as ${\displaystyle 3/4+4/9}$).
• You will learn some basic pre-algebra. You'll solve basic equations with variables, such as x + 5 = 9 or 2x = 18. This will progress to solving inequalities, which are very similar to equations. For example, ${\displaystyle x+5>9}$ or ${\displaystyle 2x+32<54}$ are inequalities, as the two sides of the equation are not equal to each other. As usual, move the numbers to one side of the sign and the variables to one side. ${\displaystyle x>4}$ in the 1st equation and ${\displaystyle 2x<22}$ in the 2nd one. Divide both sides by 2 in the 2nd inequality, so ${\displaystyle x<11}$.
• If you're in an advanced class, a whole year of algebra will be taught. At first, you'll review pre-algebra. Then, it will progress to more complicated algebra (e.g. ${\displaystyle x^{2}+6x+55=0}$) and you'll learn how to factor out quadratic equations and how to graph them. Graphed quadratic equations are parabolas. To factor the above example, find which 2 numbers multiply to get ${\displaystyle +5}$ and have a difference of ${\displaystyle +6}$. The numbers are ${\displaystyle 11}$ and ${\displaystyle 6}$. You know that ${\displaystyle x^{2}}$ factored is just ${\displaystyle x}$. So, the factored version is ${\displaystyle (x+5)(x+11)}$ In the end, you may learn a bit of geometry, such as the area and circumference of a circle, scaling drawings, and angle relationships.
2. 2
Learn and review seventh grade grammar and writing concepts. You will learn more grammar concepts in seventh grade than in sixth grade. You might know the basic parts of speech like nouns, verbs, adjectives, adverbs, and prepositions, but there are still more parts of speech to be taught! Your writing will also improve, making sentences flow better and letting your claims be stronger in persuasive essays.
• In grammar, you'll likely learn about misplaced modifiers and fixing run-on sentences. You will probably be able to write compound, complex, and compound-complex sentences by the end of the year in addition to simple sentences. These sentences make your writing flow better instead of just writing choppy, short simple sentences. You may also learn about the different kinds of pronouns. [3]
• In writing, you'll probably write essays. As usual, you'll write with more advanced vocabulary as each year goes on. You will write more persuasive essays as the year goes on. You'll know how to support each of your claims with strong evidence from a text, and to introduce your argument with a thesis. [4] [5]
3. 3
Review and learn seventh grade reading and vocabulary. Seventh grade reading and vocabulary will be pretty similar to what you learned in sixth grade. However, there will be new things learnt, as usual. You may learn some more complex vocabulary words, and read longer books with fascinating plots.
• For reading, read books such as The Giver by Lois Lowry, New Kid by Jerry Craft, Touching Spirit Bear by Ben Mikaelsen, Sea of Trolls by Nancy Farmer, and Where the Red Fern Grows by Wilson Rawls. [6]
• Older books suitable for seventh graders include A Diary of a Young Girl by Anne Frank, Tuck Everlasting by Natalie Babbitt, The Count of Monte Cristo by Alexandre Dumas, The Time Machine by H. G. Wells, and Invisible Man by Ralph Ellison. [7]
4. 4
Review seventh grade science. Each school district teaches a slightly different curriculum, but most teach a combination of biology, physics, chemistry, and earth science. These topics are pretty similar to what you've learned in sixth grade, although the concepts will be more complex. There will also definitely be more terms to memorize!
• For biology, you may learn about genetics and heredity. Review terms like "Punnett squares", "acquired", "traits", "alleles", "dominant", and "recessive".
• For earth science, the classification of rocks may be taught. Know some terms, such as "igneous", "metamorphic", and "sedimentary".
• In physics, you may learn about energy transfers and energy transformations (e.g. from kinetic energy to mechanical energy). Heat transfer may also be taught.
• In chemistry, you may learn different types of elements and chemical bonds. Remember how to identify reactants and products and different chemical reactions.
5. 5
Review and learn seventh grade history. Seventh grade history also depends on the school you're in, so don't expect it to be completely the same as last year. Each school focuses on a different part of history. Some may learn American history, whereas other focus more on geography, modern world history, or the medieval period of world history.
• You might learn about ancient history, such as Ancient Rome, Greece, Egypt, or China. If you're learning about Ancient China, you might get a taste of the important dynasties like the Song, Han, and Tang dynasties. [8]
• You may get to learn about different revolutions, such as the American and French revolutions.
• You may get a taste of 19th century world history during the time of the Industrial Revolution, when people began mass-producing goods in factories. This caused many people to move to the cities to find factory jobs. The people owning the factories frequently mistreated their workers, resulting in strikes.
• Many famous people such as Andrew Carnegie and John D. Rockefeller owned companies that made them one of the wealthiest people in the world at the time.
• Some schools teach early 20th century history. This includes American imperialism in the late 19th century, World War I, and various revolutions such as the Russian Revolution. The teacher will likely cover the causes of World War I, how US affected and expanded its territory, relations with countries at this time, and revolutions.
Method 2
Method 2 of 3:
1. 1
Take notes. Your teachers may not require notes for every subject, but it's still great to take notes using at least a thin notebook. The most important subjects to take notes on are math, science, and history. You can take notes in language arts too, although they'll be shorter, as you will likely only need to take notes for grammar or when reading a book for school. If you're reading an assigned book (e.g. The Diary of a Young Girl or The Giver), your teacher may require reading notes on the characters, plot, setting, and the main ideas of each chapter or section of the book. Jot down key terms and definitions. You should also provide a few examples.
• For math, draw diagrams and do a few example problems in your math notebook. If your school tells you to buy your own school supplies, buy a graph paper notebook. This will make it easier to line up numbers and draw shapes on coordinate planes. Your bar and line graphs will also be neater.
• For history, write down key events that happened, who was involved, and when they happened. For example, you could write, "11/11/1918 – World War I ends".
• Don't write complete sentences for your notes! Unless you're writing down example sentences for grammar, writing whole sentences will waste your time when you want to write down as many points as possible. It will also tire your hand out faster. Use abbreviations, semicolons, bullet points, and fragments of sentences to make your notes concise. Remember, you're not copying down the whole textbook!
2. 2
Study for tests. During earlier grade levels such as third grade, second grade, first grade, and kindergarten, you probably did not study for tests, or at least not as much. Since elementary school concepts are easier to understand and there is usually less material, kids often have an easy time remembering most of the information. As early as third grade, tests may become harder to ace without studying for at least 10 minutes daily. Addition facts are easy to memorize, but using the area of a circle formula? That information is harder to retain in your brain over a long period.
• Another benefit of studying every day is that you are prepared for pop quizzes. If you only study when you have a test coming up, you may not be prepared if there is a pop quiz. If you already study with or without tests, it becomes a habit, and you'll already know the information.
• Try studying with friends before a big test to have some fun and review together.
3. 3
Use a planner. If your school gives you a school supply list, it may already include a planner on it. If your school doesn't and expects students to buy their own supplies, buy one. A planner is essentially a booklet in which you jot down important dates and assignments. These could be related to school, or they could even be about extracurriculars or clubs. Some planners already give you a full calendar with the dates for each school year printed on it, while others allow you to make your own calendar.
• To keep track of important dates, you could write something like, "ACSL club - Fibonacci sequence project due" on the day that project is due, or "Math test (Topic 4)" on the day that test is scheduled for.
• You can also keep separate planners; one for school assignments and one for fun events or extracurriculars. This makes your schedule organized, and you won't get mixed up with what's important and what's not.
4. 4
Be organized. Make sure that on your homework folder, hopefully it has two pockets, that one pocket says "Finished" and "Not Finished". That way, you can keep track of what homework you need to complete and what homework you need to turn in.
• Label your school supplies. On each journal (try a different color for each topic) write on the front cover what subject it is for. You could also put each journal in the subject divider for their subjects. Being organized will help you be the best student you can be!
5. 5
Make a schedule, so you know when you have to work on homework after school.[9]
• Make a slot for mornings, afternoons, and evenings. Make sure to list everything in those slots on what you need to do. In the mornings, list everything you need to do in the morning. The same goes for afternoons and evenings, just make sure to write the times and what you need to do in the slots.
Method 3
Method 3 of 3:
### Social Life
1. 1
Be friendly to everyone. This is why someone who always frowns or annoys people will not get many friends. People who seem uninterested in talking to other people (the kids who always sit alone and do not contribute to conversations) will not get many friends either. Most kids like a friendly person that will help them through challenges and talk about their day with them. Say hi to someone while on the way to class, help someone with their homework, or help clean up the classroom after class. These are all ways to be friendly. Doing this will earn you at least a few friends.
• Contrary to what some kids think, being a big bully/mean girl does not earn you friends. Do not do this, as you don't want to ruin your reputation at the beginning of middle school! Sure, they may seem like friends to you on the outside, but on the inside, they may be as mean as you. They might make friends with you just to gain popularity, so they may fight with you to get the "most popular".
2. 2
Hang out with classmates during breaks and lunch. These are the best times to hang out and get to know each other! If you don't stay in study hall or attend any clubs, you can hang out with your friends during break and lunch. You can even talk while you are going to the bathroom with them. Yes, some kids can talk anywhere; during class, in the bathroom, at lunch, during clubs/study hall, and on the bus to/from school if they use the bus to get to school. That doesn't guarantee that other kids like them though; they may make others annoyed or distracted from their schoolwork. So, only chat during appropriate times. Save conversations for breaks, lunch, and after-school times!
• Whispering or talking to your pal during class or during an inappropriate time (e.g. during an assembly or during a trip to the principal's office) may get you into trouble, so don't do it often. If you do this too frequently, your reputation may downgrade, because you may be known as "too chatty" or "a distraction".
3. 3
Choose an elective. Middle schools often have a wide variety of electives for students to choose. You can choose something that you enjoy or want to learn about. It's normal for interests to jump around during middle school- there's still room for developing interests. During 8th grade and high school, however, you may want to settle down, since you'll need to decide on a career path.
• Middle schools offer a lot of electives- some include journalism, Spanish, engineering, speech and debate, visual arts, performing arts, graphic arts, orchestra, and craft electives.
4. 4
Join multiple clubs in your school. From robotics to newspaper and drama, there will probably be at least one club that you're interested in. People of all grade levels and classes can join clubs, so you will find people that are older than you there. Middle schools will usually have more clubs than elementary schools, so expect there to be variety. These clubs demonstrate your interests and help you gain friends that share the same hobbies as you. For example, if you join the art club, you will find art lovers all over the club. If you attend the robotics club, a bunch of AI enthusiasts will be there.
• Be careful not to join too many clubs, or you won't be able to get any work done. Try and stick to 2 or 3 clubs you're particularly interested in.
• During a club, you can talk to the people sitting next to you. If you're rehearsing lines (say, for performing arts), rehearse with the people in the same scene and/or act as you. Rehearse your lines a couple of times, then talk a bit to get to know each other.
5. 5
Join the student council. There are a few positions in the student council that you can run for: class representative, secretary, vice president, and president. Even though you're not the oldest, you aren't the youngest in your middle school anymore either! This means that the sixth graders will look up to you as trustworthy role models. If you feel like you're responsible and can handle being in the student council, try running! Plus, if you continually run for student council every year, you will have established that you won't give up. If you get elected for a different role each year, you might improve on your leadership skills. If you were elected as class representative in sixth grade and as secretary in seventh, you may be elected as vice president or even the president in eighth grade.
• If you're running for any of these positions, you will need a speech and possibly a campaign poster.
• Write your name in large letters on your poster and make it colorful for others to notice it and actually read what it says. You could also think of a creative slogan. For example, Whig party delegates William Henry Harrison and John Tyler had a catchy slogan in the 1840 U.S. elections; "Tippecanoe and Tyler Too", which was originally published as "Tip and Ty". [10]
6. 6
Deal with having a crush in seventh grade. This could be your first ever love interest or your third crush. You may feel overwhelmed with emotion, whether this is your first crush or your seventh, or whether you know them or not. Sometimes it may be "love at first sight", in which case you may want to check if you actually love this person for more than just their looks. For others, it takes a while to develop a crush by getting to know them better, and that's ok too! You may also be very shy about having a romantic interest, and not know how to express your feelings.
• Firstly, check if you really have a crush. If you want to hug or hold hands with the person (physical affection) and feel overly nervous or excited around them, that's a crush and not just a friend.
• Next, decide whether or not you want to tell them. Some people don't want to let anyone know that they have feelings for the person, and that's perfectly fine. Letting it out is more satisfying though, and you really get to know whether or not they like you romantically or not. Ultimately, it's up to you whether or not you want to confess your feelings.
• Understand if your crush rejects you that it is not the end of the world! Don't despair- you will meet many more romantic interests over the course of middle school, high school, and college. Take time to grieve and focus on yourself.
## Expert Q&A
200 characters left
## Tips
• Eat healthy. Don't eat too much junk food[11]
⧼thumbs_response⧽
• If you have a locker, feel free to decorate it with mirrors, notepads, and inspirational quotes! Avoid using sticky substances such as tape or glue in your locker, especially if your school doesn't allow it.
⧼thumbs_response⧽
• Doing your homework and submitting it by the due date is also a great way to have awesome grades.
⧼thumbs_response⧽
## Warnings
• If possible, avoid bullies. They will just destroy your self-esteem.
⧼thumbs_response⧽
• If you see someone hurt, or being bullied, tell an adult immediately.
⧼thumbs_response⧽
• Don't pick friends that will treat you rudely.
⧼thumbs_response⧽
• Don't be rude to others. If you are, you're not going to have a lot of friends.
⧼thumbs_response⧽
## Things You'll Need
• Pencils (both wooden or mechanical pencils are fine)
• Pens
• Highlighters
• Erasers
• Some extra colored pencils
• Some extra crayons
• A pair of scissors
• A glue stick
• A ruler
• A protractor (to measure angles with)
• A compass (to draw circles)
• A pencil case
• Plain white paper
• A graph paper notebook or individual sheets of graph paper (for math when drawing on coordinate planes)
• At least 3 notebooks (depends on how many subjects are taught)
• A planner (to keep track of assignments, tests, and due dates)
• Some folders for classwork and homework
• School books (depends on the school, so check your supply list to make sure!)
• A binder (to put papers into)
• A large backpack (a rolling backpack or a regular one can be fine)
• A reusable water bottle to prevent yourself from getting dehydrated
• Any medication you may need, like an asthma inhaler, diabetes medication and pumps, ADHD medication, an EpiPen for allergic reactions, etc.
• Period supplies (a few tampons, pads, extra underwear, wet wipes, period pain medication for cramps)
• A pack of small tissues (in case you need to wipe their hands, face, or nose)
• A small bottle of hand sanitizer (especially important during the coronavirus pandemic)
Co-authored by:
School Counselor
This article was co-authored by Ashley Pritchard, MA. Ashley Pritchard is an Academic and School Counselor at Delaware Valley Regional High School in Frenchtown, New Jersey. Ashley has over 3 years of high school, college, and career counseling experience. She has an MA in School Counseling with a specialization in Mental Health from Caldwell University and is certified as an Independent Education Consultant through the University of California, Irvine. This article has been viewed 30,932 times.
Co-authors: 20
Updated: January 1, 2023
Views: 30,932
Thanks to all authors for creating a page that has been read 30,932 times.
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2023-02-02 19:14:26
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|
https://fluxml.ai/Flux.jl/dev/models/basics/
|
# How Flux Works: Gradients and Layers
Flux's core feature is taking gradients of Julia code. The gradient function takes another Julia function f and a set of arguments, and returns the gradient with respect to each argument. (It's a good idea to try pasting these examples in the Julia terminal.)
julia> using Flux
julia> f(x) = 3x^2 + 2x + 1;
julia> df(x) = gradient(f, x)[1]; # df/dx = 6x + 2
julia> df(2)
14.0
julia> d2f(x) = gradient(df, x)[1]; # d²f/dx² = 6
julia> d2f(2)
6.0
When a function has many parameters, we can get gradients of each one at the same time:
julia> f(x, y) = sum((x .- y).^2);
julia> gradient(f, [2, 1], [2, 0])
([0.0, 2.0], [-0.0, -2.0])
These gradients are based on x and y. Flux works by instead taking gradients based on the weights and biases that make up the parameters of a model.
Machine learning often can have hundreds of parameter arrays. Instead of passing them to gradient individually, we can store them together in a structure. The simplest example is a named tuple, created by the following syntax:
julia> nt = (a = [2, 1], b = [2, 0], c = tanh);
julia> g(x::NamedTuple) = sum(abs2, x.a .- x.b);
julia> g(nt)
1
(a = [0.0, 2.0], b = [-0.0, -2.0], c = nothing)
Notice that gradient has returned a matching structure. The field dg_nt.a is the gradient for nt.a, and so on. Some fields have no gradient, indicated by nothing.
Rather than define a function like g every time (and think up a name for it), it is often useful to use anonymous functions: this one is x -> sum(abs2, x.a .- x.b). Anonymous functions can be defined either with -> or with do, and such do blocks are often useful if you have a few steps to perform:
julia> gradient((x, y) -> sum(abs2, x.a ./ y .- x.b), nt, [1, 2])
((a = [0.0, 0.5], b = [-0.0, -1.0], c = nothing), [-0.0, -0.25])
julia> gradient(nt, [1, 2]) do x, y
z = x.a ./ y
sum(abs2, z .- x.b)
end
((a = [0.0, 0.5], b = [-0.0, -1.0], c = nothing), [-0.0, -0.25])
Sometimes you may want to know the value of the function, as well as its gradient. Rather than calling the function a second time, you can call withgradient instead:
julia> Flux.withgradient(g, nt)
(val = 1, grad = ((a = [0.0, 2.0], b = [-0.0, -2.0], c = nothing),))
Flux used to handle many parameters in a different way, using the params function. This uses a method of gradient which takes a zero-argument function, and returns a dictionary through which the resulting gradients can be looked up:
julia> x = [2, 1];
julia> y = [2, 0];
julia> gs = gradient(Flux.params(x, y)) do
f(x, y)
end
julia> gs[x]
2-element Vector{Float64}:
0.0
2.0
julia> gs[y]
2-element Vector{Float64}:
-0.0
-2.0
## Building Simple Models
Consider a simple linear regression, which tries to predict an output array y from an input x.
W = rand(2, 5)
b = rand(2)
predict(x) = W*x .+ b
function loss(x, y)
ŷ = predict(x)
sum((y .- ŷ).^2)
end
x, y = rand(5), rand(2) # Dummy data
loss(x, y) # ~ 3
To improve the prediction we can take the gradients of the loss with respect to W and b and perform gradient descent.
using Flux
gs = gradient(() -> loss(x, y), Flux.params(W, b))
Now that we have gradients, we can pull them out and update W to train the model.
W̄ = gs[W]
W .-= 0.1 .* W̄
loss(x, y) # ~ 2.5
The loss has decreased a little, meaning that our prediction x is closer to the target y. If we have some data we can already try training the model.
All deep learning in Flux, however complex, is a simple generalisation of this example. Of course, models can look very different – they might have millions of parameters or complex control flow. Let's see how Flux handles more complex models.
## Building Layers
It's common to create more complex models than the linear regression above. For example, we might want to have two linear layers with a nonlinearity like sigmoid (σ) in between them. In the above style we could write this as:
using Flux
W1 = rand(3, 5)
b1 = rand(3)
layer1(x) = W1 * x .+ b1
W2 = rand(2, 3)
b2 = rand(2)
layer2(x) = W2 * x .+ b2
model(x) = layer2(σ.(layer1(x)))
model(rand(5)) # => 2-element vector
This works but is fairly unwieldy, with a lot of repetition – especially as we add more layers. One way to factor this out is to create a function that returns linear layers.
function linear(in, out)
W = randn(out, in)
b = randn(out)
x -> W * x .+ b
end
linear1 = linear(5, 3) # we can access linear1.W etc
linear2 = linear(3, 2)
model(x) = linear2(σ.(linear1(x)))
model(rand(5)) # => 2-element vector
Another (equivalent) way is to create a struct that explicitly represents the affine layer.
struct Affine
W
b
end
Affine(in::Integer, out::Integer) =
Affine(randn(out, in), randn(out))
# Overload call, so the object can be used as a function
(m::Affine)(x) = m.W * x .+ m.b
a = Affine(10, 5)
a(rand(10)) # => 5-element vector
Congratulations! You just built the Dense layer that comes with Flux. Flux has many interesting layers available, but they're all things you could have built yourself very easily.
(There is one small difference with Dense – for convenience it also takes an activation function, like Dense(10 => 5, σ).)
## Stacking It Up
It's pretty common to write models that look something like:
layer1 = Dense(10 => 5, σ)
# ...
model(x) = layer3(layer2(layer1(x)))
For long chains, it might be a bit more intuitive to have a list of layers, like this:
using Flux
layers = [Dense(10 => 5, σ), Dense(5 => 2), softmax]
model(x) = foldl((x, m) -> m(x), layers, init = x)
model(rand(10)) # => 2-element vector
Handily, this is also provided for in Flux:
model2 = Chain(
Dense(10 => 5, σ),
Dense(5 => 2),
softmax)
model2(rand(10)) # => 2-element vector
This quickly starts to look like a high-level deep learning library; yet you can see how it falls out of simple abstractions, and we lose none of the power of Julia code.
A nice property of this approach is that because "models" are just functions (possibly with trainable parameters), you can also see this as simple function composition.
m = Dense(5 => 2) ∘ Dense(10 => 5, σ)
m(rand(10))
Likewise, Chain will happily work with any Julia function.
m = Chain(x -> x^2, x -> x+1)
m(5) # => 26
## Layer Helpers
There is still one problem with this Affine layer, that Flux does not know to look inside it. This means that Flux.train! won't see its parameters, nor will gpu be able to move them to your GPU. These features are enabled by the @functor macro:
Flux.@functor Affine
Finally, most Flux layers make bias optional, and allow you to supply the function used for generating random weights. We can easily add these refinements to the Affine layer as follows, using the helper function create_bias:
function Affine((in, out)::Pair; bias=true, init=Flux.randn32)
W = init(out, in)
b = Flux.create_bias(W, bias, out)
Affine(W, b)
end
Affine(3 => 1, bias=false, init=ones) |> gpu
|
2023-02-06 02:17:40
|
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|
https://www.semanticscholar.org/paper/Anomalous-dimension-in-a-two-species-system-Vollmayr-Lee-Hanson/9248323244137e8a9ce0d5466127dca8e2d632d8
|
# Anomalous dimension in a two-species reaction–diffusion system
@article{VollmayrLee2017AnomalousDI,
title={Anomalous dimension in a two-species reaction–diffusion system},
author={Benjamin P. Vollmayr-Lee and Jack Hanson and R Scott McIsaac and Joshua D. Hellerick},
journal={Journal of Physics A: Mathematical and Theoretical},
year={2017},
volume={51}
}
• Published 17 August 2017
• Physics
• Journal of Physics A: Mathematical and Theoretical
We study a two-species reaction–diffusion system with the reactions A+A→(0,A) and A+B→A, with general diffusion constants DA and DB. Previous studies showed that for dimensions d⩽2 the B particle density decays with a nontrivial, universal exponent that includes an anomalous dimension resulting from field renormalization. We demonstrate via renormalization group methods that the scaled B particle correlation function has a distinct anomalous dimension resulting in the asymptotic scaling C~BB(r…
5 Citations
• Physics
Journal of Physics A: Mathematical and Theoretical
• 2020
We make two corrections to the renormalization group calculation presented in Vollmayr-Lee et al [1]. First, the field renormalization technique presented is not applicable for the B particle density
• Physics
• 2020
We study fluctuation effects in the two-species reaction-diffusion system A + B → O and A + A → (O, A ). In contrast to the usually assumed ordinary short-range diffusion spreading of the reactants
• Physics
Physical review. E
• 2020
A computer simulation method is developed, motivated by the technique of Mehra and Grassberger, that determines the complete probability distribution of the B particles for a given realization of the A-particle dynamics, thus providing a significant increase in the quality of statistics.
• Physics
• 2019
Two-species reaction diffusion system $$A+B\rightarrow A$$ and $$A+A\rightarrow (\emptyset ,A)$$ is studied in presence of long-range spreading. Long-range hops are described by Levy flights, i.e. by
• Mathematics
Journal of Mathematical Physics
• 2019
In this paper, we employ the renormalization group method to study the long-time asymptotics of solutions to a class of nonlinear integral equations with a generalized heat kernel. The nonlinearities
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Physical review. E, Statistical, nonlinear, and soft matter physics
• 2003
The renormalization group can be used to calculate the time dependence of the density of particles, and provides both an exact value for the exponent governing the decay of particles and an epsilon expansion for the amplitude of this power law.
• Physics
• 1997
We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A → ∅ and A → (m + 1) A, where m ≥ 1.
The diffusion-controlled reaction kA + M is known IO be strongly dependent on Ruchlations in dimensions d < d. = 2/(k - 1). We develop a field-theoretic renormalization group approach to this system
• Physics
• 2005
We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction–diffusion problems. We first investigate the physical origin of universality
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2006
The principal tool of the study is the dynamical renormalization group and it is concluded that the epsilon corrections of order two and higher are absent in the previous answer for Pt(N, Delta V) for N=1, 2, 3, 4.
• Physics
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
• 1994
The kinetics of diffusion-controlled heterogeneous single-species annihilation, where the diffusivity of each particle may be different, is investigated, and the theoretical predictions compare well with both Monte Carlo simulations and time series expansions.
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2003
This work calculates P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations, and derives an exact nonperturbative relation between the exponents: namely delta(Q)=theta(1-Q).
• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2002
The Monte Carlo results for the two-species trapping reaction A+B-->B with diffusing A and B on lattices in one, two, and three dimensions are presented, showing that the asymptotic regime has not been reached, at least for d=2 and d=3.
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2023-03-23 12:24:18
|
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|
https://socratic.org/questions/how-much-work-would-it-take-to-push-a-4-kg-weight-up-a-3-m-plane-that-is-at-an-i
|
# How much work would it take to push a 4 kg weight up a 3 m plane that is at an incline of pi / 4 ?
Apr 17, 2017
$83.156 j$
$F = m g \sin \theta$ where $m = 4 k g , g = 9.8 \frac{m}{s} ^ 2 \mathmr{and} \theta = \frac{\pi}{4}$
work $= 4 \cdot 9.8 \cdot \sin \left(\frac{\pi}{4}\right) \cdot 3 = 4 \cdot 9.8 \cdot 0.7071 \cdot 3 = 83.156 j$
|
2019-09-22 23:29:51
|
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|
https://itecnotes.com/electrical/electronic-reducing-volume-for-a-split-phase-230vac60hz-sealed-power-supply-providing-5vdc50ma/
|
# Electronic – Reducing volume for a split-phase 230VAC@60Hz sealed power supply providing 5VDC@50mA
capactivepowerpower supplysplit-phasetransformerless
I'm a hobbyist interested in re-designing and implementing a DC power supply providing a \$+5V_{DC}\pm 200mV\$ output with a current compliance of \$50mA\$ (I could live with \$30mA\$ successfully) from a \$230V_{AC}\$ @ 60Hz split-phase US mains supply. This supply will be completely sealed and there will be no access at all to any conductors outside the sealed package, except for the AC input wires. The DC is used with a microcontroller, which itself will also be inside this sealed package. So there is no way anyone can make any contact with either the DC ground or the \$+5V_{DC}\$ lines. Think of it as a brick with only the two hot AC power input wires, plus ground, leaving the brick. (It is actually a micro that accepts fiber optic signals and controls four SCRs as part of a hybrid relay system gating the \$230V_{AC}\$ for an H1/H2 output. But that's a longer story.)
My goal is to come up with an approach that uses the least volume while using reasonably available parts and/or ones I can reasonably fabricate. (Reasonable is something I can't define well, but it tends to mean that I don't want to use parts that are very expensive or very difficult to come by in hobby quantities.)
This opens up a few questions:
1. Just use a transformer to get the voltage down close, followed with a bridge, etc. I'm open to the idea of a custom wired transformer, but I suspect I'm going to run into difficulties because of the volt-seconds (Webers) that the core will have to sustain when operating with 60Hz. I think the volume is going to be huge. I need to delve into the specifics and I haven't done that, yet. The question here is: Is my intuition correct that the transformer volume will be large (compared to answers for the following questions), despite my modest current requirements?
2. Use a capacitive driven power supply. You can see an example of what I'm thinking about, given an existing design I posted here about two years ago: Split-phase \$230V_{AC}\$ transformerless capacitive power supply for \$5V@30mA\$ regulated output — series cap?. No answers developed from that one, but a similar design does work satisfactorily. A problem here is that the capacitor should be non-polar and rated for such use. These are large. This is the reference design I'd like to shrink. The question here, other than providing a reference for the other questions, is: I'm using a \$450V_{AC}\$ rated non-polar X2 capacitor now. Are there reasonable alternative capacitor choices I might consider that are smaller and yet rated for this exact kind of use?
3. Use some specialized IC designed specifically for this purpose. Those I've examined are complex to design and not so well tailored to my modest ~\$30mA\$ (to \$50mA\$) requirements, enough so that I'm once again not sure about the resulting volume. It would take quite some time to gather these up and test specific designs and given that none I've found are specifically designed for very low currents and voltages a question arises here. Are there any ICs that an experienced designer might suggest I look at, which has a promise of showing a total volume less than what is typical for designs as in (2) above and doesn't use overly exotic parts different to obtain or too expensive? (I'm looking here to narrow the work I have ahead. Not a perfect answer here. I can explore what is offered. So I'm just looking for reasonably educated IC suggestions, and not whole circuits using them.)
(In answering the above, I'm able to wind and test transformers and coils and I do have access to oscilloscopes, lab supplies, and the usual tools that a modest hobbyist might accumulate. And feel free to criticize the earlier design approach, too. I'm just a hobbyist and will take my lumps, with pleasure.)
The target volume is something smaller than \$1 \, in^3\$. The existing design uses commonly available parts.
This is the smartest offline regulator that fits your requirements for size, simplicity and low cost. The regulator only charges the cap line rectified input is <32Vdc. It is adjustable.
Since your requirements do not include cost, availability and are only looking for a prototype quantity and you don't have access to Far East suppliers, and the interface requirements to the grid and heat loss in a small thermally insulated 1" cube, your options will always be a compromise. The easiest one to adjust is volume, your artificial requirement.
The simplest is a bridge and a series self-healing X1 cap. The load Cap will act as a voltage divider from 340Vp to 8Vp which is then clamped with a 5V Zener The load is a combination of the duty cycle of pulsed current to charge the cap and linear current to discharge the cap.
A unregulated output should result RC=8*10ms for 100Hz 10% ripple. As a check your load 5V@50mA=250mW will have loss in LDO and bridge. So cap choices must ensure Vin(min) is >5V with 0.1V drop easy to match.
This is easy at 50mA but values of C escalate quickly with load current beyond 50mA making size and cost undesireable.
IF load is 300mW at 5V including LDO =50mW for a target and 250mW for a load, and 230Vac RMS/5Vdc RMS= 46 than VAR value of series X1 cap must be at least 46x1/4W ~ 12VAR @ 100Hz.
We can approximate this with VAR=1/2CV^2 @100Hz although the rectified signal is rich in harmonics. Thus 12[VAR]=E=1/2*C(230)^2 ... C(X1)=450uF. hmm Doesn't sound right.
|
2022-12-01 10:33:44
|
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|
https://piping-designer.com/index.php/properties/fluid-mechanics/2766-dewpoint-temperature?tmpl=component&print=1
|
# Dew Point Temperature
Written by Jerry Ratzlaff on . Posted in Fluid Dynamics
Dew point, abbreviated as $$T_d$$, is the temperature the air needs to be cooled to (at constant pressure) in order to achive a relative humidity of 100%.
## Dew point Temperature formula
$$\large{ T_d = \left( \frac{ RH }{ 100 } \right)^{ \frac{1}{8} } \; \left( 112 + 0.9 \; T \right) + 0.1 \; T - 112 }$$
Symbol English Metric
$$\large{ T_d }$$ = dew point temperature $$\large{F}$$ $$\large{C}$$
$$\large{ RH }$$ = relative humidity $$\large{F}$$ $$\large{C}$$
$$\large{ T }$$ = temperature $$\large{F}$$ $$\large{C}$$
|
2022-11-28 18:30:57
|
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|
https://byjus.com/question-answer/why-is-ice-at-273-k-more-effective-in-cooling-than-water-at-the-same-1/
|
Question
# Why is ice at $$273\ K$$ more effective in cooling than water at the same temperature?
Solution
## Cooling takes place when heat is removed. In the case of ice at $$273 K$$ ($$0^0C$$), it will first take the heat (the latent heat of melting) to convert itself into the water at $$273 K$$ Whereas, water at $$273 K$$ will absorb heat lesser than the latent heat of melting of ice and also its phase won't change until the temperature reaches $$373K$$ ($$100^0C$$).So, at the same temperature of $$273 K$$, a large amount of heat will be removed by ice from the surroundings than in the case of water. Hence, ice at $$273 K$$ is more effective in cooling than water at the same temperature. ScienceNCERTStandard IX
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2022-01-28 04:44:02
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http://www.sawaal.com/logical-venn-diagram-questions-and-answers/in-a-dinner-party-both-fish-and-meat-were-served-some-took-only-fish-and-some-only-meat-there-were-s_8598
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40
Q:
# In a dinner party both fish and meat were served. Some took only fish and some only meat. There were some vegetarians who did not accept either. The rest accepted both fish and meat. Which of the following Venn-diagrams correctly reflects this situation?
A) 1 B) 2 C) 3 D) 4
Explanation:
The Given situation can be represented as under:
Q:
Out of 120 students in aschool, 5% can play all the three games Cricket, Chess and Carroms. If so happens that the number of players who can play any and only two games is 30. The number of students who can play the Cricket alone is 40. What is the total number of those who can play Chess alone or Carroms alone ?
A) 45 B) 44 C) 46 D) 24
Explanation:
Given U=120
5% of 120 = 6
$\therefore$ Students who can play Chess alone or Carroms alone = 120 - (30+40+6)=44
8 35
Q:
Select from the five alternative diagrams, the one that best illustrates the relationship among the three classes : Truck, Ship, Goods
A) 1 B) 3 C) 4 D) 5
Explanation:
Truck and Ship are entirely different. But some Goods are carried by some Trucks and some Goods are carried by some Ships.
61 839
Q:
Select from four alternative diagrams, the one that best illustrates the relationship among the three classes : Pigeons, Birds, Dogs
A) 1 B) 2 C) 3 D) 4
Explanation:
All Pigeons are Birds. But,Dogs are entairly different.
57 918
Q:
Which of the following Venn- diagram correctly illustrates the relation ship among the classes : Tennis fans, Cricket players, Students
A) 1 B) 2 C) 3 D) 4
Explanation:
Some Students can be Cricket players. Some Cricket players can be Tennis fans. Some Students can be Tennis fans. So, the given items are partly related to each other.
78 2963
Q:
Which of the following Venn- diagram correctly illustrates the relation ship among the classes : Carrot, Food, Vegetables
A) a B) b C) c D) d
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2017-01-24 13:27:14
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https://blog.csdn.net/trochiluses/article/details/14517379
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# 1.git push产生冲突的形成过程
X------A
hyk@hyk-linux:~/xfstests/.git (master)
$cat refs/remotes/origin/master 56a3959a96f1b5e046b3760778fd34b4911d0516 本地refs： hyk@hyk-linux:~/xfstests/.git (master)$ cat refs/heads/master
# 2.方案一：强制覆盖
There is another common situation where you may encounter non-fast-forward rejection when you try to push, and it is possible even when you are pushing into a repository nobody else pushes into. After you push commit A yourself (in the first picture in this section), replace it with "git commit --amend" to produce commit B, and you try to push it out, because forgot that you have pushed A out already.
# 3.方案二：形成merge形式的提交历史
B---C
/ /
---X---A
# 4.方案三：形成线性的提交历史
------X-------A--------D(the diff of X and B)
• 广告
• 抄袭
• 版权
• 政治
• 色情
• 无意义
• 其他
120
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2019-01-23 22:01:07
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https://damask.mpie.de/Documentation/Plasticity?cover=print;
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The purpose of the plasticity constitutive law is to introduce in the computation the dynamics of plastic flow, which means to calculate the plastic shear strain rate of every slip system $\dot{\gamma}^\alpha$ as a function of the second Piola–Kirchhoff stress $\tnsr S$.
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2019-05-19 11:40:21
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https://qiskit.org/documentation/locale/ja_JP/stubs/qiskit.circuit.Instruction.html
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# qiskit.circuit.Instruction¶
class Instruction(name, num_qubits, num_clbits, params, duration=None, unit='dt')[ソース]
Generic quantum instruction.
Create a new instruction.
パラメータ
• name (str) – instruction name
• num_qubits (int) – instruction’s qubit width
• num_clbits (int) – instruction’s clbit width
• params (list[int|float|complex|str|ndarray|list|ParameterExpression]) – list of parameters
• duration (int or float) – instruction’s duration. it must be integer if unit is 『dt』
• unit (str) – time unit of duration
CircuitError – when the register is not in the correct format.
__init__(name, num_qubits, num_clbits, params, duration=None, unit='dt')[ソース]
Create a new instruction.
パラメータ
• name (str) – instruction name
• num_qubits (int) – instruction’s qubit width
• num_clbits (int) – instruction’s clbit width
• params (list[int|float|complex|str|ndarray|list|ParameterExpression]) – list of parameters
• duration (int or float) – instruction’s duration. it must be integer if unit is 『dt』
• unit (str) – time unit of duration
CircuitError – when the register is not in the correct format.
Methods
__init__(name, num_qubits, num_clbits, params) Create a new instruction. add_decomposition(decomposition) Add a decomposition of the instruction to the SessionEquivalenceLibrary. Assemble a QasmQobjInstruction broadcast_arguments(qargs, cargs) Validation of the arguments. c_if(classical, val) Add classical condition on register classical and value val. copy([name]) Copy of the instruction. Invert this instruction. Return True .IFF. DEPRECATED: use instruction.reverse_ops(). Return a default OpenQASM string for the instruction. Creates an instruction with gate repeated n amount of times. For a composite instruction, reverse the order of sub-instructions. soft_compare(other) Soft comparison between gates. validate_parameter(parameter) Instruction parameters has no validation or normalization.
Attributes
decompositions Get the decompositions of the instruction from the SessionEquivalenceLibrary. definition Return definition in terms of other basic gates. duration Get the duration. params return instruction params. unit Get the time unit of duration.
add_decomposition(decomposition)[ソース]
Add a decomposition of the instruction to the SessionEquivalenceLibrary.
assemble()[ソース]
Assemble a QasmQobjInstruction
broadcast_arguments(qargs, cargs)[ソース]
Validation of the arguments.
パラメータ
• qargs (List) – List of quantum bit arguments.
• cargs (List) – List of classical bit arguments.
Tuple(List, List) – A tuple with single arguments.
CircuitError – If the input is not valid. For example, the number of arguments does not match the gate expectation.
c_if(classical, val)[ソース]
Add classical condition on register classical and value val.
copy(name=None)[ソース]
Copy of the instruction.
パラメータ
name (str) – name to be given to the copied circuit, if None then the name stays the same.
a copy of the current instruction, with the name
updated if it was provided
qiskit.circuit.Instruction
property decompositions
Get the decompositions of the instruction from the SessionEquivalenceLibrary.
property definition
Return definition in terms of other basic gates.
property duration
Get the duration.
inverse()[ソース]
Invert this instruction.
If the instruction is composite (i.e. has a definition), then its definition will be recursively inverted.
Special instructions inheriting from Instruction can implement their own inverse (e.g. T and Tdg, Barrier, etc.)
a fresh instruction for the inverse
qiskit.circuit.Instruction
CircuitError – if the instruction is not composite and an inverse has not been implemented for it.
is_parameterized()[ソース]
Return True .IFF. instruction is parameterized else False
mirror()[ソース]
DEPRECATED: use instruction.reverse_ops().
a new instruction with sub-instructions
reversed.
qiskit.circuit.Instruction
property params
return instruction params.
qasm()[ソース]
Return a default OpenQASM string for the instruction.
Derived instructions may override this to print in a different format (e.g. measure q[0] -> c[0];).
repeat(n)[ソース]
Creates an instruction with gate repeated n amount of times.
パラメータ
n (int) – Number of times to repeat the instruction
Containing the definition.
qiskit.circuit.Instruction
CircuitError – If n < 1.
reverse_ops()[ソース]
For a composite instruction, reverse the order of sub-instructions.
This is done by recursively reversing all sub-instructions. It does not invert any gate.
a new instruction with
sub-instructions reversed.
qiskit.circuit.Instruction
soft_compare(other)[ソース]
Soft comparison between gates. Their names, number of qubits, and classical bit numbers must match. The number of parameters must match. Each parameter is compared. If one is a ParameterExpression then it is not taken into account.
パラメータ
other (instruction) – other instruction.
are self and other equal up to parameter expressions.
bool
property unit
Get the time unit of duration.
validate_parameter(parameter)[ソース]
Instruction parameters has no validation or normalization.
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2021-05-16 09:38:02
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https://www.astronomyclub.xyz/strange-stars/a-1.html
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## A
Uj being a single-particle potential acting on a nucleon j. The definition of Uj is important for the convergence of the BBG expansion series, where Hi is treated as a perturbation. As the system is spatially uniform, Uj is constant in space, and unperturbed nucleon states are plane waves \p). For simplicity, we omit spin indices and express nucleon momenta p in units of h. In momentum representation, the unperturbed single-particle energy is h2p2
19The linked cluster theorem, formulated in terms of diagrams, which were latter named "Goldstone diagrams", and the equation governing the wave function of a nucleon pair in nuclear matter (Bethe-Goldstone equation) were all derived during the graduate studies of Goldstone at Trinity College of the Cambridge University (Cambridge, England).
where N = n or p. Because of the isotropy of the nuclear matter, which is assumed to be spin-unpolarized, single-particle energies are independent of momentum direction and nucleon spin. However, owing to a neutron excess, the single-particle potentials for neutrons and protons are different and the G-matrix is no longer charge symmetric (Gnn = Gpp). As we will see below, Un can be expressed in terms of the G-matrix, the central quantity of the BBG theory. Its calculation is equivalent to the summation of the "ladder diagrams" of the BBG expansion series. It is performed by solving the integral equation, which can be written in the operator form:
Here, QNN' is the two-particle exclusion-principle operator, which projects particle states outside the Fermi surface, and hNN' is the Hamiltonian operator acting on uncorrelated two-particle states,
while 2 is the starting energy parameter. In the low-density limit we get hNN'\P1P2) —► 2m(pI + p2)\PIP2) , Qnn' —► 1 . (5.32)
In this case the G-matrix equation transforms into the well known equation for the scattering T-matrix, which describes the NN scattering in vacuum (see, e.g., Messiah 1961, vol. II, Chapter XIX, §14). Passing to the momentum representation, we get
(P1P2IGW (z)|PlP2) = (pIpAVnn'\PlP2) dk\ dk2 (2n) Qnn' (ki, k2)
In G-matrix elements relevant for calculating the ground-state energy, the starting energy is the sum of single-particle energies of the initial \p1p2) state:
with single-particle potentials given in terms of the G-matrix via a Hartree-Fock expression
+2 / dp3(pip3\G nn(en (pi) + en(p3))\pip3)a , (5.35a)
+2 / dp3(pip3\Gpp(ep(pi) +ep(p3))\pip3)a • (5.35b) Jp
Here, for the sake of compactness, we use the notations
\pip2)a = \pip2) - \p2pi) , J^ dp = J (2^3 @(kFN - p) • (5.36)
As we are dealing with the spin-unpolarized system, we can use the spin-averaged G-matrix. The spin degeneracy gives a factor of two in front of the integrals.
The auxiliary single-particle potential term U(p), Eq. (5.27), crucial for the convergence of the linked-cluster expansion, deserves an additional comment. The choice of U (p) for states above the Fermi surface (p > pF) has been a subject of a long debate since the formulation of the BBG theory. Eventually, the so called continuous prescription for U has been regarded as the most advantageous. According to this prescription, no energy gap is introduced between the energies of occupied (p < pF) and empty (p > pF) momentum states.20 Such a choice turns out to be particularly suitable in view of the rapid convergence of the BBG expansion (see, e.g., Baldo et al. 2000,2001). The lowest-order BBG approximation for the energy density E (without nucleon rest energy contribution) is given by the Hartree-Fock expression, where the G-matrix acts as an effective interaction. This justifies the name "Brueckner-Hartree-Fock" (BHF) approximation, used by many authors. The BHF expression for E reads
E = EFFG(nn,np) + 2 J dpi j dp2(piP2\Gpp(ep(pi) + ep(p2))|pip2)a dpi / dp2(piP2|G nn(en (Pi) + en(P2))\pip2)a n
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2020-01-23 08:18:20
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https://web2.0calc.com/questions/please-help_25299
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+0
# please help!
+1
185
2
please help!!!
Feb 16, 2019
### 2+0 Answers
#1
0
please??
Feb 16, 2019
#2
+2343
+2
If we assume that "the graph consists of three line segments," then we can generate the equation of the lines.
Of course, I could use $$y=mx+b$$, but that would require me to know the y-intercept of the first red line, and, unfortunately, the y-intercept is not clear-cut in the image.
I will use point-slope form instead; this form requires me to know the coordinates of any two points on that line. $$y-y_1=m(x-x_1)$$ is the form of point-slope form. $$(x_1,y_1)$$ is the coordinate of any one point of the line. In the image, I can pinpoint that $$(-4,4)\text{ and }(-1,-1)$$ both lie on the line I care about for this problem. Now that they are identified, I can now find the slope.
$$m=\frac{-1-4}{-1-(-4)}=\frac{-5}{3}$$
I will substitute in the point $$(-1,-1)$$ as my point.
Since we are trying to find $$f(-2)$$ , x=-2.
$$y_1=-1;m=-\frac{5}{3};x=-2;x_1=-1\\ y-y_1=m(x-x_1)\\ y-(-1)=-\frac{5}{3}(-2-(-1))$$ It is time to solve for y! $$y+1=-\frac{5}{3}*-1\\ y+\frac{3}{3}=\frac{5}{3}\\ y=\frac{2}{3}$$
$$f(-2)=\frac{2}{3}$$. You will see that this answer is consistent with the initially given diagram.
Feb 16, 2019
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2019-09-21 22:24:06
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https://ftp.aimsciences.org/article/doi/10.3934/naco.2012.2.167
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# American Institute of Mathematical Sciences
2012, 2(1): 167-185. doi: 10.3934/naco.2012.2.167
## A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection
1 Laboratory of Theoretical and Applied Computer Science (LITA), Paul Verlaine - Metz University, Ile du Saulcy, 57045, Metz, France 2 Laboratory of Theorical and Applied computer Science LITA, UFR MIM, University Paul Verlaine of Metz, Ile du Saulcy-Metz 57045, France 3 Laboratory of Mathematics. National Institute for Applied Sciences, Rouen BP 08, Place Emile Blondel F 76131, Mont Saint Aignan Cedex, France
Received March 2011 Revised October 2011 Published March 2012
In this paper, we consider a class of bilevel programming problems where the upper objective function is convex quadratic while the lower objective function and the constraints are linear. The problem is first rewritten as minimizing a convex quadratic function subject to linear constraints and one concave constraint. Then, we use an exact penalty technique to reformulate the problem as a DC program. Afterward, DCA, an efficient algorithm in nonconvex programming, is developed to solve the resulting problem.
For globally solving the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). DCA is applied to compute upper bounds, while lower bounds are calculated via a DC relaxation of the DC constraint. The proposed algorithms, DCA and BB-DCA, are compared with the Branch-and-Bound algorithm without DCA (BB) on random data. The numerical results show that the proposed algorithms are efficient.
Finally, we consider an application of portfolio selection problems with the historical data related to the prices in the market of France, Luxembourg, United States and England during the period 2000-2007. The experimental results confirm the efficiency and the rapidity of DCA.
Citation: Le Thi Hoai An, Tran Duc Quynh, Pham Dinh Tao. A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 167-185. doi: 10.3934/naco.2012.2.167
##### References:
[1] G. J. Alexander and A. M. Baptista, A Portfolio selection with a drawdown constraint, Journal of Banking & Finance, 30 (2006), 3171-3189. doi: 10.1016/j.jbankfin.2005.12.006. [2] J. F. Bard, Some properties of the bilevel programming problem, Journal of Optimization Theory and Applications, 68 (1991), 371-378. doi: 10.1007/BF00941574. [3] B. Colson, P. Marcotte and G. Savard, Bilevel programming: a survey, A Quarterly Journal of Operations Research, 3 (2005), 87-107. [4] , DC Programming and DCA:, \url{http://lita.sciences.univ-metz.fr/~lethi/DCA.html}., (). [5] T. Pham Dinh and H. A. Le Thi, Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355. [6] T. Pham Dinh and H. A. Le Thi, DC optimization algorihms for solving the trust region subproblem, SIAM J. Optimization, 8 (1998), 476-505. doi: 10.1137/S1052623494274313. [7] T. Pham Dinh, N. Nguyen Canh and H. A. Le Thi, An efficient combination of DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs, Journal of Global Optimization, 48 (2010), 595-632. doi: 10.1007/s10898-009-9507-y. [8] P. Hansen, B. Jaumard and G. Savard, New branch-and-bound rules for for linear bilevel programming, SIAM Journal on Scientific and Statistical Computing, 13 (1992), 1194-1217. doi: 10.1137/0913069. [9] R. Horst and V. T. Nguyen, DC Programming: Overview, Journal of Optimization Theory and Application, 101 (1999), 1-43. doi: 10.1023/A:1021765131316. [10] D. M. Le and V. T. Nguyen, A global optimization method for solving convex quadratic bilevel programming problems, Journal of Global Optimization, 26 (2003), 199-219. doi: 10.1023/A:1023047900333. [11] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-99. doi: 10.2307/2975974. [12] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, First edition, (1970). [13] Marc C. Steinbach, Markowitz revisited mean variance model in financial protfolio a nalysis, SIAM review, 43 (2001), 31-85. [14] H. A. Le Thi, Contribution à l'optimisation non convex and l'optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des recherches, Université de Rouen, 1997. [15] H. A. Le Thi and T. Pham Dinh, A Continuous approach for globally solving linearly constrained quadratic zero-one programming problem, Optimization, 50 (2001), 93-120. [16] H. A. Le Thi, T. Pham Dinh, N. Nguyen Canh and T. Nguyen Van, DC programming techniques for solving a class of nonlinear bilevel programs, Journal of Global Optimization, 44 (2009), 313-337. doi: 10.1007/s10898-008-9325-7. [17] H. A. Le Thi and T. Pham Dinh, The DC (difference of convex functions) Programming and DCA revisited with DC models of real world non convex optimization problems, Annals of Operations Research, 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [18] H. A. Le Thi, T. Pham Dinh and V. N. Huynh, Exact penalty and error bounds in DC programming,, to appear in Journal of Global Optimization., (). [19] H. A. Le Thi, T. Pham Dinh and V. N. Huynh, Convergence analysis of DCA for DC programming with subanalytic data, Research Report (2009), National Institute for Applied Sciences, Rouen. [20] H. A. Le Thi and T. Pham Dinh, Large scale global molecular optimization from exact distance matrices by a d.c. optimization approach, SIAM Journal on Optimization, 14 (2003), 77-114. doi: 10.1137/S1052623498342794. [21] H. A. Le Thi, T. Pham Dinh and D. M. Le, Numerical solution for optimization over the efficient set by DC optimization algorithms, Operation Research Letters, 19 (1996), 117-128. doi: 10.1016/0167-6377(96)00022-3. [22] H. Tuy, "Convex Analysis and Global Optimization," Kluwer Academic Publisher, First edition, 1997.
show all references
##### References:
[1] G. J. Alexander and A. M. Baptista, A Portfolio selection with a drawdown constraint, Journal of Banking & Finance, 30 (2006), 3171-3189. doi: 10.1016/j.jbankfin.2005.12.006. [2] J. F. Bard, Some properties of the bilevel programming problem, Journal of Optimization Theory and Applications, 68 (1991), 371-378. doi: 10.1007/BF00941574. [3] B. Colson, P. Marcotte and G. Savard, Bilevel programming: a survey, A Quarterly Journal of Operations Research, 3 (2005), 87-107. [4] , DC Programming and DCA:, \url{http://lita.sciences.univ-metz.fr/~lethi/DCA.html}., (). [5] T. Pham Dinh and H. A. Le Thi, Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355. [6] T. Pham Dinh and H. A. Le Thi, DC optimization algorihms for solving the trust region subproblem, SIAM J. Optimization, 8 (1998), 476-505. doi: 10.1137/S1052623494274313. [7] T. Pham Dinh, N. Nguyen Canh and H. A. Le Thi, An efficient combination of DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs, Journal of Global Optimization, 48 (2010), 595-632. doi: 10.1007/s10898-009-9507-y. [8] P. Hansen, B. Jaumard and G. Savard, New branch-and-bound rules for for linear bilevel programming, SIAM Journal on Scientific and Statistical Computing, 13 (1992), 1194-1217. doi: 10.1137/0913069. [9] R. Horst and V. T. Nguyen, DC Programming: Overview, Journal of Optimization Theory and Application, 101 (1999), 1-43. doi: 10.1023/A:1021765131316. [10] D. M. Le and V. T. Nguyen, A global optimization method for solving convex quadratic bilevel programming problems, Journal of Global Optimization, 26 (2003), 199-219. doi: 10.1023/A:1023047900333. [11] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-99. doi: 10.2307/2975974. [12] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, First edition, (1970). [13] Marc C. Steinbach, Markowitz revisited mean variance model in financial protfolio a nalysis, SIAM review, 43 (2001), 31-85. [14] H. A. Le Thi, Contribution à l'optimisation non convex and l'optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des recherches, Université de Rouen, 1997. [15] H. A. Le Thi and T. Pham Dinh, A Continuous approach for globally solving linearly constrained quadratic zero-one programming problem, Optimization, 50 (2001), 93-120. [16] H. A. Le Thi, T. Pham Dinh, N. Nguyen Canh and T. Nguyen Van, DC programming techniques for solving a class of nonlinear bilevel programs, Journal of Global Optimization, 44 (2009), 313-337. doi: 10.1007/s10898-008-9325-7. [17] H. A. Le Thi and T. Pham Dinh, The DC (difference of convex functions) Programming and DCA revisited with DC models of real world non convex optimization problems, Annals of Operations Research, 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [18] H. A. Le Thi, T. Pham Dinh and V. N. Huynh, Exact penalty and error bounds in DC programming,, to appear in Journal of Global Optimization., (). [19] H. A. Le Thi, T. Pham Dinh and V. N. Huynh, Convergence analysis of DCA for DC programming with subanalytic data, Research Report (2009), National Institute for Applied Sciences, Rouen. [20] H. A. Le Thi and T. Pham Dinh, Large scale global molecular optimization from exact distance matrices by a d.c. optimization approach, SIAM Journal on Optimization, 14 (2003), 77-114. doi: 10.1137/S1052623498342794. [21] H. A. Le Thi, T. Pham Dinh and D. M. Le, Numerical solution for optimization over the efficient set by DC optimization algorithms, Operation Research Letters, 19 (1996), 117-128. doi: 10.1016/0167-6377(96)00022-3. [22] H. Tuy, "Convex Analysis and Global Optimization," Kluwer Academic Publisher, First edition, 1997.
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Impact Factor:
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2022-05-19 12:46:41
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https://eduzip.com/ask/question/let-sleft-xyx-2-2xyy-2-3x-3y20-right-then-s-520119
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Mathematics
# Let $S=\left\{ { (x,y)|x^{ 2 }+2xy+y^{ 2 }-3x-3y+2=0 } \right\}$, then $S-$
consists of two parallel lines which are not coincident.
##### SOLUTION
Given,
$x^{ 2 }+2xy+y^{ 2 }-3x-3y+2=0$
or, ${(x+y)^2 }-3(x+y)+2=0$
or, $(x+y-2)(x+y-1)=0$.
or, $x+y=2$ and $x+y=1$ these are the straight lines represented by $S$ and these straight lines are parallel.
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Single Correct Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86
#### Realted Questions
Q1 Subjective Medium
Find the complement of the following angle:
$60^\circ$
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q2 Subjective Medium
Two complementary angles differ by $10^{o}$. Find the angles
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q3 Single Correct Medium
The angle which makes a linear pair with an angle of $61^o$ is of
• A. $29^o$
• B. $61^o$
• C. $122^o$
• D. $119^o$
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q4 Subjective Medium
Name the pairs of supplementary angles in the above figure:
Asked in: Mathematics - Lines and Angles
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Q5 Subjective Medium
Write down the measures of some acute angles.(give at least two examples)
Asked in: Mathematics - Lines and Angles
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2022-01-18 18:53:55
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https://en.wikipedia.org/wiki/Burning_Ship_fractal
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# Burning Ship fractal
High-quality overview image of the Burning Ship fractal
High-quality image of the large ship in the left antenna
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:
${\displaystyle z_{n+1}=(|\operatorname {Re} \left(z_{n}\right)|+i|\operatorname {Im} \left(z_{n}\right)|)^{2}+c,\quad z_{0}=0}$
in the complex plane ${\displaystyle \mathbb {C} }$ which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations.[1]
## Implementation
Animation of a continuous zoom-out to show the amount of detail for an implementation with 64 maximum iterations
The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code. Note that the typical images of the Burning Ship fractal display the ship upright: the actual fractal, and that produced by the below pseudocode, is inverted along the x-axis.
1 For each pixel (x, y) on the screen, do:
2 {
3 x = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
4 y = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
5
6
7 zx = x; // zx represents the real part of z
8 zy = y; // zy represents the imaginary part of z
9
10
11 iteration = 0
12 max_iteration = 1000
13
14 while (zx*zx + zy*zy < 4 AND iteration < max_iteration)
15 {
16 xtemp = zx*zx - zy*zy + x
17 zy = abs(2*zx*zy + y) //abs returns the absolute value
18 zx = abs(xtemp)
19
20 iteration = iteration + 1
21 }
22
23 if (iteration == max_iteration) //Belongs to the set
24 return insideColor;
25
26 return iteration * color;
27 }
## References
1. ^ Michael Michelitsch and Otto E. Rössler (1992). "The "Burning Ship" and Its Quasi-Julia Sets". In: Computers & Graphics Vol. 16, No. 4, pp. 435–438, 1992. Reprinted in Clifford A. Pickover Ed. (1998). Chaos and Fractals: A Computer Graphical Journey — A 10 Year Compilation of Advanced Research. Amsterdam, Netherlands: Elsevier. ISBN 0-444-50002-2
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2019-12-10 05:47:56
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https://pypi.org/project/zfs-tools/0.5.1/
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ZFS synchronization and snapshotting tools
# ZFS tools
Donate to support this free software
1Cw9nZu9ygknussPofMWCzmSMveusTbQvN
The ZFS backup tools will help you graft an entire ZFS pool as a filesystem into a backup machine, without having to screw around snapshot names or complicated shell commands or crontabs.
The utilities let you do this:
1. zfs-shell:
a shell that allows remote ZFS administration and nothing more
2. zsnap:
a command that snapshots a dataset or pool, then deletes old snapshots
3. zreplicate
a command that replicates an entire dataset tree using ZFS replication streams. Best used in combination with zsnap as in:
• zsnap on the local machine
• zreplicate from the local machine to the destination machine
Obsolete snapshots deleted by zsnap will be automatically purged on the destination machine by zreplicate, as a side effect of using replication streams. To inhibit this, use the --no-replication-stream option.
Run zreplicate --help for a compendium of options you may use.
4. zbackup: a command to snapshot and replicate filesystems according to their user properties. This uses zsnap and zreplicate to do the work, which is all driven by properties. For details, see this further description of zbackup.
The repository, bug tracker and Web site for this tool is at http://github.com/Rudd-O/zfs-tools. Comments to me through rudd-o@rudd-o.com.
## Setting up
Setup is rather complicated. It assumes that you already have ZFS running and vaults on both the machine you're going to back up and the machine that will be receiving the backup.
### On the machine to back up
• Install the zfs-shell command
cp zfs-shell /usr/local/sbin
chmod 755 /usr/local/sbin/zfs-shell
chown root.root /usr/local/sbin/zfs-shell
• Create a user with a home directory and shell zfs-shell
useradd -rUm -b /var/lib -s /usr/local/sbin/zfs-shell zfs
• Let sudo know that the new user can run the zfs command
zfs ALL = NOPASSWD: /usr/local/sbin/zfs
(ensure you remove the requiretty default on /etc/sudoers) (check sudoers.zfs-tools in contrib/ for an example)
• Set up a cron job to run zsnap as frequently as you want to, snapshotting the dataset you intend to replicate.
### On the backup machine
• Set up public key authentication for SSH so the backup machine may log as the user zfs (as laid out above) in the machine to be backed up.
• Create a dataset to receive the backup stream.
• Set up a cron job to fetch the dataset snapshotted by zsnap from the remote machine into the newly created dataset. You will use zreplicate for that (see below for examples).
• After the first replication, you may want to set the mountpoint attributes on the received datasets so they do not automount on the backup machine.
### Test
If all went well, you should be able to do this without issue:
(on the machine to back up)
[root@peter]
zsnap senderpool
[root@paul]
zfs create receiverpool/senderpool # <--- run this ONLY ONCE
# this should send the entire senderpool with all snapshots
# over from peter to paul, placing it in receiverpool/senderpool
(on the machine to back up)
[root@peter]
zsnap senderpool
[root@paul]
# this should send an incremental stream of senderpool
And that's it, really.
## Project details
Uploaded source
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2022-10-02 22:57:49
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https://brilliant.org/problems/fun-with-circles-2/
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# Fun with Circles
Level pending
(Not to scale)
If a circle with a diameter of 7cm rolls around a circle with a radius of 6cm so that the first circle rotates half a revolution, find the distance between the starting point and the finish point (x to 2 decimal places).
×
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2017-12-15 08:37:14
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https://www.yaclass.in/p/mathematics-state-board/class-10/geometry-11420/pythagoras-theorem-13075/re-0c42bf4c-a103-4070-affa-c62fd8736806
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LEARNATHON
III
Competition for grade 6 to 10 students! Learn, solve tests and earn prizes!
### Theory:
Statement:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Explanation:
The theorem states that in the right angled triangle $$ABC$$, $$AC^2=AB^2+BC^2$$.
Proof of the theorem:
Given:
A triangle right angled at $$B$$.
That is $$\angle ABC$$ $$=$$ $$90^{\circ}$$.
To prove:
$$AC^2=AB^2+BC^2$$
Construction:
Construct a line from $$B$$ to $$AC$$ to intersect at $$D$$ such that $$BD \perp AC$$.
Proof:
Consider the triangles $$ABC$$ and $$BDC$$.
If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
By the theorem, we have $$\Delta ABC$$ $$\sim$$ $$\Delta BDC$$.
Hence, the ratio of the corresponding sides of the triangles are equal.
That is, $$\frac{BC}{CD} = \frac{AC}{BC}$$.
This implies, $$BC^{2} = AC \times CD$$ ……$$(1)$$
Now consider the triangles $$ABC$$ and $$ABD$$.
Similarly, by the above mentioned theorem we have $$\Delta ABC$$ $$\sim$$ $$\Delta ABD$$.
Hence, the ratio of the corresponding sides of the triangles are equal.
So, $$\frac{AB}{AD} = \frac{AC}{AB}$$.
This implies, $$AB^{2} = AC \times AD$$ ……$$(2)$$
Add equations $$(1)$$ and $$(2)$$ as follows:
$$BC^2 + AB^2$$ $$=$$ $$(AC \times CD) + (AC \times AD)$$
$$=$$ $$AC (CD +AD)$$
$$=$$ $$AC \cdot AC$$
$$=$$ $$AC^2$$.
Therefore, $$AC^2 = AB^2 + BC^2$$.
Hence, the proof.
Example:
In a right angled triangle, if the measure of the hypotenuse is $$29 cm$$ and one of its sides is $$21$$ $$cm$$ then, find the length of the other side.
Solution:
Let the triangle be $$ABC$$ right angled at $$B$$.
This implies that the side $$AC$$ is the hypotenuse.
By the Pythagorean theorem, we have $$AC^2 = AB^2 + BC^2$$.
Thus, $$AB^2 = AC^2 - BC^2$$.
$$\Rightarrow AB^2 = 29^2 -21^2$$
$$= 841 - 441$$
$$= 400$$
Hence, $$AB = \sqrt{400}$$.
$$AB = 20$$
Therefore, the length of the other side is $$20$$ $$cm$$.
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2022-01-17 07:11:59
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http://mathhelpforum.com/pre-calculus/187590-domain-problem-inverse-problem.html
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# Thread: Domain Problem and Inverse Problem
1. ## Domain Problem and Inverse Problem
I need a little help here !!!
3) Find the domain of
f(x) = sqrt x / (ln (x^2 + 1) - 17) i don't know how to do this ..please use union
4) Find inverse f(x) = e^x^3 - 2 the -2 is not beside 3 it is beside x....i got cube root 3 of lnx + 2
Thanks
2. ## Re: Domain Problem and Inverse Problem
"Domain" is usally defined in terms of the GREATEST set of values. This makes it most convenient, very often, to consider a very GREAT set of values. Try starting with "All Real Numbers" and see what must be discarded.
ln(x) requires x > 0
sqrt(x) requires x >= 0
1/x requires x <> 0
What say you?
3. ## Re: Domain Problem and Inverse Problem
Originally Posted by qwerty999
I need a little help here !!!
3) Find the domain of
f(x) = sqrt x / (ln (x^2 + 1) - 17) i don't know how to do this ..please use union
4) Find inverse f(x) = e^x^3 - 2 the -2 is not beside 3 it is beside x....i got cube root 3 of lnx + 2
Thanks
Assuming you mean " $f(x)= e^{x^3}}- 2$" and not " $f(x)= (e^x)^3- 2$" which would be more simply written " $f(x)= e^{3x}- 2$", and assuming you mean "cube root of (ln(x+ 2))" and not "cube root of (ln(x)+ 2)" or "cube root of (ln(x))+ 2", yes, that is correct.
Please use parentheses to clarify what you mean!
But what in the world is "cube root 3 of "??? I assume you mean just "cube root". "third root" would mean the same. Never use the unfortunate notation "root 3" or worse "squareroot 3" which I have seen before. Both of those would mean "square root of 3 times...".
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2017-12-17 09:02:51
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http://jsmith.cis.byuh.edu/books/principles-of-general-chemistry-v1.0m/s17-solutions.html
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This is “Solutions”, chapter 13 from the book Principles of General Chemistry (v. 1.0M). For details on it (including licensing), click here.
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Chapter 13 Solutions
We explored the general properties of gases, liquids, and solids in Chapter 10 "Gases", Chapter 11 "Liquids", and Chapter 12 "Solids", respectively. Most of the discussion focused on pure substances containing a single kind of atom, molecule, or cation–anion pair. The substances we encounter in our daily lives, however, are usually mixtures rather than pure substances. Some are heterogeneous mixtures, which consist of at least two phases that are not uniformly dispersed on a microscopic scale; others are homogeneous mixtures, consisting of a single phase in which the components are uniformly distributed. (For more information about homogeneous mixtures, see Chapter 1 "Introduction to Chemistry", Section 1.3 "A Description of Matter".) Homogeneous mixtures are also called solutionsA homogeneous mixture of two or more substances in which the substances present in lesser amounts (the solutes) are dispersed uniformly throughout the substance present in greater amount (the solvent).; they include the air we breathe, the gas we use to cook and heat our homes, the water we drink, the gasoline or diesel fuel that powers engines, and the gold and silver jewelry we wear.
Beads of oil in water. When a nonpolar liquid such as oil is dispersed in a polar solvent such as water, it does not dissolve, but forms spherical beads. Oil is insoluble in water because the intermolecular interactions within the solute (oil) and the solvent (water) are stronger than the intermolecular interactions between the solute and the solvent.
Many of the concepts that we will use in our discussion of solutions were introduced in earlier chapters. In Chapter 4 "Reactions in Aqueous Solution", for example, we described reactions that occur in aqueous solution and how to use molarity to describe concentrations. In Chapter 4 "Reactions in Aqueous Solution", Chapter 7 "The Periodic Table and Periodic Trends", and Chapter 11 "Liquids", we introduced the principles that govern ion–ion and molecule–molecule interactions in pure substances; similar interactions also occur in solutions. Now we use the principles developed in those chapters to understand the factors that determine how much of one substance can dissolve in another, and how the properties of a solution differ from those of its components.
The properties of mixtures of gases were described in Chapter 10 "Gases", and the properties of certain types of solid solutions, such as alloys and doped semiconductors, were discussed in Chapter 12 "Solids". This chapter focuses on liquid solutions, aqueous or otherwise. By the end of this chapter, your understanding of solutions will enable you to explain why the radiator in your car must contain ethylene glycol to avoid damage to the engine on cold winter nights, why salt is spread on icy roads in the winter (and why it isn’t effective when the temperature is too low), why certain vitamins accumulate in your body at toxic levels while others are rapidly excreted, and how salt can be removed from seawater to provide drinking water.
13.1 Factors Affecting Solution Formation
Learning Objective
1. To understand how enthalpy and entropy changes affect solution formation.
In all solutions, whether gaseous, liquid, or solid, the substance present in the greatest amount is the solvent, and the substance or substances present in lesser amounts are the solute(s). The solute does not have to be in the same physical state as the solvent, but the physical state of the solvent usually determines the state of the solution. As long as the solute and solvent combine to give a homogeneous solution, the solute is said to be soluble in the solvent. Table 13.1 "Types of Solutions" lists some common examples of gaseous, liquid, and solid solutions and identifies the physical states of the solute and solvent in each.
Table 13.1 Types of Solutions
Solution Solute Solvent Examples
gas gas gas air, natural gas
liquid gas liquid seltzer water (CO2 gas in water)
liquid liquid liquid alcoholic beverage (ethanol in water), gasoline
liquid solid liquid tea, salt water
solid gas solid H2 in Pd (used for H2 storage)
solid solid liquid mercury in silver or gold (amalgam often used in dentistry)
Forming a Solution
The formation of a solution from a solute and a solvent is a physical process, not a chemical one. That is, both solute and solvent can be recovered in chemically unchanged forms using appropriate separation methods. For example, solid zinc nitrate dissolves in water to form an aqueous solution of zinc nitrate:
Equation 13.1
Because Zn(NO3)2 can be recovered easily by evaporating the water, this is a physical process. In contrast, metallic zinc appears to dissolve in aqueous hydrochloric acid. In fact, the two substances undergo a chemical reaction to form an aqueous solution of zinc chloride with evolution of hydrogen gas:
Equation 13.2
Zn(s) + 2H+(aq) + 2Cl(aq) → Zn2+(aq) + 2Cl(aq) + H2(g)
Note the Pattern
Dissolution of a solute in a solvent to form a solution does not involve a chemical transformation.
When the solution evaporates, we do not recover metallic zinc, so we cannot say that metallic zinc is soluble in aqueous hydrochloric acid because it is chemically transformed when it dissolves. The dissolution of a solute in a solvent to form a solution does not involve a chemical transformation.
Substances that form a single homogeneous phase in all proportions are said to be completely miscibleCapable of forming a single homogeneous phase, regardless of the proportions with which the substances are mixed. in one another. Ethanol and water are miscible, just as mixtures of gases are miscible. If two substances are essentially insoluble in each other, such as oil and water, they are immiscible. Examples of gaseous solutions that we have already discussed include Earth’s atmosphere (see Chapter 3 "Chemical Reactions") and natural gas (see Chapter 10 "Gases").
The Role of Enthalpy in Solution Formation
As we saw in Chapter 10 "Gases"Chapter 12 "Solids", energy is required to overcome the intermolecular interactions in a solute. This energy can be supplied only by the new interactions that occur in the solution, when each solute particle is surrounded by particles of the solvent in a process called solvationThe process of surrounding each solute particle with particles of solvent., or hydrationThe process of surrounding solute particles with water molecules. when the solvent is water. Thus all of the solute–solute interactions and many of the solvent–solvent interactions must be disrupted for a solution to form. In this section, we describe the role of enthalpy in this process.
Because enthalpy is a state function, we can use the same type of thermochemical cycle described in Chapter 5 "Energy Changes in Chemical Reactions" to analyze the energetics of solution formation. (For more information about state functions, see Chapter 5 "Energy Changes in Chemical Reactions", Section 5.2 "Enthalpy".) The process occurs in three discrete steps, indicated by ΔH1, ΔH2, and ΔH3 in Figure 13.1 "Enthalpy Changes That Accompany the Formation of a Solution". The overall enthalpy change in the formation of the solution (ΔHsoln) is the sum of the enthalpy changes in the three steps:
Equation 13.3
ΔHsoln = ΔH1 + ΔH2 + ΔH3
When a solvent is added to a solution, steps 1 and 2 are both endothermic because energy is required to overcome the intermolecular interactions in the solvent (ΔH1) and the solute (ΔH2). Because ΔH is positive for both steps 1 and 2, the solute–solvent interactions (ΔH3) must be stronger than the solute–solute and solvent–solvent interactions they replace in order for the dissolution process to be exothermic (ΔHsoln < 0). When the solute is an ionic solid, ΔH2 corresponds to the lattice energy that must be overcome to form a solution. As you learned in Chapter 12 "Solids", the higher the charge of the ions in an ionic solid, the higher the lattice energy. Consequently, solids that have very high lattice energies, such as MgO (−3791 kJ/mol), are generally insoluble in all solvents.
Figure 13.1 Enthalpy Changes That Accompany the Formation of a Solution
Solvation can be an exothermic or endothermic process depending on the nature of the solute and solvent. In both cases, step 1, separation of the solvent particles, is energetically uphill (ΔH1 > 0), as is step 2, separation of the solute particles (ΔH2 > 0). In contrast, energy is released in step 3 (ΔH3 < 0) because of interactions between the solute and solvent. (a) When ΔH3 is larger in magnitude than the sum of ΔH1 and ΔH2, the overall process is exothermic (ΔHsoln < 0), as shown in the thermochemical cycle. (b) When ΔH3 is smaller in magnitude than the sum of ΔH1 and ΔH2, the overall process is endothermic (ΔHsoln > 0).
As you will see in Chapter 18 "Chemical Thermodynamics", a positive value for ΔHsoln does not mean that a solution will not form. Whether a given process, including formation of a solution, occurs spontaneously depends on whether the total energy of the system is lowered as a result. Enthalpy is only one of the contributing factors. A high ΔHsoln is usually an indication that the substance is not very soluble. Instant cold packs used to treat athletic injuries, for example, take advantage of the large positive ΔHsoln of ammonium nitrate during dissolution (+25.7 kJ/mol), which produces temperatures less than 0°C (Figure 13.2 "Commercial Cold Packs for Treating Injuries").
Figure 13.2 Commercial Cold Packs for Treating Injuries
These packs contain solid NH4NO3 and water in separate compartments. When the seal between the compartments is broken, the NH4NO3 dissolves in the water. Because ΔHsoln for NH4NO3 is much greater than zero, heat is absorbed by the cold pack during the dissolution process, producing local temperatures less than 0°C.
Entropy and Solution Formation
The enthalpy change that accompanies a process is important because processes that release substantial amounts of energy tend to occur spontaneously. A second property of any system, its entropy, is also important in helping us determine whether a given process occurs spontaneously. We will discuss entropy in more detail in Chapter 18 "Chemical Thermodynamics", but for now we can state that entropy(S)The degree of disorder in a thermodynamic system. The greater the number of possible microstates for a system, the higher the entropy. is a thermodynamic property of all substances that is proportional to their degree of disorder. A perfect crystal at 0 K, whose atoms are regularly arranged in a perfect lattice and are motionless, is arbitrarily assigned an entropy of zero. In contrast, gases have large positive entropies because their molecules are highly disordered and in constant motion at high speeds.
The formation of a solution disperses molecules, atoms, or ions of one kind throughout a second substance, which generally increases the disorder and results in an increase in the entropy of the system. Thus entropic factors almost always favor formation of a solution. In contrast, a change in enthalpy may or may not favor solution formation. The London dispersion forces that hold cyclohexane and n-hexane together in pure liquids, for example, are similar in nature and strength. Consequently, ΔHsoln should be approximately zero, as is observed experimentally. Mixing equal amounts of the two liquids, however, produces a solution in which the n-hexane and cyclohexane molecules are uniformly distributed over approximately twice the initial volume. In this case, the driving force for solution formation is not a negative ΔHsoln but rather the increase in entropy due to the increased disorder in the mixture. All spontaneous processes with ΔH ≥ 0 are characterized by an increase in entropy. In other cases, such as mixing oil with water, salt with gasoline, or sugar with hexane, the enthalpy of solution is large and positive, and the increase in entropy resulting from solution formation is not enough to overcome it. Thus in these cases a solution does not form.
Note the Pattern
All spontaneous processes with ΔH ≥ 0 are characterized by an increase in entropy.
Table 13.2 "Relative Changes in Enthalpies for Different Solute–Solvent Combinations*" summarizes how enthalpic factors affect solution formation for four general cases. The column on the far right uses the relative magnitudes of the enthalpic contributions to predict whether a solution will form from each of the four. Keep in mind that in each case entropy favors solution formation. In two of the cases the enthalpy of solution is expected to be relatively small and can be either positive or negative. Thus the entropic contribution dominates, and we expect a solution to form readily. In the other two cases the enthalpy of solution is expected to be large and positive. The entropic contribution, though favorable, is usually too small to overcome the unfavorable enthalpy term. Hence we expect that a solution will not form readily.
Table 13.2 Relative Changes in Enthalpies for Different Solute–Solvent Combinations*
ΔH1 (separation of solvent molecules) ΔH2 (separation of solute particles) ΔH3 (solute–solvent interactions) ΔHsolnH1 + ΔH2H3) Result of Mixing Solute and Solvent
large; positive large; positive large; negative small; positive or negative solution will usually form
small; positive large; positive small; negative large; positive solution will not form
large; positive small; positive small; negative large; positive solution will not form
small; positive small; positive small; negative small; positive or negative solution will usually form
H1, ΔH2, and ΔH3 refer to the processes indicated in the thermochemical cycle shown in Figure 13.1 "Enthalpy Changes That Accompany the Formation of a Solution".
In all four cases, entropy increases.
In contrast to liquid solutions, the intermolecular interactions in gases are weak (they are considered to be nonexistent in ideal gases). Hence mixing gases is usually a thermally neutral process (ΔHsoln ≈ 0), and the entropic factor due to the increase in disorder is dominant (Figure 13.3 "Formation of a Solution of Two Gases"). Consequently, all gases dissolve readily in one another in all proportions to form solutions. We will return to a discussion of enthalpy and entropy in Chapter 18 "Chemical Thermodynamics", where we treat their relationship quantitatively.
Figure 13.3 Formation of a Solution of Two Gases
(top) Pure samples of two different gases are in separate bulbs. (bottom) When the connecting stopcock is opened, diffusion causes the two gases to mix together and form a solution. Even though ΔHsoln is zero for the process, the increased entropy of the solution (the increased disorder) versus that of the separate gases favors solution formation.
Example 1
Considering LiCl, benzoic acid (C6H5CO2H), and naphthalene, which will be most soluble and which will be least soluble in water?
Given: three compounds
Asked for: relative solubilities in water
Strategy:
Assess the relative magnitude of the enthalpy change for each step in the process shown in Figure 13.1 "Enthalpy Changes That Accompany the Formation of a Solution". Then use Table 13.2 "Relative Changes in Enthalpies for Different Solute–Solvent Combinations*" to predict the solubility of each compound in water and arrange them in order of decreasing solubility.
Solution:
The first substance, LiCl, is an ionic compound, so a great deal of energy is required to separate its anions and cations and overcome the lattice energy (ΔH2 is far greater than zero in Equation 13.3). Because water is a polar substance, the interactions between both Li+ and Cl ions and water should be favorable and strong. Thus we expect ΔH3 to be far less than zero, making LiCl soluble in water. In contrast, naphthalene is a nonpolar compound, with only London dispersion forces holding the molecules together in the solid state. We therefore expect ΔH2 to be small and positive. We also expect the interaction between polar water molecules and nonpolar naphthalene molecules to be weak ΔH3 ≈ 0. Hence we do not expect naphthalene to be very soluble in water, if at all. Benzoic acid has a polar carboxylic acid group and a nonpolar aromatic ring. We therefore expect that the energy required to separate solute molecules (ΔH2) will be greater than for naphthalene and less than for LiCl. The strength of the interaction of benzoic acid with water should also be intermediate between those of LiCl and naphthalene. Hence benzoic acid is expected to be more soluble in water than naphthalene but less soluble than LiCl. We thus predict LiCl to be the most soluble in water and naphthalene to be the least soluble.
Exercise
Considering ammonium chloride, cyclohexane, and ethylene glycol (HOCH2CH2OH), which will be most soluble and which will be least soluble in benzene?
Answer: The most soluble is cyclohexane; the least soluble is ammonium chloride.
Summary
Solutions are homogeneous mixtures of two or more substances whose components are uniformly distributed on a microscopic scale. The component present in the greatest amount is the solvent, and the components present in lesser amounts are the solute(s). The formation of a solution from a solute and a solvent is a physical process, not a chemical one. Substances that are miscible, such as gases, form a single phase in all proportions when mixed. Substances that form separate phases are immiscible. Solvation is the process in which solute particles are surrounded by solvent molecules. When the solvent is water, the process is called hydration. The overall enthalpy change that accompanies the formation of a solution, ΔHsoln, is the sum of the enthalpy change for breaking the intermolecular interactions in both the solvent and the solute and the enthalpy change for the formation of new solute–solvent interactions. Exothermic (ΔHsoln < 0) processes favor solution formation. In addition, the change in entropy, the degree of disorder of the system, must be considered when predicting whether a solution will form. An increase in entropy (a decrease in order) favors dissolution.
Key Takeaway
• The magnitude of the changes in both enthalpy and entropy must be considered when predicting whether a given solute–solvent combination will spontaneously form a solution.
Conceptual Problems
1. Classify each of the following as a heterogeneous mixture or homogeneous mixture. Explain your rationale in each case.
1. aqueous ammonia
2. liquid decongestant
3. vinegar
4. seawater
5. gasoline
6. fog
2. Solutions and heterogeneous mixtures are at the extreme ends of the solubility scale. Name one type of mixture that is intermediate on this scale. How are the properties of the mixture you have chosen different from those of a solution or a heterogeneous mixture?
3. Classify each process as simple dissolution or a chemical reaction.
1. a naphthalene mothball dissolving in benzene
2. a sample of a common drain cleaner that has a mixture of NaOH crystals and Al chunks dissolving in water to give H2 gas and an aqueous solution of Na+, OH, and Al3+ ions
3. an iron ship anchor slowly dissolving in seawater
4. sodium metal dissolving in liquid ammonia
4. Classify each process as simple dissolution or a chemical reaction.
1. a sugar cube dissolving in a cup of hot tea
2. SO3 gas dissolving in water to produce sulfuric acid
3. calcium oxide dissolving in water to produce a basic solution
4. metallic gold dissolving in a small quantity of liquid mercury
5. You notice that a gas is evolved as you are dissolving a solid in a liquid. Will you be able to recover your original solid by evaporation? Why or why not?
6. Why is heat evolved when sodium hydroxide pellets are dissolved in water? Does this process correspond to simple dissolution or a chemical reaction? Justify your answer.
7. Which process(es) is the simple formation of a solution, and which process(es) involves a chemical reaction?
1. mixing an aqueous solution of NaOH with an aqueous solution of HCl
2. bubbling HCl gas through water
3. adding iodine crystals to CCl4
4. adding sodium metal to ethanol to produce sodium ethoxide (C2H5ONa+) and hydrogen gas
8. Using thermochemical arguments, explain why some substances that do not form a solution at room temperature will form a solution when heated. Explain why a solution can form even when ΔHsoln is positive.
9. If you wanted to formulate a new compound that could be used in an instant cold pack, would you select a compound with a positive or negative value of ΔHsoln in water? Justify your answer.
10. Why is entropy the dominant factor in the formation of solutions of two or more gases? Is it possible for two gases to be immiscible? Why or why not?
1. Homogeneous mixtures: aqueous ammonia, liquid decongestant, vinegar, and gasoline. Heterogeneous mixtures: seawater and fog.
2. All are chemical reactions except dissolving iodine crystals in CCl4.
13.2 Solubility and Molecular Structure
Learning Objective
1. To understand the relationship between solubility and molecular structure.
When a solute dissolves, its individual atoms, molecules, or ions interact with the solvent, become solvated, and are able to diffuse independently throughout the solution (part (a) in Figure 13.4 "Dissolution and Precipitation"). This is not, however, a unidirectional process. If the molecule or ion happens to collide with the surface of a particle of the undissolved solute, it may adhere to the particle in a process called crystallization. Dissolution and crystallization continue as long as excess solid is present, resulting in a dynamic equilibrium analogous to the equilibrium that maintains the vapor pressure of a liquid. (For more information about vapor pressure, see Chapter 11 "Liquids", Section 11.3 "Unique Properties of Liquids".) We can represent these opposing processes as follows:
Equation 13.4
Although the terms precipitation and crystallization are both used to describe the separation of solid solute from a solution, crystallization refers to the formation of a solid with a well-defined crystalline structure, whereas precipitation refers to the formation of any solid phase, often one with very small particles.
Figure 13.4 Dissolution and Precipitation
(a) When a solid is added to a solvent in which it is soluble, solute particles leave the surface of the solid and become solvated by the solvent, initially forming an unsaturated solution. (b) When the maximum possible amount of solute has dissolved, the solution becomes saturated. If excess solute is present, the rate at which solute particles leave the surface of the solid equals the rate at which they return to the surface of the solid. (c) A supersaturated solution can usually be formed from a saturated solution by filtering off the excess solute and lowering the temperature. (d) When a seed crystal of the solute is added to a supersaturated solution, solute particles leave the solution and form a crystalline precipitate.
Factors Affecting Solubility
The maximum amount of a solute that can dissolve in a solvent at a specified temperature and pressure is its solubilityA measure of the how much of a solid substance remains dissolved in a given amount of a specified liquid at a specified temperature and pressure.. Solubility is often expressed as the mass of solute per volume (g/L) or mass of solute per mass of solvent (g/g), or as the moles of solute per volume (mol/L). Even for very soluble substances, however, there is usually a limit to how much solute can dissolve in a given quantity of solvent. In general, the solubility of a substance depends on not only the energetic factors we have discussed but also the temperature and, for gases, the pressure. At 20°C, for example, 177 g of NaI, 91.2 g of NaBr, 35.9 g of NaCl, and only 4.1 g of NaF dissolve in 100 g of water. At 70°C, however, the solubilities increase to 295 g of NaI, 119 g of NaBr, 37.5 g of NaCl, and 4.8 g of NaF. As you learned in Chapter 12 "Solids", the lattice energies of the sodium halides increase from NaI to NaF. The fact that the solubilities decrease as the lattice energy increases suggests that the ΔH2 term in Figure 13.1 "Enthalpy Changes That Accompany the Formation of a Solution" dominates for this series of compounds.
A solution with the maximum possible amount of solute is saturatedA solution with the maximum possible amount of a solute under a given set of conditions.. If a solution contains less than the maximum amount of solute, it is unsaturated. When a solution is saturated and excess solute is present, the rate of dissolution is exactly equal to the rate of crystallization (part (b) in Figure 13.4 "Dissolution and Precipitation"). Using the value just stated, a saturated aqueous solution of NaCl, for example, contains 35.9 g of NaCl per 100 mL of water at 20°C. We can prepare a homogeneous saturated solution by adding excess solute (in this case, greater than 35.9 g of NaCl) to the solvent (water), stirring until the maximum possible amount of solute has dissolved, and then removing undissolved solute by filtration.
Note the Pattern
The solubility of most solids increases with increasing temperature.
Because the solubility of most solids increases with increasing temperature, a saturated solution that was prepared at a higher temperature usually contains more dissolved solute than it would contain at a lower temperature. When the solution is cooled, it can therefore become supersaturatedAn unstable solution with more dissolved solute than it would normally contain under the given set of conditions. (part (c) in Figure 13.4 "Dissolution and Precipitation"). Like a supercooled or superheated liquid (see Chapter 11 "Liquids"), a supersaturated solution is unstable. Consequently, adding a small particle of the solute, a seed crystalA solid sample of a substance that can be added to a supercooled liquid or a supersaturated solution to help induce crystallization., will usually cause the excess solute to rapidly precipitate or crystallize, sometimes with spectacular results, as was shown in Figure 1.9 "The Crystallization of Sodium Acetate from a Concentrated Solution of Sodium Acetate in Water". The rate of crystallization in Equation 13.4 is greater than the rate of dissolution, so crystals or a precipitate form (part (d) in Figure 13.4 "Dissolution and Precipitation"). In contrast, adding a seed crystal to a saturated solution reestablishes the dynamic equilibrium, and the net quantity of dissolved solute no longer changes.
Because crystallization is the reverse of dissolution, a substance that requires an input of heat to form a solution (ΔHsoln > 0) releases that heat when it crystallizes from solution (ΔHcrys < 0). The amount of heat released is proportional to the amount of solute that exceeds its solubility. Two substances that have a positive enthalpy of solution are sodium thiosulfate (Na2S2O3) and sodium acetate (CH3CO2Na), both of which are used in commercial hot packs, small bags of supersaturated solutions used to warm hands (see Figure 5.13 "The High Specific Heat of Liquid Water Has Major Effects on Climate").
Interactions in Liquid Solutions
The interactions that determine the solubility of a substance in a liquid depend largely on the chemical nature of the solute (such as whether it is ionic or molecular) rather than on its physical state (solid, liquid, or gas). We will first describe the general case of forming a solution of a molecular species in a liquid solvent and then describe the formation of a solution of an ionic compound.
Solutions of Molecular Substances in Liquids
The London dispersion forces, dipole–dipole interactions, and hydrogen bonds that hold molecules to other molecules are generally weak. Even so, energy is required to disrupt these interactions. As we described in Section 13.1 "Factors Affecting Solution Formation", unless some of that energy is recovered in the formation of new, favorable solute–solvent interactions, the increase in entropy on solution formation is not enough for a solution to form.
For solutions of gases in liquids, we can safely ignore the energy required to separate the solute molecules (ΔH2 = 0) because the molecules are already separated. Thus we need to consider only the energy required to separate the solvent molecules (ΔH1) and the energy released by new solute–solvent interactions (ΔH3). Nonpolar gases such as N2, O2, and Ar have no dipole moment and cannot engage in dipole–dipole interactions or hydrogen bonding. Consequently, the only way they can interact with a solvent is by means of London dispersion forces, which may be weaker than the solvent–solvent interactions in a polar solvent. It is not surprising, then, that nonpolar gases are most soluble in nonpolar solvents. In this case, ΔH1 and ΔH3 are both small and of similar magnitude. In contrast, for a solution of a nonpolar gas in a polar solvent, ΔH1 is far greater than ΔH3. As a result, nonpolar gases are less soluble in polar solvents than in nonpolar solvents. For example, the concentration of N2 in a saturated solution of N2 in water, a polar solvent, is only 7.07 × 10−4 M compared with 4.5 × 10−3 M for a saturated solution of N2 in benzene, a nonpolar solvent.
The solubilities of nonpolar gases in water generally increase as the molecular mass of the gas increases, as shown in Table 13.3 "Solubilities of Selected Gases in Water at 20°C and 1 atm Pressure". This is precisely the trend expected: as the gas molecules become larger, the strength of the solvent–solute interactions due to London dispersion forces increases, approaching the strength of the solvent–solvent interactions.
Table 13.3 Solubilities of Selected Gases in Water at 20°C and 1 atm Pressure
Gas Solubility (M) × 10−4
He 3.90
Ne 4.65
Ar 15.2
Kr 27.9
Xe 50.2
H2 8.06
N2 7.07
CO 10.6
O2 13.9
N2O 281
CH4 15.5
Virtually all common organic liquids, whether polar or not, are miscible. The strengths of the intermolecular attractions are comparable; thus the enthalpy of solution is expected to be small (ΔHsoln ≈ 0), and the increase in entropy drives the formation of a solution. If the predominant intermolecular interactions in two liquids are very different from one another, however, they may be immiscible. For example, organic liquids such as benzene, hexane, CCl4, and CS2 (S=C=S) are nonpolar and have no ability to act as hydrogen bond donors or acceptors with hydrogen-bonding solvents such as H2O, HF, and NH3; hence they are immiscible in these solvents. When shaken with water, they form separate phases or layers separated by an interface (Figure 13.5 "Immiscible Liquids"), the region between the two layers. Just because two liquids are immiscible, however, does not mean that they are completely insoluble in each other. For example, 188 mg of benzene dissolves in 100 mL of water at 23.5°C. Adding more benzene results in the separation of an upper layer consisting of benzene with a small amount of dissolved water (the solubility of water in benzene is only 178 mg/100 mL of benzene).
Figure 13.5 Immiscible Liquids
Water is immiscible with both CCl4 and hexane. When all three liquids are mixed, they separate into three distinct layers. Because water is less dense than CCl4, the water layer floats on the CCl4. In contrast, hexane is less dense than water, so the hexane floats on the water layer. Because I2 is intensely purple and quite soluble in both CCl4 and hexane, but insoluble in water, a small amount of I2 has been added to help identify the hexane and CCl4 layers.
The solubilities of simple alcohols in water are given in Table 13.4 "Solubilities of Straight-Chain Organic Alcohols in Water at 20°C". Only the three lightest alcohols (methanol, ethanol, and n-propanol) are completely miscible with water. As the molecular mass of the alcohol increases, so does the proportion of hydrocarbon in the molecule. Correspondingly, the importance of hydrogen bonding and dipole–dipole interactions in the pure alcohol decreases, while the importance of London dispersion forces increases, which leads to progressively fewer favorable electrostatic interactions with water. Organic liquids such as acetone, ethanol, and tetrahydrofuran are sufficiently polar to be completely miscible with water yet sufficiently nonpolar to be completely miscible with all organic solvents.
Table 13.4 Solubilities of Straight-Chain Organic Alcohols in Water at 20°C
Alcohol Solubility (mol/100 g of H2O)
methanol completely miscible
ethanol completely miscible
n-propanol completely miscible
n-butanol 0.11
n-pentanol 0.030
n-hexanol 0.0058
n-heptanol 0.0008
The same principles govern the solubilities of molecular solids in liquids. For example, elemental sulfur is a solid consisting of cyclic S8 molecules that have no dipole moment. Because the S8 rings in solid sulfur are held to other rings by London dispersion forces, elemental sulfur is insoluble in water. It is, however, soluble in nonpolar solvents that have comparable London dispersion forces, such as CS2 (23 g/100 mL). In contrast, glucose contains five –OH groups that can form hydrogen bonds. Consequently, glucose is very soluble in water (91 g/120 mL of water) but essentially insoluble in nonpolar solvents such as benzene. The structure of one isomer of glucose is shown here.
Low-molecular-mass hydrocarbons with highly electronegative and polarizable halogen atoms, such as chloroform (CHCl3) and methylene chloride (CH2Cl2), have both significant dipole moments and relatively strong London dispersion forces. These hydrocarbons are therefore powerful solvents for a wide range of polar and nonpolar compounds. Naphthalene, which is nonpolar, and phenol (C6H5OH), which is polar, are very soluble in chloroform. In contrast, the solubility of ionic compounds is largely determined not by the polarity of the solvent but rather by its dielectric constant, a measure of its ability to separate ions in solution, as you will soon see.
Example 2
Identify the most important solute–solvent interactions in each solution.
1. iodine in benzene
2. aniline (C6H5NH2) in dichloromethane (CH2Cl2)
3. iodine in water
Given: components of solutions
Strategy:
Identify all possible intermolecular interactions for both the solute and the solvent: London dispersion forces, dipole–dipole interactions, or hydrogen bonding. Determine which is likely to be the most important factor in solution formation.
Solution:
1. Benzene and I2 are both nonpolar molecules. The only possible attractive forces are London dispersion forces.
2. Aniline is a polar molecule with an –NH2 group, which can act as a hydrogen bond donor. Dichloromethane is also polar, but it has no obvious hydrogen bond acceptor. Therefore, the most important interactions between aniline and CH2Cl2 are likely to be London interactions.
3. Water is a highly polar molecule that engages in extensive hydrogen bonding, whereas I2 is a nonpolar molecule that cannot act as a hydrogen bond donor or acceptor. The slight solubility of I2 in water (1.3 × 10−3 mol/L at 25°C) is due to London dispersion forces.
Exercise
Identify the most important interactions in each solution:
1. ethylene glycol (HOCH2CH2OH) in acetone
2. acetonitrile (CH3C≡N) in acetone
3. n-hexane in benzene
1. hydrogen bonding
2. London interactions
3. London dispersion forces
Hydrophilic and Hydrophobic Solutes
A solute can be classified as hydrophilicA substance attracted to water. Hydrophilic substances are polar and can form hydrogen bonds to water. (literally, “water loving”), meaning that it has an electrostatic attraction to water, or hydrophobicA substance that repels water. Hydrophobic substances do not interact favorably with water. (“water fearing”), meaning that it repels water. A hydrophilic substance is polar and often contains O–H or N–H groups that can form hydrogen bonds to water. For example, glucose with its five O–H groups is hydrophilic. In contrast, a hydrophobic substance may be polar but usually contains C–H bonds that do not interact favorably with water, as is the case with naphthalene and n-octane. Hydrophilic substances tend to be very soluble in water and other strongly polar solvents, whereas hydrophobic substances are essentially insoluble in water and soluble in nonpolar solvents such as benzene and cyclohexane.
The difference between hydrophilic and hydrophobic substances has substantial consequences in biological systems. For example, vitamins can be classified as either fat soluble or water soluble. Fat-soluble vitamins, such as vitamin A, are mostly nonpolar, hydrophobic molecules. As a result, they tend to be absorbed into fatty tissues and stored there. In contrast, water-soluble vitamins, such as vitamin C, are polar, hydrophilic molecules that circulate in the blood and intracellular fluids, which are primarily aqueous. Water-soluble vitamins are therefore excreted much more rapidly from the body and must be replenished in our daily diet. A comparison of the chemical structures of vitamin A and vitamin C quickly reveals why one is hydrophobic and the other hydrophilic.
Because water-soluble vitamins are rapidly excreted, the risk of consuming them in excess is relatively small. Eating a dozen oranges a day is likely to make you tired of oranges long before you suffer any ill effects due to their high vitamin C content. In contrast, fat-soluble vitamins constitute a significant health hazard when consumed in large amounts. For example, the livers of polar bears and other large animals that live in cold climates contain large amounts of vitamin A, which have occasionally proven fatal to humans who have eaten them.
Example 3
The following substances are essential components of the human diet:
Using what you know of hydrophilic and hydrophobic solutes, classify each as water soluble or fat soluble and predict which are likely to be required in the diet on a daily basis.
1. arginine
2. pantothenic acid
3. oleic acid
Given: chemical structures
Asked for: classification as water soluble or fat soluble; dietary requirement
Strategy:
Based on the structure of each compound, decide whether it is hydrophilic or hydrophobic. If it is hydrophilic, it is likely to be required on a daily basis.
Solution:
1. Arginine is a highly polar molecule with two positively charged groups and one negatively charged group, all of which can form hydrogen bonds with water. As a result, it is hydrophilic and required in our daily diet.
2. Although pantothenic acid contains a hydrophobic hydrocarbon portion, it also contains several polar functional groups (–OH and –CO2H) that should interact strongly with water. It is therefore likely to be water soluble and required in the diet. (In fact, pantothenic acid is almost always a component of multiple-vitamin tablets.)
3. Oleic acid is a hydrophobic molecule with a single polar group at one end. It should be fat soluble and not required daily.
Exercise
These compounds are consumed by humans: caffeine, acetaminophen, and vitamin D. Identify each as primarily hydrophilic (water soluble) or hydrophobic (fat soluble), and predict whether each is likely to be excreted from the body rapidly or slowly.
Answer: Caffeine and acetaminophen are water soluble and rapidly excreted, whereas vitamin D is fat soluble and slowly excreted.
Solid Solutions
Solutions are not limited to gases and liquids; solid solutions also exist. For example, amalgamsA solution (usually a solid solution) of a metal in liquid mercury., which are usually solids, are solutions of metals in liquid mercury. Because most metals are soluble in mercury, amalgams are used in gold mining, dentistry, and many other applications. A major difficulty when mining gold is separating very small particles of pure gold from tons of crushed rock. One way to accomplish this is to agitate a suspension of the crushed rock with liquid mercury, which dissolves the gold (as well as any metallic silver that might be present). The very dense liquid gold–mercury amalgam is then isolated and the mercury distilled away.
An alloy is a solid or liquid solution that consists of one or more elements in a metallic matrix. A solid alloy has a single homogeneous phase in which the crystal structure of the solvent remains unchanged by the presence of the solute. Thus the microstructure of the alloy is uniform throughout the sample. Examples are substitutional and interstitial alloys such as brass or solder. (For more information about alloys, see Chapter 12 "Solids", Section 12.5 "Correlation between Bonding and the Properties of Solids") Liquid alloys include sodium/potassium and gold/mercury. In contrast, a partial alloy solution has two or more phases that can be homogeneous in the distribution of the components, but the microstructures of the two phases are not the same. As a liquid solution of lead and tin is cooled, for example, different crystalline phases form at different cooling temperatures. As you learned in Chapter 12 "Solids", alloys usually have properties that differ from those of the component elements.
Network solids such as diamond, graphite, and SiO2 are insoluble in all solvents with which they do not react chemically. The covalent bonds that hold the network or lattice together are simply too strong to be broken under normal conditions. They are certainly much stronger than any conceivable combination of intermolecular interactions that might occur in solution. Most metals are insoluble in virtually all solvents for the same reason: the delocalized metallic bonding is much stronger than any favorable metal atom–solvent interactions. Many metals react with solutions such as aqueous acids or bases to produce a solution. However, as we saw in Section 13.1 "Factors Affecting Solution Formation", in these instances the metal undergoes a chemical transformation that cannot be reversed by simply removing the solvent.
Note the Pattern
Solids with very strong intermolecular bonding tend to be insoluble.
Solubilities of Ionic Substances in Liquids
Table 4.1 "Common Units of Concentration" introduced you to guidelines for predicting the solubility of ionic compounds in water. Ionic substances are generally most soluble in polar solvents; the higher the lattice energy, the more polar the solvent must be to overcome the lattice energy and dissolve the substance. Because of its high polarity, water is the most common solvent for ionic compounds. Many ionic compounds are soluble in other polar solvents, however, such as liquid ammonia, liquid hydrogen fluoride, and methanol. Because all these solvents consist of molecules that have relatively large dipole moments, they can interact favorably with the dissolved ions.
The interaction of water with Na+ and Cl ions in an aqueous solution of NaCl was illustrated in Figure 4.3 "The Dissolution of Sodium Chloride in Water". The ion–dipole interactions between Li+ ions and acetone molecules in a solution of LiCl in acetone are shown in Figure 13.6 "Ion–Dipole Interactions in the Solvation of Li". The energetically favorable Li+–acetone interactions make ΔH3 in Figure 13.1 "Enthalpy Changes That Accompany the Formation of a Solution" sufficiently negative to overcome the positive ΔH1 and ΔH2. Because the dipole moment of acetone (2.88 D), and thus its polarity, is actually larger than that of water (1.85 D), one might even expect that LiCl would be more soluble in acetone than in water. In fact, the opposite is true: 83 g of LiCl dissolve in 100 mL of water at 20°C, but only about 4.1 g of LiCl dissolve in 100 mL of acetone. This apparent contradiction arises from the fact that the dipole moment is a property of a single molecule in the gas phase. A more useful measure of the ability of a solvent to dissolve ionic compounds is its dielectric constant (ε)A constant that expresses the ability of a bulk substance to decrease the electrostatic forces between two charged particles., which is the ability of a bulk substance to decrease the electrostatic forces between two charged particles. By definition, the dielectric constant of a vacuum is 1. In essence, a solvent with a high dielectric constant causes the charged particles to behave as if they have been moved farther apart. At 25°C, the dielectric constant of water is 80.1, one of the highest known, and that of acetone is only 21.0. Hence water is better able to decrease the electrostatic attraction between Li+ and Cl ions, so LiCl is more soluble in water than in acetone. This behavior is in contrast to that of molecular substances, for which polarity is the dominant factor governing solubility.
Note the Pattern
A solvent’s dielectric constant is the most useful measure of its ability to dissolve ionic compounds. A solvent’s polarity is the dominant factor in dissolving molecular substances.
Figure 13.6 Ion–Dipole Interactions in the Solvation of Li+ Ions by Acetone, a Polar Solvent
It is also possible to dissolve ionic compounds in organic solvents using crown ethersCyclic polyether with four or more oxygen atoms separated by two or three carbon atoms. All crown ethers have a central cavity that can accommodate a metal ion coordinated to the ring of oxygen atoms., cyclic compounds with the general formula (OCH2CH2)n. Crown ethers are named using both the total number of atoms in the ring and the number of oxygen atoms. Thus 18-crown-6 is an 18-membered ring with six oxygen atoms (part (a) in Figure 13.7 "Crown Ethers and Cryptands"). The cavity in the center of the crown ether molecule is lined with oxygen atoms and is large enough to be occupied by a cation, such as K+. The cation is stabilized by interacting with lone pairs of electrons on the surrounding oxygen atoms. Thus crown ethers solvate cations inside a hydrophilic cavity, whereas the outer shell, consisting of C–H bonds, is hydrophobic. Crown ethers are useful for dissolving ionic substances such as KMnO4 in organic solvents such as isopropanol [(CH3)2CHOH] (Figure 13.8 "Effect of a Crown Ether on the Solubility of KMnO"). The availability of crown ethers with cavities of different sizes allows specific cations to be solvated with a high degree of selectivity.
Figure 13.7 Crown Ethers and Cryptands
(a) The potassium complex of the crown ether 18-crown-6. Note how the cation is nestled within the central cavity of the molecule and interacts with lone pairs of electrons on the oxygen atoms. (b) The potassium complex of 2,2,2-cryptand, showing how the cation is almost hidden by the cryptand. Cryptands solvate cations via lone pairs of electrons on both oxygen and nitrogen atoms.
Figure 13.8 Effect of a Crown Ether on the Solubility of KMnO4 in Isopropanol (2-Propanol)
(a) Normally KMnO4, which is intensely purple, is completely insoluble in isopropanol, which has a relatively low dielectric constant. (b) In the presence of a small amount of 18-crown-6, KMnO4 dissolves in isopropanol, as shown by the reddish-purple color caused by permanganate ions in solution.
CryptandsConsisting of three $(–OCH2CH2O–)n$ chains connected by two nitrogen atoms, cryptands have a central cavity that can encapsulate a metal ion coordinated to the oxygen and nitrogen atoms. (from the Greek kryptós, meaning “hidden”) are compounds that can completely surround a cation with lone pairs of electrons on oxygen and nitrogen atoms (part (b) in Figure 13.7 "Crown Ethers and Cryptands"). The number in the name of the cryptand is the number of oxygen atoms in each strand of the molecule. Like crown ethers, cryptands can be used to prepare solutions of ionic compounds in solvents that are otherwise too nonpolar to dissolve them.
Summary
The solubility of a substance is the maximum amount of a solute that can dissolve in a given quantity of solvent; it depends on the chemical nature of both the solute and the solvent and on the temperature and pressure. When a solution contains the maximum amount of solute that can dissolve under a given set of conditions, it is a saturated solution. Otherwise, it is unsaturated. Supersaturated solutions, which contain more dissolved solute than allowed under particular conditions, are not stable; the addition of a seed crystal, a small particle of solute, will usually cause the excess solute to crystallize. A system in which crystallization and dissolution occur at the same rate is in dynamic equilibrium. The solubility of a substance in a liquid is determined by intermolecular interactions, which also determine whether two liquids are miscible. Solutes can be classified as hydrophilic (water loving) or hydrophobic (water fearing). Vitamins with hydrophilic structures are water soluble, whereas those with hydrophobic structures are fat soluble. Many metals dissolve in liquid mercury to form amalgams. Covalent network solids and most metals are insoluble in nearly all solvents. The solubility of ionic compounds is largely determined by the dielectric constant (ε) of the solvent, a measure of its ability to decrease the electrostatic forces between charged particles. Solutions of many ionic compounds in organic solvents can be dissolved using crown ethers, cyclic polyethers large enough to accommodate a metal ion in the center, or cryptands, compounds that completely surround a cation.
Key Takeaway
• The strength of intramolecular bonding determines the solubility of a solute in a given solvent.
Conceptual Problems
1. If a compound is only slightly soluble in a particular solvent, what are the relative strengths of the solvent–solvent and solute–solute interactions versus the solute–solvent interactions?
2. Predict whether each of the following sets of conditions favors formation of a solution:
Intermolecular Attractive Forces (Solute) Intermolecular Attractive Forces (Solvent) ΔHsoln
London dispersion hydrogen bonding slightly positive
dipole–dipole hydrogen bonding very negative
ionic dipole–dipole slightly positive
ionic London dispersion positive
3. Arrange the following liquids in order of increasing solubility in water: t-butanol [(CH3)3COH], benzene, ammonia, and heptane. Justify your answer.
4. Which compound in each pair will be more soluble in water? Explain your reasoning in each case.
1. toluene (C7H8) or ethyl ether (C2H5OC2H5)
2. chloroform (CHCl3) or acetone (CH3COCH3)
3. carbon tetrachloride (CCl4) or tetrahydrofuran (C4H8O)
4. CaCl2 or CH2Cl2
5. Which compound in each pair will be more soluble in benzene? Explain your reasoning in each case.
1. cyclohexane or methanol
2. I2 or MgCl2
3. methylene chloride (CH2Cl2) or acetic acid
6. Two water-insoluble compounds—n-decylamine [CH3(CH2)9NH2] and n-decane—can be separated by the following procedure: The compounds are dissolved in a solvent such as toluene that is immiscible with water. When adding an aqueous HCl solution to the mixture and stirring vigorously, the HCl reacts with one of the compounds to produce a salt. When the stirring is stopped and the mixture is allowed to stand, two layers are formed. At this point, each layer contains only one of the two original compounds. After the layers are separated, adding aqueous NaOH to the aqueous layer liberates one of the original compounds, which can then be removed by stirring with a second portion of toluene to extract it from the water.
1. Identify the compound that is present in each layer following the addition of HCl. Explain your reasoning.
2. How can the original compounds be recovered from the toluene solution?
7. Bromine and iodine are both soluble in CCl4, but bromine is much more soluble. Why?
8. A solution is made by mixing 50.0 mL of liquid A with 75.0 mL of liquid B. Which is the solute, and which is the solvent? Is it valid to assume that the volume of the resulting solution will be 125 mL? Explain your answer.
9. The compounds NaI, NaBr, and NaCl are far more soluble in water than NaF, a substance that is used to fluoridate drinking water. In fact, at 25°C the solubility of NaI is 184 g/100 mL of water, versus only 4.2 g/100 mL of water for NaF. Why is sodium iodide so much more soluble in water? Do you expect KCl to be more soluble or less soluble in water than NaCl?
10. When water is mixed with a solvent with which it is immiscible, the two liquids usually form two separate layers. If the density of the nonaqueous solvent is 1.75 g/mL at room temperature, sketch the appearance of the heterogeneous mixture in a beaker and label which layer is which. If you were not sure of the density and the identity of the other liquid, how might you be able to identify which is the aqueous layer?
11. When two liquids are immiscible, the addition of a third liquid can occasionally be used to induce the formation of a homogeneous solution containing all three.
1. Ethylene glycol (HOCH2CH2OH) and hexane are immiscible, but adding acetone [(CH3)2CO] produces a homogeneous solution. Why does adding a third solvent produce a homogeneous solution?
2. Methanol and n-hexane are immiscible. Which of the following solvents would you add to create a homogeneous solution—water, n-butanol, or cyclohexane? Justify your choice.
12. Some proponents of vitamin therapy for combating illness encourage the consumption of large amounts of fat-soluble vitamins. Why can this be dangerous? Would it be as dangerous to consume large amounts of water-soluble vitamins? Why or why not?
13. Why are most metals insoluble in virtually all solvents?
14. Because sodium reacts violently with water, it is difficult to weigh out small quantities of sodium metal for a reaction due to its rapid reaction with small amounts of moisture in the air. Would a Na/Hg amalgam be as sensitive to moisture as metallic sodium? Why or why not? A Na/K alloy is a liquid at room temperature. Will it be more or less sensitive to moisture than solid Na or K?
15. Dental amalgams often contain high concentrations of Hg, which is highly toxic. Why isn’t dental amalgam toxic?
16. Arrange 2,2,3-trimethylpentane, 1-propanol, toluene (C7H8), and dimethyl sulfoxide [(CH3)2S=O] in order of increasing dipole moment. Explain your reasoning.
17. Arrange acetone, chloroform, cyclohexane, and 2-butanol in order of increasing dielectric constant. Explain your reasoning.
18. Dissolving a white crystalline compound in ethanol gave a blue solution. Evaporating the ethanol from the solution gave a bluish-crystalline product, which slowly transformed into the original white solid on standing in the air for several days. Explain what happened. How does the mass of the initial bluish solid compare with the mass of the white solid finally recovered?
19. You have been asked to develop a new drug that could be used to bind Fe3+ ions in patients who suffer from iron toxicity, allowing the bound iron to be excreted in the urine. Would you consider a crown ether or a cryptand to be a reasonable candidate for such a drug? Explain your answer.
20. Describe two different situations in which fractional crystallization will not work as a separation technique when attempting to isolate a single compound from a mixture.
21. You have been given a mixture of two compounds—A and B—and have been told to isolate pure A. You know that pure A has a lower solubility than pure B and that the solubilities of both A and B increase with temperature. Outline a procedure to isolate pure A. If B had the lower solubility, could you use the same procedure to isolate pure A? Why or why not?
1. London dispersion forces increase with increasing atomic mass. Iodine is a solid while bromine is a liquid due to the greater intermolecular interactions between the heavier iodine atoms. Iodine is less soluble than bromine in virtually all solvents because it requires more energy to separate I2 molecules than Br2 molecules.
1. A third solvent with intermediate polarity and/or dielectric constant can effectively dissolve both of the immiscible solvents, creating a single liquid phase.
2. n-butanol—it is intermediate in polarity between methanol and n-hexane, while water is more polar than either and cyclohexane is comparable to n-hexane.
2. In dental amalgam, the mercury atoms are locked in a solid phase that does not undergo corrosion under physiological conditions; hence, the mercury atoms cannot readily diffuse to the surface where they could vaporize or undergo chemical reaction.
3. Dissolve the mixture of A and B in a solvent in which they are both soluble when hot and relatively insoluble when cold, filter off any undissolved B, and cool slowly. Pure A should crystallize, while B stays in solution. If B were less soluble, it would be impossible to obtain pure A by this method in a single step, because some of the less soluble compound (B) will always be present in the solid that crystallizes from solution.
13.3 Units of Concentration
Learning Objective
1. To describe the concentration of a solution in the way that is most appropriate for a particular problem or application.
There are several different ways to quantitatively describe the concentrationThe quantity of solute that is dissolved in a particular quantity of solvent or solution. of a solution. For example, molarity was introduced in Chapter 4 "Reactions in Aqueous Solution" as a useful way to describe solution concentrations for reactions that are carried out in solution. Mole fractions, introduced in Chapter 10 "Gases", are used not only to describe gas concentrations but also to determine the vapor pressures of mixtures of similar liquids. Example 4 reviews the methods for calculating the molarity and mole fraction of a solution when the masses of its components are known.
Example 4
Commercial vinegar is essentially a solution of acetic acid in water. A bottle of vinegar has 3.78 g of acetic acid per 100.0 g of solution. Assume that the density of the solution is 1.00 g/mL.
1. What is its molarity?
2. What is its mole fraction?
Given: mass of substance and mass and density of solution
Asked for: molarity and mole fraction
Strategy:
A Calculate the number of moles of acetic acid in the sample. Then calculate the number of liters of solution from its mass and density. Use these results to determine the molarity of the solution.
B Determine the mass of the water in the sample and calculate the number of moles of water. Then determine the mole fraction of acetic acid by dividing the number of moles of acetic acid by the total number of moles of substances in the sample.
Solution:
1. A The molarity is the number of moles of acetic acid per liter of solution. We can calculate the number of moles of acetic acid as its mass divided by its molar mass. The volume of the solution equals its mass divided by its density. The calculations follow:
This result makes intuitive sense. If 100.0 g of aqueous solution (equal to 100 mL) contains 3.78 g of acetic acid, then 1 L of solution will contain 37.8 g of acetic acid, which is a little more than $12$ mole. Keep in mind, though, that the mass and volume of a solution are related by its density; concentrated aqueous solutions often have densities greater than 1.00 g/mL.
2. B To calculate the mole fraction of acetic acid in the solution, we need to know the number of moles of both acetic acid and water. The number of moles of acetic acid is 0.0629 mol, as calculated in part (a). We know that 100.0 g of vinegar contains 3.78 g of acetic acid; hence the solution also contains (100.0 g − 3.78 g) = 96.2 g of water. We have
The mole fraction X of acetic acid is the ratio of the number of moles of acetic acid to the total number of moles of substances present:
This answer makes sense, too. There are approximately 100 times as many moles of water as moles of acetic acid, so the ratio should be approximately 0.01.
Exercise
A solution of HCl gas dissolved in water (sold commercially as “muriatic acid,” a solution used to clean masonry surfaces) has 20.22 g of HCl per 100.0 g of solution, and its density is 1.10 g/mL.
1. What is its molarity?
2. What is its mole fraction?
1. 6.10 M HCl
2. XHCl = 0.111
The concentration of a solution can also be described by its molality (m)The number of moles of solute present in exactly 1 kg of solvent., the number of moles of solute per kilogram of solvent:
Equation 13.5
Molality, therefore, has the same numerator as molarity (the number of moles of solute) but a different denominator (kilogram of solvent rather than liter of solution). For dilute aqueous solutions, the molality and molarity are nearly the same because dilute solutions are mostly solvent. Thus because the density of water under standard conditions is very close to 1.0 g/mL, the volume of 1.0 kg of H2O under these conditions is very close to 1.0 L, and a 0.50 M solution of KBr in water, for example, has approximately the same concentration as a 0.50 m solution.
Another common way of describing concentration is as the ratio of the mass of the solute to the total mass of the solution. The result can be expressed as mass percentageThe ratio of the total mass of the solute to the total mass of the solution., parts per million (ppm)Milligrams of solute per kilogram of solvent., or parts per billion (ppb)Micrograms of solute per kilogram of solvent.:
Equation 13.6
Equation 13.7
Equation 13.8
In the health sciences, the concentration of a solution is typically expressed as parts per thousand (ppt)Grams of solute per kilogram of solvent, primarily used in the health sciences., indicated as a proportion. For example, adrenalin, the hormone produced in high-stress situations, is available in a 1:1000 solution, or one gram of adrenalin per 1000 g of solution.
The labels on bottles of commercial reagents often describe the contents in terms of mass percentage. Sulfuric acid, for example, is sold as a 95% aqueous solution, or 95 g of H2SO4 per 100 g of solution. Parts per million and parts per billion are used to describe concentrations of highly dilute solutions. These measurements correspond to milligrams and micrograms of solute per kilogram of solution, respectively. For dilute aqueous solutions, this is equal to milligrams and micrograms of solute per liter of solution (assuming a density of 1.0 g/mL).
Example 5
Several years ago, millions of bottles of mineral water were contaminated with benzene at ppm levels. This incident received a great deal of attention because the lethal concentration of benzene in rats is 3.8 ppm. A 250 mL sample of mineral water has 12.7 ppm of benzene. Because the contaminated mineral water is a very dilute aqueous solution, we can assume that its density is approximately 1.00 g/mL.
1. What is the molarity of the solution?
2. What is the mass of benzene in the sample?
Given: volume of sample, solute concentration, and density of solution
Asked for: molarity of solute and mass of solute in 250 mL
Strategy:
A Use the concentration of the solute in parts per million to calculate the molarity.
B Use the concentration of the solute in parts per million to calculate the mass of the solute in the specified volume of solution.
Solution:
1. A To calculate the molarity of benzene, we need to determine the number of moles of benzene in 1 L of solution. We know that the solution contains 12.7 ppm of benzene. Because 12.7 ppm is equivalent to 12.7 mg/1000 g of solution and the density of the solution is 1.00 g/mL, the solution contains 12.7 mg of benzene per liter (1000 mL). The molarity is therefore
2. B We are given that there are 12.7 mg of benzene per 1000 g of solution, which is equal to 12.7 mg/L of solution. Hence the mass of benzene in 250 mL (250 g) of solution is
Exercise
The maximum allowable concentration of lead in drinking water is 9.0 ppb. What is the molarity of Pb2+ in a 9.0 ppb aqueous solution? Use your calculated concentration to determine how many grams of Pb2+ are in an 8 oz glass of water.
Answer: 4.3 × 10−8 M; 2 × 10−6 g
How do chemists decide which units of concentration to use for a particular application? Although molarity is commonly used to express concentrations for reactions in solution or for titrations, it does have one drawback—molarity is the number of moles of solute divided by the volume of the solution, and the volume of a solution depends on its density, which is a function of temperature. Because volumetric glassware is calibrated at a particular temperature, typically 20°C, the molarity may differ from the original value by several percent if a solution is prepared or used at a significantly different temperature, such as 40°C or 0°C. For many applications this may not be a problem, but for precise work these errors can become important. In contrast, mole fraction, molality, and mass percentage depend on only the masses of the solute and solvent, which are independent of temperature.
Mole fraction is not very useful for experiments that involve quantitative reactions, but it is convenient for calculating the partial pressure of gases in mixtures, as we saw in Chapter 10 "Gases". As you will learn in Section 13.5 "Colligative Properties of Solutions", mole fractions are also useful for calculating the vapor pressures of certain types of solutions. Molality is particularly useful for determining how properties such as the freezing or boiling point of a solution vary with solute concentration. Because mass percentage and parts per million or billion are simply different ways of expressing the ratio of the mass of a solute to the mass of the solution, they enable us to express the concentration of a substance even when the molecular mass of the substance is unknown. Units of ppb or ppm are also used to express very low concentrations, such as those of residual impurities in foods or of pollutants in environmental studies.
Table 13.5 "Different Units for Expressing the Concentrations of Solutions*" summarizes the different units of concentration and typical applications for each. When the molar mass of the solute and the density of the solution are known, it becomes relatively easy with practice to convert among the units of concentration we have discussed, as illustrated in Example 6.
Table 13.5 Different Units for Expressing the Concentrations of Solutions*
Unit Definition Application
molarity (M) moles of solute/liter of solution (mol/L) Used for quantitative reactions in solution and titrations; mass and molecular mass of solute and volume of solution are known.
mole fraction (X) moles of solute/total moles present (mol/mol) Used for partial pressures of gases and vapor pressures of some solutions; mass and molecular mass of each component are known.
molality (m) moles of solute/kg of solvent (mol/kg) Used in determining how colligative properties vary with solute concentration; masses and molecular mass of solute are known.
mass percentage (%) [mass of solute (g)/mass of solution (g)] × 100 Useful when masses are known but molecular masses are unknown.
parts per thousand (ppt) [mass of solute/mass of solution] × 103 (g solute/kg solution) Used in the health sciences, ratio solutions are typically expressed as a proportion, such as 1:1000.
parts per million (ppm) [mass of solute/mass of solution] × 106 (mg solute/kg solution) Used for trace quantities; masses are known but molecular masses may be unknown.
parts per billion (ppb) [mass of solute/mass of solution] × 109 (µg solute/kg solution) Used for trace quantities; masses are known but molecular masses may be unknown.
*The molarity of a solution is temperature dependent, but the other units shown in this table are independent of temperature.
Example 6
Vodka is essentially a solution of pure ethanol in water. Typical vodka is sold as “80 proof,” which means that it contains 40.0% ethanol by volume. The density of pure ethanol is 0.789 g/mL at 20°C. If we assume that the volume of the solution is the sum of the volumes of the components (which is not strictly correct), calculate the following for the ethanol in 80-proof vodka.
1. the mass percentage
2. the mole fraction
3. the molarity
4. the molality
Given: volume percent and density
Asked for: mass percentage, mole fraction, molarity, and molality
Strategy:
A Use the density of the solute to calculate the mass of the solute in 100.0 mL of solution. Calculate the mass of water in 100.0 mL of solution.
B Determine the mass percentage of solute by dividing the mass of ethanol by the mass of the solution and multiplying by 100.
C Convert grams of solute and solvent to moles of solute and solvent. Calculate the mole fraction of solute by dividing the moles of solute by the total number of moles of substances present in solution.
D Calculate the molarity of the solution: moles of solute per liter of solution. Determine the molality of the solution by dividing the number of moles of solute by the kilograms of solvent.
Solution:
The key to this problem is to use the density of pure ethanol to determine the mass of ethanol (CH3CH2OH), abbreviated as EtOH, in a given volume of solution. We can then calculate the number of moles of ethanol and the concentration of ethanol in any of the required units. A Because we are given a percentage by volume, we assume that we have 100.0 mL of solution. The volume of ethanol will thus be 40.0% of 100.0 mL, or 40.0 mL of ethanol, and the volume of water will be 60.0% of 100.0 mL, or 60.0 mL of water. The mass of ethanol is obtained from its density:
If we assume the density of water is 1.00 g/mL, the mass of water is 60.0 g. We now have all the information we need to calculate the concentration of ethanol in the solution.
1. B The mass percentage of ethanol is the ratio of the mass of ethanol to the total mass of the solution, expressed as a percentage:
2. C The mole fraction of ethanol is the ratio of the number of moles of ethanol to the total number of moles of substances in the solution. Because 40.0 mL of ethanol has a mass of 31.6 g, we can use the molar mass of ethanol (46.07 g/mol) to determine the number of moles of ethanol in 40.0 mL:
Similarly, the number of moles of water is
The mole fraction of ethanol is thus
3. D The molarity of the solution is the number of moles of ethanol per liter of solution. We already know the number of moles of ethanol per 100.0 mL of solution, so the molarity is
4. The molality of the solution is the number of moles of ethanol per kilogram of solvent. Because we know the number of moles of ethanol in 60.0 g of water, the calculation is again straightforward:
Exercise
A solution is prepared by mixing 100.0 mL of toluene with 300.0 mL of benzene. The densities of toluene and benzene are 0.867 g/mL and 0.874 g/mL, respectively. Assume that the volume of the solution is the sum of the volumes of the components. Calculate the following for toluene.
1. mass percentage
2. mole fraction
3. molarity
4. molality
1. mass percentage toluene = 24.8%
2. Xtoluene = 0.219
3. 2.35 M toluene
4. 3.59 m toluene
Summary
The concentration of a solution is the quantity of solute in a given quantity of solution. It can be expressed in several ways: molarity (moles of solute per liter of solution); mole fraction, the ratio of the number of moles of solute to the total number of moles of substances present; mass percentage, the ratio of the mass of the solute to the mass of the solution times 100; parts per thousand (ppt), grams of solute per kilogram of solution; parts per million (ppm), milligrams of solute per kilogram of solution; parts per billion (ppb), micrograms of solute per kilogram of solution; and molality (m), the number of moles of solute per kilogram of solvent.
Key Takeaway
• Different units are used to express the concentrations of a solution depending on the application.
Key Equations
molality
mass percentage
parts per million
parts per billion
Conceptual Problems
1. Does the molality have the same numerical value as the molarity for a highly concentrated aqueous solution of fructose (C6H12O6) (approximately 3.2 M)? Why or why not?
2. Explain why the molality and molarity of an aqueous solution are not always numerically identical. Will the difference between the two be greater for a dilute or a concentrated solution? Explain your answer.
3. Under what conditions are molality and molarity likely to be equal? Is the difference between the two greater when water is the solvent or when the solvent is not water? Why?
4. What is the key difference between using mole fraction or molality versus molarity to describe the concentration of a solution? Which unit(s) of concentration is most appropriate for experiments that must be carried out at several different temperatures?
5. An experiment that relies on very strict control of the reaction stoichiometry calls for adding 50.0 mL of a 0.95 M solution of A to 225 mL of a 1.01 M solution of B, followed by heating for 1 h at 60°C. To save time, a student decided to heat solution B to 60°C before measuring out 225 mL of solution B, transferring it to the flask containing solution A, and proceeding normally. This change in procedure caused the yield of product to be less than usual. How could such an apparently minor change in procedure have resulted in a decrease in the yield?
Numerical Problems
1. Complete the following table for aqueous solutions of the compounds given.
Compound Molarity (M) Solution Density (g/mL) Mole Fraction (X)
H2SO4 18.0 1.84
CH3COOH 1.00 7.21 × 10−3
KOH 3.60 1.16
2. Complete the following table for each compound given.
Compound Mass (g) Volume of Solution (mL) Molarity (M)
Na2SO4 7.80 225
KNO3 125 1.27
NaO2CCH3 18.64 0.95
3. How would you prepare 100.0 mL of an aqueous solution with 0.40 M KI? a solution with 0.65 M NaCN?
4. Calculate the molality of a solution with 775 mg of NaCl in 500.0 g of water. Do you expect the molarity to be the same as the molality? Why or why not?
5. What is the molarity of each solution?
1. 12.8 g of glucose (C6H12O6) in water, total volume 150.0 mL
2. 9.2 g of Na3PO4 in water, total volume 200.0 mL
3. 843 mg of I2 in EtOH, total volume 150.0 mL
6. A medication used to treat abnormal heart rhythms is labeled “Procainamide 0.5 g/250 cc.” Express this concentration in parts per thousand.
7. Meperidine is a medication used for pain relief. A bottle of meperidine is labeled as 50 mg/mL. Express this concentration in parts per thousand.
8. An aqueous solution that is 4.61% NaOH by mass has a density of 1.06 g/mL. Calculate the molarity of the solution, the mole fraction of NaOH, and the molality of the solution.
9. A solution of concentrated phosphoric acid contains 85.0% H3PO4 by mass and has a density of 1.684 g/mL. Calculate the following.
1. the molarity of the solution
2. the mole fraction of H3PO4
3. the molality of the solution
10. A solution of commercial concentrated nitric acid is 16 M HNO3 and has a density of 1.42 g/mL. What is the percentage of HNO3 in the solution by mass? What is the molality?
11. A commercial aqueous ammonia solution contains 28.0% NH3 by massand has a density of 0.899 g/mL. Calculate the following.
1. the molarity
2. the mole fraction
12. Concentrated, or glacial, acetic acid is pure acetic acid and has a density of 1.053 g/mL. It is widely used in organic syntheses, in the manufacture of rayon and plastics, as a preservative in foods, and occasionally to treat warts. What volume of glacial acetic acid is required to prepare 5.00 L of a 1.75 M solution of acetic acid in ethanol?
13. Solutions of sodium carbonate decahydrate, also known as washing soda, are used as skin cleansers. The solubility of this compound in cold water is 21.52 g/100 mL. If a saturated solution has a density of 1.20 g/mL, what is its molarity? What is the mole fraction of sodium carbonate decahydrate in this solution?
14. Hydrogen peroxide (H2O2) is usually sold over the counter as an aqueous solution that is 3% by mass. Assuming a solution density of 1.01 g/mL, what is the molarity of hydrogen peroxide? What is the molar concentration of a solution that is 30% hydrogen peroxide by mass (density = 1.112 g/mL)? How would you prepare 100.0 mL of a 3% solution from the 30% solution?
15. Determine the concentration of a solution with 825 mg of Na2HPO4 dissolved in 450.0 mL of H2O at 20°C in molarity, molality, mole fraction, and parts per million. Assume that the density of the solution is the same as that of water. Which unit of concentration is most convenient for calculating vapor pressure changes? Why?
16. How many moles of Cl are there in 25.0 mL of a 0.15 M CaCl2 solution?
17. How many moles of Na+ are there in 25.0 g of a 1.33 × 10−3 m Na2HPO4 solution? What is the sodium concentration of this solution in ppb?
18. How many grams of copper are there in 30.0 mL of a 0.100 M CuSO4 solution?
19. How many grams of nitrate ion are there in 75.0 g of a 1.75 × 10−4 m Pb(NO3)2 solution? What is the nitrate concentration of the solution in ppb?
20. How many milliliters of a 0.750 M solution of K2CrO4 are required to deliver 250 mg of chromate ion?
21. How many milliliters of a 1.95 × 10−6 M solution of Ag3PO4 are required to deliver 14.0 mg of Ag+?
22. Iron reacts with bromine according to the following equation:
2Fe(s) + 3Br2(aq) → 2FeBr3(aq)
How many milliliters of a 5.0 × 10−2 M solution of bromine in water are required to react completely with 750.0 mg of iron metal?
23. Aluminum reacts with HCl according to the following equation:
2Al(s) + 6HCl(aq) → 2AlCl3(aq) + 3H2(g)
If 25.0 mL of a solution of HCl in water is required to react completely with 1.05 g of aluminum metal, what is the molarity of the HCl solution?
24. The precipitation of silver chloride is a diagnostic test for the presence of chloride ion. If 25.0 mL of 0.175 M AgNO3 are required to completely precipitate the chloride ions from 10.0 mL of an NaCl solution, what was the original concentration of NaCl?
25. Barium sulfate is virtually insoluble. If a 10.0 mL solution of 0.333 M Ba(NO3)2 is stirred with 40.0 mL of a 0.100 M Na2SO4, how many grams of barium sulfate will precipitate? Which reactant is present in excess? What is its final concentration?
1. Compound Molarity (M) Solution Density (g/mL) Mole Fraction (X)
H2SO4 18.0 1.84 0.82
CH3COOH 0.393 1.00 7.21 × 10−3
KOH 3.60 1.16 6.33 × 10 −2
2. 100.0 ml of 0.40 M KI: dissolve 6.64 g of KI in enough water to make 100.0 mL of solution; 100.0 ml of 0.65 M NaCN: dissolve 3.18 g of NaCN in enough water to make 100.0 mL of solution.
1. 0.474 M glucose
2. 0.28 M Na3PO4
3. 0.0221 M I2
1. 14.6 M
2. X = 0.510
3. 57.7 m
1. 14.8 M
2. X = 0.292
3. The molarity is 0.745 M, and the mole fraction is 0.0134.
4. The molarity is 0.0129 M, the molality is 0.0129 m, the mole fraction is 2.33 × 10−4, and the solution contains 1830 ppm Na2HPO4. Mole fraction is most useful for calculating vapor pressure, because Raoult’s law states that the vapor pressure of a solution containing a non-volatile solute is equal to the mole fraction of solvent times the vapor pressure of the pure solvent. The mole fraction of the solvent is just one minus the mole fraction of solute.
5. 6.65 × 10−5 mol sodium; 6.14 × 104 ppb
6. 1.63 × 10−3 g; 2.17 × 104 ppb
7. 2.22 × 104 mL or 22.2 L
8. 4.68 M HCl
9. 0.777 g BaSO4; Na2SO4; 0.0134 M Na2SO4
13.4 Effects of Temperature and Pressure on Solubility
Learning Objective
1. To understand the relationship among temperature, pressure, and solubility.
Experimentally it is found that the solubility of most compounds depends strongly on temperature and, if a gas, on pressure as well. As we shall see, the ability to manipulate the solubility by changing the temperature and pressure has several important consequences.
Effect of Temperature on the Solubility of Solids
Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature" shows plots of the solubilities of several organic and inorganic compounds in water as a function of temperature. Although the solubility of a solid generally increases with increasing temperature, there is no simple relationship between the structure of a substance and the temperature dependence of its solubility. Many compounds (such as glucose and CH3CO2Na) exhibit a dramatic increase in solubility with increasing temperature. Others (such as NaCl and K2SO4) exhibit little variation, and still others (such as Li2SO4) become less soluble with increasing temperature.
Figure 13.9 Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature
Solubility may increase or decrease with temperature; the magnitude of this temperature dependence varies widely among compounds.
Notice in particular the curves for NH4NO3 and CaCl2. The dissolution of ammonium nitrate in water is endothermic (ΔHsoln = +25.7 kJ/mol), whereas the dissolution of calcium chloride is exothermic (ΔHsoln = −68.2 kJ/mol), yet Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature" shows that the solubility of both compounds increases sharply with increasing temperature. In fact, the magnitudes of the changes in both enthalpy and entropy for dissolution are temperature dependent. Because the solubility of a compound is ultimately determined by relatively small differences between large numbers, there is generally no good way to predict how the solubility will vary with temperature.
The variation of solubility with temperature has been measured for a wide range of compounds, and the results are published in many standard reference books. Chemists are often able to use this information to separate the components of a mixture by fractional crystallizationThe separation of compounds based on their relative solubilities in a given solvent., the separation of compounds on the basis of their solubilities in a given solvent. For example, if we have a mixture of 150 g of sodium acetate (CH3CO2Na) and 50 g of KBr, we can separate the two compounds by dissolving the mixture in 100 g of water at 80°C and then cooling the solution slowly to 0°C. According to the temperature curves in Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature", both compounds dissolve in water at 80°C, and all 50 g of KBr remains in solution at 0°C. Only about 36 g of CH3CO2Na are soluble in 100 g of water at 0°C, however, so approximately 114 g (150 g − 36 g) of CH3CO2Na crystallizes out on cooling. The crystals can then be separated by filtration. Thus fractional crystallization allows us to recover about 75% of the original CH3CO2Na in essentially pure form in only one step.
Fractional crystallization is a common technique for purifying compounds as diverse as those shown in Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature" and from antibiotics to enzymes. For the technique to work properly, the compound of interest must be more soluble at high temperature than at low temperature, so that lowering the temperature causes it to crystallize out of solution. In addition, the impurities must be more soluble than the compound of interest (as was KBr in this example) and preferably present in relatively small amounts.
Effect of Temperature on the Solubility of Gases
The solubility of gases in liquids decreases with increasing temperature, as shown in Figure 13.10 "Solubilities of Several Common Gases in Water as a Function of Temperature at Partial Pressure of 1 atm". Attractive intermolecular interactions in the gas phase are essentially zero for most substances. When a gas dissolves, it does so because its molecules interact with solvent molecules. Because heat is released when these new attractive interactions form, dissolving most gases in liquids is an exothermic process (ΔHsoln < 0). Conversely, adding heat to the solution provides thermal energy that overcomes the attractive forces between the gas and the solvent molecules, thereby decreasing the solubility of the gas.The phenomenon is similar to that involved in the increase in vapor pressure of a pure liquid with increasing temperature, as discussed in Chapter 11 "Liquids". In the case of vapor pressure, however, it is attractive forces between solvent molecules that are being overcome by the added thermal energy when the temperature is increased.
Figure 13.10 Solubilities of Several Common Gases in Water as a Function of Temperature at Partial Pressure of 1 atm
The solubilities of all gases decrease with increasing temperature.
The decrease in the solubilities of gases at higher temperatures has both practical and environmental implications. Anyone who routinely boils water in a teapot or electric kettle knows that a white or gray deposit builds up on the inside and must eventually be removed. The same phenomenon occurs on a much larger scale in the giant boilers used to supply hot water or steam for industrial applications, where it is called “boiler scale,” a deposit that can seriously decrease the capacity of hot water pipes (Figure 13.11 "Boiler Scale in a Water Pipe"). The problem is not a uniquely modern one: aqueducts that were built by the Romans 2000 years ago to carry cold water from alpine regions to warmer, drier regions in southern France were clogged by similar deposits. The chemistry behind the formation of these deposits is moderately complex and will be described in more detail in Chapter 17 "Solubility and Complexation Equilibriums", but the driving force is the loss of dissolved CO2 from solution. Hard water contains dissolved Ca2+ and HCO3 (bicarbonate) ions. Calcium bicarbonate [Ca(HCO3)2] is rather soluble in water, but calcium carbonate (CaCO3) is quite insoluble. A solution of bicarbonate ions can react to form carbon dioxide, carbonate ion, and water:
Equation 13.9
2HCO3(aq) → CO22−(aq) + H2O(l) + CO2(aq)
Heating the solution decreases the solubility of CO2, which escapes into the gas phase above the solution. In the presence of calcium ions, the carbonate ions precipitate as insoluble calcium carbonate, the major component of boiler scale.
Figure 13.11 Boiler Scale in a Water Pipe
Calcium carbonate (CaCO3) deposits in hot water pipes can significantly reduce pipe capacity. These deposits, called boiler scale, form when dissolved CO2 is driven into the gas phase at high temperatures.
In thermal pollution, lake or river water that is used to cool an industrial reactor or a power plant is returned to the environment at a higher temperature than normal. Because of the reduced solubility of O2 at higher temperatures (Figure 13.10 "Solubilities of Several Common Gases in Water as a Function of Temperature at Partial Pressure of 1 atm"), the warmer water contains less dissolved oxygen than the water did when it entered the plant. Fish and other aquatic organisms that need dissolved oxygen to live can literally suffocate if the oxygen concentration of their habitat is too low. Because the warm, oxygen-depleted water is less dense, it tends to float on top of the cooler, denser, more oxygen-rich water in the lake or river, forming a barrier that prevents atmospheric oxygen from dissolving. Eventually even deep lakes can be suffocated if the problem is not corrected. Additionally, most fish and other nonmammalian aquatic organisms are cold-blooded, which means that their body temperature is the same as the temperature of their environment. Temperatures substantially greater than the normal range can lead to severe stress or even death. Cooling systems for power plants and other facilities must be designed to minimize any adverse effects on the temperatures of surrounding bodies of water.
A similar effect is seen in the rising temperatures of bodies of water such as the Chesapeake Bay, the largest estuary in North America, where global warming has been implicated as the cause (For more information on global warming, see Chapter 5 "Energy Changes in Chemical Reactions", Section 5.5 "Energy Sources and the Environment".) For each 1.5°C that the bay’s water warms, the capacity of water to dissolve oxygen decreases by about 1.1%. Many marine species that are at the southern limit of their distributions have shifted their populations farther north. In 2005, the eelgrass, which forms an important nursery habitat for fish and shellfish, disappeared from much of the bay following record high water temperatures. Presumably, decreased oxygen levels decreased populations of clams and other filter feeders, which then decreased light transmission to allow the eelsgrass to grow. The complex relationships in ecosystems such as the Chesapeake Bay are especially sensitive to temperature fluctuations that cause a deterioration of habitat quality.
Effect of Pressure on the Solubility of Gases: Henry’s Law
External pressure has very little effect on the solubility of liquids and solids. In contrast, the solubility of gases increases as the partial pressure of the gas above a solution increases. This point is illustrated in Figure 13.12 "A Model Depicting Why the Solubility of a Gas Increases as the Partial Pressure Increases at Constant Temperature", which shows the effect of increased pressure on the dynamic equilibrium that is established between the dissolved gas molecules in solution and the molecules in the gas phase above the solution. Because the concentration of molecules in the gas phase increases with increasing pressure, the concentration of dissolved gas molecules in the solution at equilibrium is also higher at higher pressures.
Figure 13.12 A Model Depicting Why the Solubility of a Gas Increases as the Partial Pressure Increases at Constant Temperature
(a) When a gas comes in contact with a pure liquid, some of the gas molecules (purple spheres) collide with the surface of the liquid and dissolve. When the concentration of dissolved gas molecules has increased so that the rate at which gas molecules escape into the gas phase is the same as the rate at which they dissolve, a dynamic equilibrium has been established, as depicted here. This equilibrium is entirely analogous to the one that maintains the vapor pressure of a liquid. (For more information on vapor pressure, see Chapter 11 "Liquids", Section 11.3 "Unique Properties of Liquids".) (b) Increasing the pressure of the gas increases the number of molecules of gas per unit volume, which increases the rate at which gas molecules collide with the surface of the liquid and dissolve. (c) As additional gas molecules dissolve at the higher pressure, the concentration of dissolved gas increases until a new dynamic equilibrium is established.
The relationship between pressure and the solubility of a gas is described quantitatively by Henry’s lawAn equation that quantifies the relationship between the pressure and the solubility of a gas: $C=kP.$, which is named for its discoverer, the English physician and chemist, William Henry (1775–1836):
Equation 13.10
C = kP
where C is the concentration of dissolved gas at equilibrium, P is the partial pressure of the gas, and k is the Henry’s law constant, which must be determined experimentally for each combination of gas, solvent, and temperature. Although the gas concentration may be expressed in any convenient units, we will use molarity exclusively. The units of the Henry’s law constant are therefore mol/(L·atm) = M/atm. Values of the Henry’s law constants for solutions of several gases in water at 20°C are listed in Table 13.6 "Henry’s Law Constants for Selected Gases in Water at 20°C".
As the data in Table 13.6 "Henry’s Law Constants for Selected Gases in Water at 20°C" demonstrate, the concentration of a dissolved gas in water at a given pressure depends strongly on its physical properties. For a series of related substances, London dispersion forces increase as molecular mass increases. Thus among the elements of group 18, the Henry’s law constants increase smoothly from He to Ne to Ar. The table also shows that O2 is almost twice as soluble as N2. Although London dispersion forces are too weak to explain such a large difference, O2 is paramagnetic and hence more polarizable than N2, which explains its high solubility.
Table 13.6 Henry’s Law Constants for Selected Gases in Water at 20°C
Gas Henry’s Law Constant [mol/(L·atm)] × 10−4
He 3.9
Ne 4.7
Ar 15
H2 8.1
N2 7.1
O2 14
CO2 392
Gases that react chemically with water, such as HCl and the other hydrogen halides, H2S, and NH3, do not obey Henry’s law; all of these gases are much more soluble than predicted by Henry’s law. For example, HCl reacts with water to give H+(aq) and Cl(aq), not dissolved HCl molecules, and its dissociation into ions results in a much higher solubility than expected for a neutral molecule.
Note the Pattern
Gases that react with water do not obey Henry’s law.
Henry’s law has important applications. For example, bubbles of CO2 form as soon as a carbonated beverage is opened because the drink was bottled under CO2 at a pressure greater than 1 atm. When the bottle is opened, the pressure of CO2 above the solution drops rapidly, and some of the dissolved gas escapes from the solution as bubbles. Henry’s law also explains why scuba divers have to be careful to ascend to the surface slowly after a dive if they are breathing compressed air. At the higher pressures under water, more N2 from the air dissolves in the diver’s internal fluids. If the diver ascends too quickly, the rapid pressure change causes small bubbles of N2 to form throughout the body, a condition known as “the bends.” These bubbles can block the flow of blood through the small blood vessels, causing great pain and even proving fatal in some cases.
Due to the low Henry’s law constant for O2 in water, the levels of dissolved oxygen in water are too low to support the energy needs of multicellular organisms, including humans. To increase the O2 concentration in internal fluids, organisms synthesize highly soluble carrier molecules that bind O2 reversibly. For example, human red blood cells contain a protein called hemoglobin that specifically binds O2 and facilitates its transport from the lungs to the tissues, where it is used to oxidize food molecules to provide energy. The concentration of hemoglobin in normal blood is about 2.2 mM, and each hemoglobin molecule can bind four O2 molecules. Although the concentration of dissolved O2 in blood serum at 37°C (normal body temperature) is only 0.010 mM, the total dissolved O2 concentration is 8.8 mM, almost a thousand times greater than would be possible without hemoglobin. Synthetic oxygen carriers based on fluorinated alkanes have been developed for use as an emergency replacement for whole blood. Unlike donated blood, these “blood substitutes” do not require refrigeration and have a long shelf life. Their very high Henry’s law constants for O2 result in dissolved oxygen concentrations comparable to those in normal blood.
Example 7
The Henry’s law constant for O2 in water at 25°C is 1.27 × 10−3 M/atm, and the mole fraction of O2 in the atmosphere is 0.21. Calculate the solubility of O2 in water at 25°C at an atmospheric pressure of 1.00 atm.
Given: Henry’s law constant, mole fraction of O2, and pressure
Strategy:
A Use Dalton’s law of partial pressures to calculate the partial pressure of oxygen. (For more information about Dalton’s law of partial pressures, see Chapter 10 "Gases", Section 10.5 "Mixtures of Gases".)
B Use Henry’s law to calculate the solubility, expressed as the concentration of dissolved gas.
Solution:
A According to Dalton’s law, the partial pressure of O2 is proportional to the mole fraction of O2:
PA = XAPt = (0.21)(1.00 atm) = 0.21 atm
B From Henry’s law, the concentration of dissolved oxygen under these conditions is
Exercise
To understand why soft drinks “fizz” and then go “flat” after being opened, calculate the concentration of dissolved CO2 in a soft drink
1. bottled under a pressure of 5.0 atm of CO2.
2. in equilibrium with the normal partial pressure of CO2 in the atmosphere (approximately 3 × 10−4 atm).
The Henry’s law constant for CO2 in water at 25°C is 3.4 × 10−2 M/atm.
1. 0.17 M
2. 1 × 10−5 M
Summary
The solubility of most substances depends strongly on the temperature and, in the case of gases, on the pressure. The solubility of most solid or liquid solutes increases with increasing temperature. The components of a mixture can often be separated using fractional crystallization, which separates compounds according to their solubilities. The solubility of a gas decreases with increasing temperature. Henry’s law describes the relationship between the pressure and the solubility of a gas.
Key Takeaway
• The solubility of a solid may increase or decrease with increasing temperature, whereas the solubility of a gas decreases with an increase in temperature and a decrease in pressure.
Conceptual Problems
1. Use the kinetic molecular theory of gases discussed in Chapter 10 "Gases" to explain why the solubility of virtually all gases in liquids decreases with increasing temperature.
2. An industrial plant uses water from a nearby stream to cool its reactor and returns the water to the stream after use. Over a period of time, dead fish start to appear downstream from the plant, but there is no evidence for any leaks of potentially toxic chemicals into the stream. What other factor might be causing the fish to die?
3. One manufacturer’s instructions for setting up an aquarium specify that if boiled water is used, the water must be cooled to room temperature and allowed to stand overnight before fish are added. Why is it necessary for the water to stand?
4. Using a carbonated beverage as an example, discuss the effect of temperature on the “fizz.” How does the “foaminess” of a carbonated beverage differ between Los Angeles, California, and Denver, Colorado?
5. A common laboratory technique for degassing a solvent is to place it in a flask that is sealed to the atmosphere and then evacuate the flask to remove any gases above the liquid. Why is this procedure effective? Why does the temperature of the solvent usually decrease substantially during this process?
1. When water is boiled, all of the dissolved oxygen and nitrogen are removed. When the water is cooled to room temperature, it initially contains very little dissolved oxygen. Allowing the water to stand overnight allows oxygen in the air to dissolve, so that the fish will not suffocate.
2. Evacuating the flask to remove gases decreases the partial pressure of oxygen above the solution. According to Henry’s law, the solubility of any gas decreases as its partial pressure above the solution decreases. Consequently, dissolved oxygen escapes from solution into the gas phase, where it is removed by the vacuum pump. Filling the flask with nitrogen gas and repeating this process several times effectively removes almost all of the dissolved oxygen. The temperature of the solvent decreases because some solvent evaporates as well during this process. The heat that is required to evaporate some of the liquid is initially removed from the rest of the solvent, decreasing its temperature.
Numerical Problems
1. The solubility of CO2 in water at 0°C and 1 atm is 0.335 g/100 g of H2O. At 20°C and 1 atm, the solubility of CO2 in water is 0.169 g/100 g of H2O.
1. What volume of CO2 would be released by warming 750 g of water saturated with CO2 from 0°C to 20°C?
2. What is the value of the Henry’s law constant for CO2 under each set of conditions?
2. The solubility of O2 in 100 g of H2O at varying temperatures and a pressure of 1 atm is given in the following table:
Solubility (g) Temperature (°C)
0.0069 0
0.0054 10
0.0043 20
1. What is the value of the Henry’s law constant at each temperature?
2. Does Henry’s law constant increase or decrease with increasing temperature?
3. At what partial pressure of O2 would the concentration of O2 in water at 0°C be the same as the concentration in water at 20°C at a partial pressure of 1 atm?
4. Assuming that air is 20% O2 by volume, at what atmospheric pressure would the O2 concentration be the same at 20°C as it is at atmospheric pressure and 0°C?
1. 0.678 L CO2
2. k0°C = 7.61 × 10−2 M/atm, k20°C = 3.84 × 10−2 M/atm
13.5 Colligative Properties of Solutions
Learning Objective
1. To describe the relationship between solute concentration and the physical properties of a solution.
Many of the physical properties of solutions differ significantly from those of the pure substances discussed in earlier chapters, and these differences have important consequences. For example, the limited temperature range of liquid water (0°C–100°C) severely limits its use. Aqueous solutions have both a lower freezing point and a higher boiling point than pure water. Probably one of the most familiar applications of this phenomenon is the addition of ethylene glycol (“antifreeze”) to the water in an automobile radiator. This solute lowers the freezing point of the water, preventing the engine from cracking in very cold weather from the expansion of pure water on freezing. Antifreeze also enables the cooling system to operate at temperatures greater than 100°C without generating enough pressure to explode.
Changes in the freezing point and boiling point of a solution depend primarily on the number of solute particles present rather than the kind of particles. Such properties of solutions are called colligative propertiesA property of a solution that depends primarily on the number of solute particles rather than the kind of solute particles. (from the Latin colligatus, meaning “bound together” as in a quantity). As we will see, the vapor pressure and osmotic pressure of solutions are also colligative properties.
When we determine the number of particles in a solution, it is important to remember that not all solutions with the same molarity contain the same concentration of solute particles. Consider, for example, 0.01 M aqueous solutions of sucrose, NaCl, and CaCl2. Because sucrose dissolves to give a solution of neutral molecules, the concentration of solute particles in a 0.01 M sucrose solution is 0.01 M. In contrast, both NaCl and CaCl2 are ionic compounds that dissociate in water to yield solvated ions. As a result, a 0.01 M aqueous solution of NaCl contains 0.01 M Na+ ions and 0.01 M Cl ions, for a total particle concentration of 0.02 M. Similarly, the CaCl2 solution contains 0.01 M Ca2+ ions and 0.02 M Cl ions, for a total particle concentration of 0.03 M.These values are correct for dilute solutions, where the dissociation of the compounds to form separately solvated ions is complete. At higher concentrations (typically >1 M), especially with salts of small, highly charged ions (such as Mg2+ or Al3+), or in solutions with less polar solvents, dissociation to give separate ions is often incomplete (see Figure 13.21 "Ion Pairs"). The sum of the concentrations of the dissolved solute particles dictates the physical properties of a solution. In the following discussion, we must therefore keep the chemical nature of the solute firmly in mind.
Vapor Pressure of Solutions and Raoult’s Law
Adding a nonvolatile solute, one whose vapor pressure is too low to measure readily, to a volatile solvent decreases the vapor pressure of the solvent. We can understand this phenomenon qualitatively by examining Figure 13.13 "A Model Depicting Why the Vapor Pressure of a Solution of Glucose Is Less Than the Vapor Pressure of Pure Water", which is a schematic diagram of the surface of a solution of glucose in water. In an aqueous solution of glucose, a portion of the surface area is occupied by nonvolatile glucose molecules rather than by volatile water molecules. As a result, fewer water molecules can enter the vapor phase per unit time, even though the surface water molecules have the same kinetic energy distribution as they would in pure water. At the same time, the rate at which water molecules in the vapor phase collide with the surface and reenter the solution is unaffected. The net effect is to shift the dynamic equilibrium between water in the vapor and the liquid phases, decreasing the vapor pressure of the solution compared with the vapor pressure of the pure solvent.
Figure 13.13 A Model Depicting Why the Vapor Pressure of a Solution of Glucose Is Less Than the Vapor Pressure of Pure Water
(a) When water or any volatile solvent is in a closed container, water molecules move into and out of the liquid phase at the same rate in a dynamic equilibrium. (b) If a nonvolatile solute such as glucose is added, some fraction of the surface area is occupied by solvated solute molecules. As a result, the rate at which water molecules evaporate is decreased, although initially their rate of condensation is unchanged. (c) When the glucose solution reaches equilibrium, the concentration of water molecules in the vapor phase, and hence the vapor pressure, is less than that of pure water.
Figure 13.14 "Transfer of Water to a Beaker Containing a Glucose Solution" shows two beakers, one containing pure water and one containing an aqueous glucose solution, in a sealed chamber. We can view the system as having two competing equilibria: water vapor will condense in both beakers at the same rate, but water molecules will evaporate more slowly from the glucose solution because fewer water molecules are at the surface. Eventually all of the water will evaporate from the beaker containing the liquid with the higher vapor pressure (pure water) and condense in the beaker containing the liquid with the lower vapor pressure (the glucose solution). If the system consisted of only a beaker of water inside a sealed container, equilibrium between the liquid and vapor would be achieved rather rapidly, and the amount of liquid water in the beaker would remain constant.
Figure 13.14 Transfer of Water to a Beaker Containing a Glucose Solution
(top) One beaker contains an aqueous solution of glucose, and the other contains pure water. If they are placed in a sealed chamber, the lower vapor pressure of water in the glucose solution results in a net transfer of water from the beaker containing pure water to the beaker containing the glucose solution. (bottom) Eventually, all of the water is transferred to the beaker that has the glucose solution.
If the particles of a solute are essentially the same size as those of the solvent and both solute and solvent have roughly equal probabilities of being at the surface of the solution, then the effect of a solute on the vapor pressure of the solvent is proportional to the number of sites occupied by solute particles at the surface of the solution. Doubling the concentration of a given solute causes twice as many surface sites to be occupied by solute molecules, resulting in twice the decrease in vapor pressure. The relationship between solution composition and vapor pressure is therefore
Equation 13.11
$P A = X A P A 0$
where PA is the vapor pressure of component A of the solution (in this case the solvent), XA is the mole fraction of A in solution, and $PA0$ is the vapor pressure of pure A. Equation 13.11 is known as Raoult’s lawAn equation that quantifies the relationship between solution composition and vapor pressure: $PA=XAPA0.$, after the French chemist who developed it. If the solution contains only a single nonvolatile solute (B), then XA + XB = 1, and we can substitute XA = 1 − XB to obtain
Equation 13.12
$P A = ( 1 − X B ) P A 0 = P A 0 − X B P A 0$
Rearranging and defining $ΔPA =PA0 − PA,$ we obtain a relationship between the decrease in vapor pressure and the mole fraction of nonvolatile solute:
Equation 13.13
$P A 0 − P A = Δ P A = X B P A 0$
We can solve vapor pressure problems in either of two ways: by using Equation 13.11 to calculate the actual vapor pressure above a solution of a nonvolatile solute, or by using Equation 13.13 to calculate the decrease in vapor pressure caused by a specified amount of a nonvolatile solute.
Example 8
Ethylene glycol (HOCH2CH2OH), the major ingredient in commercial automotive antifreeze, increases the boiling point of radiator fluid by lowering its vapor pressure. At 100°C, the vapor pressure of pure water is 760 mmHg. Calculate the vapor pressure of an aqueous solution containing 30.2% ethylene glycol by mass, a concentration commonly used in climates that do not get extremely cold in winter.
Given: identity of solute, percentage by mass, and vapor pressure of pure solvent
Asked for: vapor pressure of solution
Strategy:
A Calculate the number of moles of ethylene glycol in an arbitrary quantity of water, and then calculate the mole fraction of water.
B Use Raoult’s law to calculate the vapor pressure of the solution.
Solution:
A A 30.2% solution of ethylene glycol contains 302 g of ethylene glycol per kilogram of solution; the remainder (698 g) is water. To use Raoult’s law to calculate the vapor pressure of the solution, we must know the mole fraction of water. Thus we must first calculate the number of moles of both ethylene glycol (EG) and water present:
The mole fraction of water is thus
B From Raoult’s law (Equation 13.11), the vapor pressure of the solution is
Alternatively, we could solve this problem by calculating the mole fraction of ethylene glycol and then using Equation 13.13 to calculate the resulting decrease in vapor pressure:
The same result is obtained using either method.
Exercise
Seawater is an approximately 3.0% aqueous solution of NaCl by mass with about 0.5% of other salts by mass. Calculate the decrease in the vapor pressure of water at 25°C caused by this concentration of NaCl, remembering that 1 mol of NaCl produces 2 mol of solute particles. The vapor pressure of pure water at 25°C is 23.8 mmHg.
Answer: 0.45 mmHg. This may seem like a small amount, but it constitutes about a 2% decrease in the vapor pressure of water and accounts in part for the higher humidity in the north-central United States near the Great Lakes, which are freshwater lakes. The decrease therefore has important implications for climate modeling.
Even when a solute is volatile, meaning that it has a measurable vapor pressure, we can still use Raoult’s law. In this case, we calculate the vapor pressure of each component separately. The total vapor pressure of the solution (PT) is the sum of the vapor pressures of the components:
Equation 13.14
$P T = P A + P B = X A P A 0 + X B P B 0$
Because XB = 1 − XA for a two-component system,
Equation 13.15
$P T = X A P A 0 + ( 1 − X A ) P B 0$
Thus we need to specify the mole fraction of only one of the components in a two-component system. Consider, for example, the vapor pressure of solutions of benzene and toluene of various compositions. At 20°C, the vapor pressures of pure benzene and toluene are 74.7 and 22.3 mmHg, respectively. The vapor pressure of benzene in a benzene–toluene solution is
Equation 13.16
$P C 6 H 6 = X C 6 H 6 P C 6 H 6 0$
and the vapor pressure of toluene in the solution is
Equation 13.17
$P C 6 H 5 CH 3 = X C 6 H 5 CH 3 P C 6 H 5 CH 3 0$
Equation 13.16 and Equation 13.17 are both in the form of the equation for a straight line: y = mx + b, where b = 0. Plots of the vapor pressures of both components versus the mole fractions are therefore straight lines that pass through the origin, as shown in Figure 13.15 "Vapor Pressures of Benzene–Toluene Solutions". Furthermore, a plot of the total vapor pressure of the solution versus the mole fraction is a straight line that represents the sum of the vapor pressures of the pure components. Thus the vapor pressure of the solution is always greater than the vapor pressure of either component.
Figure 13.15 Vapor Pressures of Benzene–Toluene Solutions
Plots of the vapor pressures of benzene (C6H6) and toluene (C6H5CH3) versus the mole fractions at 20°C are straight lines. For a solution like this, which approximates an ideal solution, the total vapor pressure of the solution (Pt) is the sum of the vapor pressures of the components.
A solution of two volatile components that behaves like the solution in Figure 13.15 "Vapor Pressures of Benzene–Toluene Solutions" is an ideal solutionA solution that obeys Raoult’s law., which is defined as a solution that obeys Raoult’s law. Like an ideal gas, an ideal solution is a hypothetical system whose properties can be described in terms of a simple model. Mixtures of benzene and toluene approximate an ideal solution because the intermolecular forces in the two pure liquids are almost identical in both kind and magnitude. Consequently, the change in enthalpy on solution formation is essentially zero (ΔHsoln ≈ 0), which is one of the defining characteristics of an ideal solution.
Note the Pattern
Ideal solutions and ideal gases are both simple models that ignore intermolecular interactions.
Most real solutions, however, do not obey Raoult’s law precisely, just as most real gases do not obey the ideal gas law exactly. Real solutions generally deviate from Raoult’s law because the intermolecular interactions between the two components A and B differ. We can distinguish between two general kinds of behavior, depending on whether the intermolecular interactions between molecules A and B are stronger or weaker than the A–A and B–B interactions in the pure components. If the A–B interactions are stronger than the A–A and B–B interactions, each component of the solution exhibits a lower vapor pressure than expected for an ideal solution, as does the solution as a whole. The favorable A–B interactions effectively stabilize the solution compared with the vapor. This kind of behavior is called a negative deviation from Raoult’s law. Systems stabilized by hydrogen bonding between two molecules, such as acetone and ethanol, exhibit negative deviations from Raoult’s law. Conversely, if the A–B interactions are weaker than the A–A and B–B interactions yet the entropy increase is enough to allow the solution to form, both A and B have an increased tendency to escape from the solution into the vapor phase. The result is a higher vapor pressure than expected for an ideal solution, producing a positive deviation from Raoult’s law. In a solution of CCl4 and methanol, for example, the nonpolar CCl4 molecules interrupt the extensive hydrogen bonding network in methanol, and the lighter methanol molecules have weaker London dispersion forces than the heavier CCl4 molecules. Consequently, solutions of CCl4 and methanol exhibit positive deviations from Raoult’s law.
Example 9
For each system, compare the intermolecular interactions in the pure liquids and in the solution to decide whether the vapor pressure will be greater than that predicted by Raoult’s law (positive deviation), approximately equal to that predicted by Raoult’s law (an ideal solution), or less than the pressure predicted by Raoult’s law (negative deviation).
1. cyclohexane and ethanol
2. methanol and acetone
3. n-hexane and isooctane (2,2,4-trimethylpentane)
Given: identity of pure liquids
Asked for: predicted deviation from Raoult’s law
Strategy:
Identify whether each liquid is polar or nonpolar, and then predict the type of intermolecular interactions that occur in solution.
Solution:
1. Liquid ethanol contains an extensive hydrogen bonding network, and cyclohexane is nonpolar. Because the cyclohexane molecules cannot interact favorably with the polar ethanol molecules, they will disrupt the hydrogen bonding. Hence the A–B interactions will be weaker than the A–A and B–B interactions, leading to a higher vapor pressure than predicted by Raoult’s law (a positive deviation).
2. Methanol contains an extensive hydrogen bonding network, but in this case the polar acetone molecules create A–B interactions that are stronger than the A–A or B–B interactions, leading to a negative enthalpy of solution and a lower vapor pressure than predicted by Raoult’s law (a negative deviation).
3. Hexane and isooctane are both nonpolar molecules (isooctane actually has a very small dipole moment, but it is so small that it can be ignored). Hence the predominant intermolecular forces in both liquids are London dispersion forces. We expect the A–B interactions to be comparable in strength to the A–A and B–B interactions, leading to a vapor pressure in good agreement with that predicted by Raoult’s law (an ideal solution).
Exercise
For each system, compare the intermolecular interactions in the pure liquids with those in the solution to decide whether the vapor pressure will be greater than that predicted by Raoult’s law (positive deviation), approximately equal to that predicted by Raoult’s law (an ideal solution), or less than the pressure predicted by Raoult’s law (negative deviation):
1. benzene and n-hexane
2. ethylene glycol and CCl4
3. acetic acid and n-propanol
1. approximately equal
2. positive deviation (vapor pressure greater than predicted)
3. negative deviation (vapor pressure less than predicted)
Boiling Point Elevation
Recall from Chapter 11 "Liquids" that the normal boiling point of a substance is the temperature at which the vapor pressure equals 1 atm. If a nonvolatile solute lowers the vapor pressure of a solvent, it must also affect the boiling point. Because the vapor pressure of the solution at a given temperature is less than the vapor pressure of the pure solvent, achieving a vapor pressure of 1 atm for the solution requires a higher temperature than the normal boiling point of the solvent. Thus the boiling point of a solution is always greater than that of the pure solvent. We can see why this must be true by comparing the phase diagram for an aqueous solution with the phase diagram for pure water (Figure 13.16 "Phase Diagrams of Pure Water and an Aqueous Solution of a Nonvolatile Solute"). The vapor pressure of the solution is less than that of pure water at all temperatures. Consequently, the liquid–vapor curve for the solution crosses the horizontal line corresponding to P = 1 atm at a higher temperature than does the curve for pure water.
Note the Pattern
The boiling point of a solution with a nonvolatile solute is always greater than the boiling point of the pure solvent.
Figure 13.16 Phase Diagrams of Pure Water and an Aqueous Solution of a Nonvolatile Solute
The vaporization curve for the solution lies below the curve for pure water at all temperatures, which results in an increase in the boiling point and a decrease in the freezing point of the solution.
The magnitude of the increase in the boiling point is related to the magnitude of the decrease in the vapor pressure. As we have just discussed, the decrease in the vapor pressure is proportional to the concentration of the solute in the solution. Hence the magnitude of the increase in the boiling point must also be proportional to the concentration of the solute (Figure 13.17 "Vapor Pressure Decrease and Boiling Point Increase as Functions of the Mole Fraction of a Nonvolatile Solute"). We can define the boiling point elevation (ΔTb)The difference between the boiling point of a solution and the boiling point of the pure solvent. as the difference between the boiling points of the solution and the pure solvent:
Equation 13.18
$Δ T b = T b − T b 0$
where Tb is the boiling point of the solution and $Tb0$ is the boiling point of the pure solvent. We can express the relationship between ΔTb and concentration as follows:
Equation 13.19
ΔTb = mKb
where m is the concentration of the solute expressed in molality, and Kb is the molal boiling point elevation constant of the solvent, which has units of °C/m. Table 13.7 "Boiling Point Elevation Constants (" lists characteristic Kb values for several commonly used solvents.
Figure 13.17 Vapor Pressure Decrease and Boiling Point Increase as Functions of the Mole Fraction of a Nonvolatile Solute
For relatively dilute solutions, the magnitude of both properties is proportional to the solute concentration.
Table 13.7 Boiling Point Elevation Constants (Kb) and Freezing Point Depression Constants (Kf) for Some Solvents
Solvent Boiling Point (°C) Kb (°C/m) Freezing Point (°C) Kf (°C/m)
acetic acid 117.90 3.22 16.64 3.63
benzene 80.09 2.64 5.49 5.07
d-(+)-camphor 207.4 4.91 178.8 37.8
carbon disulfide 46.2 2.42 −112.1 3.74
carbon tetrachloride 76.8 5.26 −22.62 31.4
chloroform 61.17 3.80 −63.41 4.60
nitrobenzene 210.8 5.24 5.70 6.87
water 100.00 0.51 0.00 1.86
The concentration of the solute is typically expressed as molality rather than mole fraction or molarity for two reasons. First, because the density of a solution changes with temperature, the value of molarity also varies with temperature. If the boiling point depends on the solute concentration, then by definition the system is not maintained at a constant temperature. Second, molality and mole fraction are proportional for relatively dilute solutions, but molality has a larger numerical value (a mole fraction can be only between zero and one). Using molality allows us to eliminate nonsignificant zeros.
According to Table 13.7 "Boiling Point Elevation Constants (", the molal boiling point elevation constant for water is 0.51°C/m. Thus a 1.00 m aqueous solution of a nonvolatile molecular solute such as glucose or sucrose will have an increase in boiling point of 0.51°C, to give a boiling point of 100.51°C at 1.00 atm. The increase in the boiling point of a 1.00 m aqueous NaCl solution will be approximately twice as large as that of the glucose or sucrose solution because 1 mol of NaCl produces 2 mol of dissolved ions. Hence a 1.00 m NaCl solution will have a boiling point of about 101.02°C.
Example 10
In Example 8, we calculated that the vapor pressure of a 30.2% aqueous solution of ethylene glycol at 100°C is 85.1 mmHg less than the vapor pressure of pure water. We stated (without offering proof) that this should result in a higher boiling point for the solution compared with pure water. Now that we have seen why this assertion is correct, calculate the boiling point of the aqueous ethylene glycol solution.
Given: composition of solution
Strategy:
Calculate the molality of ethylene glycol in the 30.2% solution. Then use Equation 13.19 to calculate the increase in boiling point.
Solution:
From Example 8, we know that a 30.2% solution of ethylene glycol in water contains 302 g of ethylene glycol (4.87 mol) per 698 g of water. The molality of the solution is thus
From Equation 13.19, the increase in boiling point is therefore
The boiling point of the solution is thus predicted to be 104°C. With a solute concentration of almost 7 m, however, the assumption of a dilute solution used to obtain Equation 13.19 may not be valid.
Exercise
Assume that a tablespoon (5.00 g) of NaCl is added to 2.00 L of water at 20.0°C, which is then brought to a boil to cook spaghetti. At what temperature will the water boil?
Answer: 100.04°C, or 100°C to three significant figures. (Recall that 1 mol of NaCl produces 2 mol of dissolved particles. The small increase in temperature means that adding salt to the water used to cook pasta has essentially no effect on the cooking time.)
Freezing Point Depression
The phase diagram in Figure 13.16 "Phase Diagrams of Pure Water and an Aqueous Solution of a Nonvolatile Solute" shows that dissolving a nonvolatile solute in water not only raises the boiling point of the water but also lowers its freezing point. The solid–liquid curve for the solution crosses the line corresponding to P = 1 atm at a lower temperature than the curve for pure water.
We can understand this result by imagining that we have a sample of water at the normal freezing point temperature, where there is a dynamic equilibrium between solid and liquid. Water molecules are continuously colliding with the ice surface and entering the solid phase at the same rate that water molecules are leaving the surface of the ice and entering the liquid phase. If we dissolve a nonvolatile solute such as glucose in the liquid, the dissolved glucose molecules will reduce the number of collisions per unit time between water molecules and the ice surface because some of the molecules colliding with the ice will be glucose. Glucose, though, has a very different structure than water, and it cannot fit into the ice lattice. Consequently, the presence of glucose molecules in the solution can only decrease the rate at which water molecules in the liquid collide with the ice surface and solidify. Meanwhile, the rate at which the water molecules leave the surface of the ice and enter the liquid phase is unchanged. The net effect is to cause the ice to melt. The only way to reestablish a dynamic equilibrium between solid and liquid water is to lower the temperature of the system, which decreases the rate at which water molecules leave the surface of the ice crystals until it equals the rate at which water molecules in the solution collide with the ice.
By analogy to our treatment of boiling point elevation, the freezing point depression (ΔTf)The difference between the freezing point of a pure solvent and the freezing point of the solution. is defined as the difference between the freezing point of the pure solvent and the freezing point of the solution:
Equation 13.20
$Δ T f = T f 0 − T f$
where $Tf0$ is the freezing point of the pure solvent and Tf is the freezing point of the solution. The order of the terms is reversed compared with Equation 13.18 to express the freezing point depression as a positive number. The relationship between ΔTf and the solute concentration is given by an equation analogous to Equation 13.19:
Equation 13.21
ΔTf = mKf
where m is the molality of the solution and Kf is the molal freezing point depression constant for the solvent (in units of °C/m). Like Kb, each solvent has a characteristic value of Kf (see Table 13.7 "Boiling Point Elevation Constants ("). Freezing point depression depends on the total number of dissolved nonvolatile solute particles, just as with boiling point elevation. Thus an aqueous NaCl solution has twice as large a freezing point depression as a glucose solution of the same molality.
People who live in cold climates use freezing point depression to their advantage in many ways. For example, salt is used to melt ice and snow on roads and sidewalks, ethylene glycol is added to engine coolant water to prevent an automobile engine from being destroyed, and methanol is added to windshield washer fluid to prevent the fluid from freezing.
Note the Pattern
The decrease in vapor pressure, increase in boiling point, and decrease in freezing point of a solution versus a pure liquid all depend on the total number of dissolved nonvolatile solute particles.
Example 11
In colder regions of the United States, NaCl or CaCl2 is often sprinkled on icy roads in winter to melt the ice and make driving safer. Use the data in Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature" to estimate the concentrations of two saturated solutions at 0°C, one of NaCl and one of CaCl2, and calculate the freezing points of both solutions to see which salt is likely to be more effective at melting ice.
Given: solubilities of two compounds
Asked for: concentrations and freezing points
Strategy:
A Estimate the solubility of each salt in 100 g of water from Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature". Determine the number of moles of each in 100 g and calculate the molalities.
B Determine the concentrations of the dissolved salts in the solutions. Substitute these values into Equation 13.21 to calculate the freezing point depressions of the solutions.
Solution:
A From Figure 13.9 "Solubilities of Several Inorganic and Organic Solids in Water as a Function of Temperature", we can estimate the solubilities of NaCl and CaCl2 to be about 36 g and 60 g, respectively, per 100 g of water at 0°C. The corresponding concentrations in molality are
The lower formula mass of NaCl more than compensates for its lower solubility, resulting in a saturated solution that has a slightly higher concentration than CaCl2.
B Because these salts are ionic compounds that dissociate in water to yield two and three ions per formula unit of NaCl and CaCl2, respectively, the actual concentrations of the dissolved species in the two saturated solutions are 2 × 6.2 m = 12 m for NaCl and 3 × 5.4 m = 16 m for CaCl2. The resulting freezing point depressions can be calculated using Equation 13.21:
Because the freezing point of pure water is 0°C, the actual freezing points of the solutions are −22°C and −30°C, respectively. Note that CaCl2 is substantially more effective at lowering the freezing point of water because its solutions contain three ions per formula unit. In fact, CaCl2 is the salt usually sold for home use, and it is also often used on highways.
Because the solubilities of both salts decrease with decreasing temperature, the freezing point can be depressed by only a certain amount, regardless of how much salt is spread on an icy road. If the temperature is significantly below the minimum temperature at which one of these salts will cause ice to melt (say −35°C), there is no point in using salt until it gets warmer.
Exercise
Calculate the freezing point of the 30.2% solution of ethylene glycol in water whose vapor pressure and boiling point we calculated in Example 13.8 and Example 13.10.
Example 12
Arrange these aqueous solutions in order of decreasing freezing points: 0.1 m KCl, 0.1 m glucose, 0.1 m SrCl2, 0.1 m ethylene glycol, 0.1 m benzoic acid, and 0.1 m HCl.
Given: molalities of six solutions
Strategy:
A Identify each solute as a strong, weak, or nonelectrolyte, and use this information to determine the number of solute particles produced.
B Multiply this number by the concentration of the solution to obtain the effective concentration of solute particles. The solution with the highest effective concentration of solute particles has the largest freezing point depression.
Solution:
A Because the molal concentrations of all six solutions are the same, we must focus on which of the substances are strong electrolytes, which are weak electrolytes, and which are nonelectrolytes to determine the actual numbers of particles in solution. KCl, SrCl2, and HCl are strong electrolytes, producing two, three, and two ions per formula unit, respectively. Benzoic acid is a weak electrolyte (approximately one particle per molecule), and glucose and ethylene glycol are both nonelectrolytes (one particle per molecule).
B The molalities of the solutions in terms of the total particles of solute are: KCl and HCl, 0.2 m; SrCl2, 0.3 m; glucose and ethylene glycol, 0.1 m; and benzoic acid, 0.1–0.2 m. Because the magnitude of the decrease in freezing point is proportional to the concentration of dissolved particles, the order of freezing points of the solutions is: glucose and ethylene glycol (highest freezing point, smallest freezing point depression) > benzoic acid > HCl = KCl > SrCl2.
Exercise
Arrange these aqueous solutions in order of increasing freezing points: 0.2 m NaCl, 0.3 m acetic acid, 0.1 m CaCl2, and 0.2 m sucrose.
Answer: 0.2 m NaCl (lowest freezing point) < 0.3 m acetic acid ≈ 0.1 m CaCl2 < 0.2 m sucrose (highest freezing point)
In biological systems, freezing plant and animal tissues produces ice crystals that rip cells apart, causing severe frostbite and degrading the quality of fish or meat. How, then, can living organisms survive in freezing climates, where we might expect that exposure to freezing temperatures would be fatal? Many organisms that live in cold climates are able to survive at temperatures well below freezing by synthesizing their own chemical antifreeze in concentrations that prevent freezing. Such substances are typically small organic molecules with multiple –OH groups analogous to ethylene glycol.
Colligative properties can also be used to determine the molar mass of an unknown compound. One method that can be carried out in the laboratory with minimal equipment is to measure the freezing point of a solution with a known mass of solute. This method is accurate for dilute solutions (≤1% by mass) because changes in the freezing point are usually large enough to measure accurately and precisely. By comparing Kb and Kf values in Table 13.7 "Boiling Point Elevation Constants (", we see that changes in the boiling point are smaller than changes in the freezing point for a given solvent. Boiling point elevations are thus more difficult to measure precisely. For this reason, freezing point depression is more commonly used to determine molar mass than is boiling point elevation. Because of its very large value of Kf (37.8°C/m), d-(+)-camphor (Table 13.7 "Boiling Point Elevation Constants (") is often used to determine the molar mass of organic compounds by this method.
Example 13
A 7.08 g sample of elemental sulfur is dissolved in 75.0 g of CS2 to create a solution whose freezing point is −113.5°C. Use these data to calculate the molar mass of elemental sulfur and thus the formula of the dissolved Sn molecules (i.e., what is the value of n?).
Given: masses of solute and solvent and freezing point
Asked for: molar mass and number of S atoms per molecule
Strategy:
A Use Equation 13.20, the measured freezing point of the solution, and the freezing point of CS2 from Table 13.7 "Boiling Point Elevation Constants (" to calculate the freezing point depression. Then use Equation 13.21 and the value of Kf from Table 13.7 "Boiling Point Elevation Constants (" to calculate the molality of the solution.
B From the calculated molality, determine the number of moles of solute present.
C Use the mass and number of moles of the solute to calculate the molar mass of sulfur in solution. Divide the result by the molar mass of atomic sulfur to obtain n, the number of sulfur atoms per mole of dissolved sulfur.
Solution:
A The first step is to calculate the freezing point depression using Equation 13.20:
$Δ T f = T f 0 − T f = − 112.1 ° C − ( − 113.5 ° C ) = 1.4 ° C$
Then Equation 13.21 gives
$m = Δ T f K f = 1.4 ° C 3.74 ° C / m = 0.37 m$
B The total number of moles of solute present in the solution is
C We now know that 0.708 g of elemental sulfur corresponds to 0.028 mol of solute. The molar mass of dissolved sulfur is thus
The molar mass of atomic sulfur is 32 g/mol, so there must be 260/32 = 8.1 sulfur atoms per mole, corresponding to a formula of S8.
Exercise
One of the byproducts formed during the synthesis of C60 is a deep red solid containing only carbon. A solution of 205 mg of this compound in 10.0 g of CCl4 has a freezing point of −23.38°C. What are the molar mass and most probable formula of the substance?
Osmotic Pressure
Osmotic pressure is a colligative property of solutions that is observed using a semipermeable membraneA barrier with pores small enough to allow solvent molecules to pass through but not solute molecules or ions., a barrier with pores small enough to allow solvent molecules to pass through but not solute molecules or ions. The net flow of solvent through a semipermeable membrane is called osmosisThe net flow of solvent through a semipermeable membrane. (from the Greek osmós, meaning “push”). The direction of net solvent flow is always from the side with the lower concentration of solute to the side with the higher concentration.
Osmosis can be demonstrated using a U-tube like the one shown in Figure 13.18 "Osmotic Pressure", which contains pure water in the left arm and a dilute aqueous solution of glucose in the right arm. A net flow of water through the membrane occurs until the levels in the arms eventually stop changing, which indicates that equilibrium has been reached. The osmotic pressure (Π)The pressure difference between the two sides of a semipermeable membrane that separates a pure solvent from a solution prepared from the same solvent. of the glucose solution is the difference in the pressure between the two sides, in this case the heights of the two columns. Although the semipermeable membrane allows water molecules to flow through in either direction, the rate of flow is not the same in both directions because the concentration of water is not the same in the two arms. The net flow of water through the membrane can be prevented by applying a pressure to the right arm that is equal to the osmotic pressure of the glucose solution.
Figure 13.18 Osmotic Pressure
(a) A dilute solution of glucose in water is placed in the right arm of a U-tube, and the left arm is filled to the same height with pure water; a semipermeable membrane separates the two arms. Because the flow of pure solvent through the membrane from left to right (from pure water to the solution) is greater than the flow of solvent in the reverse direction, the level of liquid in the right tube rises. (b) At equilibrium, the pressure differential, equal to the osmotic pressure of the solution (Πsoln), equalizes the flow rate of solvent in both directions. (c) Applying an external pressure equal to the osmotic pressure of the original glucose solution to the liquid in the right arm reverses the flow of solvent and restores the original situation.
Just as with any other colligative property, the osmotic pressure of a solution depends on the concentration of dissolved solute particles. Osmotic pressure obeys a law that resembles the ideal gas equation:
Equation 13.22
$Π = n R T V = M R T$
where M is the number of moles of solute per unit volume of solution (i.e., the molarity of the solution), R is the ideal gas constant, and T is the absolute temperature. As shown in Example 14, osmotic pressures tend to be quite high, even for rather dilute solutions.
Example 14
When placed in a concentrated salt solution, certain yeasts are able to produce high internal concentrations of glycerol to counteract the osmotic pressure of the surrounding medium. Suppose that the yeast cells are placed in an aqueous solution containing 4.0% NaCl by mass; the solution density is 1.02 g/mL at 25°C.
1. Calculate the osmotic pressure of a 4.0% aqueous NaCl solution at 25°C.
2. If the normal osmotic pressure inside a yeast cell is 7.3 atm, corresponding to a total concentration of dissolved particles of 0.30 M, what concentration of glycerol must the cells synthesize to exactly balance the external osmotic pressure at 25°C?
Given: concentration, density, and temperature of NaCl solution; internal osmotic pressure of cell
Asked for: osmotic pressure of NaCl solution and concentration of glycerol needed
Strategy:
A Calculate the molarity of the NaCl solution using the formula mass of the solute and the density of the solution. Then calculate the total concentration of dissolved particles.
B Use Equation 13.22 to calculate the osmotic pressure of the solution.
C Subtract the normal osmotic pressure of the cells from the osmotic pressure of the salt solution to obtain the additional pressure needed to balance the two. Use Equation 13.22 to calculate the molarity of glycerol needed to create this osmotic pressure.
Solution:
1. A The solution contains 4.0 g of NaCl per 100 g of solution. Using the formula mass of NaCl (58.44 g/mol) and the density of the solution (1.02 g/mL), we can calculate the molarity:
Because 1 mol of NaCl produces 2 mol of particles in solution, the total concentration of dissolved particles in the solution is (2)(0.70 M) = 1.4 M.
B Now we can use Equation 13.22 to calculate the osmotic pressure of the solution:
2. C If the yeast cells are to exactly balance the external osmotic pressure, they must produce enough glycerol to give an additional internal pressure of (34 atm − 7.3 atm) = 27 atm. Glycerol is a nonelectrolyte, so we can solve Equation 13.22 for the molarity corresponding to this osmotic pressure:
In solving this problem, we could also have recognized that the only way the osmotic pressures can be the same inside the cells and in the solution is if the concentrations of dissolved particles are the same. We are given that the normal concentration of dissolved particles in the cells is 0.3 M, and we have calculated that the NaCl solution is effectively 1.4 M in dissolved particles. The yeast cells must therefore synthesize enough glycerol to increase the internal concentration of dissolved particles from 0.3 M to 1.4 M—that is, an additional 1.1 M concentration of glycerol.
Exercise
Assume that the fluids inside a sausage are approximately 0.80 M in dissolved particles due to the salt and sodium nitrite used to prepare them. Calculate the osmotic pressure inside the sausage at 100°C to learn why experienced cooks pierce the semipermeable skin of sausages before boiling them.
Because of the large magnitude of osmotic pressures, osmosis is extraordinarily important in biochemistry, biology, and medicine. Virtually every barrier that separates an organism or cell from its environment acts like a semipermeable membrane, permitting the flow of water but not solutes. The same is true of the compartments inside an organism or cell. Some specialized barriers, such as those in your kidneys, are slightly more permeable and use a related process called dialysisA process that uses a semipermeable membrane with pores large enough to allow small solute molecules and solvent molecules to pass through but not large solute molecules., which permits both water and small molecules to pass through but not large molecules such as proteins.
The same principle has long been used to preserve fruits and their essential vitamins over the long winter. High concentrations of sugar are used in jams and jellies not for sweetness alone but because they greatly increase the osmotic pressure. Thus any bacteria not killed in the cooking process are dehydrated, which keeps them from multiplying in an otherwise rich medium for bacterial growth. A similar process using salt prevents bacteria from growing in ham, bacon, salt pork, salt cod, and other preserved meats. The effect of osmotic pressure is dramatically illustrated in Figure 13.19 "Effect on Red Blood Cells of the Surrounding Solution’s Osmotic Pressure", which shows what happens when red blood cells are placed in a solution whose osmotic pressure is much lower or much higher than the internal pressure of the cells.
Figure 13.19 Effect on Red Blood Cells of the Surrounding Solution’s Osmotic Pressure
(a) When red blood cells are placed in a dilute salt solution having the same osmotic pressure as the intracellular fluid, the rate of flow of water into and out of the cells is the same and their shape does not change. (b) When cells are placed in distilled water whose osmotic pressure is less than that of the intracellular fluid, the rate of flow of water into the cells is greater than the rate of flow out of the cells. The cells swell and eventually burst. (c) When cells are placed in a concentrated salt solution with an osmotic pressure greater than that of the intracellular fluid, the rate of flow of water out of the cells is greater than the rate of flow into the cells. The cells shrivel and become so deformed that they cannot function.
In addition to capillary action, trees use osmotic pressure to transport water and other nutrients from the roots to the upper branches. (For more information about capillary action, see Chapter 11 "Liquids", Section 11.3 "Unique Properties of Liquids".) Evaporation of water from the leaves results in a local increase in the salt concentration, which generates an osmotic pressure that pulls water up the trunk of the tree to the leaves.
Finally, a process called reverse osmosisA process that uses the application of an external pressure greater than the osmotic pressure of a solution to reverse the flow of solvent through the semipermeable membrane. can be used to produce pure water from seawater. As shown schematically in Figure 13.20 "Desalinization of Seawater by Reverse Osmosis", applying high pressure to seawater forces water molecules to flow through a semipermeable membrane that separates pure water from the solution, leaving the dissolved salt behind. Large-scale desalinization plants that can produce hundreds of thousands of gallons of freshwater per day are common in the desert lands of the Middle East, where they supply a large proportion of the freshwater needed by the population. Similar facilities are now being used to supply freshwater in southern California. Small, hand-operated reverse osmosis units can produce approximately 5 L of freshwater per hour, enough to keep 25 people alive, and are now standard equipment on US Navy lifeboats.
Figure 13.20 Desalinization of Seawater by Reverse Osmosis
(top) When the pressure applied to seawater equals its osmotic pressure (Πsoln), there is no net flow of water across the semipermeable membrane. (bottom) The application of pressure greater than the osmotic pressure of seawater forces water molecules to flow through the membrane, leaving behind a concentrated salt solution. In desalinization plants, seawater is continuously introduced under pressure and pure water is collected, so the process continues indefinitely.
Colligative Properties of Electrolyte Solutions
Thus far we have assumed that we could simply multiply the molar concentration of a solute by the number of ions per formula unit to obtain the actual concentration of dissolved particles in an electrolyte solution. We have used this simple model to predict such properties as freezing points, melting points, vapor pressure, and osmotic pressure. If this model were perfectly correct, we would expect the freezing point depression of a 0.10 m solution of sodium chloride, with 2 mol of ions per mole of NaCl in solution, to be exactly twice that of a 0.10 m solution of glucose, with only 1 mol of molecules per mole of glucose in solution. In reality, this is not always the case. Instead, the observed change in freezing points for 0.10 m aqueous solutions of NaCl and KCl are significantly less than expected (−0.348°C and −0.344°C, respectively, rather than −0.372°C), which suggests that fewer particles than we expected are present in solution.
The relationship between the actual number of moles of solute added to form a solution and the apparent number as determined by colligative properties is called the van’t Hoff factor(i)The ratio of the apparent number of particles in solution to the number predicted by the stoichiometry of the salt. and is defined as follows:Named for Jacobus Hendricus van’t Hoff (1852–1911), a Dutch chemistry professor at the University of Amsterdam who won the first Nobel Prize in Chemistry (1901) for his work on thermodynamics and solutions.
Equation 13.23
Note the Pattern
As the solute concentration increases the van’t Hoff factor decreases.
The van’t Hoff factor is therefore a measure of a deviation from ideal behavior. The lower the van’t Hoff factor, the greater the deviation. As the data in Table 13.8 "van’t Hoff Factors for 0.0500 M Aqueous Solutions of Selected Compounds at 25°C" show, the van’t Hoff factors for ionic compounds are somewhat lower than expected; that is, their solutions apparently contain fewer particles than predicted by the number of ions per formula unit. As the concentration of the solute increases, the van’t Hoff factor decreases because ionic compounds generally do not totally dissociate in aqueous solution. Instead, some of the ions exist as ion pairsA cation and anion that are in intimate contact in solution rather than separated by solvent and that migrates in solution as a single unit., a cation and an anion that for a brief time are associated with each other without an intervening shell of water molecules (Figure 13.21 "Ion Pairs"). Each of these temporary units behaves like a single dissolved particle until it dissociates. Highly charged ions such as Mg2+, Al3+, SO42−, and PO43− have a greater tendency to form ion pairs because of their strong electrostatic interactions. The actual number of solvated ions present in a solution can be determined by measuring a colligative property at several solute concentrations.
Figure 13.21 Ion Pairs
In concentrated solutions of electrolytes like NaCl, some of the ions form neutral ion pairs that are not separated by solvent and diffuse as single particles.
Table 13.8 van’t Hoff Factors for 0.0500 M Aqueous Solutions of Selected Compounds at 25°C
Compound i (measured) i (ideal)
glucose 1.0 1.0
sucrose 1.0 1.0
NaCl 1.9 2.0
HCl 1.9 2.0
MgCl2 2.7 3.0
FeCl3 3.4 4.0
Ca(NO3)2 2.5 3.0
AlCl3 3.2 4.0
MgSO4 1.4 2.0
Example 15
A 0.0500 M aqueous solution of FeCl3 has an osmotic pressure of 4.15 atm at 25°C. Calculate the van’t Hoff factor i for the solution.
Given: solute concentration, osmotic pressure, and temperature
Strategy:
A Use Equation 13.22 to calculate the expected osmotic pressure of the solution based on the effective concentration of dissolved particles in the solvent.
B Calculate the ratio of the observed osmotic pressure to the expected value. Multiply this number by the number of ions of solute per formula unit, and then use Equation 13.23 to calculate the van’t Hoff factor.
Solution:
A If FeCl3 dissociated completely in aqueous solution, it would produce four ions per formula unit [Fe3+(aq) plus 3Cl(aq)] for an effective concentration of dissolved particles of 4 × 0.0500 M = 0.200 M. The osmotic pressure would be
B The observed osmotic pressure is only 4.15 atm, presumably due to ion pair formation. The ratio of the observed osmotic pressure to the calculated value is 4.15 atm/4.89 atm = 0.849, which indicates that the solution contains (0.849)(4) = 3.40 particles per mole of FeCl3 dissolved. Alternatively, we can calculate the observed particle concentration from the osmotic pressure of 4.15 atm:
The ratio of this value to the expected value of 0.200 M is 0.170 M/0.200 M = 0.850, which again gives us (0.850)(4) = 3.40 particles per mole of FeCl3 dissolved. From Equation 13.23, the van’t Hoff factor for the solution is
Exercise
Calculate the van’t Hoff factor for a 0.050 m aqueous solution of MgCl2 that has a measured freezing point of −0.25°C.
Answer: 2.7 (versus an ideal value of 3)
Summary
The colligative properties of a solution depend on only the total number of dissolved particles in solution, not on their chemical identity. Colligative properties include vapor pressure, boiling point, freezing point, and osmotic pressure. The addition of a nonvolatile solute (one without a measurable vapor pressure) decreases the vapor pressure of the solvent. The vapor pressure of the solution is proportional to the mole fraction of solvent in the solution, a relationship known as Raoult’s law. Solutions that obey Raoult’s law are called ideal solutions. Most real solutions exhibit positive or negative deviations from Raoult’s law. The boiling point elevation (ΔTb) and freezing point depression (ΔTf) of a solution are defined as the differences between the boiling and freezing points, respectively, of the solution and the pure solvent. Both are proportional to the molality of the solute. When a solution and a pure solvent are separated by a semipermeable membrane, a barrier that allows solvent molecules but not solute molecules to pass through, the flow of solvent in opposing directions is unequal and produces an osmotic pressure, which is the difference in pressure between the two sides of the membrane. Osmosis is the net flow of solvent through such a membrane due to different solute concentrations. Dialysis uses a semipermeable membrane with pores that allow only small solute molecules and solvent molecules to pass through. In more concentrated solutions, or in solutions of salts with highly charged ions, the cations and anions can associate to form ion pairs, which decreases their effect on the colligative properties of the solution. The extent of ion pair formation is given by the van’t Hoff factor (i), the ratio of the apparent number of particles in solution to the number predicted by the stoichiometry of the salt.
Key Takeaway
• The total number of nonvolatile solute particles determines the decrease in vapor pressure, increase in boiling point, and decrease in freezing point of a solution versus the pure solvent.
Key Equations
Henry’s law
Equation 13.10: C = kP
Raoult’s law
Equation 13.11: $PA=XAPA0$
vapor pressure lowering
Equation 13.13: $PA0−PA=ΔP=XBPA0$
vapor pressure of a system containing two volatile components
Equation 13.15: $PT=XAPA0+(1−XA)PB0$
boiling point elevation
Equation 13.19: ΔTb = mKb
freezing point depression
Equation 13.21: ΔTf = mKf
osmotic pressure
Equation 13.22: $Π=nRTV=MRT$
van’t Hoff factor
Conceptual Problems
1. Why does the vapor pressure of a solvent decrease when adding a nonvolatile solute?
2. Does seawater boil at the same temperature as distilled water? If not, which has the higher boiling point? Explain your answer.
3. Which will be more soluble in benzene—O2 or HCl? Will H2S or HCl be more soluble in water? Explain your reasoning in each case.
4. Will the vapor pressure of a solution of hexane and heptane have an ideal vapor pressure curve (i.e., obey Raoult’s law)? Explain your answer. What properties of two liquids determine whether a solution of the two will exhibit an ideal behavior?
5. Predict whether the following mixtures will exhibit negative, zero, or positive deviations from Raoult’s law. Explain your reasoning in each case.
1. carbon tetrachloride and heptane
2. methanol and tetrahydrofuran (C4H8O)
3. acetone [(CH3)2C=CO] and dichloromethane
4. hexane and methanol
6. Why are deviations from the ideal behavior predicted by Raoult’s law more common for solutions of liquids than are deviations from the ideal behavior predicted by the ideal gas law for solutions of gases?
7. Boiling point elevation is proportional to the molal concentration of the solute. Is it also proportional to the molar concentration of the solution? Why or why not?
8. Many packaged foods in sealed bags are cooked by placing the bag in boiling water. How could you reduce the time required to cook the contents of the bag using this cooking method?
9. If the costs per kilogram of ethylene glycol and of ethanol were the same, which would be the more cost-effective antifreeze?
10. Many people get thirsty after eating foods such as ice cream or potato chips that have a high sugar or salt content, respectively. Suggest an explanation for this phenomenon.
11. When two aqueous solutions with identical concentrations are separated by a semipermeable membrane, no net movement of water occurs. What happens when a solute that cannot penetrate the membrane is added to one of the solutions? Why?
12. A solution injected into blood vessels must have an electrolyte concentration nearly identical to that found in blood plasma. Why? What would happen if red blood cells were placed in distilled water? What would happen to red blood cells if they were placed in a solution that had twice the electrolyte concentration of blood plasma?
13. If you were stranded on a desert island, why would drinking seawater lead to an increased rate of dehydration, eventually causing you to die of thirst?
14. What is the relationship between the van’t Hoff factor for a compound and its lattice energy?
Numerical Problems
1. Hemoglobin is the protein that is responsible for the red color of blood and for transporting oxygen from the lungs to the tissues. A solution with 11.2 mg of hemoglobin per mL has an osmotic pressure of 2.9 mmHg at 5°C. What is the molecular mass of hemoglobin?
2. To determine the molar mass of the antifreeze protein from the Arctic right-eye flounder, the osmotic pressure of a solution containing 13.2 mg of protein per mL was measured and found to be 21.2 mmHg at 10°C. What is the molar mass of the protein?
3. What is the osmotic pressure at 21.0°C of 13.5 mL of a solution with 1.77 g of sucrose (C12H22O11)?
4. A solution of NaNO3 is generated by dissolving 1.25 g of NaNO3 in enough water to give a final volume of 25.0 mL. What is the osmotic pressure of this sample at 25.0°C?
5. Which would have the lower vapor pressure—an aqueous solution that is 0.12 M in glucose or one that is 0.12 M in CaCl2? Why?
6. What is the total particle concentration expected for each aqueous solution? Which would produce the highest osmotic pressure?
1. 0.35 M KBr
2. 0.11 M MgSO4
3. 0.26 M MgCl2
4. 0.24 M glucose (C6H12O6)
7. The boiling point of an aqueous solution of sodium chloride is 100.37°C. What is the molality of the solution? How many grams of NaCl are present in 125 g of the solution?
8. Calculate the boiling point of a solution of sugar prepared by dissolving 8.4 g of glucose (C6H12O6) in 250 g of water.
9. At 37°C, the vapor pressure of 300.0 g of water was reduced from 0.062 atm to 0.058 atm by the addition of NaBr. How many grams of NaBr were added?
10. How many grams of KCl must be added to reduce the vapor pressure of 500.0 g of H2O from 17.5 mmHg to 16.0 mmHg at 20.0°C?
11. How much NaCl would you have to add to 2.0 L of water at a mountain lodge at an elevation of 7350 ft, where the pressure is 0.78 atm and the boiling point of water is 94°C, to get the water to boil at the same temperature as in New Orleans, Louisiana, where the pressure is 1.00 atm?
12. You have three solutions with the following compositions: 12.5 g of KCl in 250 mL of water, 12.5 g of glucose in 400 mL of water, and 12.5 g of MgCl2 in 350 mL of water. Which will have the highest boiling point?
13. Assuming the price per kilogram is the same, which is a better salt to use for deicing wintry roads—NaCl or MgCl2? Why? Would magnesium chloride be an effective deicer at a temperature of −8°C?
14. How many grams of KNO3 must be added to water to produce the same boiling point elevation as a solution of 2.03 g of MgCl2 in a total volume of 120.0 mL of solution, assuming complete dissociation? If the van’t Hoff factor for MgCl2 at this concentration is 2.73, how much KNO3 would be needed?
15. Calculate the quantity of each compound that would need to be added to lower the freezing point of 500.0 mL of water by 1.0°C: KBr, ethylene glycol, MgBr2, ethanol. Assume that the density of water is 1.00 g/cm3.
16. The melting point depression of biphenyl (melting point = 69.0°C) can be used to determine the molecular mass of organic compounds. A mixture of 100.0 g of biphenyl and 2.67 g of naphthalene (C10H8) has a melting point of 68.50°C. If a mixture of 1.00 g of an unknown compound with 100.0 g of biphenyl has a melting point of 68.86°C, what is the molar mass of the unknown compound?
17. Four solutions of urea in water were prepared, with concentrations of 0.32 m, 0.55 m, 1.52 m, and 3.16 m. The freezing points of these solutions were found to be −0.595°C, −1.02°C, −2.72°C, and −5.71°C, respectively. Graphically determine the freezing point depression constant for water. A fifth solution made by dissolving 6.22 g of urea in 250.0 g of water has a freezing point of −0.75°C. Use these data to determine the molar mass of urea.
18. The term osmolarity has been used to describe the total solute concentration of a solution (generally water), where 1 osmole (Osm) is equal to 1 mol of an ideal, nonionizing molecule.
1. What is the osmolarity of a 1.5 M solution of glucose? a 1.5 M solution of NaCl? a 1.5 M solution of CaCl2?
2. What is the relationship between osmolarity and the concentration of water?
3. What would be the direction of flow of water through a semipermeable membrane separating a 0.1 M solution of NaCl and a 0.1 M solution of CaCl2?
19. At 40°C, the vapor pressures of pure CCl4 and cyclohexane are 0.2807 atm and 0.2429 atm, respectively. Assuming ideal behavior, what is the vapor pressure of a solution with a CCl4 mole fraction of 0.475? What is the mole fraction of cyclohexane in the vapor phase? The boiling points of CCl4 and cyclohexane are 76.8°C and 80.7°C, respectively.
20. A benzene/toluene solution with a mole fraction of benzene of 0.6589 boils at 88°C at 1 atm. The vapor pressures of pure benzene and toluene at this temperature are 1.259 atm and 0.4993 atm, respectively. What is the composition of the vapor above the boiling solution at this temperature?
21. Plot the vapor pressure of the solution versus composition for the system CCl4–CH3CN at 45°C, given the following experimental data:
$XCCl4$ (liquid) 0.035 0.375 0.605 0.961 $XCCl4$ (vapor) 0.18 0.543 0.594 0.8 Total P (atm) 0.326 0.48 0.488 0.414
1. 6.7 × 104 amu
2. 9.24 atm
3. The CaCl2 solution will have a lower vapor pressure, because it contains three times as many particles as the glucose solution.
4. 0.36 m NaCl, 2.6 g NaCl
5. 60 g NaBr
6. 700 g NaCl
7. MgCl2 produces three particles in solution versus two for NaCl, so the same molal concentration of MgCl2 will produce a 50% greater freezing point depression than for NaCl. Nonetheless, the molar mass of MgCl2 is 95.3 g/mol versus 48.45 g/mol for NaCl. Consequently, a solution containing 1 g NaCl per 1000 g H2O will produce a freezing point depression of 0.064°C versus 0.059°C for a solution containing 1 g MgCl2 per 1000 g H2O. Thus, given equal cost per gram, NaCl is more effective. Yes, MgCl2 would be effective at −8°C; a 1.43 m solution (136 g per 1000 g H2O) would be required.
8.
kf = 1.81(°C·kg)/mol, molecular mass of urea = 60.0 g/mol
13.6 Aggregate Particles in Aqueous Solution
Learning Objective
1. To distinguish between true solutions and solutions with aggregate particles.
Suspensions and colloids are two common types of mixtures whose properties are in many ways intermediate between those of true solutions and heterogeneous mixtures. A suspensionA heterogeneous mixture of particles with diameters of about 1 µm that are distributed throughout a second phase and that separate from the dispersing phase on standing. is a heterogeneous mixture of particles with diameters of about 1 µm (1000 nm) that are distributed throughout a second phase. Common suspensions include paint, blood, and hot chocolate, which are solid particles in a liquid, and aerosol sprays, which are liquid particles in a gas. If the suspension is allowed to stand, the two phases will separate, which is why paints must be thoroughly stirred or shaken before use. A colloidA heterogeneous mixture of particles with diameters of about 2–500 nm that are distributed throughout a second phase and do not separate from the dispersing phase on standing. is also a heterogeneous mixture, but the particles of a colloid are typically smaller than those of a suspension, generally in the range of 2 to about 500 nm in diameter. Colloids include fog and clouds (liquid particles in a gas), milk (solid particles in a liquid), and butter (solid particles in a solid). Other colloids are used industrially as catalysts. Unlike in a suspension, the particles in a colloid do not separate into two phases on standing. The only combination of substances that cannot produce a suspension or a colloid is a mixture of two gases because their particles are so small that they always form true solutions. The properties of suspensions, colloids, and solutions are summarized in Table 13.9 "Properties of Liquid Solutions, Colloids, and Suspensions".
Table 13.9 Properties of Liquid Solutions, Colloids, and Suspensions
Type of Mixture Approximate Size of Particles (nm) Characteristic Properties Examples
solution < 2 not filterable; does not separate on standing; does not scatter visible light air, white wine, gasoline, salt water
colloid 2–500 scatters visible light; translucent or opaque; not filterable; does not separate on standing smoke, fog, ink, milk, butter, cheese
suspension 500–1000 cloudy or opaque; filterable; separates on standing muddy water, hot cocoa, blood, paint
Colloids and Suspensions
Colloids were first characterized in about 1860 by Thomas Graham, who also gave us Graham’s law of diffusion and effusion. Although some substances, such as starch, gelatin, and glue, appear to dissolve in water to produce solutions, Graham found that they diffuse very slowly or not at all compared with solutions of substances such as salt and sugar. Graham coined the word colloid (from the Greek kólla, meaning “glue”) to describe these substances, as well as the words solA dispersion of solid particles in a liquid or solid. and gelA semisolid sol in which all of the liquid phase has been absorbed by the solid particles. to describe certain types of colloids in which all of the solvent has been absorbed by the solid particles, thus preventing the mixture from flowing readily, as we see in Jell-O. Two other important types of colloids are aerosolsA dispersion of solid or liquid particles in a gas., which are dispersions of solid or liquid particles in a gas, and emulsions, which are dispersions of one liquid in another liquid with which it is immiscible.
Colloids share many properties with solutions. For example, the particles in both are invisible without a powerful microscope, do not settle on standing, and pass through most filters. However, the particles in a colloid scatter a beam of visible light, a phenomenon known as the Tyndall effectThe phenomenon of scattering a beam of visible light.,The effect is named after its discoverer, John Tyndall, an English physicist (1820–1893). whereas the particles of a solution do not. The Tyndall effect is responsible for the way the beams from automobile headlights are clearly visible from the side on a foggy night but cannot be seen from the side on a clear night. It is also responsible for the colored rays of light seen in many sunsets, where the sun’s light is scattered by water droplets and dust particles high in the atmosphere. An example of the Tyndall effect is shown in Figure 13.22 "Tyndall Effect, the Scattering of Light by Colloids".
Figure 13.22 Tyndall Effect, the Scattering of Light by Colloids
Both cylinders contain a solution of red food coloring in water, but a small amount of gelatin has been added to the cylinder on the right to form a colloidal suspension of gelatin particles. The beam of light goes straight through the true solution on the left, but the light beam is scattered by the colloid on the right.
Although colloids and suspensions can have particles similar in size, the two differ in stability: the particles of a colloid remain dispersed indefinitely unless the temperature or chemical composition of the dispersing medium is changed. The chemical explanation for the stability of colloids depends on whether the colloidal particles are hydrophilic or hydrophobic.
Most proteins, including those responsible for the properties of gelatin and glue, are hydrophilic because their exterior surface is largely covered with polar or charged groups. Starch, a long-branched polymer of glucose molecules, is also hydrophilic. A hydrophilic colloid particle interacts strongly with water, resulting in a shell of tightly bound water molecules that prevents the particles from aggregating when they collide. Heating such a colloid can cause aggregation because the particles collide with greater energy and disrupt the protective shell of solvent. Moreover, heat causes protein structures to unfold, exposing previously buried hydrophobic groups that can now interact with other hydrophobic groups and cause the particles to aggregate and precipitate from solution. When an egg is boiled, for example, the egg white, which is primarily a colloidal suspension of a protein called albumin, unfolds and exposes its hydrophobic groups, which aggregate and cause the albumin to precipitate as a white solid.
In some cases, a stable colloid can be transformed to an aggregated suspension by a minor chemical modification. Consider, for example, the behavior of hemoglobin, a major component of red blood cells. Hemoglobin molecules normally form a colloidal suspension inside red blood cells, which typically have a “donut” shape and are easily deformed, allowing them to squeeze through the capillaries to deliver oxygen to tissues. In a common inherited disease called sickle-cell anemia, one of the amino acids in hemoglobin that has a hydrophilic carboxylic acid side chain (glutamate) is replaced by another amino acid that has a hydrophobic side chain (valine, Figure 5.16 "The Structures of 10 Amino Acids"). Under some conditions, the abnormal hemoglobin molecules can aggregate to form long, rigid fibers that cause the red blood cells to deform, adopting a characteristic sickle shape that prevents them from passing through the capillaries (Figure 13.23 "Sickle-Cell Anemia"). The reduction in blood flow results in severe cramps, swollen joints, and liver damage. Until recently, many patients with sickle-cell anemia died before the age of 30 from infection, blood clots, or heart or kidney failure, although individuals with the sickle-cell genetic trait are more resistant to malaria than are those with “normal” hemoglobin.
Figure 13.23 Sickle-Cell Anemia
The characteristic shape of sickled red blood cells is the result of fibrous aggregation of hemoglobin molecules inside the cell.
Figure 13.24 Formation of New Land by the Destabilization of a Colloid Suspension
This satellite photograph shows the Mississippi River delta from New Orleans (top) to the Gulf of Mexico (bottom). Where seawater mixes with freshwater from the Mississippi River, colloidal clay particles in the river water precipitate (tan area).
Aggregation and precipitation can also result when the outer, charged layer of a particle is neutralized by ions with the opposite charge. In inland waterways, clay particles, which have a charged surface, form a colloidal suspension. High salt concentrations in seawater neutralize the charge on the particles, causing them to precipitate and form land at the mouths of large rivers, as seen in the satellite view in Figure 13.24 "Formation of New Land by the Destabilization of a Colloid Suspension". Charge neutralization is also an important strategy for precipitating solid particles from gaseous colloids such as smoke, and it is widely used to reduce particulate emissions from power plants that burn fossil fuels.
Emulsions
EmulsionsA dispersion of one liquid phase in another liquid with which it is immiscible. are colloids formed by the dispersion of a hydrophobic liquid in water, thereby bringing two mutually insoluble liquids, such as oil and water, in close contact. Various agents have been developed to stabilize emulsions, the most successful being molecules that combine a relatively long hydrophobic “tail” with a hydrophilic “head”:
Examples of such emulsifying agents include soaps, which are salts of long-chain carboxylic acids, such as sodium stearate [CH3(CH2)16CO2Na+], and detergents, such as sodium dodecyl sulfate [CH3(CH2)11OSO3Na+], whose structures are as follows:
When you wash your laundry, the hydrophobic tails of soaps and detergents interact with hydrophobic particles of dirt or grease through dispersion forces, dissolving in the interior of the hydrophobic particle. The hydrophilic group is then exposed at the surface of the particle, which enables it to interact with water through ion–dipole forces and hydrogen bonding. This causes the particles of dirt or grease to disperse in the wash water and allows them to be removed by rinsing. Similar agents are used in the food industry to stabilize emulsions such as mayonnaise.
A related mechanism allows us to absorb and digest the fats in buttered popcorn and French fries. To solubilize the fats so that they can be absorbed, the gall bladder secretes a fluid called bile into the small intestine. Bile contains a variety of bile salts, detergent-like molecules that emulsify the fats.
Micelles
Detergents and soaps are surprisingly soluble in water in spite of their hydrophobic tails. The reason for their solubility is that they do not, in fact, form simple solutions. Instead, above a certain concentration they spontaneously form micellesA spherical or cylindrical aggregate of detergents or soaps in water that minimizes contact between the hydrophobic tails of the detergents or soaps and water., which are spherical or cylindrical aggregates that minimize contact between the hydrophobic tails and water. In a micelle, only the hydrophilic heads are in direct contact with water, and the hydrophobic tails are in the interior of the aggregate (part (a) in Figure 13.25 "Micelles and a Phospholipid Bilayer").
Figure 13.25 Micelles and a Phospholipid Bilayer
(a) Soaps and detergents, which contain a single hydrophobic tail on each molecule, form spherical micelles with the intertwined tails in the interior and the hydrophilic head groups on the exterior. (b) Phospholipids, which have two hydrophobic tails, tend to form extended double layers in which the hydrophobic tails are sandwiched between the hydrophilic head groups.
A large class of biological molecules called phospholipidsA large class of biological, detergent-like molecules that have a hydrophilic head and two hydrophobic tails and that form bilayers. consists of detergent-like molecules with a hydrophilic head and two hydrophobic tails, as can be seen in the molecule of phosphatidylcholine. The additional tail results in a cylindrical shape that prevents phospholipids from forming a spherical micelle. Consequently, phospholipids form bilayersA two-dimensional sheet consisting of a double layer of phospholipid molecules arranged tail to tail., extended sheets consisting of a double layer of molecules. As shown in part (b) in Figure 13.25 "Micelles and a Phospholipid Bilayer", the hydrophobic tails are in the center of the bilayer, where they are not in contact with water, and the hydrophilic heads are on the two surfaces, in contact with the surrounding aqueous solution.
A cell membraneA mixture of phospholipids that form a phospholipid bilayer around the cell. is essentially a mixture of phospholipids that form a phospholipid bilayer. One definition of a cellA collection of molecules, capable of reproducing itself, that is surrounded by a phospholipid bilayer. is a collection of molecules surrounded by a phospholipid bilayer that is capable of reproducing itself. The simplest cells are bacteria, which consist of only a single compartment surrounded by a single membrane. Animal and plant cells are much more complex, however, and contain many different kinds of compartments, each surrounded by a membrane and able to carry out specialized tasks.
Summary
A suspension is a heterogeneous mixture of particles of one substance distributed throughout a second phase; the dispersed particles separate from the dispersing phase on standing. In contrast, the particles in a colloid are smaller and do not separate on standing. A colloid can be classified as a sol, a dispersion of solid particles in a liquid or solid; a gel, a semisolid sol in which all of the liquid phase has been absorbed by the solid particles; an aerosol, a dispersion of solid or liquid particles in a gas; or an emulsion, a dispersion of one liquid phase in another. A colloid can be distinguished from a true solution by its ability to scatter a beam of light, known as the Tyndall effect. Hydrophilic colloids contain an outer shell of groups that interact favorably with water, whereas hydrophobic colloids have an outer surface with little affinity for water. Emulsions are prepared by dispersing a hydrophobic liquid in water. In the absence of a dispersed hydrophobic liquid phase, solutions of detergents in water form organized spherical aggregates called micelles. Phospholipids are a class of detergent-like molecules that have two hydrophobic tails attached to a hydrophilic head. A bilayer is a two-dimensional sheet consisting of a double layer of phospholipid molecules arranged tail to tail with a hydrophobic interior and a hydrophilic exterior. Cells are collections of molecules that are surrounded by a phospholipid bilayer called a cell membrane and are able to reproduce themselves.
Key Takeaway
• Colloids and suspensions are mixtures whose properties are in many ways intermediate between those of true solutions and heterogeneous mixtures.
Conceptual Problems
1. How does a colloid differ from a suspension? Which has a greater effect on solvent properties, such as vapor pressure?
2. Is homogenized milk a colloid or a suspension? Is human plasma a colloid or a suspension? Justify your answers.
3. How would you separate the components of an emulsion of fat dispersed in an aqueous solution of sodium chloride?
13.7 End-of-Chapter Material
Application Problems
Problems marked with a ♦ involve multiple concepts.
1. ♦ Scuba divers utilize high-pressure gas in their tanks to allow them to breathe under water. At depths as shallow as 100 ft (30 m), the pressure exerted by water is 4.0 atm. At 25°C the values of Henry’s law constants for N2, O2, and He in blood are as follows: N2 = 6.5 × 10−4 mol/(L·atm), O2 = 1.28 × 10−3 mol/(L·atm), and He = 3.7 × 10−4 mol/(L·atm).
1. What would be the concentration of nitrogen and oxygen in blood at sea level where the air is 21% oxygen and 79% nitrogen?
2. What would be the concentration of nitrogen and oxygen in blood at a depth of 30 m, assuming that the diver is breathing compressed air?
2. ♦ Many modern batteries take advantage of lithium ions dissolved in suitable electrolytes. Typical batteries have lithium concentrations of 0.10 M. Which aqueous solution has the higher concentration of ion pairs: 0.08 M LiCl or 1.4 M LiCl? Why? Does an increase in the number of ion pairs correspond to a higher or lower van’t Hoff factor? Batteries rely on a high concentration of unpaired Li+ ions. Why is using a more concentrated solution not an ideal strategy in this case?
3. Hydrogen sulfide, which is extremely toxic to humans, can be detected at a concentration of 2.0 ppb. At this level, headaches, dizziness, and nausea occur. At higher concentrations, however, the sense of smell is lost, and the lack of warning can result in coma and death can result. What is the concentration of H2S in milligrams per liter at the detection level? The lethal dose of hydrogen sulfide by inhalation for rats is 7.13 × 10−4 g/L. What is this lethal dose in ppm? The density of air is 1.2929 g/L.
4. One class of antibiotics consists of cyclic polyethers that can bind alkali metal cations in aqueous solution. Given the following antibiotics and cation selectivities, what conclusion can you draw regarding the relative sizes of the cavities?
Antibiotic Cation Selectivity
nigericin K+ > Rb+ > Na+ > Cs+ > Li+
lasalocid Ba2+ >> Cs+ > Rb+, K+ > Na+, Ca2+, Mg2+
5. Phenylpropanolamine hydrochloride is a common nasal decongestant. An aqueous solution of phenylpropanolamine hydrochloride that is sold commercially as a children’s decongestant has a concentration of 6.67 × 10−3 M. If a common dose is 1.0 mL/12 lb of body weight, how many moles of the decongestant should be given to a 26 lb child?
6. The “freeze-thaw” method is often used to remove dissolved oxygen from solvents in the laboratory. In this technique, a liquid is placed in a flask that is then sealed to the atmosphere, the liquid is frozen, and the flask is evacuated to remove any gas and solvent vapor in the flask. The connection to the vacuum pump is closed, the liquid is warmed to room temperature and then refrozen, and the process is repeated. Why is this technique effective for degassing a solvent?
7. Suppose that, on a planet in a galaxy far, far away, a species has evolved whose biological processes require even more oxygen than we do. The partial pressure of oxygen on this planet, however, is much less than that on Earth. The chemical composition of the “blood” of this species is also different. Do you expect their “blood” to have a higher or lower value of the Henry’s law constant for oxygen at standard temperature and pressure? Justify your answer.
8. A car owner who had never taken general chemistry decided that he needed to put some ethylene glycol antifreeze in his car’s radiator. After reading the directions on the container, however, he decided that “more must be better.” Instead of using the recommended mixture (30% ethylene glycol/70% water), he decided to reverse the amounts and used a 70% ethylene glycol/30% water mixture instead. Serious engine problems developed. Why?
9. The ancient Greeks produced “Attic ware,” pottery with a characteristic black and red glaze. To separate smaller clay particles from larger ones, the powdered clay was suspended in water and allowed to settle. This process yielded clay fractions with coarse, medium, and fine particles, and one of these fractions was used for painting. Which size of clay particles forms a suspension, which forms a precipitate, and which forms a colloidal dispersion? Would the colloidal dispersion be better characterized as an emulsion? Why or why not? Which fraction of clay particles was used for painting?
10. The Tyndall effect is often observed in movie theaters, where it makes the beam of light from the projector clearly visible. What conclusions can you draw about the quality of the air in a movie theater where you observe a large Tyndall effect?
11. Aluminum sulfate is the active ingredient in styptic pencils, which can be used to stop bleeding from small cuts. The Al3+ ions induce aggregation of colloids in the blood, which facilitates formation of a blood clot. How can Al3+ ions induce aggregation of a colloid? What is the probable charge on the colloidal particles in blood?
12. ♦ The liver secretes bile, which is essential for the digestion of fats. As discussed in Chapter 5 "Energy Changes in Chemical Reactions", fats are biomolecules with long hydrocarbon chains. The globules of fat released by partial digestion of food particles in the stomach and lower intestine are too large to be absorbed by the intestine unless they are emulsified by bile salts, such as glycocholate. Explain why a molecule like glycocholate is effective at creating an aqueous dispersion of fats in the digestive tract.
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2017-09-22 22:38:30
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http://math.stackexchange.com/questions/2872/how-do-you-take-the-derivative-of-a-variable-with-respect-to-a-different-variabl
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How do you take the derivative of a variable with respect to a different variable:
How can I use the chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ to differentiate an expression like $\frac{d}{dx}(y^3)$ ? The solution in the book is
\begin{equation*} \frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}. \end{equation*}
I'm not sure how they got it though.
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Wait... partial or total? One is different from the other... – J. M. Aug 20 '10 at 17:32
If it was $\partial(y^3)/\partial x$ isn't the result trivially 0? – kennytm Aug 20 '10 at 17:37
I believe total.. This is where I got the expression from en.wikibooks.org/wiki/Calculus/… – Tony Johnson Aug 20 '10 at 17:38
@Tony: Isn't the answer already in the link you've quoted...? – kennytm Aug 20 '10 at 17:39
Kenny: Exactly. Tony: In that case, you should always think of the other variables as functions of x; thus, you are in fact taking the derivative of $y(x)^3$. Then you apply the chain rule. – J. M. Aug 20 '10 at 17:51
In the problem, you want to differentiate the expression $\color{green}y^{\color{blue}3}$ with the variable $\color{red}x$. But there is no information on what $\color{green}y$ is. So we must assume $\color{green}y$ is an arbitrary function, i.e. $\color{green}y := \color{green}y(\color{red}x)$.
Now the crucial point is to note that the 3 itself is also a function, i.e. we can define $\color{blue}z(\color{green}y) = \color{green}y^{\color{blue}3}$.
In this way, we could apply the chain rule (some names changed to be consistent with the problem), $$\frac{d \color{blue} z}{d \color{red} x} = \frac{d \color{blue} z}{d \color{green} y} \cdot \frac{d \color{green} y}{d \color{red} x}$$ i.e. explictly, $$\frac{d \Bigl( \color{blue} z\bigl(\color{green}y(\color{red}x)\bigr) \Bigr)}{d \color{red} x} = \frac{d \Bigl( \color{blue} z \bigl(\color{green}y\color{silver}{(x)}\bigr) \Bigr) }{d\bigl( \color{green}y\color{silver}{(x)}\bigr)} \cdot \frac{d \bigl( \color{green} y (\color{red}x)\bigr) }{d \color{red} x}$$ In $d\color{blue}z/d\color{green}y$, since no other variables (like $\color{red}x$) is directly involved, we could treat $\color{green}y$ as a variable instead of an implicit function. In $d\color{green}y/d\color{red}x$, as the function $\color{green}y(\color{red}x)$ is arbitrary, the differentiated form is the best representation already. Therefore, the expression above simplifies to $$\frac{d \left( \color{green}y^{\color{blue}3} \right) }{d \color{red} x} = \underbrace{ \color{teal}{ 3 \color{green}y^2 } }_{d(\color{green}y^{\color{blue}3})/d\color{green}y} \cdot \frac{d \color{green} y}{d \color{red} x}$$
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Yea, in intro calculus books they typically leave out the exact definition of the functions... So I'm not sure I understand where you got them. Or how they apply to the operators in question. – Tony Johnson Aug 20 '10 at 17:40
Ok, I see. That helps a lot. Thanks! – Tony Johnson Aug 20 '10 at 18:42
I think there is a bit of a mix up over notation - specifically, the problem is using $y$ in two different contexts.
The definition of the chain rule that you need (and what you have correctly stated):
$$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$
Now, for a composite function $y(u(x)) = [u(x)]^3$ (here, $y$ is the 'cubing' function), in order to take the derivative of $y$ with respect to $x$ we first take the derivative of $y$ w.r.t. $u$, then since $u$ is a function of $x$ (and should be written as $u(x)$ at this point) we use the chain rule as follows:
$\frac{dy}{du} = 3u^{2}$, and $\frac{du}{dx} = u'(x)$ since we don't have the expression of the function $u(x)$ itself. So it looks like $$\frac{d}{dx}([u(x)]^3) = 3[u(x)]^{2}u'(x).$$
This looks different because I have chosen to relabel the function $y^3$ as $u^{3}$ in order for the stated chain rule to 'look right.' Alternatively, we could state the chain rule as $\frac{dh}{dx} = \frac{dh}{dy}\frac{dy}{dx}$ where $h = h(y(x)) = [y(x)]^3$. Then $\frac{dh}{dy} = 3y^2$ and $\frac{dy}{dx}$ is just $y'$ again because we don't have a specific expression for that function. Here we would have $\frac{dh}{dx} = 3y^2y'$.
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So if I understand you correctly, then when the book states that the answer is $\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}$, $y$ is a function of x (if understood implicitly) such that $y=u(x)$? – Tony Johnson Aug 20 '10 at 18:31
No, the book is correct with the $\frac{dy}{dx}$ at the end. The book is making its mistake (IMO) when it declares the chain rule for the derivative of a composite function $y$ w.r.t. $x$ and then asks for the derivative of $y^3$ w.r.t. $x$. In this terminology the function $y^3$ of the problem is the function $y$ in the statement of the chain rule. – Tom Stephens Aug 20 '10 at 18:37
After today you will be able to distinguish these things and make the proper substitutions without even thinking about it. The book is essentially teaching you an example of multiplication using $2 \cdot 2=4$ (so there can be confusion since $2+2=4$ as well...) – Tom Stephens Aug 20 '10 at 18:40
This helps also. Thanks! – Tony Johnson Aug 20 '10 at 18:43
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2016-07-28 04:52:49
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http://limbury.de/cuda-matrix-multiplication-github.html
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## Cuda Matrix Multiplication Github
It is a good habit to check the dimensions of the matrix to see what is going on. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. Matrix Multiplication using CUDA C++. Support or Contact. Matrix elements are integer within the range [0, 16). On the other hand, so far as I know, there is only one L4T version. as_tensor([-0. 1024x1024 on GPU. I've tried lots of open sourced matmul kernels on github, but the best one I found was still about 5 times. The number of columns of Matrix A. In the following 1D kernel, cuda. Seismic modeling of complex stratified reservoirs. First of all, you have to know that none of the big guys. This is an algorithm performed on GPUs due to the parallel nature of matrix multiplication. We propose optimization of based on ELLPACK from two aspects: (1) enhanced performance for the dense vector by reducing cache misses, and (2) reduce accessed. This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). Matrix Multiplication using GPU (CUDA) Cuda Matrix Implementation using Global and Shared memory. But we can't do all of this in OpenCL nor in CUDA. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. Matrix-Matrix Multiplication on the GPU with Nvidia CUDA In the previous article we discussed Monte Carlo methods and their implementation in CUDA, focusing on option pricing. Matrix multiplication is one of the most well-known and widely-used linear algebra operations, and is frequently used to demonstrate the high-performance computing capabilities of GPUs. Since I had some background in CUDA, this was similarly derived from the common paradigm in parallel computing for a GPU. GitHub Gist: instantly share code, notes, and snippets. - GitHub - debowin/cuda-tiled-matrix-multiplication: Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. has a very low computation-data ratio and its performance is mainly bound by the memory bandwidth. If condition is true then. * Host code. 3: CUDA Operations Task 3. Support or Contact. Batched matrix-vector multiplication. With Numpy. Matrix Multiplication for CUDA explanation. Authors and Contributors. 1: Parallelization Task 3. Authors and Contributors. However, cublas is column-dominated matrix, vertically stacking matrix requires that all elements in. Batched matrix-vector multiplication. 3: CUDA Operations Task 3. the github page but an example of matrix multiplication would be: # convert matrix to gpuMatrix object gpuA <- gpuMatrix(A) gpuB <- gpuMatrix(B) # matrix mutliplication gpuC <- gpuA %*% gpuB Also, if a user is looking in to GPGPU, they are likely dealing with 'big data' so this package is intended to be used in concert with the. as_tensor([-0. Search for jobs related to Cuda matrix multiplication github or hire on the world's largest freelancing marketplace with 20m+ jobs. The image below shows the computation with 3x3 windows. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. It is a good habit to check the dimensions of the matrix to see what is going on. device("cuda:0") cur_mat = torch. Search: Cuda Matrix Multiplication Github. We analyse acoustic streaming flows using an arbitrary Lagrangian Eulerian (ALE) perspective. matrix multiplication in CUDA, this is a toy program for learning CUDA, some functions are reusable for other purposes. Optimize matrix multiplication. Maybe my expectations were a bit too high. About Matrix Cuda Github Multiplication. Since I had some background in CUDA, this was similarly derived from the common paradigm in parallel computing for a GPU. It's free to sign up and bid on jobs. as_tensor([-0. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. 26 seconds. 1: Parallelization Task 3. * Host code. Turbidite reservoirs in deep-water depositional systems, such as the oil fields in the offshore Gulf of Mexico and North Sea, are becoming an important exploration target in the petroleum industry. I’d like to share a bit of my experience on working in OpenCL through Nim. Pull requests. We performed the operations on both CPU and different GPUs and compare their results based on the time required for calculations and also calculated their CPU to GPU ratio. Working with OpenCL and Cuda in Nim. This takes a very long time ¶. Terminology: Host (a CPU and host memory), device (a GPU and device memory). GitHub; Twitter; Guides Parallel Computation Fusing Operations GPU Programming On this page Tasks Task 3. After matrix multiplication the prepended 1 is removed. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. This is an algorithm performed on GPUs due to the parallel nature of matrix multiplication. device("cuda:0") cur_mat = torch. Invoke a kernel. NASA Astrophysics Data System (ADS) Lai, Hung-Liang. Is is worth noting that we tried to use gemm in another context with a matrix of size (n,m) where m >> n multiplied bu another matrix of small size; but here the disparity of sizes and the data layout caused very poor. Matrix elements are integer within the range [0, 16). This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). This takes a very long time ¶. Matrix Multiplication using CUDA C++. Pull requests. 4 with OpenCL support. So far, I don't quite understand where this bug. Time elapsed on matrix multiplication of 1024x1024. In this article, we discuss the performance modeling and optimization of Sparse Matrix-Vector Multiplication ( ) on NVIDIA GPUs using CUDA. 1024x1024 on GPU: 13. One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. Matrix Multiplication on GPGPU in CUDA is an analytical project in which we compute the multiplication of higher order matrices. 5: Training Task 3. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. matrix-cuda. Arraymancer is a tensor library I’m writing from the ground up in Nim. 15 seconds to 0. Maybe my expectations were a bit too high. The code works well when the matrix size is less than 320*320 and requesting block size to be 32*32. But when the matrix size exceeds 320, like 321, the matrix product produced by GPU is not equal to the result by CPU. But we can't do all of this in OpenCL nor in CUDA. It has been written for clarity of exposition to illustrate various OpenCL programming principles, not with the goal of providing the most performant generic kernel for matrix multiplication. “pip install” with one of the wheels from this repo. 2007-01-01. USGS Publications Warehouse. There are currently 3 options to get tensorflow without with CUDA 11: Use the nightly version; pip install tf-nightly-gpu==2. Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts - GitHub - kberkay/Cuda-Matrix-Multiplication: Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts. device("cuda:0") cur_mat = torch. The number of lines of Matrix B. * * This sample implements matrix multiplication as described in Chapter 3 * of the programming guide. OpenCL Matrix Multiplication This sample implements matrix multiplication and is exactly the same as Chapter 6 of the programming guide. dev20201028. Invoke a kernel. We propose optimization of based on ELLPACK from two aspects: (1) enhanced performance for the dense vector by reducing cache misses, and (2) reduce accessed. Maybe my expectations were a bit too high. Time elapsed on matrix multiplication of 1024x1024. Instead of storing each matrix in a 2D array, we use 1D layout to ease the data transfer between CPU and GPU. Cuda-Matrix-Multiplication. Figure 2&3 show the details of combined matrix multiplication: Figure 2. After matrix multiplication the appended 1 is removed. unsqueeze(0) cur_vec = torch. PubMed Central. On the other hand, so far as I know, there is only one L4T version. If we multiply 6 seconds by 1000 we get 6,000 seconds to complete the matrix multiplication in python, which is a little over 4 days. 2017-01-01. Cuda-Matrix-Multiplication Matrix Multiplication on GPGPU in CUDA is an analytical project in which we compute the multiplication of higher order matrices. * Matrix multiplication: C = A * B. Time elapsed on matrix multiplication of 1024x1024. The number of columns of Matrix A. 2: Matrix Multiplication Task 3. Batched Sparse Matrix Multiplication for Accelerating Graph Convolutional Networks Yusuke Nagasaka†, Akira Nukada†, Kojima Ryosuke‡, Satoshi Matsuoka ,† †Tokyo Institute of Technology ‡Kyoto University RIKEN Center for Computational Science. * It has been written for clarity of exposition to illustrate various CUDA * programming principles, not with the goal of providing the most * performant generic kernel for matrix multiplication. Search: Cuda Matrix Multiplication Github. device("cuda:0") cur_mat = torch. Batched matrix-vector multiplication. The difference between them is very tiny, like the scale of 1e-5. : It is apparent that W,I,O on the left corresponds to a,b, and o on the right, respectively. gridsize(1) returns the length of the grid of threads: [ ]. Sample code in adding 2 numbers with a GPU. I've tried lots of open sourced matmul kernels on github, but the best one I found was still about 5 times. Instead of storing each matrix in a 2D array, we use 1D layout to ease the data transfer between CPU and GPU. 15 seconds to 0. Maybe my expectations were a bit too high. matrix multiplication in CUDA, this is a toy program for learning CUDA, some functions are reusable for other purposes. There are currently 3 options to get tensorflow without with CUDA 11: Use the nightly version; pip install tf-nightly-gpu==2. GitHub Gist: instantly share code, notes, and snippets. : It is apparent that W,I,O on the left corresponds to a,b, and o on the right, respectively. seismic demand models: Topics by Science. Search: Cuda Matrix Multiplication Github. Matrix Multiplication code on GPU with CUDA. PubMed Central. as_tensor([-0. * * This sample implements matrix multiplication as described in Chapter 3 * of the programming guide. But we can't do all of this in OpenCL nor in CUDA. Convert Division to Multiplication; Put More Things into The Table; Homework for DCS316 Multi-core Programming. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. We will especially look at a method called "tiling," which is used to reduce global memory accesses by taking advantage of the shared memory on the GPU. After some struggles, I made them to work, but then got disappointed when I saw my kernels are 10 times slower than cuBLAS GEMM kernels. unsqueeze(0) cur_vec = torch. 3 last December, I just released the new v0. * Matrix multiplication: C = A * B. Matrix Multiplication using CUDA C++. If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. 1024 1024 1024. In this post I'm going to show you how you can multiply two arrays on a CUDA device with CUBLAS. Time elapsed on matrix multiplication of 1024x1024. 3 last December, I just released the new v0. One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. The difference between them is very tiny, like the scale of 1e-5. Arraymancer is a tensor library I’m writing from the ground up in Nim. This will allow us to: (1) schedule instructions for maximum ILP, (2) save precious registers to increase register tiling, (3) use 32-bit addresses, and (4) ensure that there are no register bank-conflicts. In this article, we discuss the performance modeling and optimization of Sparse Matrix-Vector Multiplication ( ) on NVIDIA GPUs using CUDA. So the dimensions of $\bs{C}$ are ($3 \times 1$). Contribute to cvryn7/Matrix-Multiplication-With-Tiling-CUDA development by creating an account on GitHub. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. as_tensor([-0. First of all, you have to know that none of the big guys. Since I had some background in CUDA, this was similarly derived from the common paradigm in parallel computing for a GPU. GitHub Gist: instantly share code, notes, and snippets. Matrix Multiplication using CUDA C++. : It is apparent that W,I,O on the left corresponds to a,b, and o on the right, respectively. USGS Publications Warehouse. matmul differs from dot in two important ways: Multiplication by scalars is not allowed, use * instead. About Matrix Cuda Github Multiplication. a) Insert the elements at matrix1 using two for loops:. The input follows this pattern: The number of lines of Matrix A. Matrix multiplication tutorial¶ This tutorial demonstrates how to use Kernel Tuner to test and tune kernels, using matrix multiplication as an example. However, cublas is column-dominated matrix, vertically stacking matrix requires that all elements in. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. please type in m n and k. Arraymancer is a tensor library I’m writing from the ground up in Nim. I started to learn CUDA last year, and started writing matrix multiplication kernels as a learning project. 1024x1024 on GPU: 13. GitHub; Twitter; Guides Parallel Computation Fusing Operations GPU Programming On this page Tasks Task 3. 2) Read row,column numbers of matrix1, matrix2 and check column number of matrix1= row number of matrix2. check its L4T info "JetPack never installs to the Jetson. as_tensor([-0. Remember that was 1/1000 of the dataset. Search for jobs related to Cuda matrix multiplication github or hire on the world's largest freelancing marketplace with 20m+ jobs. The difference between them is very tiny, like the scale of 1e-5. 0 -MATH LIBRARIES TURING Large FFT & 16-GPU Strong Scaling Symmetric Eigensolver & Cholesky Performance cuSPARSE Sparse-Dense Matrix Multiply Performance. Matrix Multiplication on GPGPU in CUDA is an analytical project in which we compute the multiplication of higher order matrices. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. Matrix-Vector Multiplication parallel program in CUDA - matVecMul. as_tensor([-0. With Numpy. Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. Seismic modeling of complex stratified reservoirs. Each CUDA thread corresponds to an element of C and compute its result. Pull requests. Matrix multiplication tutorial¶ This tutorial demonstrates how to use Kernel Tuner to test and tune kernels, using matrix multiplication as an example. Problem Description. After spending awhile last Friday trying to vectorize a loop of a small matrix-vector multiplication for every pixel of an image, I gave up and decided to just write it as a DLM. OpenCL Matrix Multiplication This sample implements matrix multiplication and is exactly the same as Chapter 6 of the programming guide. Instead of storing each matrix in a 2D array, we use 1D layout to ease the data transfer between CPU and GPU. So far, we've been working with one-dimensional arrays, making use of a 1D grid of threads. seismic demand models: Topics by Science. matrix multiplication in CUDA, this is a toy program for learning CUDA, some functions are reusable for other purposes. Matrix multiplication is a key computation within many scientific applications, We are releasing our CUTLASS source code on GitHub as an initial exposition of CUDA GEMM techniques that will evolve into a template library API. Search for jobs related to Cuda matrix multiplication github or hire on the world's largest freelancing marketplace with 20m+ jobs. Turbidite reservoirs in deep-water depositional systems, such as the oil fields in the offshore Gulf of Mexico and North Sea, are becoming an important exploration target in the petroleum industry. test results following tests were carried out on a Tesla M2075 card [[email protected] liu]$. But when the matrix size exceeds 320, like 321, the matrix product produced by GPU is not equal to the result by CPU. In the following 1D kernel, cuda. This takes a very long time ¶. 0 -MATH LIBRARIES TURING Large FFT & 16-GPU Strong Scaling Symmetric Eigensolver & Cholesky Performance cuSPARSE Sparse-Dense Matrix Multiply Performance. matmul differs from dot in two important ways: Multiplication by scalars is not allowed, use * instead. Matrix multiplication is one of the most well-known and widely-used linear algebra operations, and is frequently used to demonstrate the high-performance computing capabilities of GPUs. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. Probabilistic seismic demand analysis using advanced ground motion intensity measures. Simon McIntosh-Smith, University of Bristol. Each CUDA thread corresponds to an element of C and compute its result. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. gridsize(1) returns the length of the grid of threads: [ ]. 5: Training Task 3. Only L4T or packages, so there isn't any JetPack version associated with the Jetson other than possibly the install GUI front end was JetPack of a certain version. please type in m n and k. Terminology: Host (a CPU and host memory), device (a GPU and device memory). Batched Sparse Matrix Multiplication for Accelerating Graph Convolutional Networks Yusuke Nagasaka†, Akira Nukada†, Kojima Ryosuke‡, Satoshi Matsuoka ,† †Tokyo Institute of Technology ‡Kyoto University RIKEN Center for Computational Science. This is an algorithm performed on GPUs due to the parallel nature of matrix multiplication. So the dimensions of$\bs{C}$are ($3 \times 1$). as_tensor([-0. Overall, we reduce 8 matrix multiplication to 2 for both Rh and Wx. If we multiply 6 seconds by 1000 we get 6,000 seconds to complete the matrix multiplication in python, which is a little over 4 days. Sparse matrix multiplication shows up in many places, and in Python, it's often handy to use a sparse matrix representation for memory purposes. Matrix Multiplication using CUDA C++. GitHub Gist: instantly share code, notes, and snippets. 1024 1024 1024. 3 last December, I just released the new v0. As you can see to calculate 50 of these using python for loops took us 5. Cuda support was added in v0. After matrix multiplication the appended 1 is removed. Pull requests. Matrix-Matrix Multiplication on the GPU with Nvidia CUDA In the previous article we discussed Monte Carlo methods and their implementation in CUDA, focusing on option pricing. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts - GitHub - kberkay/Cuda-Matrix-Multiplication: Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts. Improved performance of sparse matrix-vector multiplication allows applications using these operations to perform better and/or handle increased data resolution. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. 0 -MATH LIBRARIES TURING Large FFT & 16-GPU Strong Scaling Symmetric Eigensolver & Cholesky Performance cuSPARSE Sparse-Dense Matrix Multiply Performance. Authors and Contributors. About Matrix Cuda Github Multiplication. test results following tests were carried out on a Tesla M2075 card [[email protected] liu]$. Sparse matrix multiplication shows up in many places, and in Python, it's often handy to use a sparse matrix representation for memory purposes. Search: Cuda Matrix Multiplication Github. One thing nice about the newest version of Python 3 is the @ operator, which takes two matrices and multiplies them. please type in m n and k. Experiment making things run faster. After some struggles, I made them to work, but then got disappointed when I saw my kernels are 10 times slower than cuBLAS GEMM kernels. Matrix-Vector Multiplication parallel program in CUDA - matVecMul. 5: Training Task 3. CUDA Matrix Multiplication with Shared Memory. width + col) typedef struct {int width; int height; float * elements; int stride. I started to learn CUDA last year, and started writing matrix multiplication kernels as a learning project. In the following 1D kernel, cuda. The difference between them is very tiny, like the scale of 1e-5. 2017-01-01. Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts - GitHub - kberkay/Cuda-Matrix-Multiplication: Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts. A typical approach to this will be to create three arrays on CPU (the host in CUDA terminology), initialize them, copy the arrays on GPU (the device on CUDA terminology), do the actual matrix multiplication on GPU and finally copy the result on CPU. Each CUDA thread corresponds to an element of C and compute its result. 3 last December, I just released the new v0. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. Compute the entropy for each point of a 2D matrix using a 5x5 window. 4: CUDA Matrix Multiplication. 3: CUDA Operations Task 3. Matrix-Vector Multiplication parallel program in CUDA - matVecMul. Optimize matrix multiplication. * It has been written for clarity of exposition to illustrate various CUDA * programming principles, not with the goal of providing the most * performant generic kernel for matrix multiplication. USGS Publications Warehouse. We will especially look at a method called "tiling," which is used to reduce global memory accesses by taking advantage of the shared memory on the GPU. With extra registers, we can further increase the tile-sizes and get better performance. 1: Parallelization Task 3. Cuda support was added in v0. The warp tile structure may be implemented with the CUDA Warp Matrix Multiply-Accumulate API (WMMA) introduced. Working with OpenCL and Cuda in Nim. Time elapsed on matrix multiplication of 1024x1024. Matrix multiplication is one of the most well-known and widely-used linear algebra operations, and is frequently used to demonstrate the high-performance computing capabilities of GPUs. In this article, we discuss the performance modeling and optimization of Sparse Matrix-Vector Multiplication ( ) on NVIDIA GPUs using CUDA. The values of Matrix A. Experiment making things run faster. GitHub; Twitter; Guides Parallel Computation Fusing Operations GPU Programming On this page Tasks Task 3. So far, I don't quite understand where this bug. unsqueeze(0) cur_vec = torch. Tothong, P. Matrix Multiplication using GPU (CUDA) Cuda Matrix Implementation using Global and Shared memory. We propose optimization of based on ELLPACK from two aspects: (1) enhanced performance for the dense vector by reducing cache misses, and (2) reduce accessed. The input follows this pattern: The number of lines of Matrix A; The number of columns of Matrix A; The number of lines of Matrix B; The number of columns of Matrix B; The values of Matrix A; The values of Matrix B. as_tensor([-0. GitHub; Twitter; Guides Parallel Computation Fusing Operations GPU Programming On this page Tasks Task 3. device("cuda:0") cur_mat = torch. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. 2) Read row,column numbers of matrix1, matrix2 and check column number of matrix1= row number of matrix2. 1024x1024 on GPU. Each CUDA thread corresponds to an element of C and compute its result. parallel-computing cuda gpgpu matrix-multiplication high. Matrix Multiplication using CUDA C++. GitHub; Twitter; Guides Parallel Computation Fusing Operations GPU Programming On this page Tasks Task 3. * It has been written for clarity of exposition to illustrate various CUDA * programming principles, not with the goal of providing the most * performant generic kernel for matrix multiplication. Matrix Multiplication using GPU (CUDA) Cuda Matrix Implementation using Global and Shared memory. 4: CUDA Matrix Multiplication. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. dev20201028. as_tensor([-0. USGS Publications Warehouse. The following figure verifies, in hexadecimal representation, that the matrix multiplication module works as intended. The difference between them is very tiny, like the scale of 1e-5. Batched matrix-vector multiplication. NASA Astrophysics Data System (ADS) Lai, Hung-Liang. test results following tests were carried out on a Tesla M2075 card [[email protected] liu]$. h * Author:- Robert Hochberg * January 24, 2012 * Author note: Based nearly entirely on the code from the CUDA C Programming Guide */ #include // Matrices are stored in row-major order: // M(row, col) = *(M. 2017-01-01. Terminology: Host (a CPU and host memory), device (a GPU and device memory). 26 seconds. 1024x1024 on GPU. If we multiply 6 seconds by 1000 we get 6,000 seconds to complete the matrix multiplication in python, which is a little over 4 days. As you can see to calculate 50 of these using python for loops took us 5. Search: Cuda Matrix Multiplication Github. Define a cudaFlow for Matrix Multiplication. @LeifWickland , i am not that expertise in Cuda parallel programming , but i had tried this code on some data sets and was returning a correct result , now the important is to reduce the execution time , and here i can't say that this code is the best , so if in both cases (the correctness of the code and for better execution time ) feel free please to suggest a modulation so it will be useful. 1: Parallelization Task 3. Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. matrix multiplication; CUDA; parallelism; Let's talk about tiled matrix multiplication today. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. matrix multiplication in CUDA, this is a toy program for learning CUDA, some functions are reusable for other purposes. 3: CUDA Operations Task 3. About Matrix Cuda Github Multiplication. PubMed Central. Matrix elements are integer within the range [0, 16). Cuda-Matrix-Multiplication. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. After some struggles, I made them to work, but then got disappointed when I saw my kernels are 10 times slower than cuBLAS GEMM kernels. * Host code. Working with OpenCL and Cuda in Nim. The code works well when the matrix size is less than 320*320 and requesting block size to be 32*32. In this project, I applied GPU Computing and the parallel programming model CUDA to solve the diffusion equation. The Numpy function dot() can be used to compute the matrix product (or dot product. OpenCL Matrix Multiplication This sample implements matrix multiplication and is exactly the same as Chapter 6 of the programming guide. 3 last December, I just released the new v0. device("cuda:0") cur_mat = torch. This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). Matrix multiplication tutorial¶ This tutorial demonstrates how to use Kernel Tuner to test and tune kernels, using matrix multiplication as an example. First of all, you have to know that none of the big guys. : It is apparent that W,I,O on the left corresponds to a,b, and o on the right, respectively. Maybe my expectations were a bit too high. We will especially look at a method called "tiling," which is used to reduce global memory accesses by taking advantage of the shared memory on the GPU. Turbidite reservoirs in deep-water depositional systems, such as the oil fields in the offshore Gulf of Mexico and North Sea, are becoming an important exploration target in the petroleum industry. Simon McIntosh-Smith, University of Bristol. matrix multiplication in CUDA, this is a toy program for learning CUDA, some functions are reusable for other purposes. Support or Contact. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. Multidimensional datasets : Matrix multiplication. On the other hand, so far as I know, there is only one L4T version. Matrix Multiplication In Java – Using For Loop 1) Condition for multiplication of two matrices is -1st matrix column number equal to 2nd matrix row number. Batched Sparse Matrix Multiplication for Accelerating Graph Convolutional Networks Yusuke Nagasaka†, Akira Nukada†, Kojima Ryosuke‡, Satoshi Matsuoka ,† †Tokyo Institute of Technology ‡Kyoto University RIKEN Center for Computational Science. 3: CUDA Operations Task 3. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. The input follows this pattern: The number of lines of Matrix A; The number of columns of Matrix A; The number of lines of Matrix B; The number of columns of Matrix B; The values of Matrix A; The values of Matrix B. After matrix multiplication the prepended 1 is removed. It has been written for clarity of exposition to illustrate various OpenCL programming principles, not with the goal of providing the most performant generic kernel for matrix multiplication. device("cuda:0") cur_mat = torch. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. unsqueeze(0) cur_vec = torch. Nama, Nitesh; Huang, Tony Jun; Costanzo, Francesco. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. After spending awhile last Friday trying to vectorize a loop of a small matrix-vector multiplication for every pixel of an image, I gave up and decided to just write it as a DLM. - GitHub - debowin/cuda-tiled-matrix-multiplication: Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. 1024 1024 1024. 4: CUDA Matrix Multiplication Task 3. Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. The code works well when the matrix size is less than 320*320 and requesting block size to be 32*32. I’d like to share a bit of my experience on working in OpenCL through Nim. check its L4T info "JetPack never installs to the Jetson. It has been written for clarity of exposition to illustrate various OpenCL programming principles, not with the goal of providing the most performant generic kernel for matrix multiplication. Tom Deakin. device("cuda:0") cur_mat = torch. Batched matrix-vector multiplication. The code works well when the matrix size is less than 320*320 and requesting block size to be 32*32. The Numpy function dot() can be used to compute the matrix product (or dot product. The input follows this pattern: The number of lines of Matrix A; The number of columns of Matrix A; The number of lines of Matrix B; The number of columns of Matrix B; The values of Matrix A; The values of Matrix B. Matrix multiplication is one of the most well-known and widely-used linear algebra operations, and is frequently used to demonstrate the high-performance computing capabilities of GPUs. If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. The warp tile structure may be implemented with the CUDA Warp Matrix Multiply-Accumulate API (WMMA) introduced. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. About Matrix Cuda Github Multiplication. seismic demand models: Topics by Science. Instead of storing each matrix in a 2D array, we use 1D layout to ease the data transfer between CPU and GPU. So far, I don't quite understand where this bug. GitHub Gist: instantly share code, notes, and snippets. Tothong, P. as_tensor([-0. Matrix Multiplication code on GPU with CUDA. 0 -MATH LIBRARIES TURING Large FFT & 16-GPU Strong Scaling Symmetric Eigensolver & Cholesky Performance cuSPARSE Sparse-Dense Matrix Multiply Performance. test results following tests were carried out on a Tesla M2075 card [[email protected] liu]$. One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. GitHub Gist: instantly share code, notes, and snippets. USGS Publications Warehouse. The number of columns of Matrix A. Optimize matrix multiplication. It is a good habit to check the dimensions of the matrix to see what is going on. Batched Sparse Matrix Multiplication for Accelerating Graph Convolutional Networks Yusuke Nagasaka†, Akira Nukada†, Kojima Ryosuke‡, Satoshi Matsuoka ,† †Tokyo Institute of Technology ‡Kyoto University RIKEN Center for Computational Science. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. Multidimensional datasets : Matrix multiplication. But when the matrix size exceeds 320, like 321, the matrix product produced by GPU is not equal to the result by CPU. Seismic modeling of complex stratified reservoirs. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. device("cuda:0") cur_mat = torch. Terminology: Host (a CPU and host memory), device (a GPU and device memory). check its L4T info "JetPack never installs to the Jetson. In this project, I applied GPU Computing and the parallel programming model CUDA to solve the diffusion equation. Batched matrix-vector multiplication. matrix-cuda. This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). Search: Cuda Matrix Multiplication Github. This is an algorithm performed on GPUs due to the parallel nature of matrix multiplication. Today, we take a step back from finance to introduce a couple of essential topics, which will help us to write more advanced (and efficient!) programs in the future. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. We will especially look at a method called "tiling," which is used to reduce global memory accesses by taking advantage of the shared memory on the GPU. 1024x1024 on GPU. Cuda support was added in v0. The values of Matrix A. 2: Matrix Multiplication Task 3. There are currently 3 options to get tensorflow without with CUDA 11: Use the nightly version; pip install tf-nightly-gpu==2. dev20201028. So far, I don't quite understand where this bug. Allocate & initialize the host data. Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. Matrix-Matrix Multiplication on the GPU with Nvidia CUDA In the previous article we discussed Monte Carlo methods and their implementation in CUDA, focusing on option pricing. ) Profiling OpenCL programs. device("cuda:0") cur_mat = torch. Experiment making things run faster. Tothong, P. Since I had some background in CUDA, this was similarly derived from the common paradigm in parallel computing for a GPU. please type in m n and k. Sample code in adding 2 numbers with a GPU. So the dimensions of $\bs{C}$ are ($3 \times 1$). Time elapsed on matrix multiplication of 1024x1024. seismic demand models: Topics by Science. After matrix multiplication the prepended 1 is removed. 1024x1024 on GPU. This will allow us to: (1) schedule instructions for maximum ILP, (2) save precious registers to increase register tiling, (3) use 32-bit addresses, and (4) ensure that there are no register bank-conflicts. Tom Deakin. The difference between them is very tiny, like the scale of 1e-5. as_tensor([-0. There are currently 3 options to get tensorflow without with CUDA 11: Use the nightly version; pip install tf-nightly-gpu==2. So far, we've been working with one-dimensional arrays, making use of a 1D grid of threads. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. matrix multiplication; CUDA; parallelism; Let's talk about tiled matrix multiplication today. Arraymancer is a tensor library I’m writing from the ground up in Nim. 3 last December, I just released the new v0. Search: Cuda Matrix Multiplication Github. After spending awhile last Friday trying to vectorize a loop of a small matrix-vector multiplication for every pixel of an image, I gave up and decided to just write it as a DLM. Simon McIntosh-Smith, University of Bristol. It is a good habit to check the dimensions of the matrix to see what is going on. Acoustic streaming: an arbitrary Lagrangian– Eulerian perspective. Nama, Nitesh; Huang, Tony Jun; Costanzo, Francesco. * Host code. Indeed, the matrix product multiplied a matrix by its transpose, operation that is heavily optimized on GPU but not on CPU. device("cuda:0") cur_mat = torch. Simon McIntosh-Smith, University of Bristol. I started to learn CUDA last year, and started writing matrix multiplication kernels as a learning project. About Matrix Cuda Github Multiplication. 1024x1024 on GPU. As you can see to calculate 50 of these using python for loops took us 5. please type in m n and k. In this post I'm going to show you how you can multiply two arrays on a CUDA device with CUBLAS. Terminology: Host (a CPU and host memory), device (a GPU and device memory). as_tensor([-0. After spending awhile last Friday trying to vectorize a loop of a small matrix-vector multiplication for every pixel of an image, I gave up and decided to just write it as a DLM. Pull requests. /* Filename: multShare. Arraymancer is a tensor library I’m writing from the ground up in Nim. If condition is true then. matrix-cuda. If we multiply 6 seconds by 1000 we get 6,000 seconds to complete the matrix multiplication in python, which is a little over 4 days. This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). unsqueeze(0) cur_vec = torch. The number of columns of Matrix A. About Matrix Cuda Github Multiplication. Optimized Parallel Tiled Approach to perform Matrix Multiplication by taking advantage of the lower latency, higher bandwidth shared memory within GPU thread blocks. 0 -MATH LIBRARIES TURING Large FFT & 16-GPU Strong Scaling Symmetric Eigensolver & Cholesky Performance cuSPARSE Sparse-Dense Matrix Multiply Performance. Cuda support was added in v0. We can see in this example that the shape of $\bs{A}$ is ($3 \times 2$) and the shape of $\bs{b}$ is ($2 \times 1$). In this project, I applied GPU Computing and the parallel programming model CUDA to solve the diffusion equation. Is is worth noting that we tried to use gemm in another context with a matrix of size (n,m) where m >> n multiplied bu another matrix of small size; but here the disparity of sizes and the data layout caused very poor. We will especially look at a method called "tiling," which is used to reduce global memory accesses by taking advantage of the shared memory on the GPU. After matrix multiplication the appended 1 is removed. In order to do combined matrix multiplication correctly, we need to stack 4 matrix vertically. Cuda-Matrix-Multiplication Matrix Multiplication on GPGPU in CUDA is an analytical project in which we compute the multiplication of higher order matrices. Improved performance of sparse matrix-vector multiplication allows applications using these operations to perform better and/or handle increased data resolution. * It has been written for clarity of exposition to illustrate various CUDA * programming principles, not with the goal of providing the most * performant generic kernel for matrix multiplication. Tom Deakin. We analyse acoustic streaming flows using an arbitrary Lagrangian Eulerian (ALE) perspective. 🐛 Bug The matrix multiplication operator can't get correct results on 3090 !! To Reproduce mini code sample: import torch device = torch. matrix-cuda. The image below shows the computation with 3x3 windows. The following figure verifies, in hexadecimal representation, that the matrix multiplication module works as intended. Define a cudaFlow for Matrix Multiplication. unsqueeze(0) cur_vec = torch. Define a cudaFlow for Matrix Multiplication. 5: Training Task 3. please type in m n and k. Matrix Multiplication using GPU (CUDA) Cuda Matrix Implementation using Global and Shared memory. The code works well when the matrix size is less than 320*320 and requesting block size to be 32*32. Matrix multiplication is a key computation within many scientific applications, We are releasing our CUTLASS source code on GitHub as an initial exposition of CUDA GEMM techniques that will evolve into a template library API. elements + row * M. Arraymancer is a tensor library I’m writing from the ground up in Nim. Matrix Multiplication code on GPU with CUDA. Sparse matrix multiplication shows up in many places, and in Python, it's often handy to use a sparse matrix representation for memory purposes. Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts - GitHub - kberkay/Cuda-Matrix-Multiplication: Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts. 1: Parallelization Task 3. On the other hand, so far as I know, there is only one L4T version. seismic demand models: Topics by Science. This takes a very long time ¶. In order to do combined matrix multiplication correctly, we need to stack 4 matrix vertically. * It has been written for clarity of exposition to illustrate various CUDA * programming principles, not with the goal of providing the most * performant generic kernel for matrix multiplication. We performed the operations on both CPU and different GPUs and compare their results based on the time required for calculations and also calculated their CPU to GPU ratio. 4: CUDA Matrix Multiplication Task 3. "CUDA Tutorial" Mar 6, 2017. /* Filename: multShare. Improved performance of sparse matrix-vector multiplication allows applications using these operations to perform better and/or handle increased data resolution. For my image sizes of 1024 by 1024 pixels (actually two images of that size), the run time went from 3. Convert a simple CUDA application to OpenCL (program TBA). Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. elements + row * M. Turbidite reservoirs in deep-water depositional systems, such as the oil fields in the offshore Gulf of Mexico and North Sea, are becoming an important exploration target in the petroleum industry. Matrix Multiplication using CUDA C++. About Matrix Cuda Github Multiplication. Allocate & initialize the device data. Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts - GitHub - kberkay/Cuda-Matrix-Multiplication: Matrix Multiplication on GPU using Shared Memory considering Coalescing and Bank Conflicts. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. Porting CUDA to OpenCL. After spending awhile last Friday trying to vectorize a loop of a small matrix-vector multiplication for every pixel of an image, I gave up and decided to just write it as a DLM. Search: Cuda Matrix Multiplication Github. GitHub Gist: instantly share code, notes, and snippets. The difference between them is very tiny, like the scale of 1e-5. Matrix multiplication is one of the most well-known and widely-used linear algebra operations, and is frequently used to demonstrate the high-performance computing capabilities of GPUs. After some struggles, I made them to work, but then got disappointed when I saw my kernels are 10 times slower than cuBLAS GEMM kernels. First of all, you have to know that none of the big guys. Matrix-Vector Multiplication parallel program in CUDA - matVecMul. Matrix Multiplication code on GPU with CUDA. One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. matrix multiplication; CUDA; parallelism; Let's talk about tiled matrix multiplication today. For my image sizes of 1024 by 1024 pixels (actually two images of that size), the run time went from 3. So far, we've been working with one-dimensional arrays, making use of a 1D grid of threads. dev20201028. Nama, Nitesh; Huang, Tony Jun; Costanzo, Francesco. We can see in this example that the shape of $\bs{A}$ is ($3 \times 2$) and the shape of $\bs{b}$ is ($2 \times 1$). Cuda support was added in v0. device("cuda:0") cur_mat = torch. as_tensor([-0. Indeed, the matrix product multiplied a matrix by its transpose, operation that is heavily optimized on GPU but not on CPU. 2007-01-01. The number of lines of Matrix B. If we multiply 6 seconds by 1000 we get 6,000 seconds to complete the matrix multiplication in python, which is a little over 4 days. After matrix multiplication the prepended 1 is removed. One thing nice about the newest version of Python 3 is the @ operator, which takes two matrices and multiplies them. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. GitHub Gist: instantly share code, notes, and snippets. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. Terminology: Host (a CPU and host memory), device (a GPU and device memory). The Numpy function dot() can be used to compute the matrix product (or dot product. So far, we've been working with one-dimensional arrays, making use of a 1D grid of threads. grid(1) returns a single index that identifies the position of the thread in the grid, and and cuda. Tothong, P. In a row-major layout, an element (x, y) in the 2D matrix can be addressed at x * width + y in the transformed 1D layout. Matrix Multiplication code on GPU with CUDA. matrix-cuda. PubMed Central. Cuda-Matrix-Multiplication. Probabilistic seismic demand analysis using advanced ground motion intensity measures. Today, we take a step back from finance to introduce a couple of essential topics, which will help us to write more advanced (and efficient!) programs in the future. please type in m n and k. unsqueeze(0) cur_vec = torch. This will allow us to: (1) schedule instructions for maximum ILP, (2) save precious registers to increase register tiling, (3) use 32-bit addresses, and (4) ensure that there are no register bank-conflicts. This project is a part of my thesis focusing on researching and applying the general-purpose graphics processing unit (GPGPU) in high performance computing. Working with OpenCL and Cuda in Nim. Indeed, the matrix product multiplied a matrix by its transpose, operation that is heavily optimized on GPU but not on CPU. unsqueeze(0) cur_vec = torch. We performed the operations on both CPU and different GPUs and compare their results based on the time required for calculations and also calculated their CPU to GPU ratio. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. So the dimensions of $\bs{C}$ are ($3 \times 1$). This sample code adds 2 numbers together with a GPU: Define a kernel (a function to run on a GPU). One of the objectives in performance-based earthquake engineering is to quantify the seismic reliability of a structure at a site. It has been written for clarity of exposition to illustrate various OpenCL programming principles, not with the goal of providing the most performant generic kernel for matrix multiplication. If condition is true then. /* Filename: multShare. PubMed Central. Since I had some background in CUDA, this was similarly derived from the common paradigm in parallel computing for a GPU. GitHub Gist: instantly share code, notes, and snippets. Turbidite reservoirs in deep-water depositional systems, such as the oil fields in the offshore Gulf of Mexico and North Sea, are becoming an important exploration target in the petroleum industry. 15 seconds to 0. But we can't do all of this in OpenCL nor in CUDA. : It is apparent that W,I,O on the left corresponds to a,b, and o on the right, respectively. NASA Astrophysics Data System (ADS) Lai, Hung-Liang. Matrix Multiplication using CUDA C++. However, cublas is column-dominated matrix, vertically stacking matrix requires that all elements in. Nama, Nitesh; Huang, Tony Jun; Costanzo, Francesco. 26 seconds. Cuda-Matrix-Multiplication. After some struggles, I made them to work, but then got disappointed when I saw my kernels are 10 times slower than cuBLAS GEMM kernels.
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2021-12-09 03:14:51
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https://ksp-kos.github.io/KOS_DOC/structures/misc/steeringmanager.html
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# SteeringManager¶
See the cooked control tuning explanation for information to help with tuning the steering manager. It’s important to read that section first to understand which setting below is affecting which portion of the steering system.
The SteeringManager is a bound variable, not a suffix to a specific vessel. This prevents access to the SteeringManager of other vessels. You can access the steering manager as shown below:
// Display the ship facing, target facing, and world coordinates vectors.
SET STEERINGMANAGER:SHOWFACINGVECTORS TO TRUE.
// Change the torque calculation to multiply the available torque by 1.5.
SET STEERINGMANAGER:ROLLTORQUEFACTOR TO 1.5.
structure SteeringManager
Suffix Type Description
PITCHPID PIDLoop The PIDLoop for the pitch rotational velocity PID.
YAWPID PIDLoop The PIDLoop for the yaw rotational velocity PID.
ROLLPID PIDLoop The PIDLoop for the roll rotational velocity PID.
ENABLED boolean Returns true if the SteeringManager is currently controlling the vessel
TARGET Direction The direction that the vessel is currently steering towards
RESETPIDS() none Called to call RESET on all steering PID loops.
RESETTODEFAULT() none Called to reset all steering tuning parameters.
SHOWFACINGVECTORS boolean Enable/disable display of ship facing, target, and world coordinates vectors.
SHOWANGULARVECTORS boolean Enable/disable display of angular rotation vectors
SHOWSTEERINGSTATS boolean Enable/disable printing of the steering information on the terminal
WRITECSVFILES boolean Enable/disable logging steering to csv files.
PITCHTS scalar (s) Settling time for the pitch torque calculation.
YAWTS scalar (s) Settling time for the yaw torque calculation.
ROLLTS scalar (s) Settling time for the roll torque calculation.
MAXSTOPPINGTIME scalar (s) The maximum amount of stopping time to limit angular turn rate.
ROLLCONTROLANGLERANGE scalar (deg) The maximum value of ANGLEERROR for which to control roll.
ANGLEERROR scalar (deg) The angle between vessel:facing and target directions
PITCHERROR scalar (deg) The angular error in the pitch direction
YAWERROR scalar (deg) The angular error in the yaw direction
ROLLERROR scalar (deg) The angular error in the roll direction
PITCHTORQUEADJUST scalar (kN) Additive adjustment to pitch torque (calculated)
YAWTORQUEADJUST scalar (kN) Additive adjustment to yaw torque (calculated)
ROLLTORQUEADJUST scalar (kN) Additive adjustment to roll torque (calculated)
PITCHTORQUEFACTOR scalar Multiplicative adjustment to pitch torque (calculated)
YAWTORQUEFACTOR scalar Multiplicative adjustment to yaw torque (calculated)
ROLLTORQUEFACTOR scalar Multiplicative adjustment to roll torque (calculated)
Warning
New in version v0.20.1: The suffixes SHOWRCSVECTORS and SHOWTHRUSTVECTORS were deprecated with the move to using stock torque calculation with KSP 1.1.
SteeringManager:PITCHPID
Type: PIDLoop Get only
Returns the PIDLoop object responsible for calculating the target angular velocity in the pitch direction. This allows direct manipulation of the gain parameters, and other components of the PIDLoop structure. Changing the loop’s MAXOUTPUT or MINOUTPUT values will have no effect as they are overwritten every physics frame. They are set to limit the maximum turning rate to that which can be stopped in a MAXSTOPPINGTIME seconds (calculated based on available torque, and the ship’s moment of inertia).
SteeringManager:YAWPID
Type: PIDLoop Get only
Returns the PIDLoop object responsible for calculating the target angular velocity in the yaw direction. This allows direct manipulation of the gain parameters, and other components of the PIDLoop structure. Changing the loop’s MAXOUTPUT or MINOUTPUT values will have no effect as they are overwritten every physics frame. They are set to limit the maximum turning rate to that which can be stopped in a MAXSTOPPINGTIME seconds (calculated based on available torque, and the ship’s moment of inertia).
SteeringManager:ROLLPID
Type: PIDLoop Get only
Returns the PIDLoop object responsible for calculating the target angular velocity in the roll direction. This allows direct manipulation of the gain parameters, and other components of the PIDLoop structure. Changing the loop’s MAXOUTPUT or MINOUTPUT values will have no effect as they are overwritten every physics frame. They are set to limit the maximum turning rate to that which can be stopped in a MAXSTOPPINGTIME seconds (calculated based on available torque, and the ship’s moment of inertia).
Note
The SteeringManager will ignore the roll component of steering until after both the pitch and yaw components are close to being correct. In other words it will try to point the nose of the craft in the right direction first, before it makes any attempt to roll the craft into the right orientation. As long as the pitch or yaw is still far off from the target aim, this PIDloop won’t be getting used at all.
SteeringManager:ENABLED
Type: boolean Get only
Returns true if the SteeringManager is currently controlling the vessel steering.
SteeringManager:TARGET
Type: Direction Get only
Returns direction that the is currently being targeted. If steering is locked to a vector, this will return the calculated direction in which kOS chose an arbitrary roll to go with the vector. If steering is locked to “kill”, this will return the vessel’s last facing direction.
SteeringManager:RESETPIDS()
Returns: none
Resets the integral sum to zero for all six steering PID Loops.
SteeringManager:RESETTODEFAULT()
Returns: none
Resets the various tuning parameters of the SteeringManager to their default values as if the ship had just been loaded. This internally will also call SteeringManager:RESETPIDS.
SteeringManager:SHOWFACINGVECTORS
Type: boolean Get/Set
Setting this suffix to true will cause the steering manager to display graphical vectors (see VecDraw) representing the forward, top, and starboard of the facing direction, as well as the world x, y, and z axis orientation (centered on the vessel). Setting to false will hide the vectors, as will disabling locked steering.
SteeringManager:SHOWANGULARVECTORS
Type: boolean Get/Set
Setting this suffix to true will cause the steering manager to display graphical vectors (see VecDraw) representing the current and target angular velocities in the pitch, yaw, and roll directions. Setting to false will hide the vectors, as will disabling locked steering.
SteeringManager:SHOWSTEERINGSTATS
Type: boolean Get/Set
Setting this suffix to true will cause the steering manager to clear the terminal screen and print steering data each update.
SteeringManager:WRITECSVFILES
Type: boolean Get/Set
Setting this suffix to true will cause the steering manager log the data from all 6 PIDLoops calculating target angular velocity and target torque. The files are stored in the [KSP Root]GameDatakOSPluginsPluginDatakOS folder, with one file per loop and a new file created for each new manager instance (i.e. every launch, every revert, and every vessel load). These files can grow quite large, and add up quickly, so it is recommended to only set this value to true for testing or debugging and not normal operation.
SteeringManager:PITCHTS
Type: scalar Get/Set
Represents the settling time for the PID calculating pitch torque based on target angular velocity. The proportional and integral gain is calculated based on the settling time and the moment of inertia in the pitch direction. Ki = (moment of inertia) * (4 / (settling time)) ^ 2. Kp = 2 * sqrt((moment of inertia) * Ki).
SteeringManager:YAWTS
Type: scalar Get/Set
Represents the settling time for the PID calculating yaw torque based on target angular velocity. The proportional and integral gain is calculated based on the settling time and the moment of inertia in the yaw direction. Ki = (moment of inertia) * (4 / (settling time)) ^ 2. Kp = 2 * sqrt((moment of inertia) * Ki).
SteeringManager:ROLLTS
Type: scalar Get/Set
Represents the settling time for the PID calculating roll torque based on target angular velocity. The proportional and integral gain is calculated based on the settling time and the moment of inertia in the roll direction. Ki = (moment of inertia) * (4 / (settling time)) ^ 2. Kp = 2 * sqrt((moment of inertia) * Ki).
SteeringManager:MAXSTOPPINGTIME
Type: scalar (s) Get/Set
This value is used to limit the turning rate when calculating target angular velocity. The ship will not turn faster than what it can stop in this amount of time. The maximum angular velocity about each axis is calculated as: (max angular velocity) = MAXSTOPPINGTIME * (available torque) / (moment of inertia).
Note
This setting affects all three of the rotational velocity PID’s at once (pitch, yaw, and roll), rather than affecting the three axes individually one at a time.
SteeringManager:ROLLCONTROLANGLERANGE
Type: scalar (deg) Get/Set
The maximum value of ANGLEERROR for which kOS will attempt to respond to error along the roll axis. If this is set to 5 (the default value), the facing direction will need to be within 5 degrees of the target direction before it actually attempts to roll the ship. Setting the value to 180 will effectivelly allow roll control at any error amount. When ANGLEERROR is greater than this value, kOS will only attempt to kill all roll angular velocity. The value is clamped between 180 and 1e-16.
SteeringManager:ANGLEERROR
Type: scalar (deg) Get only
The angle between the ship’s facing direction forward vector and the target direction’s forward. This is the combined pitch and yaw error.
SteeringManager:PITCHERROR
Type: scalar (deg) Get only
The pitch angle between the ship’s facing direction and the target direction.
SteeringManager:YAWERROR
Type: scalar (deg) Get only
The yaw angle between the ship’s facing direction and the target direction.
SteeringManager:ROLLERROR
Type: scalar (deg) Get only
The roll angle between the ship’s facing direction and the target direction.
SteeringManager:PITCHTORQUEADJUST
Type: scalar (kNm) Get/Set
You can set this value to provide an additive bias to the calculated available pitch torque used in the pitch torque PID. (available torque) = ((calculated torque) + PITCHTORQUEADJUST) * PITCHTORQUEFACTOR.
SteeringManager:YAWTORQUEADJUST
Type: scalar (kNm) Get/Set
You can set this value to provide an additive bias to the calculated available yaw torque used in the yaw torque PID. (available torque) = ((calculated torque) + YAWTORQUEADJUST) * YAWTORQUEFACTOR.
SteeringManager:ROLLTORQUEADJUST
Type: scalar (kNm) Get/Set
You can set this value to provide an additive bias to the calculated available roll torque used in the roll torque PID. (available torque) = ((calculated torque) + ROLLTORQUEADJUST) * ROLLTORQUEFACTOR.
SteeringManager:PITCHTORQUEFACTOR
Type: scalar (kNm) Get/Set
You can set this value to provide an multiplicative factor bias to the calculated available pitch torque used in the torque PID. (available torque) = ((calculated torque) + PITCHTORQUEADJUST) * PITCHTORQUEFACTOR.
SteeringManager:YAWTORQUEFACTOR
Type: scalar (kNm) Get/Set
You can set this value to provide an multiplicative factor bias to the calculated available yaw torque used in the torque PID. (available torque) = ((calculated torque) + YAWTORQUEADJUST) * YAWTORQUEFACTOR.
SteeringManager:ROLLTORQUEFACTOR
Type: scalar (kNm) Get/Set
You can set this value to provide an multiplicative factor bias to the calculated available roll torque used in the torque PID. (available torque) = ((calculated torque) + ROLLTORQUEADJUST) * ROLLTORQUEFACTOR.
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2019-10-22 23:53:39
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http://spie.org/Publications/Journal/10.1117/1.JPE.7.047501
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### Journal of Photonics for Energy • new
Electrical characteristics of funnel-shaped silicon nanowire solar cells
Author(s): Ghada Yassin Abdel-Latif; Mohamed Farhat O. Hameed; Mohamed Hussein; Maher Abdel Razzak; Salah S. A. Obayya
Format Member Price Non-Member Price
PDF \$20.00 \$25.00
Paper Abstract
The electrical characteristics of funnel-shaped silicon nanowire (SiNW) solar cells are introduced and numerically analyzed. The funnel-shaped NW consists of a cylinder over a conical unit. Its aim is to maximize the optical absorption over a large wavelength range and hence the electrical efficiency by increasing the number of resonance wavelengths or by enlarging the resonance wavelength range. The conical part has different radii in the axial direction, which increases the number of resonance wavelengths. Further, the coupling between the supported modes by the upper cylinder and the lower tapered cone offers multiple optical resonances required for broadband absorption. The optical characteristics and generation rates through the studied design are obtained using 3-D finite difference time domain. However, the electrical properties are calculated using finite element via the Lumerical device software package. In this regard, radial and axial junctions are examined for the suggested design and compared with the conventional cylindrical SiNW counterpart. In this investigation, short circuit current density, open circuit voltage, fill factor, and power conversion efficiency (PCE) are simulated to quantify the optoelectronic performance of the reported design. Furthermore, the effects of the doping concentration and carrier lifetime on the performance of the funnel-shaped design are reported. The proposed SiNWs offer PCE and short circuit density of 12.7% and $27.6 mA / cm 2$ , respectively, for the axial junction. However, the funnel design with core–shell junction shows an efficiency and short-circuit current ( $J sc$ ) of 14.13% and $31.94 mA / cm 2$ , respectively. Therefore, the suggested design has higher efficiency than 6.4% and 9.6% of the conventional cylindrical SiNWs according to the axial and core shell junctions, respectively.
Paper Details
Date Published: 5 December 2017
PDF: 15 pages
J. Photon. Energy 7(4) 047501 doi: 10.1117/1.JPE.7.047501
Published in: Journal of Photonics for Energy Volume 7, Issue 4
Show Author Affiliations
Ghada Yassin Abdel-Latif, Mansoura Univ. (Egypt)
Mohamed Farhat O. Hameed, Ctr. for Photonics and Smart Materials, Zewail City of Science and Technology (Egypt)
Univ. of Science and Technology, Zewail City of Science and Technology (Egypt)
Mansoura Univ. (Egypt)
Mohamed Hussein, Ctr. for Photonics and Smart Materials, Zewail City of Science and Technology (Egypt)
Ain Shams Univ. (Egypt)
Maher Abdel Razzak, Mansoura Univ. (Egypt)
Salah S. A. Obayya, Mansoura Univ. (Egypt)
Ctr. for Photonics and Smart Materials, Zewail City of Science and Technology (Egypt)
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2018-04-25 01:32:57
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http://math.stackexchange.com/tags/rational-numbers/hot
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# Tag Info
## Hot answers tagged rational-numbers
5
$a)$ it amounts to solving in $\mathbb{Z}: x^2+y^2=3z^2$. You have that $x^2+y^2 = 0 \pmod 3 \to x = y = 0 \pmod 3$, and you get back the original one using descending method, and this proves $x = y = z = 0$, but this means the first equation $x^2+y^2 = 3$ has no rational solutions.
4
First note that: $x\to \sin(x)$ is locally a bijection on the interval you have given , so it's rational for an infinite number of arguments, in particular whenever it's evaluated on $\arcsin(r)$ for rational $r$ Another way of solving your problem is assuming that $\sin(x)$ and $\cos(x)$ are together rationals and here we will come up with the Pythagorean ...
3
The only nice $x$ in this range for which $\sin x$ is rational are $0$, $\pi/6$, and $\pi/2$. Every other preimage of a rational number is neither rational nor a rational multiple of $\pi$. (In fact I'm almost sure that if $\sin x$ is rational and not in $\{-1,-\frac12,0,\frac12,1\}$, then $x$ is algebraically independent of $\pi$).
3
I don't know if this is what you are looking for, but if we work on the unit circle, $sinx$ is the y-coordinate of any point $(x,y)$. Then $a^2+b^2=1$ , where $a=cosx, b=sinx$ . Then $$b=\sqrt{1-a^2} \in \mathbb Q$$ iff $(1-a^2)= \frac {p^2}{q^2}$ , so $y$ as the coordinate of a point is rational iff the x coordinate is of the form $\sqrt { 1-p^2/q^2}$ ...
3
Set $\,x=\sqrt{\dfrac{4-2\sqrt 3\strut}7}$, and eliminate progressively the radicals. You'll obtain the equation: $$49x^4-56x^2+4=0$$ If $x$ is rational, say $x=\dfrac pq$, $p$ and $q$ coprime, we know $p$ is a divisor of $4$, and $q$ a divisor of $49$, whence the statement.
3
What you want to prove is: given an irrational number $b$, then one of the following numbers : $$r^{\frac{1}{b}}\ \ \ \ \ \ r\in \Bbb Q$$ is irrational, this seems to be an attainable result for what we know nowadays about the transcendence of numbers, we have for example the following theorem: Six Exponentials Theorem:Let $(x_1,x_2)$ and ...
2
I am not quite sure where Fubini factors in, but your question can be answered by a slightly stronger statement. Proposition Let $A, B\subset \mathbb{R}$ be Lebesgue measurable sets of positive measure, then the difference set $A - B$ contains an interval. To wit, if $A - B$ contains an interval, it must contain a rational number, which would ...
2
Firstly, you should not say "infinite integers" if you mean "infinitely many integers". If one admits such a thing as an infinite integer, and $n$ is an infinite integer, then $n$ and $n+1$ are infinite integers, but they are not infinitely many, since there are only two of them. This is standard usage in mathematics, regardless of what usages may prevail ...
2
Let $r \in L_2$, i.e. $0 < r < 2$. We want to show that there exist $r_1, r_2 \in L_\sqrt2$ such that $r = r_1 \cdot r_2$. Since $\dfrac 2r > 1$ we can choose $v \in \mathbb N$ such that $$1 + \frac 1v < \frac{2}{r} \, . \tag 1$$ Then define $$u = \max \{ x \in \mathbb N \mid \frac{x^2}{v^2} < 2 \} \tag 2$$ so that $$... 1 According to the Wikipedia article on Pisot–Vijayaraghavan numbers, this is true. In fact, if x is algebraic (not necessarily rational) then it has to be a Pisot number, and in particular an algebraic integer. A rational integer is of course a bona fide integer. 1 One can prove that$$ Trd (\mathbb{Q}(S)) \aleph_0 + 1 \le \dim (\mathbb{Q}(S)) \le (Trd (\mathbb{Q}(S)) +1) \aleph_0 .$$In particular \dim (\mathbb{Q}(S)) = Trd (\mathbb{Q}(S)) \aleph_0 if Trd (\mathbb{Q}(S)) is non-zero. Let S' \subset S be a transcendence basis of \mathbb{Q}(S) then S' is algebraically indepenent over the rationals and ... 1 For N an integer, the general result is that if x^2+y^2=N has rational solutions, then it has at least one integer solution. 1 As shown in this answer, n can be written as the sum of two squares if and only if, in the prime factorization of n, each prime that is \equiv3\pmod4 appears with even exponent. If x^z+y^2=3z^2, then 3 appears with odd exponent. Thus, there are no rational solutions of$$ \left(\frac xz\right)^2+\left(\frac yz\right)^2=3\tag{1} As noted, ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-04-19 14:34:19
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http://tex.stackexchange.com/tags/chemfig/hot?filter=year
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# Tag Info
17
The following examples uses that the tikz code for the arc uses a center node named arccenter. The tikz option argument for the \draw command of the arc can be used with option late options to put a label in the center: \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{ N**[0,-144,dash pattern=on 2pt off 2pt, late ...
16
Here are the fish you wanted, that I caught with texdoc chemfig. \documentclass{article} \usepackage{chemfig} \begin{document} \setatomsep{2em} \setdoublesep{.3em} \renewcommand*\printatom[1]{\ensuremath{\mathsf{#1}}} \chemfig[line width=1pt] { HO-*6(-=-(-(-[::90]CH_3)(-[::-90]CH_3)-*6(-=-(-OH)=-=))=-=) } \end{document} Now your turn to fetch the ...
12
This is not fully automatic (you need to specify the rotation for the molecule and the corresponding \chemmove command) and the TikZ code can most likely be improved but it may be a start. It places two invisible bonds where I've marked the respective ends with chemfigs @{<node name>} syntax. They are used to draw the rectangle later. ...
10
I introduce \lewis with 7 arguments, and I apologize that I don't know the precise naming conventions for chemistry. (EDITED to specify valence in only one place). Arguments: #1 Core atom #2 Top electrons #3 Right electrons #4 bottom electrons #5 left electrons #6 valence #7 inner electron shells \documentclass{article} ...
9
Here are a different versions using different chemistry packages. Which one you want to use is up to you... chemfig, provides \startscheme, \stopscheme, \chemfig, \lewis, \Lewis, \chemname ... ; mhchem, provides \ce{}; chemformula (part of the chemmacros bundle), provides \ch{} with the !(<below>)(<formula>) syntax and \chlewis; bohr, provides ...
9
Something like this? MWE: \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{P(=[2,0.7]O)(-[:-30,0.8]C_{5}H_{11})(-[:150,0.8]C_{5}H_{11}O)(-[:210,0.8]C_{5}H_{11}O)} \end{document} To specify an angle you have to use the notation [:<angle>], while to specify a custom length for the bonds you have to use [,<length>], so ...
8
chemfig allows to add explizit node names to either bonds or atoms in its formulae by using the @{<name>} syntax. These names can be used in a tikzpicture with the options remember picture, overlay to draw the curved arrows. chemfig provides the wrapper \chemmove for this. So a combination of chemfig and TikZ can be used to draw the schemes. (BTW: the ...
7
Here are two ideas: \documentclass{article} \usepackage{chemfig} \definesubmol{ring}{(-[::-60]=^[::60]-[::60])=_[::60]-[::-60]=_[::-60]} \definesubmol{ring2}{(**6(------))-[,,,,draw=none]-[,,,,draw=none]} \begin{document} \chemfig{-!{ring}-!{ring}-!{ring}} \chemfig{-!{ring2}-!{ring2}-!{ring2}} \end{document}
7
The \chemfig command has two optional arguments. The manual says this: The \chemfig command takes two optional arguments; their syntax is as follows: \chemfig[<opt1>][<opt2>]{<molecule code>} The first optional argument <opt1> contains tikz instructions which will be passed to the tikzpicture environment in which the ...
7
I actually figured this issue out by digging through the 83 pages of the chemfig manual. Here's the code I've changed and the following result: \chemleft[\chemfig{\lewis{,H}\pol{+}-\lewis{2:,O}\pol{-}(-[6]\lewis{,H}\pol{+})-\lewis{,H}\pol{+}}\ind\ind\chemright]^{+}
7
I'd use an invisible bond pointing to the center of the ring (with a relative angle) to place the plus. Something like (-[::126,,,,draw=none]\oplus), possibly scaled a bit. On the other hand I do like Heiko's answer better than mine :) \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{ R-[:36]N **[216,360,dash pattern=on 2pt off ...
6
Yes, I have an idea: \documentclass{article} \usepackage{chemfig} \begin{document} \definesubmol\cc{**6(---!\ff-!\ee--)} \definesubmol\dd{**6(-!\ee--!\ff---)} \definesubmol\ee{**6(-----)} \definesubmol\ff[(-^{-}OOC)]{(-COO^{-})} \chemfig{Cu^+(-[1]N([0,.5]!\cc))(-[3]N([:180,.5]!\dd))(-[5]N([:180,.5]!\cc))(-[7]N([:0,.5]!\dd))} \end{document}
6
My suggestions for the three points: just use math mode instead of a \chemfig command. write the N before you start the ring: N*5(-----) I would pass a suitable TikZ style to the node containing the text. This can be achieved with the \arrow command: \arrow(<start node>[<options>]--<end node>[<options>]). ...
6
I've contacted Christian Tellechea, the maintainer of the chemfig package. Hi Christian, there is currently a discussion about bound joints in chemfig on tex.stackexchange: Ugly bond joints in chemfig Are you aware of that problem? Is it a chemfig-problem or a TikZ problem? I would really appreciate it if you could participate in the ...
6
It works with two set of curly braces (changed the color to blue): \documentclass{article} \usepackage{color} \usepackage{chemfig} \begin{document} \chemfig{ [:30]HO-*6(-=-(-=[::-60]-[::60](=[2]O)-[:-30]{\color{red}O}> *6(--(([6]<OH)-(-[:30]OH)=[6] {{\color{blue}O}} )--(<:OH)-(<HO)-) )=-(-HO)=) } \end{document} ...
6
I was really a conflict between babel with czech and chemfig. I found the following solution on LaTeX community: \documentclass[oneside,czech]{book} \usepackage{babel} \usepackage{chemfig} \usepackage{etoolbox} \pretocmd\schemestart{\shorthandoff{-}}{}{} \apptocmd\schemestop{\shorthandon{-}}{}{} \begin{document} \schemestart A \arrow(aa--bb) B ...
5
I have tried to "fix" the problem. It is not really a "bugfix" (since there is no bug) but a dirty workaround. It seems to work : The beta version needs more testing. If you can't wait (or want to test it), you can download it here. The zip file contains the package source itself (chemfig.tex), a small test file (test.tex) and the pdf manual, compiled ...
5
I've found a way. It was rather simple, but I've hoped that there is a special command for such thing, because there is one in chemdraw. Here is the code: \documentclass[12pt,a4paper]{article} \usepackage{graphicx} \usepackage{chemfig,tikz} \begin{document} \begin{figure}[h] \centering { \setatomsep{2em} ...
5
Chemfig's bonds have an optional argument that takes several parameters one of them being a factor to scale the bond length: <bond>[<angle spec>,<length factor>,<other parameters>] So you can just add that option to the bonds you want shorter: \documentclass[a4paper,11pt]{scrartcl} \usepackage[utf8]{inputenc} ...
5
You can: \documentclass{article} \usepackage{chemfig} \makeatletter \newcommand*\forcelen[1]{#1/\CF@atom@sep} \makeatother \begin{document} These bonds are exactly 5mm long: \chemfig{-[,\forcelen{5mm}]-[:60,\forcelen{5mm}]} \end{document} If both atoms are not empty, the argument of \forcelen is not the length of the bond. It is the distance between the ...
5
I propose the following solution. I've made a number of changes: First of all I find your use of enumerate a bit strange. It looks like you want equation numbers for your reactions. If that's it then I'd use an equation environment. With the class option leqno its numbers will be placed on the left. I'd still use chemfig's \schemestart ... \schemestop ...
5
chemfig's molecules can get a default rotation by specifying an angle as option first in the molecule: \chemfig{[:<angle>]...} Choosing the right angle will do the trick here: \documentclass{article} \usepackage{chemfig} \definesubmol{imidazole}{N*5(=-{NH}-(-)=-)} % imidazole ring \begin{document} ...
4
Here is how to draw the 2 first arrows: \documentclass{article} \usepackage{chemfig} \begin{document} \chemfig{*6((-[@{a1}]H_2@{a2}\Lewis{26,N})=[@{r1}]-[@{r2}]=(-N=N-*6(=-=(-OH)-=-))-=-=)} \chemmove[-stealth,shorten <=1pt, shorten >=1pt]{% \draw(a2)..controls +(90:5mm) and +(135:5mm)..(a1);% first arrow \draw(r1)..controls +(225:12mm) and ...
4
I defined TikZ decorations: \documentclass{article} \usepackage{chemfig} \usetikzlibrary{decorations} \makeatletter \newdimen\mystartshorten \newdimen\myendshorten \mystartshorten0pt \myendshorten0pt \pgfdeclaredecoration{sdbond}{initial}{ \state{initial}[width=\pgfdecoratedremainingdistance,next state=final] { { ...
4
chemfig's \Lewis uses \printatom (a chemfig macro) internally. You have to redefine it: \documentclass{beamer} \usepackage[T1]{fontenc} \usepackage{chemfig} \usepackage{chemmacros} \chemsetup[chemformula]{font-shape=sf} \renewcommand*\printatom[1]{\ensuremath{\mathsf{#1}}} \begin{document} \begin{frame} \begin{reactions} Cl-Cl ...
4
»not very elegant« is not a very precise description of what's wrong... First of all I'd use a list for the, well, list, i.e., enumerate. The labels can easily be adjusted with enumitem. Then I'd make the arrows longer using the last optional argument of the \arrow command: \arrow[<angle>,<length factor>] I'd also shorten the bond length a ...
4
I don't normally advertize the tabu package (see this post for reasons why) but in this case it is very handy. It scales the table horizontally to textwidth per default if you use its X type columns. Those columns have an optional argument for specifying the ratio of one column to another: X[1]X[2] would mean that the second column has a width twice as much ...
4
Perhaps something like this is what you want: \documentclass[12pt]{report} \usepackage{chemfig} %these for bonds in chemfig \setdoublesep{0.35700 em} % 'Bond Spacing' \setatomsep{1.78500 em} % 'Fixed Length' \setbondoffset{0.18265 em} % 'Margin Width' \newcommand{\bondwidth}{0.06642 em} % 'Line Width' \setbondstyle{line width = \bondwidth} %Drawing ...
4
You typed \chemfig{...Cl[@{op,.25}]...} which implies an optional argument to the »Cl« atom. There are no optional arguments to atoms, though! This means chemfig interprets the quare brackets as atoms, i.e., it typesets them. This leads to the “little box” you see. There are two legal places for the @{...} syntax: a zero size node on a bond using the ...
4
In a bind, you can slap it on after the fact with a \stackinset. The syntax here means that the inset item (a bold +) is placed 10pt to the right of center, and 15pt above center on the underlying \chemfig. \documentclass{article} \usepackage{chemfig} \usepackage{stackengine} \begin{document} \stackinset{c}{10pt}{c}{15pt}{\textbf{+}}{% \chemfig{ ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2014-07-25 05:28:05
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https://socratic.org/questions/how-do-you-simplify-8-1-2-2-3
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# How do you simplify (8^(-1/2))^(-2/3)?
Jun 16, 2016
${\left({8}^{- \frac{1}{2}}\right)}^{- \frac{2}{3}} = 2$
#### Explanation:
As ${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$
${\left({8}^{- \frac{1}{2}}\right)}^{- \frac{2}{3}}$
= ${8}^{\left(- \frac{1}{2}\right) \times \left(- \frac{2}{3}\right)}$
= ${8}^{\frac{1}{3}}$
but $8 = {2}^{3}$, hence above is equal to
= ${\left({2}^{3}\right)}^{\frac{1}{3}}$
= ${2}^{3 \times \frac{1}{3}}$
= ${2}^{1} = 2$
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2020-03-29 06:59:33
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https://artofproblemsolving.com/wiki/index.php?title=1983_AHSME_Problems/Problem_2&diff=next&oldid=78570
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# Difference between revisions of "1983 AHSME Problems/Problem 2"
## Problem
Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \ \text{cm}$ from $P$?
$\textbf{(A)} \ 1 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ 4 \qquad \textbf{(E)} \ 8$
## Solution
The points $3 \ \text{cm}$ away from $P$ can be represented as a circle centered at $P$ with radius $3 \ \text{cm}$. The maximum number of intersection points of two circles is $\boxed{(\text{B}) \; 2}$
1983 AHSME (Problems • Answer Key • Resources) Preceded byProblem 1 Followed byProblem 3 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions
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2021-10-19 09:26:20
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https://www.gamedev.net/forums/topic/329667-color-depths/
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# Color depths
## Recommended Posts
storage 175
I've been looking around for a while, but I can't find any information (maybe I'm just stupid, could be) on how many bits for everything is used at different "bits per pixel". I'm wondering if my "table" below is correct,a nd what the ?'s should be.
red green blue alpha depth buffer stencil buffer
16 BPP 5 6 5 0(?) 16(?) ?
24 BPP 8 8 8 0(?) 24(?) ?
32 BPP 8 8 8 8 24(?) 8
I'm trying to make my engine (which uses GLFW for window creation and input handling) use bits per pixel instead of having the user write the depth of every color manually. Thanks in advance! :) Sorry if I'm totally lost, it's very possible I am.
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Syranide 375
Actually, you are right in a sense, and yet not.
Because, at least when using Direct3D, you can select what each depth should contain, such as R5G5B6 (for R:5 B:5 G:5), or perhaps X1R5G5B5, or you chould use 24D8S (depth:24 stencil:8)... and so on.
Check the Direct3D for documentation on backbuffers and a like and you will see.
So as you pointed out, the bits are just spread out in the different fields according to the format.
from DX documentation:
D3DFMT_D24S8 - 32-bit z-buffer bit depth using 24 bits for the depth channel and 8 bits for the stencil channel.
D3DFMT_R8G8B8 - 24-bit RGB pixel format with 8 bits per channel.
D3DFMT_A8R8G8B8 - 32-bit ARGB pixel format with alpha, using 8 bits per channel.
D3DFMT_X8R8G8B8 - 32-bit RGB pixel format, where 8 bits are reserved for each color.
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JohnBolton 1372
There are several possible formats for each bit depth. Here is the MSDN page that describes all the pixel formats that D3D supports: D3DFORMAT
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Melekor 379
GLFW lets you specify all of those components seperately in glfwOpenWindow. If the video card driver doesn't support the combination you supplied (unlikely unless you did something wacky) then GLFW will choose the closest mode.
There is no table that maps bpp to the depth and stencil buffer, there isn't really any relation between the two. Is there a reason you need anything other than the standard 32 bpp color, 32bpp depth screen format?
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2017-08-18 18:34:40
|
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https://zbmath.org/?q=an%3A1154.11036
|
# zbMATH — the first resource for mathematics
Partition identities arising from theta function identities. (English) Zbl 1154.11036
This paper is devoted to a study of the combinatorics of identities that generally have the form $f(q) + f(-q) = g(q)$ or $f(q) - f(-q) = qh(q),$ where $$f(q)$$ is some infinite product which is essentially a modular function. The motivation is one such identity of H. M. Farkas and I. Kra [Contemp. Math. 251, 197–203 (2000; Zbl 1050.11086)], $(-q;q^2)_{\infty}(-q^7;q^{14})_{\infty} - (q;q^2)_{\infty}(q^7;q^{14})_{\infty} = 2q(-q^2;q^2)_{\infty}(-q^{14};q^{28})_{\infty},$ which, along with its combinatorial interpretation, was popularized and generalized by S. O. Warnaar [J. Comb. Theory, Ser. A 110, No. 1, 43–52 (2005; Zbl 1101.11046)]. Here we have used the usual basic hypergeometric notation.
Of course if $$f(q)$$ is a modular function then both the sum $$f(q) + f(-q)$$ and the difference $$f(q) - f(-q)$$ will both be so as well. The authors identify around 15 cases where this sum or difference is an infinite product and offer combinatorial interpretations in terms of colored partitions. For example, we record their Theorem 3.4:
Let $$A(N)$$ denote the number of partitions of $$2N+1$$ into odd parts that are not multiples of $$3$$, with each having two colors, say orange and blue. Let $$B(N)$$ denote the number of partitions of $$2N$$ into four distinct colors, with two colors, say red and green, appearing at most once and only in multiples of $$2$$, one color, say pink, appearing at most once and only in multiples of $$4$$, and the remaining color, say violet, appearing at most once and only in multiples of $$12$$. Then for $$N \geq 2$$, we have $$A(N) = 2B(N)$$.
This follows from the identity $\begin{split} \frac{1}{(q;q^6)_{\infty}^2(q^5;q^6)_{\infty}^2} - \frac{1}{(-q;q^6)_{\infty}^2(-q^5;q^6)_{\infty}^2} = 4q(-q^2;q^2)_{\infty}^4(-q^4;q^4)_{\infty}(-q^{12};q^{12})_{\infty}.\end{split}$ The authors’ proofs of such identities rely heavily on work of H. Schröter [De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, Regiomonti, Königsberg, 1854].
##### MSC:
11P83 Partitions; congruences and congruential restrictions 05A17 Combinatorial aspects of partitions of integers 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
Full Text:
##### References:
[1] Farkas, H. M., Kra, I.: Partitions and theta constant identities, in The Mathematics of Leon Ehrenpreis, Contemp. Math. No. 251, American Mathematical Society, Providence, RI, 2000, 197–203 · Zbl 1050.11086 [2] Schröter, H.: De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, Regiomonti, 1854 [3] Ramanujan, S.: Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957 · Zbl 0138.24201 [4] Baruah, N. D., Berndt, B. C.: Partition identities and Ramanujan’s modular equations. J. Combin. Theory Ser. A, 114, 1024–1045 (2007) · Zbl 1206.11132 [5] Berndt, B. C.: Partition-theoretic interpretations of certain modular equations of Schröter, Russell, and Ramanujan. Ann. of Combin., to appear · Zbl 1131.05008 [6] Berndt, B. C.: Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991 · Zbl 0733.11001 [7] Berndt, B. C.: Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998 · Zbl 0886.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-10-20 17:38:25
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https://lammps.sandia.gov/doc/fix_nve_eff.html
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# fix nve/eff command
## Syntax
fix ID group-ID nve/eff
• ID, group-ID are documented in fix command
• nve/eff = style name of this fix command
## Examples
fix 1 all nve/eff
## Description
Perform constant NVE integration to update position and velocity for nuclei and electrons in the group for the electron force field model. V is volume; E is energy. This creates a system trajectory consistent with the microcanonical ensemble.
The operation of this fix is exactly like that described by the fix nve command, except that the radius and radial velocity of electrons are also updated.
Restart, fix_modify, output, run start/stop, minimize info:
No information about this fix is written to binary restart files. None of the fix_modify options are relevant to this fix. No global or per-atom quantities are stored by this fix for access by various output commands. No parameter of this fix can be used with the start/stop keywords of the run command. This fix is not invoked during energy minimization.
## Restrictions
This fix is part of the USER-EFF package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.
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2019-04-18 10:35:09
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https://helpingwithmath.com/math-calculators/standard-deviation-calculator/
|
Home » Math Calculators » Standard Deviation Calculator
# Standard Deviation Calculator
Enter Information
Results
Fill the calculator form and click on Calculate button to get result here
Count: 0
Sum: 0
Mean: 0
Sum of Differences2: 0
Population
Variance: 0
Standard Deviation: 0
Sample
Variance: 0
Standard Deviation: 0
Differences:Every Number minus the Mean 0
Differences2:Square of each difference 0
## What is Variance?
The variance of a variable X is the arithmetic mean of the squares of all derivatives of X from the arithmetic mean of the observations and is denoted by Var ( X ) or σ 2. In other words, The variance measures the average degree to which each point differs from the mean—the average of all data points. Let us now learn about some properties of variance.
## What is Standard Deviation?
The positive square root of the variance of a variate X is known as its standard deviation and is denoted by σ. In other words, Standard deviation looks at how to spread out a group of numbers is from the mean, by looking at the square root of the variance.
Thus,
Standard Deviation = $\sqrt{Var ( X )}$
It is important to note here that, Standard deviation and variance both are measures that tell how spread out the numbers is. While variance gives us a rough idea of spread, the standard deviation is more concrete, giving us the exact distances from the mean.
## Variance and Standard Deviation of Individual Observations
If x1, x2, x3, ……, xn are n values of a variable X, then,
Variance ( X ) = $\frac{\sum_{i=1}^{n}(x_1-\overline{x})^2}{2}$
Also,
Variance ( X ) = [$\frac{1}{N} \sum_{i=1}^{n}f_i x_i$ ] 2
Another formula for variance is given by
Variance ( X ) = $\frac{1}{N}$ [ $\sum{f_i d_i^2}$ ] – [$\frac{1}{N} \sum_{i=1}^{n}f_i d_i$ ] 2
In the case of individual observations, variance and standard deviation may be computed by applying any of the above three formulas. We can now define the algorithm for finding the variance when deviations are taken from the actual mean. The algorithm is as follows –
### Algorithm for finding the variance when deviations are taken from the actual mean
1. Compute the mean $\overline{X}$ of the given observations x1, x2, x3, x4, ……. xn.
2. Take the deviations of the observations from the mean i.e. to find xi – $\overline{X}$ ; I = 1, 2, 3, ………, n.
3. Square the deviations obtained in the above step and obtain the sum
$\frac{\sum_{i=1}^{n}(x_1-\overline{x})^2}{2}$ .
4. Divide the sum $\frac{\sum_{i=1}^{n}(x_1-\overline{x})^2}{2}$ obtained in the above step by n. this gives us the variance of X.
Let us understand this through an example.
Example Compute the variance and the standard deviation of the following observations of the following data –
65, 68, 58, 44, 48, 45, 60, 62, 60, 50
Solution We have been given the data – 65, 68, 58, 44, 48, 45, 60, 62, 60, 50. We are required to find the variance and the standard deviation of the given data.
Let $\overline{X}$ be the mean of the given set of observations. Then,
$\overline{X}$ = $\frac{65+ 68+ 58+ 44 + 48+ 45 + 60+ 62+ 60 + 50}{10}$ = $\frac{560}{10}$ = 56
Let us now compute the variance of the given data
We can see that number of observations is 10. Therefore, n = 10
Variance = $\frac{\sum_{i=1}^{n}(x_1-\overline{x})^2}{n}$ = $\frac{662}{10}$ = 66.2
Hence, Standard Deviation = $\sqrt{Variance}$ = $\sqrt{66.2}$ = 8.13
Therefore, for the given set of observations,
Variance = 66.2 and Standard Deviation = 8.13
## How to find variance and standard variance using the standard deviation calculator?
Finding the variance and the standard deviation using our standard deviation calculator is quite simple. You just need to enter the values for which the variance and the standard deviation needs to be calculated. Below are the steps to be followed for this purpose –
Step 1 – The first step is to locate the box where we need to enter the values for which we wish to find the variance and the standard deviation. These values should be separated by commas. Below is the snapshot of the option that would be displayed for this purpose –
Step 2 – The next step is to enter the values. Let us for example, consider the example that we discussed above where we had the values 65, 68, 58, 44, 48, 45, 60, 62, 60, 50 for which we wanted to compute the variance and the standard deviation. Let us enter the same values in the standard deviation calculator. Below is the snapshot of how the values would be entered –
Step 3 – Now that we have entered the values, the next step is to compute the variance and the standard variation. For this purpose, we just need to click on the “ calculate “ button. As soon as we will click on this button, we can see the result obtained on the right-hand side of the values that we had entered in the previous steps. Below is a snapshot of how the selection would look like when we will click on the “ calculate “ button
Let us see in detail the results that have been provided to us.
Count – This is the number of values that we provided for computation.
Sum – This is the sum of the values which we provided for computation.
Mean – This is the mean of the values which we provided for computation. Note that this value is the same as the one we had obtained in the form of $\overline{X}$ in the above example.
Sum of differences – This is the value of (xi – $\overline{X}$ ) 2 that we had computed when finding the variance and standard deviation.
The other terms are variance, standard deviation, difference etc. which are all the same as were computed when the same was computed using the formula. So, we can clearly see that the result obtained through the calculator as well the result obtained through the formula are the same. What’s more, the calculator also displays the formula as well as the steps involved in the calculation. The repeated use of this calculator allows you to not only check the results but also understand the formula and steps involved in finding the sides of a right angled triangle, thus allowing you to have a stronghold on the concept itself.
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2022-05-18 22:26:37
|
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https://www.jiskha.com/questions/1514452/a-spherical-balloon-is-losing-air-at-the-rate-of-2-cubic-inches-per-minute-how-fast-is
|
# calculus
A spherical balloon is losing air at the rate of 2 cubic inches per minute. How fast is the radius of the ballon shrinking when the radius is 8 inches.
1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩
1. dV = surface area * dr
so
dr/dt = dV/dt/(4 pi r^2)
dr/dt = 2 /(4 pi *64)
1. 👍
2. 👎
3. ℹ️
4. 🚩
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2022-05-26 18:37:25
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https://nemeth.aphtech.org/lesson8.1
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- Use 6 Dot Entry Switch to UEB Math Tutorial
Lesson 8.1: Real Numbers
The Multiplication Dot
Another way that multiplication is shown in print is with the multiplication dot. In braille, the multiplication dot uses one cell, dots one six, which is the second cell of the symbol that is used for the multiplication cross. In fact, the hollow dot is another modification of the dots one six, preceding it with dots four six in the first cell. Since the multiplication dot is never used in spatial arrangements, it is only necessary to learn how to use it in horizontal mathematical expressions.
As with any sign of operation, when brailling a multiplication dot, do not space between the multiplication dot and the symbols with which it is used. The guidelines and rules for using this symbol are exactly the same as those for the multiplication cross. Here are some examples of simple multiplication using the multiplication dot.
Most texts do not display the multiplication dot followed by the decimal point because of the possibility of confusion for the print reader. This is particularly troublesome with handwritten material. Thus 3.7⋅0.2 would probably be displayed as (3.7)(0.2) using parentheses to indicate multiplication.
Just as with any other sign of operation, the multiplication dot can be used with other symbols.
The most common use of the radical symbol, dots three four five, is to represent square roots, a root of the second degree. The radical sign is also used to indicate radicals of a higher degree, such as cubed or fourth roots; these will be presented later. The terms square root, root, and radical are used interchangeably.
The radical symbol is paired with the termination indicator, dots one two four five six, when it is followed by a value. When it stands alone, the termination indicator is not used. As the name indicates (implies), termination indicator is strictly a symbol used as a convention in braille and has no corresponding symbol in print. When paired with the radical sign, the termination indicator is used to mark the end of the radicand, the value following the radical sign. In print, a horizontal line extends from the radical sign and over the values, expressions, variables, or abbreviations which comprise the radicand. This line is called the vinculum and serves the same purpose as the termination indicator.
A radical is to be brailled in a horizontal manner except in two circumstances. It is displayed vertically if it is represented in a spatial arrangement. It is displayed vertically if the root is displayed above the vinculum when showing the process of extracting a root. The spatial procedures used with long division are applied in a similar manner to root extractions and the termination symbol is not used.
Example 1
$5\cdot 2\cdot 10$
⠼⠢⠡⠆⠡⠂⠴
Example 2
$6\cdot 2=12$
⠼⠖⠡⠆⠀⠨⠅⠀⠼⠂⠆
Example 3
$3.7\cdot 0.2$
⠼⠒⠨⠶⠡⠴⠨⠆
Example 4
$\frac{1}{2}\cdot \frac{2}{3}$
⠹⠂⠌⠆⠼⠡⠹⠆⠌⠒⠼
Example 5
$\left(2.5\right)\left(3.2\right)$
⠷⠆⠨⠢⠾⠷⠒⠨⠆⠾
Example 6
$\sqrt{4}$
⠜⠲⠻
Example 7
$-\sqrt{4}$
⠤⠜⠲⠻
Example 8
$\sqrt{3+6}$
⠜⠒⠬⠖⠻
Example 9
$5\sqrt{3}$
⠼⠢⠜⠒⠻
Example 10
$7\sqrt{3}-2\sqrt{3}$
⠼⠶⠜⠒⠻⠤⠆⠜⠒⠻
Example 11
$8\sqrt{2}÷4\sqrt{2}$
⠼⠦⠜⠆⠻⠨⠌⠲⠜⠆⠻
Example 12
$\frac{8\sqrt{2}}{4\sqrt{2}}$
⠹⠦⠜⠆⠻⠌⠲⠜⠆⠻⠼
Example 13
$\sqrt{\frac{12}{3}}$
⠜⠹⠂⠆⠌⠒⠼⠻
Example 14
$\sqrt{2}\cdot \sqrt{3}$
⠜⠆⠻⠡⠜⠒⠻
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2020-04-07 23:13:38
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https://socratic.org/questions/the-half-life-of-a-radioactive-element-is-30-seconds-in-what-period-of-time-woul
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# The half life of a radioactive element is 30 seconds. In what period of time would the activity of the sample be reduced to one - sixteenth of the original activity?
Feb 17, 2018
Working formula: $A = {A}_{0} \cdot {e}^{-} \left(k t\right)$
#### Explanation:
You can simplify the above working formula to the following form:
$k t = \ln \left({A}_{0} / A\right)$
where ${A}_{0}$ is the activity of the element at time $t = 0$ and $A$ is the activity at a given time.
Now, we know that the formula for half-life (which can be obtained from the above equation itself) is given by:
${t}_{\text{1/2}} = \frac{\ln 2}{k}$.
So, here the half-life is $30$ seconds. Using this information, we find the decay constant $k$.
$k = \ln \frac{2}{\text{30 s}}$
Now, using this value of $k$, we get the time at which the activity of the element becomes $\frac{1}{16}$ of original activity, i.e
$A = {A}_{0} / 16$
So
$t = \frac{1}{\ln \frac{2}{\text{30 s}}} \cdot \ln \left({A}_{0} / \left({A}_{0} / 16\right)\right)$
$t = \left(\frac{\text{30 s}}{\ln} 2\right) \cdot \ln \left({2}^{4}\right)$
$t = \left(\frac{\text{30 s}}{\ln} 2\right) \cdot 4 \ln 2$
$t = \text{30 s} \cdot 4$
$t = \text{120 s}$
Therefore, the activity of the radioactive element becomes one-sixteenth of its original activity at $120$ seconds.
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2020-01-18 18:19:52
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|
http://mathhelpforum.com/statistics/149636-probability-basics-some-examples-its-application.html
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# Math Help - Probability: The basics with some examples of its application
1. ## Probability: The basics with some examples of its application
Probability: The basics with some examples of its application
I was inspired to create this thread after reading Chris L T521’s “Differential Equations Tutorial”.
While reading this thread, assumed is a basic knowledge of fractions, percentages and decimal numbers.
The basic idea of probability is applying a number to the chance of an event occurring. The number we use is always between and including zero and one. Zero representing no chance of an event occurring with one representing the event in question being certain to happen.
Let’s look at some notation to describe what was just discussed. We shorthand the “probability of event A occurring” as $P(A)$
Given the idea that a probability is always between zero and one then $0\leq P(A) \leq 1$
Now we have talked about what a probability is and the restrictions associated let’s define how we find a probability.
Let’s say we have a universal set (the set of everything in question) of numbers or objects and call it $\epsilon$ . The probability of event $A$ occurring is defined as:
$P(A) = \frac{\text{The number of As in the set}}{\text{The total number of entries in the set}}=\frac{n(A)}{n(\epsilon )}$
Now let’s apply this.
Example 1: Consider an unbiased die with the numbers one to six.
a) Find the probability of rolling a 5
b) Find the probability of rolling a prime number
c) Find the probability of rolling an 8
We have $\epsilon = \{1,2,3,4,5,6\}$ and $n(\epsilon ) = 6$
a) $P(5) = \frac{\text{The number of 5s in the set}}{\text{The total number of entries in the set}}=\frac{n(5)}{n(\epsilon )} =\frac{1}{6}$
b) $P(prime) = \frac{\text{The number of prime numbers in the set}}{\text{The total number of entries in the set}}=\frac{n(prime)}{n(\epsilon )} =\frac{3}{6}$
c) There are no 8’s on the dice in question so
$P(8) = \frac{\text{The number of 8s in the set}}{\text{The total number of entries in the set}}=\frac{n(8)}{n(\epsilon )} =\frac{0}{6}=0$
Example 2: Consider a deck of cards with no jokers.
a) Find the probability of drawing an ace
b) Find the probability of drawing a picture card
c) Find the probability of drawing a red card
d) Find the probability of drawing a black queen
As there are 52 cards in a deck so $n(\epsilon ) = 52$
a) $P(ace) = \frac{\text{The number of aces in the set}}{\text{The total number of entries in the set}}=\frac{n(aces)}{n(\epsilon )} =\frac{4}{52}$
b) $P(picture~ card) =\frac{n(picture ~cards)}{n(\epsilon )} =\frac{12}{52}$
c) $P(red~ card) =\frac{n(red~ cards)}{n(\epsilon )} =\frac{26}{52}$
d) There are four queens in the deck, one in each suit but we are only after the black ones.
$P(black~ queen) =\frac{n(black ~queens)}{n(\epsilon )} =\frac{2}{52}$
This concludes the first part of a series of posts on probability
Next I plan on looking at complementary probability, intersection and union of sets and conditional probability.
2. Probability: Intersection and Union of Sets
Hi again, in my last post I introduced 'the idea of' and 'how to calculate' a probability. I welcome any feedback on this thread at anytime via PM.
A set is a collection of objects often identified by a certain characteristic. The notation used for listing a set are these curly brackets. $\{,\}$ . Sets are often named for the convenience of not having to re-write or repeat a big sentence or group of numbers.
Here's some examples.
The set of numbers on a six sided die (let's call it set A) is $A = \{1,2,3,4,5,6\}$
The set of prime numbers between 1 and 10 (let's call it set B) is $B = \{2,3,5,7\}$
Intersection of sets. The intersection of sets is a very important idea in probability. It describes objects that are 'shared' between 2 or more sets.
The notation for intersection is $\cap$ so the intersection between set $A$ and $B$ is written as $A\cap B$
Now let's find a solution for $A\cap B$ from the two sets defined above.
Recall $A = \{1,2,3,4,5,6\}$ and $B = \{2,3,5,7\}$ , the intersection is simply elements of each set that can be found in both. By inspection we can conclude $A\cap B = \{2,3,5\}$ as these elements are in both A and B.
Union of sets. Equally as important idea of probability is the union of sets. The union describes objects that are in all sets.
The notation for union is $\cup$ so the union of set $A$ and $B$ is written as $A\cup B$
From our example above to find the union of A and B we need list everything (without repetition) that appears in both sets.
By inspection we can conclude $A\cup B = \{1,2,3,4,5,6,7\}$ as these elements are in both A or B.
Here's some more examples on intersection and union. I have hidden the answers in case you wanted to have a go yourself.
1. Consider the sets $A = \{1,2,3,4,5,6,7\}$ and $B = \{3,4,5,6,7,8,9\}$
a) find the intersection of these sets
b) find the union of these sets
Spoiler:
By inspection
a) $A\cap B = \{3,4,5,6,7\}$
b) $A \cup B = \{1,2,3,4,5,6,7,8,9\}$
2. Consider the set of letters that make up the name "Victoria" and the set of letters that make up the name "Catherine".
a) find the intersection of these sets
b) find the union of these sets
Spoiler:
Calling these sets 'V' and 'C' respectively.
No need to repeat any letters in our sets and assuming we are not case sensitive.
$V = \{V,i,c,t,o,r,a \}$
$C = \{C,a,t,h,e,r,i,n \}$
a) $V \cap C= \{ C,i,a,t,r\}$
b) $V \cup C= \{ V,i,c,t,o,r,a,h,e,n\}$
3. Consider the sets $M = \{1,4,9,16\}$ , $N = \{2,4,6,8,10\}$ and $P = \{1,2,4,8,16\}$
a) find the intersection of sets M and N
b) find the intersection of sets M and P
c) find the intersection of sets M and N and P
d) find the union of sets M and N
e) find the union of sets M and P
f) find the union of sets M and N and P
Spoiler:
Once again by inspection
a) $M\cap N = \{4\}$
b) $M\cap P = \{1,4,16\}$
c) $M\cap N\cap P = \{4\}$
d) $M\cup N = \{1,2,4,6,8,9,10,16\}$
e) $M\cup P = \{1,2,4,8,9,16\}$
f) $M\cup N\cup P = \{1,2,4,6,8,9,10,16\}$
*Notice set $P$ is contained in $M\cup N$
Next post will talk about complementary probability.
3. Before we go onto the next topic here's a quick recap on ideas explored in posts #1 and #2.
Consider a particular set of colours sold by a paint shop denoted by the set
$C = \{\text{Red},\text{Blue},\text{Yellow},\text{Orang e},\text{Green},\text{Purple}\}$
The following table represents the number of paint cans in stock for each colour.
$\begin{array}{|c|c|}
\text{Colour} & \text{\# in stock} \\ \hline
\text{Red} & 10 \\ \text{Yellow} & 5 \\ \text{Blue} & 20
\\ \text{Orange} & 1 \\ \text{Green} & 3 \\ \text{Purple} & 1\\ \hline \end{array}$
It is important to note the total number of paint cans in stock is the sum of the second column $10+5+20+1+3+1 = 40$
Now let's find some probabilities
1. Find the probability of picking a can of Yellow paint
2. Find the probability of picking a can of Red paint
3. Find the probability of picking a can of paint which is not Blue
Spoiler:
Using some letter abbreviations to make life easier
1. $P(\text{Yellow}) = P(Y)= \frac{5}{40} = \frac{1}{8}$
2. $P(R) = \frac{10}{40} = \frac{1}{4}$
3. In this question we need to add together all cans that are not blue
$P(\text{Not B} ) = \frac{10+5+1+3+1}{40} = \frac{20}{40}= \frac{1}{2}$
Well I hope that was easy enough because it's time to introduce Complementary Probability.
Complementary Probability
The complement of an event is defined to be everything that is not the event in question.
In the last set of questions we were asked Find the probability of picking a can of paint which is not Blue. This is the complement of everything that is blue.
We note this where 'B = Blue' and the probability of finding a can that is blue is $P(B)$ then the 'complement of blue' i.e (not being blue) is $P(B')$
The following rule can be applied whenever we are asked to find the complement of an event.
$P(B)+P(B')=1$ and $P(B')=1-P(B)$
Which means the probability of picking a blue can plus the probability of picking a blue can which is not blue is equal to one. This always is correct as we have picked out every single can!
So revisiting the question Find the probability of picking a can of paint which is not Blue. and using our new rule we can arrive at our answer somewhat quicker
3. Find the probability of picking a can of paint which is not Blue
$P(B')=1-P(B) = 1- \frac{20}{40}= 1- \frac{1}{2}= \frac{2}{2}- \frac{1}{2}= \frac{1}{2}$
This can become very helpful when we are given very large sets.
Now it's time to find some complmentary probabilities, from our table above:
1. Find the probability of picking a can of paint which is not Yellow
2. Find the probability of picking a can of paint which is not Red
3. Find the probability of picking a can of paint which is not Green
Spoiler:
$P(Y')=1-P(Y) = 1- \frac{5}{40}= 1- \frac{1}{8}= \frac{8}{8}-\frac{1}{8}= \frac{7}{8}$
$P(R')=1-P(R) = 1- \frac{10}{40}= 1- \frac{1}{4}= \frac{4}{4}-\frac{1}{4}=\frac{3}{4}$
$P(G')=1-P(G) = 1- \frac{3}{40}= \frac{40}{40}-\frac{3}{40}= \frac{37}{40}$
Next topic will be conditional probability.
4. Conditional Probability
Thus far we have seen some straight forward examples on how to calculate a probability with a given scenario like a set of numbers or from a table.
Often in life we need to find the probability of a certain event happening given a set of conditions. For example let's say we have a deck of 52 cards. To find the probability of a certain card being drawn i.e 'find the probability of drawing a king' we can apply the same logic from example 2 in post #1. We get
$P(K) = \frac{\text{The number of Kings in the deck}}{\text{The total number of cards in the deck}}=\frac{4}{52} \approx 0.077$
This is simple enough. But what if there is a condition set on the deck? What if the question was 'find the probability of drawing a king given you draw a red card' ?
To negotiate this we apply the rule of conditional probability. The notation for using conditional probability is $P(A/B)$ which means find the 'probability of A given B'. The rule is:
$P(A/B)= \frac{\text{The probability of the intersection between A and B }}{\text{The probability of B}}=\frac{P(A\cap B)}{P(B)}$
We can now use this new rule to answer the question 'find the probability of drawing a king given you draw a red card'.
We need to find $P(R)$ where R represents a red card.
It is easy to conculde that because half of the cards in a deck is red then $P(R) = \frac{26}{52}= \frac{1}{2}$
We need to find $P(K\cap R)$ where K is a king and R is a red card.
Recall that ths is simply how many red kings are in the deck then $P(K\cap R)=\frac{2}{52} = \frac{1}{26}$
In conclusion $P(K/R) = \frac{P(K\cap R)}{P( R)}= \frac{\frac{1}{26}}{\frac{1}{2}} = \frac{1}{26}\times \frac{2}{1} = \frac{1}{13}$
1. Given a fair 6 sided die, find the probability of rolling a 2 given a prime number is rolled.
2. Given a deck of 52 cards, find the probability of drawing a picture card given a heart number is drawn.
3. A spinner is divided into even sections and the letters from the word 'Probability' (letters are repeated) are placed in each section. Find the probability of spinnig up an 'i' given a vowel is spun.
Spoiler:
1. $\frac{\frac{1}{6}}{\frac{3}{6}} = \frac{1}{3}$
2. $\frac{\frac{3}{52}}{\frac{13}{52}} = \frac{3}{13}$
3. $\frac{\frac{2}{11}}{\frac{4}{11}} = \frac{1}{3}$
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2014-08-30 05:52:37
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http://www.ck12.org/book/CK-12-Algebra-I-Concepts-Honors/r11/section/7.2/
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# 7.2: Multiplication of Polynomials
Difficulty Level: Advanced Created by: CK-12
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Practice Multiplying Polynomials
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Jack was asked to frame a picture. He was told that the width of the frame was to be 5 inches longer than the glass width and the height of the frame was to be 7 inches longer than the glass height. Jack measures the glass and finds the height to width ratio is 4:3. Write the expression to determine the area of the picture frame.
### Guidance
To multiply polynomials you will need to use the distributive property. Recall that the distributive property says that if you start with an expression like 3(5x+2)\begin{align*}3(5x+2)\end{align*}, you can simplify it by multiplying both terms inside the parentheses by 3 to get a final answer of 15x+6\begin{align*}15x+6\end{align*}.
When multiplying polynomials, you will need to use the distributive property more than once for each problem.
#### Example A
Find the product: (x+6)(x+5)\begin{align*}(x+6)(x+5)\end{align*}
Solution: To answer this question you will use the distributive property. The distributive property would tell you to multiply x\begin{align*}x\end{align*} in the first set of parentheses by everything inside the second set of parentheses , then multiply 6 in the first set of parentheses by everything in the second set of parentheses . Here is what that looks like:
#### Example B
Find the product: (2x+5)(x3)\begin{align*}(2x+5)(x-3)\end{align*}
Solution: Again, use the distributive property. The distributive property tells you to multiply 2x\begin{align*}2x\end{align*} in the first set of parentheses by everything inside the second set of parentheses , then multiply 5 in the first set of parentheses by everything in the second set of parentheses . Here is what that looks like:
#### Example C
Find the product: (4x+3)(2x2+3x5)\begin{align*}(4x+3)(2x^2+3x-5)\end{align*}
Soltuion: Even though at first this question may seem different, you can still use the distributive property to find the product. The distributive property tells you to multiply 4x\begin{align*}4x\end{align*} in the first set of parentheses by everything inside the second set of parentheses, then multiply 3 in the first set of parentheses by everything in the second set of parentheses. Here is what that looks like:
#### Concept Problem Revisited
Jack was asked to frame a picture. He was told that the width of the frame was to be 5 inches longer than the glass width and the height of the frame was to be 7 inches longer than the glass height. Jack measures the glass and finds the height to width ratio is 4:3. Write the expression to determine the area of the picture frame.
What is known?
• The width is 5 inches longer than the glass
• The height is 7 inches longer than the glass
• The glass has a height to width ratio of 4:3
The equations:
• The height of the picture frame is 4x+7\begin{align*}4x + 7\end{align*}
• The width of the picture frame is 3x+5\begin{align*}3x + 5\end{align*}
The formula:
AreaAreaAreaArea=w×h=(3x+5)(4x+7)=12x2+21x+20x+35=12x2+41x+35\begin{align*}\text{Area} &= w \times h \\ \text{Area} &= (3x + 5) (4x + 7) \\ \text{Area} &= 12x^2 + 21x + 20x + 35 \\ \text{Area} &= 12x^2 + 41x + 35\end{align*}
### Vocabulary
Distributive Property
The distributive property states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, in the expression: \begin{align*}{\color{red}\frac{2}{3}} ({\color{blue}x + 5})\end{align*}, the distributive property states that the product of a number \begin{align*}\left({\color{red}\frac{2}{3}}\right)\end{align*} and a sum \begin{align*}({\color{blue}x+5})\end{align*} is equal to the sum of the individual products of the number \begin{align*}\left({\color{red}\frac{2}{3}}\right)\end{align*} and the addends \begin{align*}({\color{blue}x}\end{align*} and \begin{align*}{\color{blue}5})\end{align*}.
Like Terms
Like terms refers to terms in which the degrees match and the variables match. For example \begin{align*}3x\end{align*} and \begin{align*}4x\end{align*} are like terms. Like terms are also known as similar terms.
### Guided Practice
1. Find the product: \begin{align*}(x+3)(x-6)\end{align*}
2. Find the product: \begin{align*}(2x+5)(3x^2-2x-7)\end{align*}
3. An average football field has the dimensions of 160 ft by 360 ft. If the expressions to find these dimensions were \begin{align*}(3x+7)\end{align*} and \begin{align*}(7x+3),\end{align*} what value of \begin{align*}x\end{align*} would give the dimensions of the football field?
1. \begin{align*}(x+3)(x-6)\end{align*}
2. \begin{align*}(2x + 5)(3x^2 - 2x - 7)\end{align*}
3. \begin{align*}\text{Area} = l \times w\end{align*}
\begin{align*}\text{Area} &= 360 \times 160 \\ (7x+3) &= 360 \\ 7x &= 360 - 3 \\ 7x &= 357 \\ x &= 51 \\ \\ (3x +7) &= 160 \\ 3x &= 160 - 7 \\ 3x &= 153 \\ x &= 51 \end{align*}
The value of \begin{align*}x\end{align*} that satisfies these expressions is 51.
### Practice
Use the distributive property to find the product of each of the following polynomials:
1. \begin{align*}(x+4)(x+6)\end{align*}
2. \begin{align*}(x+3)(x+5)\end{align*}
3. \begin{align*}(x+7)(x-8)\end{align*}
4. \begin{align*}(x-9)(x-5)\end{align*}
5. \begin{align*}(x-4)(x-7)\end{align*}
6. \begin{align*}(x+3)(x^2+x+5)\end{align*}
7. \begin{align*}(x+7)(x^2-3x+6)\end{align*}
8. \begin{align*}(2x+5)(x^2-8x+3)\end{align*}
9. \begin{align*}(2x-3)(3x^2+7x+6)\end{align*}
10. \begin{align*}(5x-4)(4x^2-8x+5)\end{align*}
11. \begin{align*}9a^2(6a^3+3a+7)\end{align*}
12. \begin{align*}-4s^2(3s^3+7s^2+11)\end{align*}
13. \begin{align*}(x+5)(5x^3+2x^2+3x+9)\end{align*}
14. \begin{align*}(t-3)(6t^3+11t^2+22)\end{align*}
15. \begin{align*}(2g-5)(3g^3+9g^2+7g+12)\end{align*}
### Notes/Highlights Having trouble? Report an issue.
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### Vocabulary Language: English
TermDefinition
distributive property The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
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2017-04-29 06:46:12
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http://universeinproblems.com/index.php/Generation_and_detection_of_gravitational_waves
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# Generation and detection of gravitational waves
### Problem 1: Mass scales
Show that effectiveness of gravitational wave production increases with the increase of mass.
### Problem 2: The intangibility
What is the reason for the very low efficiency of gravitational waves' production, i.e. conversion of mechanical energy into that of gravitational waves?
### Problem 3: Single particle effects
What happens to a single particle as a gravitational wave passes through?
### Problem 4: Waves vs static field
One may wonder, how it is possible to infer the presence of an astronomical body by the gravitational waves that it emits, when it is clearly not possible to sense its much larger stationary (essentially Newtonian) gravitational potential. What gives us hope to overcome this problem?
The next three problems are inspired by S.Hughes, Listening to the Universe with gravitational-wave astronomy, arXiv:astro-ph/0210481
### Problem 5: Listening to the Universe
What could be the origin of this poetic terminology?
### Problem 6: Measuring infinitesimal distances
The most promising sources (neutron binaries, supernovae) of gravitational waves should give the amplitudes of the order of $h\sim 10^{-21}$. For every kilometer of baseline $L$ we need to be able to measure a distance shift of $\Delta L$ better than $10^{-16}cm$. How can we possibly hope to measure an effect that is $\sim 10^{12}$ times smaller than the wavelength of visible light (all interferometers use optical lasers)?
### Problem 7: Surface effects
The atoms on the surface of the interferometers’ test mass mirrors oscillate with an amplitude $\delta l_{atom} =\sqrt{\frac{kT}{m\omega^{2}}} \sim {10}^{-10}cm$ at room temperature $T$, with $m$ the atomic mass, and with a vibrational frequency $\omega \sim {10}^{14}s^{-1}$. This amplitude is huge relative to the effect of the gravitational waves. Why doesn't it wash out the gravitational wave?
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2017-03-28 23:25:20
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https://ncatlab.org/davidcorfield/show/Davidson%20on%20events
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# David Corfield Davidson on events
From ‘Actions and Events’ (OUP):
### Actions, Reasons and Causes
• (1) ‘I turned on the light’ and (2) ‘I wanted to turn on the light/the light on’
There’s a specificity to an action satisfying (1) that isn’t part of what’s wanted in (2), e.g, I did it with my nose because my hands were full. If (1) is a type, $T$, then an element is an achieving. (2) just is a wish for a state type to be inhabited, ‘I wanted Inhab(T)’. Its elements are warrants of the wanting.
(Should we say that the pro-attitude is towards a state, and the belief is that an activity will bring about the state, so that we explain an activity [xACT〈manner〉 (y)] by invoking a wish for a state [x 〈STATE〉] and a belief in a method of accomplishment [[X ACT〈manner〉] CAUSE [BECOME y 〈STATE〉]]?)
The pro-attitude can be used in an explanation since it is ‘logically independent’ of the action.
• What is action A under description d?
d must refer to a type, and A be an element. But then how can A belong to more than one type. (Coercion, subtyping?) So there is an action which is: ‘I flipped the switch’, ‘I turned on the light’, ‘I alerted the prowler’. Then ‘I flipped the switch with my nose, at midnight, …’ - events can be endlessly specified.
Do we need an event A, then p(A), q(A), r(A),… as the different descriptions? Then we can’t have p(A) caused q(A), but we can have my wish to turn on the light caused my flipping the switch caused the light to come on/prowler to be alerted.
Perhaps this depends on whether A is an activity or an accomplishment. An accomplishment has projections to its activity and change of state. ‘John brushed the crumbs off the table’ to the brushing and the removal of the crumbs.
### The Logical Form of Action Sentences
• ‘I crossed the Channel in 15 hours’, ‘That was slow’, ‘I swam’
Perhaps fast/quick issues, but an adverb applied to an achievement (e) (Event types) may not survive projection to (d): ‘I swam quickly across the Channel’, ‘I crossed the Channel quickly’. Similar issue to ‘Grundy was a short basketball player, but a tall man.’
• ‘Jones buttered the toast in the bathroom with a knife at midnight’
Problem if we take this as a five place predicate. How will it imply ‘Jones buttered the toast’, etc.?
• An action can be described in many ways ‘I am writing my name on a piece of paper with the intention of writing a cheque with the intention of paying off my debts.’
This action also counts as my writing my name, my paying off my debts, etc. Kenny errs by redescribing all events as accomplishments, someone brought some state of affairs about.
Last revised on July 12, 2016 at 07:05:45. See the history of this page for a list of all contributions to it.
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2021-09-26 09:32:39
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https://calculator.academy/fault-current-calculator/
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Enter the voltage and resistance of a short circuit into the calculator to determine the fault current.
## Fault Current Formula
The following formula is used to calculate a fault current.
I = V/R
• Where I is the current (amps)
• V is the voltage (volts)
• R is the resistance (ohms)
## Fault Curret Definition
A fault current is defined as the maximum current available at any given point in a distribution system.
In the case of a fault, the resistance gets very small which causes the current to become extremely large. The example problems below show this effect.
## How to calculate fault current?
Example Problem #1.
First, determine the voltage of the system. For this example, we will use 120 volts as the voltage.
Next, determine the resistance. In practice, this would be based on the total resistance in the distribution system. For this problem, we will say the resistance is .25 ohms.
Finally, use the formula from Ohm’s Law to calculate the fault current.
I = V/R
= 120/.25
= 480 amps.
Example Problem #2.
In this next example, we will take a look to see how much a smaller resistance affects the current.
Again, we will use the same voltage as above, 120 V.
In this problem, the resistance is now reduced to only .05 ohms.
Using the formula as in example 1:
I = V/R
= 120/.05
= 2,400 amps.
This is nearly 5 times the current of example 1.
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2021-09-21 14:10:13
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https://www.cheenta.com/inequality-of-a-product-expression-i-s-i-b-math-2011-subjective-problem-3/
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# Inequality of a product expression | ISI BMath 2011 Problem 3
This is a subjective problem number 3 from ISI BMath 2011 based on inequality of a product expression. Try out this problem.
Problem: Inequality of a product expression
For $\mathbf{n\in\mathbb{N}}$ prove that $\mathbf{\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}}$
Discussion
Note that $\mathbf{ \frac{2n}{2n+1} \ge \frac{2n-1}{2n} }$ since simple cross multiplication gives $\mathbf{ 4n^2 \ge 4n^2 - 1 }$ which is true for all $\mathbf{n\in\mathbb{N}}$
Hence $\mathbf{\frac{1}{2} \le \frac{2}{3} , \frac{3}{4} \le \frac{4}{5} }$ etc.
Let $\mathbf{ x = \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n} \implies x \le \frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdots\frac{2n}{2n+1} \implies x^2 \le \frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\frac{4}{5}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\cdot\frac{2n}{2n+1}}$
All the terms cancel cross wise except 2n+1.
Thus $\mathbf{ x = \frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n} \le \frac{1}{2n+1} }$
(proved)
Special Note
It is possible to show (by induction) a much stronger result; $\mathbf{\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{3n+1}}}$
Our ISI CMI Program
How to use invariance in Combinatorics - ISI Entrance Problem - Video
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### Cheenta. Passion for Mathematics
Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
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2021-06-24 05:46:41
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https://mathzsolution.com/how-is-the-godels-completeness-theorem-not-a-tautology/
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# How is the Gödel’s Completeness Theorem not a tautology?
As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would become clearer for me, if I stated what I think the Gödel’s Completeness Theorem is about (after studying various materials including Wikipedia it seems redundant for me) and someone knowledgeable would clarify my misconceptions. So here it goes:
As I understand, if we have a set $U$ with a particular structure (functions, relations etc.) we can interpret it (through a particular signature, e.g. group signature $\{ e,\cdot \}$ ), as a model $\mathfrak{A}$ for a certain mathematical theory $\mathcal{T}$ (a theory being a set of axioms and its consequences). The theory is satisfied by $\mathfrak{A}$ only if $U$’s structure satisfies the axioms.
Enter Gödel’s theorem: For every first order theory $\mathcal{T}$ :
$$\left( \exists \textrm{model } \mathfrak{A}: \mathfrak{A} \models \mathcal{T} \right) \iff \mathcal{T} \textrm{ is consistent}$$
So I’m confused. Isn’t $\mathcal{T}$ being consistent a natural requirement which implicates that a set $U$ with a corresponding structure always exists (because of the ZFC’s set theory freedom in constructing sets as we please without any concerns regarding what constitutes the set)? And that in turn always allows us to create a model $\mathfrak{A}$ with an interpretation of the signature of the theory $\mathcal{T}$ in terms of $U$’s structure?
Where am I making mistakes? What concepts do I need to understand better in order to be able to properly comprehend this theorem and what model theory is and is not about? Please help!
It may help to look at things from a more general perspective. Presentations that focus on just first-order logic may obscure the fact that specific choices are implicit in the definitions of first-order logic; the general perspective highlights these choices. I want to write this up in detail, as a reference.
### General “logics”
We define a particular type of general “logic” with negation. This definition is intended to be very general. In particular, it accommodates much broader types of “syntax” and “semantics” than first-order logic.
A general “logic” will consist of:
• A set of “sentences” $L$. These do not have to be sentences in the sense of first-order logic, they can be any set of objects.
• A function $N: L \to L$ that assigns to each $x \in L$ a “negation” or “denial” $N(x)$.
• A set of “deductive rules”, which are given as a closure operation on the powerset of $L$. So we have a function $c: 2^L \to 2^L$ such that
1. $S \subseteq c(S)$ for each $S \subseteq L$
2. $c(c(S)) = c(S)$ for each $S \subseteq L$
3. If $S \subseteq S’$ then $c(S) \subseteq c(S’)$.
• A set of “models” $M$. These do not have to be structures in the sense of first-order logic. The only assumption is that each $m \in M$ comes with a set $v_m \subseteq L$ of sentences that are “satisfied” (in some sense) by $M$:
1. If $S \subseteq L$ and $x \in v_m$ for each $x \in S$ then $y \in v_m$ for each $y \in c(S)$
2. There is no $m \in M$ and $x \in L$ with $x \in v_m$ and $N(x) \in v_m$
The exact nature of the “sentences”, “deductive rules”, and “models”, and the definition of a model “satisfying” a sentence are irrelevant, as long as they satisfy the axioms listed above. These axioms are compatible with both classical and intuitionistic logic. They are also compatible with infinitary logics such as $L_{\omega_1, \omega}$, with modal logics, and other logical systems.
The main restriction in a general “logic” is that we have included a notion of negation or denial in the definition of a general “logic” so that we can talk about consistency.
• We say that a set $S \subseteq L$ is syntactically consistent if there is no $x \in L$ such that $x$ and $N(x)$ are both in $c(S)$.
• We say $S$ is semantically consistent if there is an $m \in M$ such that $x \in v_m$ for all $x \in S$.
The definition of a general “logic” is designed to imply that each semantically consistent theory is syntactically consistent.
### First-order logic as a general logic
To see how the definition of a general “logic” works, here is how to view first-order logic in any fixed signature as a general “logic”. Fix a signature $\sigma$.
• $L$ will be the set of all $\sigma$-sentences.
• $N$ will take a sentence $x$ and return $\lnot x$, the canonical negation of $x$.
• $c$ will take $S \subseteq L$ and return the set of all $\sigma$-sentences provable from $S$.
• $M$ will be the set of all $\sigma$-structures. For each $m \in M$, $v_m$ is given by the usual Tarski definition of truth.
With these definitions, syntactic consistency and semantic consistency in the general sense match up with syntactic consistency and semantic consistency as usually defined for first-order logic.
### The completeness theorem
Gödel’s completeness theorem simply says that, if we treat first-order logic in a fixed signature as a general “logic” (as above) then syntactic consistency is equivalent to semantic consistency.
The benefit of the general perspective is that we can see how things could go wrong if we change just one part of the interpretation of first-order logic with signature $\sigma$ as a general “logic”:
1. If we were to replace $c$ with a weaker operator, syntactic consistency may not imply semantic consistency. For example, we could take $c(S) = S$ for all $S$. Then there would be syntactically consistent theories that have no model. In practical terms, making $c$ weaker means removing deduction rules.
2. If we were to replace $M$ with a smaller class of models, syntactic consistency may not imply semantic consistency. For example, if we we take $M$ to be just the set of finite $\sigma$-structures, there are syntactically consistent theories that have no model. In practical terms, making $M$ smaller means excluding some structures from consideration.
3. If we were to replace $c$ with a stronger closure operator, semantic consistency may not imply syntactic consistency. For example, we could take $c(S) = L$ for all $S$. Then there would be semantically consistent theories that are syntactically inconsistent. In practical terms, making $c$ stronger means adding new deduction rules.
On the other hand, some changes would preserve the equivalence of syntactic and semantic consistency. For example, if we take $M$ to be just the set of finite or countable $\sigma$-structures, we can still prove the corresponding completeness theorem for first-order logic. In this sense, the choice of $M$ to be the set of all $\sigma$-structures is arbitrary.
### Other completeness theorems
We say that the “completeness theorem” for a general “logic” is the theorem that syntactic consistency is equivalent to semantic consistency in that logic.
• There is a natural completeness theorem for intuitionistic first-order logic. Here we let $c$ be the closure operator derived from any of the usual deductive systems for intuitionistic logic, and let $M$ be the set of Kripke models.
• There is a completeness theorem for second-order logic (in a fixed signature) with Henkin semantics. Here we let $c$ be the closure operator derived from the usual deductive system for second-order logic, and let $M$ be the set of Henkin models. On the other hand, if we let $M$ be the set of all “full” models, the corresponding completeness theorem fails, because this class of models is too small.
• There are similar completeness theorems for propositional and first-order modal logics using Kripke frames.
In each of those three cases, the historical development began with a deductive system, and the corresponding set of models was identified later. But, in other cases, we may begin with a set of models and look for a deductive system (including, in this sense, a set of axioms) that leads to a generalized completeness theorem. This is related to a common problem in model theory, which is to determine whether a given class of structures is “axiomatizable”.
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2022-10-05 01:55:16
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https://deepai.org/publication/on-optical-flow-models-for-variational-motion-estimation
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# On Optical Flow Models for Variational Motion Estimation
The aim of this paper is to discuss and evaluate total variation based regularization methods for motion estimation, with particular focus on optical flow models. In addition to standard L^2 and L^1 data fidelities we give an overview of different variants of total variation regularization obtained from combination with higher order models and a unified computational optimization approach based on primal-dual methods. Moreover, we extend the models by Bregman iterations and provide an inverse problems perspective to the analysis of variational optical flow models. A particular focus of the paper is the quantitative evaluation of motion estimation, which is a difficult and often underestimated task. We discuss several approaches for quality measures of motion estimation and apply them to compare the previously discussed regularization approaches.
## Authors
• 20 publications
• 6 publications
• 1 publication
• ### Joint Large-Scale Motion Estimation and Image Reconstruction
10/31/2016 ∙ by Hendrik Dirks, et al. ∙ 0
• ### Lifting methods for manifold-valued variational problems
Lifting methods allow to transform hard variational problems such as seg...
08/10/2019 ∙ by Thomas Vogt, et al. ∙ 0
• ### Total Variation Minimization and Graph Cuts for Moving Objects Segmentation
In this paper, we are interested in the application to video segmentatio...
09/18/2006 ∙ by Florent Ranchin, et al. ∙ 0
• ### First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite u...
12/13/2014 ∙ by Martin Burger, et al. ∙ 0
• ### Fast Piecewise-Affine Motion Estimation Without Segmentation
Current algorithmic approaches for piecewise affine motion estimation ar...
02/06/2018 ∙ by Denis Fortun, et al. ∙ 0
• ### Decomposition of Optical Flow on the Sphere
We propose a number of variational regularisation methods for the estima...
12/16/2013 ∙ by Clemens Kirisits, et al. ∙ 0
• ### Statistical Inverse Formulation of Optical Flow with Uncertainty Quantification
Optical flow refers to the visual motion observed between two consecutiv...
11/04/2016 ∙ by Jie Sun, et al. ∙ 0
##### This week in AI
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## 1 Introduction
Motion estimation is a crucial task in many different areas. On the one hand, motion estimation is important in medical and biological contexts where the goal is e.g. to track moving cells or to detect the motion of organs. On the other hand, it is also important in the automotive sector. Nowadays, there are many approaches that use motion estimation to make driving saver, by detecting both dangers from the outside of a car and also inattentiveness of a driver, which expresses e.g. in slower movement or closing eyes due to tiredness. These are only two of very many further applications of motion estimation.
Motion estimation generally arises in the context of image sequences , depending on a spatial position and a time . For real applications there exists only the discrete counterpart of , which is a set of images recorded at time steps . There also exist a variety of characteristics we have to consider when estimating motion:
• A digital image can suffer from low resolution, low contrast, different gray levels and noise.
• The temporal resolution is strongly connected to the underlying motion. For too large time steps we might loose correspondence between consecutive images (e.g. a very fast car might only be visible in one image).
• A natural image often contains a set of moving objects with different speeds. A sufficient model should simultaneously be able to detect small and large movements in the same sequence. On the other hand, for the static background no motion should be detected.
• For many biological applications, we have to consider the fact that the illumination is constant, but fluorescence of the observed object can be inhomogeneous in space, or might even underlie changes over time.
#### 1.0.1 Optical Flow and Real Motion
When looking at image sequences and moving objects we directly speak of motion. This is a false implication since only projections of the real 3-dimensional motion fields are recorded by an image recording device (and in particular the human eye).
To emphasize this fact, we consider a camera recording traffic on a highway. The camera is only able to follow 2-dimensional paths on the image domain , which is the projection of the real 3-dimensional path . Thus, already one degree of information gets lost here. This problem is even worse since we are not able to measure the 2-dimensional motion field directly. On images only the apparent motion (or optical flow) is visible, that is displacements of intensities. Unfortunately the apparent motion and the 2-dimensional motion field are two fundamentally different properties (a detailed discussion of this problem can be found in [32]). The Barber’s pole example is often used to underline this difference. The pole simply rotates counterclockwise, but optical devices (e.g. camera, eye) can only detect gray values tending upwards and consequently the optical flow points upwards.
The aim of an optical flow model is to detect the real motion only from the intensities of the given images. Performing flow estimation we have to deal with an inverse problem. We need to find the right motion that leads to the actual appearance of a specific image.
## 2 Models
One of the most common techniques to formally link intensity variations and motion is the optical flow constraint. Based on the assumption that the image intensity is constant along a trajectory with
we get using the chain-rule
0=dudt=∂u∂t+n∑i=1∂u∂xidxidt=ut+∇u⋅v. (1)
The last equation is generally known as the optical flow constraint. This optical flow constraint can also be regarded as a linear inverse problem. Therefore, we rewrite (1) by changing the positions of and to get
(∇u)T⋅v=−ut. (2)
We define via
Av=(∇u)T⋅v and g=−ut. (3)
Hence, the constraint fulfills the inverse problem , with or depending on the norm of the data fidelity. A more detailed introduction into the optical flow problem can be found in for example in [1] or [5]. We will discuss this issue in more detail in the following sections.
### 2.1 Variational Models with Gradient Regularization
The optical flow constraint constitutes in every point one equation, but in the context of motion estimation from images we usually have two or three spatial dimensions. Consequently, the problem is massively underdetermined. However, it is possible to estimate the motion using a variational model
where represents the so-called data term and incorporates the optical flow constraint in a suitable norm. The second part models additional a-priori knowledge on and is denoted as regularizer. The parameter regulates between data term and regularizer.
Possible choices for the data term are
D1(u,v):=12∥v⋅∇u+ut∥22,
or
D2(u,v):=∥v⋅∇u+ut∥1.
The quadratic L norm can be interpreted as solving the optical flow constraint in a least-squares sense inside the image domain . On the other hand, taking the L
norm enforces the optical flow constraint linearly and is able to handle outliers more robust.
The regularizer has to be chosen such that the a-priori knowledge is modeled in a reasonable way. If the solution is expected to be smooth, a quadratic L norm on the gradient of is chosen and we have
R1(v):=12∥∇v∥22.
Another possible approach is to choose the total variation (TV) of if we expect piecewise constant parts of motion. In the finite dimensional setting this can be written as
R2(v):=∥∇v∥1.
Taking
D1(u,v)=12∥v⋅∇u+ut∥22 and R1(v)=12∥∇v∥22
results in the very well known model of Horn and Schunck [18], 1981. With efficient primal-dual schemes to minimize L norms [12], L-TV optical flow models with
D2(u,v)=∥v⋅∇u+ut∥1 and
became very popular [36, 25]. This model was further improved by Werlberger et el [35], where the classical TV regularizer was replaced with a Huber norm. In this context, let us refer to a recent survey by Becker, Petra and Schnörr [5].
### 2.2 Extension of the Regularizer
The previously described regularizations are able to produce either smooth velocity fields (L regularization) or sharp edges (TV regularization). However, in realistic applications we often need a combination of both characteristics. To calculate velocity fields that combine smoothness and sharp edges, we need to extent the regularization term.
A possibility to In the following approach we want to make use of the potential of TV regularization to calculate sharp edges. To incorporate the possibility to get smooth parts, we need an additional primal variable , which will be needed for a second regularization term that allows for smooth transitions. We link the variables and by forcing
∇v−w≈0.
If we penalize this difference with an L norm and minimize it for , we get a simple TV regularization, which is shifted by . This way we ensure to keep characteristics of a TV regularization for our results. To combine this with the characteristics of another regularization technique, we apply a second regularizer to and also minimize everything for . This procedure leads us to the following extended regularizer:
R(v,w)=α0d∑i=1∥∇vi−w∥1+α1S(w). (4)
The term is the additional term, which should be chosen in a way that it produces smooth flow fields. The weighting parameters and on the one hand determine the relation between the regularization parts and the optical flow constraints. On the other hand they also determine the relation between the two regularization parts itself and this way adjust the proportion of smooth parts and edges in the result. If is large, the constant parts and edges outweigh the smooth parts; if it is small, the smooth parts outweigh the constant parts and edges. Note that a functional of the form (4) leads to a regularization of via
~R(v)=infwR(v,w) (5)
Similar to the TV term in a standard model, we can choose between different evaluations of the norm in the above formulation. Since this is a shifted TV norm, we once more have the possibility to use the anisotropic variant.
#### 2.2.1 L1-Tv/l2 Optical Flow Model
As we have already seen before, a regularization with the L norm leads to smooth approximations of a flow field. In the following approach, we use L of as a norm for the additional regularization term , which leads us to
R(v,w)=α0d∑i=1∥∇vi−w∥1+α12∥w∥22. (6)
Together with
D(u,v)=∥v⋅∇u+ut∥1
we get an L-TV/L model. This model is a combination between the previously described Horn-Schunck model and the L-TV model and thus contains characteristics of both kinds of flow fields. The regularizer of this model is very similar to the Huber function [19], which is defined as
Hϵ(r)={r22ϵ0≤|r|≤ϵ|r|−ϵ2ϵ<|r|.
The anisotropic variant of an optical flow model with Huber regularization has also been discussed in [35]. For another formulation of a related regularizer see e.g. [33] or [8].
Even if the Huber function can be solved directly, we want to be able to compare the different models by using the same algorithm. Therefore, later on we we will use the same the algorithm as for the other approaches.
#### 2.2.2 L1-TV/TV Optical Flow Model
Another possibility to consider smoothness in a flow estimation is to incorporate a higher order derivative. In the setting of (4), we can choose as TV of , which leads us to
R(v,w)=α0d∑i=1∥∇vi−w∥1+α1∥∇w∥1. (7)
Since the first term of the regularization leads to being close to the gradient of , the last term considers a term, which is close to the second gradient of . For the additional TV term in this model, we can again choose between the isotropic and the anisotropic variant, i.e. equivalently to those in Section 2.1. Overall, this leads to a huge variety of possible evaluations for this model.
By using
D(u,v)=∥v⋅∇u+ut∥1
as a data term we get an L-TV/TV model, which we will later use for the numerical realization.
The regularizer we get with the described approach is a primal realization of a specific case of a Total Generalized Variation (TGV) regularizer [7]. See [28, 27] for further analysis of the TGV regularizer applied to motion estimation models.
### 2.3 Bregman Iterations
A well-known drawback of TV regularization in combination with L data fidelity is the loss of contrast, which means that the difference between intensities in the result is lowered. However, edges are well recovered. Concerning optical flow, this shows up in a reduced velocity of motion. A possibility to overcome this drawback is to apply contrast-enhancing Bregman iterations [23]. For the case of higher order regularization term additional systematic bias, e.g. related to slopes, is corrected in the same way (cf. [6]). For this sake, we need the Bregman Distance of the regularizer , which is defined as
DbR(v,v∗)=R(v)−R(v∗)−⟨b,v−v∗⟩,b∈∂R(v∗),
for . This distance replaces the former regularization term in the respective models. This way it is possible to re-enhance the contrast. Thanks to the condition that is in the subgradient of , it is ensured, that the edges stay at the correct positions. The Bregman iteration iteratively computes as a solution of
minvD(u,v)+αDbnR(v,vn). (8)
Afterwards, the Bregman variable is updated by setting
bn+1=bn−1α(ut+∇u⋅vn)∇u.
Applying Bregman iterations to the optical flow problem with L data fidelity can equivalently adding the Bregman variable to the data. For the n-th Bregman iteration we can use the extended data term , which results in an additive constant for the optimality condition. So far Bregman iterations have not been investigated as an iterative regularization method for optical flow. A previous investigation [17] was only considering (split) Bregman methods as a minimization method for the original variational model, not investigating the possibility to reduce systematic bias. As we shall see below (see Figure 3) the Bregman iteration can yield improvements in parts where other variational methods underestimate motion (see e.g. the right block in the rubber whale data set below) and produce competitive results (see Table 2). It might thus nicely complement other approaches.
In the case of an data fidelity, the simple modification of the data term is not possible and the iteration needs to be carried out via the original form (8). Due to the additional linear term in comparison to the other only linear terms in the functional, it is not guaranteed that the functional is coercive, hence the well-posedness of the iteration is not guaranteed in the infinite-dimensional setting. In any case, it is not sure that the Bregman iteration yields additional benefit, since the L data fidelity does not introduce systematic bias. We leave this issue to future research.
## 3 Analysis
In the following we briefly comment on some aspects in the analysis of variational problems for optical flow, with special focus on particular aspects related to the recent extensions of total variation. We are able to generalize existing results for the Horn-Schunck and -TV to other data terms, focusing in particular on . This can be done under the same conditions on the image gradient as previously used, but the proof needs some adaption to take care of the constant functions in the null-space of the gradient. The coercivity related to the latter needs a different proof compared to previous results (cf. [30]), while on the other hand we give a more concise proof as in [16] and avoid too strong regularity conditions. The understanding of the basic structure of the proof then immediately allows a generalization to existence results for the different recent variants of total variation regularization, whose analysis has not been studied in the optical flow setting. Finally, we will discuss another problem hardly studied in the optical flow literature previously, namely the quantitative estimation of errors in the motion estimate when changing image data.
### 3.1 Existence of Minimizers
The basic issue of existence of minimizers for optical flow problems has been considered by various authors (cf. e.g. [1, 16]) using methods of calculus of variations. With the assumption and it is straightforward to see that the operator and data defined in (3) are bounded for and with standard convexity arguments lower semicontinuity of the data fidelities follow (for assumptions can even be relaxed). The associated data terms can all be shown to be L-coercive on subspaces of in the case of the Horn-Schunck model and of
for the other models. The subspaces are given by functions with mean value zero for standard models, while they are given by functions of vanishing mean and first moments in the case of the TV-TV and TV-L
regularizer. Hence, in order to obtain coercivity and thus existence of minimizers it is a natural question whether the data term yields coercivity on the whole space, i.e. boundedness of the data fidelity further implies bounds on the mean (and potentially the first moments) of . Obviously this cannot be true without further assumptions on (which is seen immediately for constant ). In the case of the Horn-Schunck model this question was answered in [30], who showed coercivity for the original Horn-Schunck model (see also [20] for a detailed discussion). His proof heavily uses the quadratic structure of the data fidelity as well as the fact that functions in the kernel of the regularization term are constant, hence it can be generalized to the L-TV model, but neither to extensions of the TV regularization nor to L-fidelities. However, the result can easily be generalized as we show in the following:
###### Lemma 3.1.
Let , arbitrary, be a bounded domain and let be a sequence in , . Let and such that the functions
are linearly independent. If there exists a sequence of vectors
such that is bounded in and
then is bounded in
###### Proof.
Under the above assumptions it obviously suffices to prove that the sequence is bounded in .We have
∥ut+∇u⋅vn∥Lp =∥ut+∇u⋅(cn+vn)−∇u⋅cn∥Lp ≥∥∇u⋅cn∥Lp−∥ut+∇u⋅(cn+vn)∥Lp.
The boundedness of of imply that is bounded in . Now assume that is unbounded, then there exists a subsequence (again denoted by with growing to infinity in a monotone way. Then is well-defined and bounded, moreover tends to zero in . Hence, has a convergent subsequence with limit satisfying , in particular . By continuity of the norm we find
∥∇u⋅d∥Lp=0,
which contradicts the linear independence of the functions and thus yields the assertion. ∎
With the Poincare-Wirtinger inequality and the embedding of into for and in arbitrary dimension, we obtain the coercivity of the total variation on functions of mean value zero. Now we choose equal to the mean value of and obtain coercivity of the respective L-TV model in . With a standard (weak) lower semicontinuity argument for the convex functionals we thus deduce the following existence result:
###### Theorem 3.1.
Let be a bounded domain and let either or . Let and such that the functions are linearly independent. Then there exists a minimizer for the L-TV model in the cases for arbitrary and for .
###### Lemma 3.2.
Let , arbitrary, be a bounded domain and let be a sequence in , . Let and such that the functions are linearly independent. If there exists a sequence of vectors and matrices such that is bounded in and
then is bounded in
###### Proof.
Under the above assumptions it obviously suffices to prove that the sequence is bounded in and the sequence is bounded in . We have
∥ut+∇u⋅vn∥Lp =∥ut+∇u⋅(cn+Anx+vn)−∇u⋅cn∥Lp ≥∥∇u⋅cn∥Lp−∥ut+∇u⋅(cn+Anx+vn)∥Lp.
The boundedness of of imply that is bounded in . A limiting procedure as in the proof of Lemma 3.1 yields a nontrivial limit respectively such that
∥∇u⋅(d+BX)∥Lp=0,
which contradicts the linear independence assumption and thus yields the assertion. ∎
Using the coercivity of the effective functionals on the space of BV-functions with vanishing mean and first moments (cf. [6] in the TV/TV case and [11] in the TV/L case) and a similar proof as above we obtain the following result:
###### Theorem 3.2.
Let be a bounded domain and let either or . Let and such that the functions are linearly independent. Then there exists a minimizer for the L-TV/TV model and the L-TV/L model in the cases for arbitrary and for .
A major restriction of the existing existence analysis is the fact that some regularity of and needs to be assumed, which might not be met by solutions of transport equations, in particular for noisy versions of the images (note however that many results in the literature assume even stronger regularity, e.g. images of class as in [16]). A detailed analysis of such issues remains a relevant question for future work. The case of stochastic noise in the image data might even enforce different approaches, e.g. Bayesian models (cf. [31]).
Let us finally mention that uniqueness of minimizers cannot be expected for total variation models, since neither the regularization nor the data term are strictly convex (due to the nullspace of the map ). In the case of an L data fidelity one can at least derive uniqueness of the component normal to the image level sets.
### 3.2 Quantitative Estimates
Quantitative estimates have hardly been investigated for optical flow models in the past, which is probably due to the nonuniqueness of the flow reconstruction. Hence, an estimate to a ground truth seems out of reach in most cases. Nonetheless, it seems interesting to take a look at the optical flow problem from the perspective of error estimation for inverse problems (cf.
[10]). Let us consider the case of L data fidelities for simplicity, we start from the optimality condition
(ut+∇u⋅v)∇u+αb=0,∂b∈∂R(v). (9)
The obvious first question to ask is the robustness of solutions for errors in the data , a second question is related to the behaviour as tends to zero. Let us first sketch the derivation of quantitative error estimates in the case of data perturbations, for this sake let be the solution of the variational model with data and . Then we have
∇u⋅(v−~v)∇u+α(b−~b)=~ut∇~u−ut∇u+∇~u⋅~v∇~u−∇u⋅~v∇u.
A duality product with and an elementary calculation yields
∥∇u⋅(v−~v)∥2L2+αDbR(~v,v)+αD~bR(v,~v)= ⟨∇~u~ut−∇uut+∇~u⋅~v∇~u−∇u⋅~v∇u,v−~v⟩
Using appropriate duality estimates on the right-hand side one immediately obtains estimates for the Bregman distances and the error in the velocity component parallel to in terms of errors in the data. To do so, let us further rewrite the right-hand side to obtain
∥∇u⋅(v−~v)∥2L2+αDbR(~v,v)+αD~bR(v,~v)= ⟨~ut+∇~u⋅~v−ut−∇u⋅~v,∇u⋅(v−~v)⟩+ ⟨(∇~u−∇u)(~ut+∇~u⋅~v),v−~v⟩
and with Young’s inequality we find
12∥∇u⋅(v−~v)∥2L2+αDbR(~v,v)+αD~bR(v,~v)≤ 12∥(~ut−ut)+(∇~u−∇u)⋅~v∥2L2+ ⟨(∇~u−∇u)(~ut+∇~u⋅~v),v−~v⟩
The two error terms on the right-hand side can be interpreted well. The first term measures the difference of the image data in a norm related to the optical flow model, it can further be related to the consistency in the optical flow constraint with estimated velocity . The second term creates more trouble, since due to it can contain terms in the direction perpendicular to , in which case it cannot be related to the first term on the left-hand side. Note that when viewing the optical flow estimation as an inverse problem this term is indeed related to the model error, i.e. the error in the forward operator . Even if this term can be estimated somehow by the Bregman distances on the left-hand side (e.g. in the case of the Horn-Schunck functional simply by Young’s inequality), it will lead to an error term of order . With the inherent smallness of we hence conclude that errors in the image gradient will strongly influence the result. This can e.g. be interpreted in terms of the accuracy in the discrete approximation of the image gradient. The quality in the estimated flow indeed depends heavily on the gradient approximation in numerical results. This is illustrated in Table 2 via an evaluation of errors for different choices of difference quotients for .
In the case of negligible error in the image gradient we can give an improved estimate:
###### Proposition 3.1.
Let and be weakly differentiable image data such that and . Then the estimate
12∥∇u⋅(v−~v)∥2L2+αDbR(~v,v)+αD~bR(v,~v)≤12∥~ut−ut∥2L2 (10)
holds for and being the solution of
minv12∥v⋅∇u+ut∥2L2+αR(v),
with data respectively .
Note that we see in any estimate that the velocity parallel to the image gradient (normal to level lines) can be estimated robustly from the image error (with constants independent of ), while we obtain hardly information perpendicular to the gradient. This is just a mathematical consequence of the aperture problem in motion estimation. In the perpendicular direction we obtain some stability of the velocity via the Bregman distance terms, but note that the error will grow inversely proportional to . We mention that to leading order in a small time step we have
∥∇u⋅(v−~v)∥L2≈∥u(⋅+τv(⋅))−u(⋅+τ~v(⋅))∥L2.
Hence, the error measure is strongly related to the end-point error frequently used in practical evaluations of motion estimation, an issue we will discuss in detail below.
The limit is the one usually studied in inverse problems with strong emphasis on the appropriate treatment of instabilities. In the case of motion estimation the underdeterminedness is the more crucial issue. Thus, the key question is to understand what solutions are obtained in the limit, respectively which kind of flows can be reconstructed well. The limiting solutions are determined by the constrained problem
minvR(v)subject to ut+∇u⋅v=0. (11)
The solutions to be reconstructed well are those satisfying a source condition (cf. [10]), i.e. the optimality condition for the constrained problem in a Lagrangian duality setting. This is equivalent to
b=w∇u∈∂R(v) (12)
for some . Under this condition one can construct data , such that the constrained solution also satisfies the unconstrained variational problem with regularization parameter , thus error estimation reduces to the above case. From (12) we can give a mathematical reason why it is it is useful to have images with a large gradient, which corresponds to the usual intuition. The data perturbation is a multiple of , hence error estimates depend on the norm of , which is however inversely proportional to the norm of . A more detailed study of (12) depends on the specific regularization terms and is an interesting question for further research.
## 4 Numerical Solution
In the following we discuss the numerical solution of the variational models discussed above. We start with a unified discussion of primal-dual iterative methods and then proceed to discretization issues.
### 4.1 Primal-Dual Algorithm
The concept of duality offers an efficient way of minimizing the introduced variational models. Let us for the numerical application consider two finite dimensional vector spaces and equipped with a scalar product and a norm . We furthermore consider a continuous linear operator . Now, class of variational problems in this section can be written as
minx∈XG(v)+F(Kv), (13)
with are proper, convex and lower semi-continuous functionals. In our application, usually incorporates the data term, whereas denotes the regularizer. In the following, we denote Equation as the primal problem. The primal problem is often hard to minimize or yields very slow algorithms. Instead of the primal problem, we can equivalently optimize the so-called primal-dual problem:
minx∈Xmaxy∈Y⟨Kv,y⟩+G(v)−F∗(y). (14)
Here refers to the convex conjugate of , which can be easily calculated for the observed regularizers. The very well known algorithm of Chambolle and Pock dedicates to this class of problems [12]. For , a pair and initial value we obtain the following iterative scheme to solve the saddle-point problem :
yk+1 =proxσF∗(yk+σK^xk) (15) vk+1 =proxτG(vk−τK∗yk+1) (16) ^vk+1 =2vk+1−vk (17)
Here, denotes the proximal or resolvent operator
proxτG(y)=(I+τ∂G)−1(y):=argminv{∥v−y∥222+τG(v)},
which can be interpreted as a compromise between minimizing and being close to the input argument . Suitable choices of and were chosen according to [26] and are given in Section 4.2 for each algorithm. The primal-dual methods are efficient if the proximal problem can be solved efficiently. In [12] the authors demonstrated an application of the primal-dual algorithm to a L-TV optical flow problem. In the following we will extend this application to each of the introduced models and provide solutions to the occurring proximal problems.
In Table 4.1 we transferred our models to the notation of the primal problem (13). Let us begin with the proximal problems for the primal part , for a more detailed description we refer to [13].
##### L2 data term:
For the L data term we have to solve for the following proximal problem:
argminv⎧⎪⎨⎪⎩∥∥v−~vk+1∥∥222+τ2∥v⋅∇u+ut∥22⎫⎪⎬⎪⎭.
This yields a linear optimality system, which can be directly calculated. The solution is given by , where is a linear function also incorporating the scalar terms and .
##### L1 data term:
In case of the L optical flow term, the proximal problem reads
argminv⎧⎪⎨⎪⎩∥∥v−~vk+1∥∥222+τ∥v⋅∇u+ut∥1⎫⎪⎬⎪⎭.
Defining the affine linear function , the problem above represents a affine-linear L optimization problem. It can be shown that the solution is given by
v=~vk+1+⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩τ∇uif ρ(~vk+1)<−τ∥∇u∥22−τ∇uif ρ(~vk+1)>τ∥∇u∥22−ρ(~vk+1)∥∇u∥22∇u% else .
##### L2 regularization:
Denoting the proximal problems for the dual part requires calculating the convex conjugate of the dual part first. For the L regularization term we calculate and obtain the proximal problem
argminv⎧⎪⎨⎪⎩∥∥y−~yk+1∥∥222+τ12α∥y∥22⎫⎪⎬⎪⎭.
Consequently, the proximal problem can be solved straightaway.
##### TV regularization:
In case of the total variation and the higher order regularization we always have L norms involved. Similar to the standard TV regularization () we have , where denotes the indicator function of the L unit ball. The corresponding proximal problem then reads
argminv⎧⎪⎨⎪⎩∥∥y−~yk+1∥∥222+ταδB(L∞)(y/α)⎫⎪⎬⎪⎭.
The solution is given by a point-wise projection of onto the unit ball corresponding to the chosen vector norm. The solution is will be discussed in more detail in Section 4.2.
### 4.2 Discretization and Parameters
In the context of discretization we have to consider two different aspects. On the one hand we have to approximate the image derivatives and , which are passed to the algorithm as static variables afterwards. One the other hand, we have to discretize the operators and coming from the primal-dual iteration.
Starting with the image discretization, we consider the image domain to consist of the regular grid:
{(t,i,j):t=0,1,i=0,…,nx,j=0,…,ny}
Since we are interested in the velocity field between and we have to evaluate the derivatives of at . In our observation, a mixed scheme consisting of a forward difference for the time derivative and central differences for the spatial derivatives turned out to give the best results. Stability in terms of is not required, due to the fact that they act as constants in the model. Consequently, we obtain the following scheme:
ui,jt =u1,i,j−u0,i,j ui,jx ={u0,i+1,j−u0,i−1,jif i>0 % and i0 % and j
For the sake of simplicity, we omitted the time-dependency in the derivatives.
The operators in our model only consist of gradients for the spatial regularization and the corresponding divergences. We propose a forward discretization with Neumann boundary conditions for the gradient and consequently obtain a backward difference for the divergence to keep a discrete adjoint structure. The resulting scheme reads
vi,jx ={vi+1,j−vi,jif i0yi,j1if i=0−yi−1,j1if i=nx⎫⎪ ⎪⎬⎪ ⎪⎭+⎧⎪ ⎪⎨⎪ ⎪⎩yi,j2−yi,j−12if j>0yi,j2if j=0−yi,j−12if j=ny⎫⎪ ⎪⎬⎪ ⎪⎭
The primal-dual algorithms moreover requires suitable parameters and such that . It has been shown in [26] that a diagonal preconditioning technique can be used to find without calculating . According to [26], Lemma 2, we choose for the gradient models from Section 2.1 and for the extended models from Section 2.2.
In the previous section, the solution of a proximal problem involving an indicator function of the L unit ball of was given as a point-wise projection. To conceive this, let us begin by writing down the convex set corresponding to :
{y:∥y/α∥∞≤1}.
Here, refers to the discrete maximum over all elements in the image domain:
∥y/α∥∞=max(i,j)∣∣yi,j/α∣∣.
An interesting aspect is how to evaluate the vector norm . Choosing
∣∣yi,j∣∣:=∣∣y1i,j∣∣+∣∣y2i,j∣∣+∣∣y3i,j∣∣+∣∣y4i,j∣∣
results in the so-called anisotropic variant, whereas
∣∣∇vi,j∣∣:=√(y1i,j)2+(y2i,j)2+(y3i,j)2+(y4i,j)2
is denoted as the fully isotropic one. The anisotropic total variation is more suitable when dealing with quadratic structures, the isotropic variant better fits to circular structures. Depending on the chosen vector norm for (anisotropic, fully isotropic), we obtain a different dual inner norm . For the anisotropic case, that refers to the vectorial L norm, we get the maximum norm for the dual variable:
∣∣yi,j∣∣=max{∣∣y1,1i,j∣∣,∣∣y1,2i,j∣∣,∣∣y2,1i,j∣∣,∣∣y2,2i,j∣∣}.
Since the fully isotropic case refers to the self-dual Euclidean norm, we get in this case
∣∣yi,j∣∣=∥∥yi,j∥∥2=√(y1,1i,j)2+(y1,2i,j)2+(y2,1i,j)2+(y2,2i,j)2.
Transferring this to the optimization problem above, we get in the anisotropic case a the point-wise projection of onto :
yk+1=min(α,max(−α,~yk+1)),
and for the isotropic case a point-wise projection of onto the L unit ball:
yk+1i,j=~yk+1i,jmax(1,∥∥~yk+1i,j/α∥∥2).
For isotropic and anisotropic variants of the extended models, the norms are achieved equivalently. However, it has to be considered that for the L-TV/TV model there are independent norms for both parts of the regularization. Using the partially isotropic variant also results in independent norms for the different components of . For the sake of simplicity we will restrict ourselves to the fully isotropic cases for all appearing total variations in the numerical realizations.
## 5 Results
### 5.1 Error Measures for Velocity Fields
Finding error measures for velocity fields is a delicate problem which can be motivated by the following simple example. Consider two images, each consisting of only pixels of the following form:
u1=⎛⎜⎝110000000⎞⎟⎠,u2=⎛⎜⎝000110000⎞⎟⎠. (18)
From our point of view we seek for a velocity field that transforms into . Unfortunately, it is unclear which velocity field underlies this motion. Examples of possible solutions are
1. Just move the pixels at position and by 1 to the bottom, hence
v1=⎛⎜⎝110000000⎞⎟⎠,v2=⎛⎜⎝000000000⎞⎟⎠.
2. Move the pixels at position and by 1 to the bottom and change their position:
v1=⎛⎜⎝110000000⎞⎟⎠,v2=⎛⎜⎝1−10000000⎞⎟⎠.
3. Move the whole image by 1 to the bottom which gives
v1=⎛⎜⎝111111111⎞⎟⎠,v2=⎛⎜⎝000000000⎞⎟⎠.
4. Move the pixels at position and by 1 to the bottom and exchange some pixels in the third column
v1=⎛⎜⎝11100100−2⎞⎟⎠,v2=⎛⎜⎝000000000⎞⎟⎠.
All these velocity fields produce the same result, that is they all fulfill the optical flow constraint . This results from the fact that for given images the optical flow constraint states for every point only one equation for a 2-dimensional velocity field . The solution is unique iff there exists a unique bijection between and . This requires both images to consist of the same intensity values, which furthermore have to be unique. Unfortunately, this assumption is far from reality, since regions of constant intensity are characteristic for background or objects. Hence, for practical problems we have a highly underdetermined system and consequently a huge variety of possible underlying velocity fields .
As we will see later an error measure always explicitly or implicitly favors one of these possible results over the others. Those error measures, which explicitly prefer one result, expect a given ground truth field and measure a distance from the calculated velocity field to the given ground truth field. This strategy is questionable because in real world examples we usually do not have a ground truth velocity field, and artificial examples can usually be generated by several different ground truth fields. Hence, setting the ground truth directly favors a subjective result. This is at least from the mathematical viewpoint questionable.
#### 5.1.1 Absolute Endpoint Error
Despite the fact that explicit error measures might be problematic they are often used in the literature and we will also use them to evaluate our algorithms. In [2] two explicit error measures have been presented. The most intuitive one is the average endpoint error (aee), proposed in [24], which is the vector-wise Euclidean norm of the difference vector . The difference is divided by and we have
aee:=1|Ω|∫Ω√(v1
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https://mersenneforum.org/showthread.php?s=6377a0b5b33946f695e9dcbeaf5fb4a0&t=23916&page=2
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mersenneforum.org > News Doubts, scientific breakthroughs and conspiracy theories about Wagstaff Conjecture (from Lucky13)
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2018-12-11, 23:30 #12
kriesel
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
4,457 Posts
Quote:
Originally Posted by GP2 Assuming the plot is accurate, analysis of the pixels places the new exponent between 82.60M and 82.75M 82.65M. We could do better than that of course. Simply identify the software that created the plot and start plotting points, varying the exponent value until you come up with a pixel-by-pixel reproduction of the plotted box pattern, including the pale shading and blur (sorry, I don't know the correct graphics terminology).
How close does your method get to the other, known p values nearby, say M49 and M50?
2018-12-11, 23:37 #13
kriesel
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
4,457 Posts
Quote:
Originally Posted by ewmayer I would be interested in some discussion of the statistical significance ascribable to the apparent trendline break of the last dozen M-prime exponents. Chris Caldwell's "this graph is amazingly linear" is getting more untenable with each new sooner-than-expected find.
It does look less and less like a fit. On the other hand, pretty much anything that is first magnitude sorted, then run through a log function and then graphed on a log scale will look pretty linear. I did the same with the set of populations of Wisconsin villages, and got a rather linear looking plot. ln(ln(x)) is very flattening. Plot the actual ordered Mersenne prime values (not exponents) on a linear scale, and, HOCKEY STICK!
2018-12-11, 23:41 #14
Uncwilly
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
22·2,161 Posts
Quote:
Originally Posted by GP2 Assuming the plot is accurate, analysis of the pixels places the new exponent between 82.60M and 82.75M 82.65M.
Your former range and other clues (based upon at least 1 being interpreted) point to 5 exponents. Your updated range limits that to <5.
Last fiddled with by Uncwilly on 2018-12-11 at 23:42
2018-12-12, 00:00 #15 GP2 Sep 2003 50248 Posts Methodology for my estimate: Magnify the graph about 5x in a Paint program. Under magnification, we see that the lines that draw the square-box plot points are not always sharp, but rather they are often blurred and smeared into two or even three adjacent lines of varying intensity. This reflects the fact that the exact center of the desired plot point usually does not map exactly onto an exact pixel point, so the plotting program uses blurring to compensate. So we can use the relative paleness/brightness of the components of the blurred/smeared lines to assign a fractional vertical pixel position to both the top horizontal line and the bottom horizontal line of each square-box plot point, and then average them to find the center. For the relative vertical pixel positions, with the zero reference point at the bottom horizontal bar of the square representing M46, I get: Code: bottom top center Mn log(Mn) M46 -0.2 9.1 4.45 42643801 17.56839 M47 0.3 10.0 5.15 43112609 17.57933 M48 20.2 30.0 25.1 57885161 17.87397 M49 37.05 46.7 41.875 74207281 18.12237 M50 39.8 49.4 44.6 77232917 18.16234 M51 44.3 54.0 49.15 xxxxxxxx 18.2296 to 18.2301 Last fiddled with by GP2 on 2018-12-12 at 00:04
2018-12-12, 00:26 #16
Prime95
P90 years forever!
Aug 2002
Yeehaw, FL
717110 Posts
Quote:
Originally Posted by ewmayer I would be interested in some discussion of the statistical significance ascribable to the apparent trendline break of the last dozen M-prime exponents. Chris Caldwell's "this graph is amazingly linear" is getting more untenable with each new sooner-than-expected find.
I would like such a discussion too.
I'd also like to see your graph along with a line with the slope predicted at
https://primes.utm.edu/notes/faq/NextMersenne.html. "we should get approximately a straight line with slope 1/egamma (about 0.56145948)"
2018-12-12, 00:34 #17
Batalov
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
100011101001112 Posts
I think it (that discussion) has started (since post like #10 and #14 )
It reminds me of that anecdote where someone heard about the prime and the only question that they had was "but what was the sign of the penulitmate iteration?"
Numbers, shnumbers... Ideas!
Found it --
Quote:
Originally Posted by L.Welsh (2003) After M29 was discovered, that was the very first question Dick Lehmer asked. I think it interested him more than the value of p!!
2018-12-12, 00:52 #18
Prime95
P90 years forever!
Aug 2002
Yeehaw, FL
71×101 Posts
Quote:
Originally Posted by Batalov I think it (that discussion) has started (since post like #10 and #14 )
I'm looking for analysis along the lines that Chris did here: https://primes.utm.edu/notes/faq/NextMersenne.html. The math is above my pay grade.
We have a theory that Mersenne primes "follow" a logarithmic line with a specific slope. We have data on the first 51 Mersennes. If someone with a good probability/statistics background could weigh in with an analysis of how well the 51 data points are matching the theory, then I and others here could hopefully learn a thing or two.
2018-12-12, 02:43 #19
Batalov
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9,127 Posts
Quote:
Originally Posted by Prime95 I'm looking for analysis along the lines that Chris did here: https://primes.utm.edu/notes/faq/NextMersenne.html. The math is above my pay grade. We have a theory that Mersenne primes "follow" a logarithmic line with a specific slope. We have data on the first 51 Mersennes. If someone with a good probability/statistics background could weigh in with an analysis of how well the 51 data points are matching the theory, then I and others here could hopefully learn a thing or two.
That's the one I criticized in #14. (The product of inverse probabilities, but only to a limit.)
I later googled a free version of Wagstaff (1983) (below) and with all due respect the UTM simplified explanation has more holes than the original paper. I don't think the https://primes.utm.edu/notes/faq/NextMersenne.html quite matches what the paper does.
Attached Files
1983_Wagstaff.pdf (1.35 MB, 69 views)
2018-12-12, 03:56 #20
kriesel
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
4,457 Posts
Quote:
Originally Posted by GP2 Methodology for my estimate: Magnify the graph about 5x in a Paint program. Under magnification, we see that the lines that draw the square-box plot points are not always sharp, but rather they are often blurred and smeared into two or even three adjacent lines of varying intensity. This reflects the fact that the exact center of the desired plot point usually does not map exactly onto an exact pixel point, so the plotting program uses blurring to compensate. So we can use the relative paleness/brightness of the components of the blurred/smeared lines to assign a fractional vertical pixel position to both the top horizontal line and the bottom horizontal line of each square-box plot point, and then average them to find the center. For the relative vertical pixel positions, with the zero reference point at the bottom horizontal bar of the square representing M46, I get: Code: bottom top center Mn log(Mn) M46 -0.2 9.1 4.45 42643801 17.56839 M47 0.3 10.0 5.15 43112609 17.57933 M48 20.2 30.0 25.1 57885161 17.87397 M49 37.05 46.7 41.875 74207281 18.12237 M50 39.8 49.4 44.6 77232917 18.16234 M51 44.3 54.0 49.15 xxxxxxxx 18.2296 to 18.2301
When I zoom Ernst's plot, there's no antialiasing, all the way up to 1600%.
Zoom first, then screen copy paste, and there's little or no antialiasing in the paint program input. A spreadsheet using pixel locations of X marks the center gives method error estimates provided by the knowns. You seem to be using the already known Mn values above, rather than computing estimates from the pixel locations for comparison to the knowns. What did I miss?
2018-12-12, 05:38 #21
GP2
Sep 2003
22·3·5·43 Posts
Quote:
Originally Posted by kriesel When I zoom Ernst's plot, there's no antialiasing, all the way up to 1600%. Zoom first, then screen copy paste, and there's little or no antialiasing in the paint program input. A spreadsheet using pixel locations of X marks the center gives method error estimates provided by the knowns. You seem to be using the already known Mn values above, rather than computing estimates from the pixel locations for comparison to the knowns. What did I miss?
No doubt, I didn't zoom properly before saving the image.That made things needlessly difficult.
This time, after zooming to the maximum in the PDF and saving the image, I then zoomed some more in the paint program to get nice big pixels. It is this second process that makes the antialiasing more readily visible.
I suspect that if I redid the calculations with the new pixel counts it would give a more accurate estimate, to an extra decimal place. Perhaps I'll try that tomorrow.
What estimate do you get?
2018-12-12, 06:09 #22
GP2
Sep 2003
22×3×5×43 Posts
Quote:
Originally Posted by Batalov That's the one I criticized in #14. (The product of inverse probabilities, but only to a limit.) I later googled a free version of Wagstaff (1983) (below) and with all due respect the UTM simplified explanation has more holes than the original paper. I don't think the https://primes.utm.edu/notes/faq/NextMersenne.html quite matches what the paper does.
Wagstaff's analysis involves estimating the expected number of prime factors of a Mersenne number between two bounds A and B (on the second page, page 386).
If A = 2p and B = sqrt of the Mersenne number, then you can try to estimate the total number of factors a Mersenne number of a given size would be expected to have. I tried to do that in a post a few months ago, maybe someone who actually understands the math can correct it.
However, thanks to user TJAOI and his very systematic "by k" factoring, we can be quite certain that we know all factors of bit-length 65 or less of all Mersenne numbers with prime exponent less than 1 billion. As I posted last year in the "User TJAOI thread", empirically we can see that he does not miss any factors, except for one very minor glitch in 2014 that he very quickly corrected.
So we can use our factor data to check if the actual exact number of factors smaller than B = 2^65 matches what Wagstaff's heuristic predicts, for various ranges of p.
If our factor data is in fact consistent with the Poisson process implied by Wagstaff's analysis, then we should expect that the predicted number of Mersenne primes is also consistent with it.
I really think someone here with the required math knowledge could do the analysis and maybe make a paper out of it.
Similar Threads Thread Thread Starter Forum Replies Last Post Tony Reix Wagstaff PRP Search 7 2013-10-10 01:23 paramveer Information & Answers 32 2012-01-15 06:05 davieddy Miscellaneous Math 209 2011-01-23 23:50 davieddy PrimeNet 17 2010-07-21 00:07 jasong Soap Box 11 2010-07-05 12:23
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# Mathematical Physics | enregistrements trouvés : 157
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## Post-edited ColDICE: a parallel Vlaslov-Poisson solver using moving adaptative simplicial tessellation Sousbie, Thierry (Auteur de la Conférence) | CIRM (Editeur )
In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities. In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical ...
## Post-edited Lagrange's research on the nature and propagation of sound Darrigol, Olivier (Auteur de la Conférence) | CIRM (Editeur )
The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis than is usually believed. On the physical side, Lagrange properly explained the functioning of string and air-column instruments, although he did not accept harmonic analysis as we now understand it.
Lagrange - acoustics - propagation of sound - harmonic analysis - Fourier analysis - vibrating strings - organ pipes
The most important works of the young Lagrange were two very learned memoirs on sound and its propagation. In a tour de force of mathematical analysis, he solved the relevant partial differential equations in a novel manner and he applied the solutions to a number of acoustic problems. Although Euler and d'Alembert may have been the only contemporaries who fully appreciated these memoirs, their contents anticipated much more of Fourier analysis ...
## Post-edited Inverse problems for fluid dynamics Yamamoto, Masahiro (Auteur de la Conférence) | CIRM (Editeur )
I discuss several types of inverse problems for fluid dynamics such as Navier-Stokes equations. I prove uniqueness and conditional stability for the formulations by Dirichlet-to-Neumann map and Carleman estimates. This is a joint work with Prof. O. Imanuvilov (Colorado State Univ.)
35R30
## Post-edited Peierls substitution for magnetic Bloch bands Teufel, Stefan (Auteur de la Conférence) | CIRM (Editeur )
We consider the one-particle Schrödinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , for $\epsilon\ll 1$ . For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrödinger operator to a corresponding almost invariant subspace. At leading order, our effective Hamiltonian can be interpreted as the Peierls substitution Hamiltonian widely used in physics for non-magnetic Bloch bands. However, while for non-magnetic Bloch bands the corresponding result is well understood, both on a heuristic and on a rigorous level, for magnetic Bloch bands it is not clear how to even define a Peierls substitution Hamiltonian beyond a formal expression. The source of the difficulty is a topological obstruction: In contrast to the non-magnetic case, magnetic Bloch bundles are generically not trivializable. As a consequence, Peierls substitution Hamiltonians for magnetic Bloch bands turn out to be pseudodifferential operators acting on sections of non-trivial vector bundles over a two-torus, the reduced Brillouin zone. As an application of our results we construct a family of canonical one-band Hamiltonians $H_{\theta=0}$ for magnetic Bloch bands with Chern number $\theta\in\mathbb{Z}$ that generalizes the Hofstadter model $H_{\theta=0}$ for a single non-magnetic Bloch band. It turns out that the spectrum of $H_\theta$ is independent of $\theta$ and thus agrees with the Hofstadter spectrum depicted in his famous (black and white) butterfly. However, the resulting Chern numbers of subbands, corresponding to Hall conductivities, depend on $\theta$ , and thus the models lead to different colored butterflies.
This is joint work with Silvia Freund.
We consider the one-particle Schrödinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , for $\epsilon\ll 1$ . For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrödinger operator to a co...
## Post-edited Modelling of magnetic fusion plasmas: from fluid to kinetic description: kinetic MHD Garbet, Xavier (Auteur de la Conférence) | CIRM (Editeur )
This lecture will present a short overview on kinetic MHD. The advantages and drawbacks of kinetic versus fluid modelling will be summarized. Various techniques to implement kinetic effects in the fluid description will be introduced with increasing complexity: bi-fluid effects, gyroaverage fields, Landau closures. Hybrid formulations, which combine fluid and kinetic approaches will be presented. It will be shown that these formulations raise several difficulties, including inconsistent ordering and choice of representation. The non linear dynamics of an internal kink mode in a tokamak will be used as a test bed for the various formulations. It will be shown that bi-fluid effects can explain to some extent fast plasma relaxations (reconnection), but cannot address kinetic instabilities due to energetic particles. Some results of hybrid codes will be shown. Recent developments and perspectives will be given in conclusion. This lecture will present a short overview on kinetic MHD. The advantages and drawbacks of kinetic versus fluid modelling will be summarized. Various techniques to implement kinetic effects in the fluid description will be introduced with increasing complexity: bi-fluid effects, gyroaverage fields, Landau closures. Hybrid formulations, which combine fluid and kinetic approaches will be presented. It will be shown that these formulations raise ...
## Post-edited Exact conservation laws for gyrokinetic Vlasov-Poisson equations Tronko, Natalia (Auteur de la Conférence) | CIRM (Editeur )
The momentum transport in a fusion device such as a tokamak has been in a scope of the interest during last decade. Indeed, it is tightly related to the plasma rotation and therefore its stabilization, which in its turn is essential for the confinement improvement. The intrinsic rotation, i.e. the part of the rotation occurring without any external torque is one of the possible sources of plasma stabilization.
The modern gyrokinetic theory [3] is an ubiquitous theoretical framework for lowfrequency fusion plasma description. In this work we are using the field theory formulation of the modern gyrokinetics [1]. The main attention is focussed on derivation of the momentum conservation law via the Noether method, which allows to connect symmetries of the system with conserved quantities by means of the infinitesimal space-time translations and rotations.
Such an approach allows to consistently keep the gyrokinetic dynamical reduction effects into account and therefore leads towards a complete momentum transport equation.
Elucidating the role of the gyrokinetic polarization is one of the main results of this work. We show that the terms resulting from each step of the dynamical reduction (guiding-center and gyrocenter) should be consistently taken into account in order to establish physical meaning of the transported quantity. The present work [2] generalizes previous result obtained in [4] by taking into the account purely geometrical contributions into the radial polarization.
The momentum transport in a fusion device such as a tokamak has been in a scope of the interest during last decade. Indeed, it is tightly related to the plasma rotation and therefore its stabilization, which in its turn is essential for the confinement improvement. The intrinsic rotation, i.e. the part of the rotation occurring without any external torque is one of the possible sources of plasma stabilization.
The modern gyrokinetic theory [3] ...
## Post-edited A simple HLLC-type Riemann solver for compressible non-equilibrium two-phase flows Furfaro, Damien (Auteur de la Conférence) | CIRM (Editeur )
A simple, robust and accurate HLLC-type Riemann solver for two-phase 7-equation type models is built. It involves 4 waves per phase, i.e. the three conventional right- and left-facing and contact waves, augmented by an extra "interfacial" wave. Inspired by the Discrete Equations Method (Abgrall and Saurel, 2003), this wave speed $u_I$ is assumed function only of the piecewise constant initial data. Therefore it is computed easily from these initial data. The same is done for the interfacial pressure $P_I$. Interfacial variables $u_I$ and $P_I$ are thus local constants in the Riemann problem. Thanks to this property there is no difficulty to express the non-conservative system of partial differential equations in local conservative form. With the conventional HLLC wave speed estimates and the extra interfacial speed $u_I$, the four-waves Riemann problem for each phase is solved following the same strategy as in Toro et al. (1994) for the Euler equations. As $u_I$ and $P_I$ are functions only of the Riemann problem initial data, the two-phase Riemann problem consists in two independent Riemann problems with 4 waves only. Moreover, it is shown that these solvers are entropy producing. The method is easy to code and very robust. Its accuracy is validated against exact solutions as well as experimental data. A simple, robust and accurate HLLC-type Riemann solver for two-phase 7-equation type models is built. It involves 4 waves per phase, i.e. the three conventional right- and left-facing and contact waves, augmented by an extra "interfacial" wave. Inspired by the Discrete Equations Method (Abgrall and Saurel, 2003), this wave speed $u_I$ is assumed function only of the piecewise constant initial data. Therefore it is computed easily from these ...
## Post-edited Mathematical properties of hierarchies of reduced MHD models Després, Bruno (Auteur de la Conférence) | CIRM (Editeur )
Reduced MHD models in Tokamak geometry are convenient simplifications of full MHD and are fundamental for the numerical simulation of MHD stability in Tokamaks. This presentation will address the mathematical well-posedness and the justification of the such models.
The first result is a systematic design of hierachies of well-posed reduced MHD models. Here well-posed means that the system is endowed with a physically sound energy identity and that existence of a weak solution can be proved. Some of these models will be detailed.
The second result is perhaps more important for applications. It provides understanding on the fact the the growth rate of linear instabilities of the initial (non reduced) model is lower bounded by the growth rate of linear instabilities of the reduced model.
This work has been done with Rémy Sart.
Reduced MHD models in Tokamak geometry are convenient simplifications of full MHD and are fundamental for the numerical simulation of MHD stability in Tokamaks. This presentation will address the mathematical well-posedness and the justification of the such models.
The first result is a systematic design of hierachies of well-posed reduced MHD models. Here well-posed means that the system is endowed with a physically sound energy ...
## Post-edited Numerical simulation of the Vlasov-Poisson model with an external magnetic field Filbet, Francis (Auteur de la Conférence) | CIRM (Editeur )
We present an efficient algorithm for the long time behavior of plasma simulations. We will focus on 4D drift-kinetic model, where the plasma's motion occurs in the plane perpendicular to the magnetic field and can be governed by the 2D guiding-center model. Hermite WENO reconstructions, already proposed in [1], are applied for solving the Vlasov equation. Here we consider an arbitrary computational domain with an appropriate numerical method for the treatment of boundary conditions. Then we apply this algorithm for plasma turbulence simulations. We first solve the 2D guiding-center model in a D-shape domain and investigate the numerical stability of the steady state. Then, the 4D drift-kinetic model is studied with a mixed method, i.e. the semi-Lagrangian method in linear phase and finite difference method during the nonlinear phase. Numerical results show that the mixed method is efficient and accurate in linear phase and it is much stable during the nonlinear phase. Moreover, in practice it has better conservation properties.
Keywords: Cartesian mesh - semi-Lagrangian method - Hermite WENO reconstruction - guiding-center - drift-kinetic model
We present an efficient algorithm for the long time behavior of plasma simulations. We will focus on 4D drift-kinetic model, where the plasma's motion occurs in the plane perpendicular to the magnetic field and can be governed by the 2D guiding-center model. Hermite WENO reconstructions, already proposed in [1], are applied for solving the Vlasov equation. Here we consider an arbitrary computational domain with an appropriate numerical method ...
## Post-edited The Weil algebra of a Hopf algebra Dubois-Violette, Michel (Auteur de la Conférence) | CIRM (Editeur )
We give a summary of a joint work with Giovanni Landi (Trieste University) on a non commutative generalization of Henri Cartan's theory of operations, algebraic connections and Weil algebra.
## Post-edited Geometric Langlands correspondence and topological field theory - Part 1 Ben-Zvi, David (Auteur de la Conférence) | CIRM (Editeur )
Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel. Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain ...
## Post-edited Mathematical and numerical analysis of some fluid structure interaction problems - Lecture 1 Grandmont, Céline (Auteur de la Conférence) | CIRM (Editeur )
Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic structure. The purpose of the present lecture is to present an overview of some of the mathematical and numerical difficulties that may be encountered when dealing with fluidstructure interaction problems such as the geometrical nonlinearities or the added mass effect and how one can deal with these difficulties. Many physical phenomena deal with a fluid interacting with a moving rigid or deformable structure. These kinds of problems have a lot of important applications, for instance, in aeroelasticity, biomechanics, hydroelasticity, sedimentation, etc. From the analytical point of view as well as from the numerical point of view they have been studied extensively over the past years. We will mainly focus on viscous fluid interacting with an elastic ...
## Post-edited Darcy problem and crowd motion modeling Maury, Bertrand (Auteur de la Conférence) | CIRM (Editeur )
We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction pressures, and that are set on the contact network. At the macroscopic level, a similar problem is obtained, that is set on the congested zone.
We emphasize the differences between the two settings: at the macroscopic level, a straight use of the maximum principle shows that congestion actually favors evacuation, which is in contradiction with experimental evidence. On the contrary, in the microscopic setting, the very particular structure of the discrete differential operators makes it possible to reproduce observed "Stop and Go waves", and the so called "Faster is Slower" effect.
We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction ...
## Post-edited The geometrical gyro-kinetic approximation Frénod, Emmanuel (Auteur de la Conférence) | CIRM (Editeur )
At the end of the 70', Littlejohn [1, 2, 3] shed new light on what is called the Gyro-Kinetic Approximation. His approach incorporated high-level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain. This theory has been being widely used to deduce the numerical methods for Tokamak and Stellarator simulation. Yet, it was formal from the mathematical point of view and not directly accessible for mathematicians.
This talk will present a mathematically rigorous version of the theory. The way to set out this Gyro-Kinetic Approximation consists of the building of a change of coordinates that decouples the Hamiltonian dynamical system satisfied by the characteristics of charged particles submitted to a strong magnetic field into a part that concerns the fast oscillation induced by the magnetic field and a other part that describes a slower dynamics.
This building is made of two steps. The goal of the first one, so-called "Darboux Algorithm", is to give to the Poisson Matrix (associated to the Hamiltonian system) a form that would achieve the goal of decoupling if the Hamiltonian function does not depend on one given variable. Then the second change of variables (which is in fact a succession of several ones), so-called "Lie Algorithm", is to remove the given variable from the Hamiltonian function without changing the form of the Poisson Matrix.
(Notice that, beside this Geometrical Gyro-Kinetic Approximation Theory, an alternative approach, based on Asymptotic Analysis and Homogenization Methods was developed in Frenod and Sonnendrücker [5, 6, 7], Frenod, Raviart and Sonnendrücker [4], Golse and Saint-Raymond [9] and Ghendrih, Hauray and Nouri [8].)
At the end of the 70', Littlejohn [1, 2, 3] shed new light on what is called the Gyro-Kinetic Approximation. His approach incorporated high-level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain. This theory has been being widely used to deduce the numerical methods for Tokamak and Stellarator s...
## Post-edited High energy asymptotics of the scattering matrix for Schrödinger and Dirac operators Nakamura, Shu (Auteur de la Conférence) | CIRM (Editeur )
We consider short-range perturbations of elliptic operators on $R^d$ with constant coefficients, and study the asymptotic properties of the scattering matrix as the energy tends to infinity. We give the leading terms of the symbol of the scattering matrix. The proof employs semiclassical analysis combined with a generalization of the Isozaki-Kitada theory on time-independent modifiers. We also consider scattering matrices for 2 and 3 dimensional Dirac operators. (joint work with Alexander Pushnitski (King's College London) We consider short-range perturbations of elliptic operators on $R^d$ with constant coefficients, and study the asymptotic properties of the scattering matrix as the energy tends to infinity. We give the leading terms of the symbol of the scattering matrix. The proof employs semiclassical analysis combined with a generalization of the Isozaki-Kitada theory on time-independent modifiers. We also consider scattering matrices for 2 and 3 dimensional ...
## Post-edited Macroscopic fluctuation theory. Lecture 1: Particle systems, scaling limits and large deviations Gabrielli, Davide (Auteur de la Conférence) | CIRM (Editeur )
In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the understanding of the structure of the stationary non equilibrium states. I will discuss these problems in specific examples and from two different perspectives. The stochastic microscopic and combinatorial point of view and the macroscopic variational approach where the microscopic details of the models are encoded just by the transport coefficients. In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the ...
## Post-edited School on the mathematics of strings theory : introduction Kashani-Poor, Amir-Kian (Auteur de la Conférence) | CIRM (Editeur )
This school consists of an array of courses which at first glance may seem to have little in common. The underlying structure relating gauge theory to enumerative geometry to number theory is string theory. In this short introduction, we will attempt to give a schematic overview of how the various topics covered in this school fit into this overarching framework.
## Post-edited Emergent anyons in quantum Hall physics Rougerie, Nicolas (Auteur de la Conférence) | CIRM (Editeur )
Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.
I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.
Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.
I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I ...
## Post-edited Phase retrieval in infinite dimensions Daubechies, Ingrid (Auteur de la Conférence) | CIRM (Editeur )
Retrieving an arbitrary signal from the magnitudes of its inner products with the elements of a frame is not possible in infinite dimensions. Under certain conditions, signals can be retrieved satisfactorily however.
##### Codes MSC
Nuage de mots clefs ici
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2017-11-19 00:58:16
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https://too-meta.neocities.org/anki/analysis/1574989335311/back/
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Show Question
Math and science::Analysis::Tao::07. Series
Cauchy criterion, harmonic series and the Riemann-zeta function
Let $$(a_n)_{n=1}^{\infty}$$ be a decreasing sequence of non-negative real numbers. Then the series $$\sum_{n=1}^{\infty}a_n$$ is convergent if and only if the series
$\sum_{k=0}^{\infty} 2^ka_{2^k} = a_1 + 2a_2 + 4a_4 + 8a_8 + ...$
is convergent. This is the Cauchy criterion.
Harmonic series
The Cauchy criterion can be used to show that the Harmonic series, $$\sum_{n=1}^{\infty}\frac{1}{n}$$, is divergent. Yet $$\sum_{n=1}^{\infty}\frac{1}{n^2}$$ is convergent.
Riemann-zeta function
The quantity $$\sum_{n=1}^{\infty}\frac{1}{q^n}$$ is called the Riemann-zeta function of q and is denoted by $$\zeta(q)$$. This function is very important in number theory in particular for investigating the distribution of primes. There is a famous unsolved problem regarding this function called the Riemann hypothesis.
An interesting feature of this criterion is that it only uses a small number of elements of the sequence $$a_n$$ (namely, those elements whose index is a power of 2, $$n = 2_k \text{ for some integer k > 0}$$) in order to deterimen whether the whole series is convergent or not.
Intuition
There is a visual/geometric understanding that makes the mechanism of the proof clear. See the drawing below.
The Cauchy Criterion introduces the idea of bunching together segments of the sum to make a new sum, and by clever bunching we can see if the series is decreasing fast enough for the sum to converge.
I think it is correct to say that if a series diverges, then there will exist a bunching of the sum such that each bunch is greater than some constant. This idea can be seen a bit more clearly if you try and create a clever bunching for the sum $$\sum_{n=1}^{\infty}\frac{1}{n^2}$$—you will find that there is no bunching that will work to form a constant, instead the bunches will keep getting smaller.
Proof
The proof of this is not short. The book has a more consise approach than what I took. The first half of my approach is:
The choice of $$2^k$$ seems quite natural when tackling the harmonic series. Try to group the harmonic series elements as above such that they sum to 1. You end up with $$1\times \frac{1}{1}, 2\times \frac{1}{2}, 4\times \frac{1}{4}...$$. Already, this coves $$1+2+4 = 7$$ elements, so the next will be the 8th whose value is $$\frac{1}{8}$$ requiring a grouping of 8 elements, bringing us up to 15, and so on.
Divergence of the harmonic series
The proof of the following proposition is an example of the application of the Cauchy criterion.
Let $$q > 0$$ be a rational number. Then the series $$\sum_{n=1}^{\infty} \frac{1}{n^q}$$ is convergent when $$q > 1$$ and divergent when $$q \le 1$$.
A particular divergent case of this series is the harmonic series: $$\sum_{n=1}^{\infty} \frac{1}{n}$$. In comparison, $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ is convergent.
Proof
The sequence $$(\frac{1}{n^q})_{n=1}^{\infty}$$ is non-negative and decreasing (by Lemma 5.6.9 d), and so the Cauchy criterion applies. Thus, the sequence is convergent if and only if
$\sum_{k=0}^{\infty}2^k \frac{1}{(2^k)^q}$
is convergent. By the laws of exponentiation we can rewrite this as
$\sum_{k=0}^{\infty}(2^{1-q})^k$
This is a geometric series ($$\sum_{k=0}^{\infty}x^k$$) and converges if and only if $$|x| < 1$$. Thus our original $$\sum_{n=1}^{\infty} \frac{1}{n^q}$$ will converge if and only if $$|2^{1-q}| < 1$$, which happens if and only if $$q > 1$$ (why?).
p174
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2023-02-07 01:47:20
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https://projecteuclid.org/euclid.aoms/1177699376
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## The Annals of Mathematical Statistics
### The Sequential Compound Decision Problem with $m \times n$ Finite Loss Matrix
J. Van Ryzin
#### Abstract
Consideration of a sequence of statistical decision problems having identical generic structure constitutes a sequential compound decision problem. The risk of a sequential compound decision problem is defined as the average risk of the component problems. In the case where the component decisions are between two fully specified distributions $P_1$ and $P_2, P_1 \neq P_2$, Samuel (Theorem 2 of [9]) gives a sequential decision function whose risk is bounded from above by the risk of a best "simple" procedure based on knowing the proportion of component problems in which $P_2$ is the governing distribution plus a sequence of positive numbers converging to zero uniformly in the space of parameter-valued sequences as the number of problems increases. Related results are abstracted by Hannan in [2] for the sequential compound decision problem where the parameter space in the component problem is finite. The decision procedures in both instances rely on the technique of "artificial randomization," which was introduced and effectively used by Hannan in [1] for sequential games in which player I's space is finite. In the game situation such randomization is necessary. However, in the compound decision problem such "artificial randomization" is not necessary as is shown in this paper. Specifically, we consider the case where each component problem consists of making one of $n$ decisions based on an observation from one of $m$ distributions. Theorems 4.1, 4.2, and 4.3 give upper bounds for the difference in the risks (the regret function) of certain sequential compound decision procedures and a best "simple" procedure which is Bayes against the empirical distribution on the component problem parameter space. None of the sequential procedures presented depend on "artificial randomization." The upper bounds in these three theorems are all of order $N^{-\frac{1}{2}}$ and are uniform in the parameter-valued sequences. All procedures depend at stage $k$ on substitution of estimates of the $k - 1$st (or $k$th) stage empirical distribution $p_{k-1}$ (or $p_k$) on the component parameter space into a Bayes solution of the component problem with respect to $p_{k-1}$ (or $p_k$). Theorem 4.1 (except in the case where the estimates are degenerate) and Theorem 4.3 when specialized to the compound testing case between $P_1$ and $P_2$ (Theorems 5.1 and 5.2) yield a threefold improvement of Samuel's results mentioned above by simultaneously eliminating the "artificial randomization," by improving the convergence rate of the upper bound of the regret function to $N^{-\frac{1}{2}}$, and by widening the class of estimates. Higher order uniform bounds on the regret function in the sequential compound testing problem are also given. The bounds in Theorems 5.3 and 5.4 (or Theorems 5.5 and 5.6) are respectively of $O((\log N)N^{-1})$ and $o(N^{-\frac{1}{2}})$ and are attained by imposing suitable continuity assumptions on the induced distribution of a certain function of the likelihood ratio of $P_1$ and $P_2$. Theorem 6.1 extends Theorems 4.1, 4.2, and 4.3 to the related "empirical Bayes" problem. Also lower bounds of equivalent or better order are given for all theorems. The next section introduces notation and preliminaries to be used in this paper and in the following paper [15].
#### Article information
Source
Ann. Math. Statist., Volume 37, Number 4 (1966), 954-975.
Dates
First available in Project Euclid: 27 April 2007
https://projecteuclid.org/euclid.aoms/1177699376
Digital Object Identifier
doi:10.1214/aoms/1177699376
Mathematical Reviews number (MathSciNet)
MR198640
Zentralblatt MATH identifier
0173.46303
JSTOR
Ryzin, J. Van. The Sequential Compound Decision Problem with $m \times n$ Finite Loss Matrix. Ann. Math. Statist. 37 (1966), no. 4, 954--975. doi:10.1214/aoms/1177699376. https://projecteuclid.org/euclid.aoms/1177699376
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2019-07-23 18:07:46
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https://cpm.embedded.rwth-aachen.de/doc/pages/viewpage.action?pageId=2293815
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# Indoor Positioning System
The Indoor Positioning System (IPS) detects the position and orientation (pose) and identity of multiple vehicles simultaneously. The poses are updated at 50 Hz.
# Optics
The Indoor Positioning System Setup works with active light sources (LEDs) on the vehicles and a camera looking down from the ceiling. The LEDs are detected based on their high brightness. The camera is set to a very short exposure (~100 microseconds). Thus, the ambient light creates a very small signal (almost black), while the LEDs still appear as white dots. The short exposure time also eliminates the problem of motion blur. At a top speed of 4 m/s the vehicle travels 0.4 mm during the exposure.
Other light sources and reflective surfaces on the vehicle can create false signals and must be covered with tape. This includes the connectors on the Raspberry Pi and the status LEDs on the motor speed controller.
# Vehicle Pose
The outer three LEDs indicate the vehicle pose and are permanently illuminated. The section Pose Calibration describes how the LED positions are related to the reference pose.
# Vehicle Identification
The central LED flashes in a pattern that is different for each vehicle. The patterns are chosen such that sampling effects do not create ambiguous signals. See our paper Vision-Based Real-Time Indoor Positioning System for Multiple Vehicles for more information.
Vehicle ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Period of flashing, in number of frames 4 7 10 13 16 7 10 13 16 19 10 13 16 19 22 13 16 19 22 25 16 19 22 25 28 LED illumination duration, in number of frames 2 2 2 2 2 5 5 5 5 5 8 8 8 8 8 11 11 11 11 11 14 14 14 14 14
# Software Architecture
The IPS software has two major components, the LED detection and the IPS pipeline. The LED detection reads images from the camera at 50 Hz and extracts the list of image coordinates for all visible LEDs. These `LedPoints` are published via DDS. The raw images are not saved or published, as this would create impractically large data volumes. The IPS pipeline processes the LED points and publishes `VehicleObservation`s, which include the vehicle's position, orientation, identity and a timestamp.
# LED Detection
The LED detection uses the OpenCV functions `cv::threshold`, `cv::findContours` and `cv::moments` to find the centers of the LEDs.
# IPS Pipeline
The IPS pipeline performs a relatively complex data processing task. To make the software easier to understand, it is broken down into independent, stateless processing steps.
The UndistortPoints function transforms the image/pixel coordinates to metric coordinates on the driving surface (floor). The calculation is based on a calibration image. See https://github.com/embedded-software-laboratory/cpm-lab/tree/master/matlab_scripts/camera_calibration
The DetectVehicles function groups and classifies points as `front`, `center`, `back_left` or `back_right`. Points that appear to not belong to a vehicle are discarded.
The Queue collects the 50 most recent samples of the vehicle points. This is necessary, as the vehicle ID can not be determined from a single image.
The DetectVehicleID function extracts the ID for each vehicle from the last second (50 frames) of vehicle points.
The PoseCalculation function calculates the vehicle reference pose based on the `front`, `back_left` and `back_right` points. The calculation is based on manually collected calibration data. See https://github.com/embedded-software-laboratory/cpm-lab/tree/master/matlab_scripts/ips_pose_calibration
The implementation can be found at https://github.com/embedded-software-laboratory/cpm-lab/tree/master/ips2/src
# Calibration
## Camera Calibration
First, other PVC canvases are removed from the floor to reveal the checkerboard canvas. Then an image of the checkerboard pattern is taken using the `PylonViewerApp`. The exposure is adjusted such that the checkerboard corners are well resolved. If the exposure is too short or long, the corners are "rounded off", which degrades the quality of the calibration.
The Matlab function `detectCheckerboardPoints` is used to determine a list of matching points `(x_floor, y_floor, x_image, y_image)`. Then a 5th order 2D polynomial is fitted to the data using linear least squares.
## Pose Calibration
The vehicle pose has the format `(x, y, yaw)`. The yaw is the rotation angle in radians of the vehicle around the vertical axis in the counter-clockwise direction. When yaw=0, the vehicle points in the x-direction. The coordinates (x, y) give the location of the vehicle's reference point. The reference point is defined as the geometric center between the front and rear axle.
To accurately implement this definition of the pose, another measurement and calibration procedure is performed. First we need a means of accurately placing the vehicle on the floor, such that the true pose is known. The vehicle is clamped into a piece of wood, which extends the vehicle's local x/y coordinate system. This makes it simple to manually align the vehicle with the checkerboard pattern.
While the IPS is running, the vehicle is placed in various poses on the floor, following a particular calibration sequence. The calibration features `(back_x, back_y, direction_x, direction_y)` are recorded from the running IPS. A linear calibration is then fitted using least squares.
# Coordinate Systems
## IPS Image Coordinates
The image coordinates correspond to the IPS image sensor pixels. They are only relevant for the IPS LED detection.
## Floor Coordinates
$x \in [0, 4.5], y \in [0, 4]$
X/Y units: Meter
Origin: corner of the PVC canvas, towards the windows on the right.
Yaw angle: Measured from the x-axis, counterclockwise.
Useful yaw equations:
`direction_x = cos(yaw)direction_y = sin(yaw)yaw = atan2(direction_y, direction_x)`
## Vehicle Coordinates
X-axis: forwards
Y-axis: left
Origin: center point between both axles.
In practice, this coordinate system is realized with a calibration jig.
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2021-10-20 14:30:50
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https://www.quizover.com/online/course/0-3-modelling-corruption-software-receiver-design-by-openstax?page=9
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# 0.3 Modelling corruption (Page 10/11)
Page 10 / 11
Thus, the impulse response describes how a system behaves directly in time, while the frequency responsedescribes how it behaves in frequency. The two descriptions are intimately related because the frequency response is theFourier transform of the impulse response. This will be used repeatedly in [link] to design filters for the manipulation (augmentation or removal)of specified frequencies.
In Exercise [link] , a system was defined to have an impulse response that is a sinc function.The Fourier transform of a sinc function in time is a rect function in frequency [link] . Hence, the frequency response of the system is a rectangle that passes all frequencies below ${f}_{c}=1/T$ and removes all frequencies above (i.e., the system is a lowpass filter).
M atlab can help to visualize the relationship between the impulse response and the frequency response.For instance, the system in convolex.m is defined via its impulse response, which is a decaying exponential. [link] shows its output when the input is a simple sum of delta functions, andExercise [link] explores the output when the input is a white noise. In freqresp.m , the behavior of this system is explained by looking at its frequency response.
Ts=1/100; time=10; % sampling interval and total time t=0:Ts:time; % create time vectorh=exp(-t); % define impulse response plotspec(h,Ts) % find and plot frequency response freqresp.m numerical example of impulse and frequency response (download file)
The output of freqresp.m is shown in [link] . The frequency response of the system (which is just the magnitude spectrum ofthe impulse response) is found using plotspec.m . In this case, the frequency response amplifies low frequenciesand attenuates other frequencies more as the frequency increases. This explains, for instance, why the output of theconvolution in Exercise [link] contained (primarily) lower frequencies, as evidenced by the slower undulations in time.
Suppose a system has an impulse response that is a $\text{sinc}$ function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Hint: center the $\text{sinc}$ in time; for instance, use h=sinc(10*(t-time/2)) .
Suppose a system has an impulse response that is a $sin$ function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Can you predict the relationship between the frequency of the sine wave and the location of the peaks in the spectrum?Hint: try h=sin(25*t) .
Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a $\text{sinc}$ function (as in Exercise [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise [link] .
Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a $\text{sin}$ function (as in Exercise [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise [link] .
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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2018-10-20 10:32:07
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-8-polynomials-and-factoring-8-6-factoring-ax-squared-bx-c-lesson-check-page-508/6
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## Algebra 1
When factoring you have to find 2 numbers that multiply to $ac$ and have a sum of $b$ In this case $ac = 24$ so the product of $p$ and $q$ would have to be = 24
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2018-10-16 23:20:06
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https://economics.stackexchange.com/questions/25399/in-perfect-competition-why-is-there-economic-loss-if-marginal-cost-marginal-r
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# In perfect competition, why is there economic loss if marginal cost > marginal revenue?
Here's a graph for reference:
In the left graph, I read from a book (CFA L1 notes) that
At any output above the quantity where $$MR = MC$$, the firm will be generating losses on its marginal production and will maximize profits by reducing output to where $$MR = MC$$."
But now consider this:
$$Q'$$ is an output level above the one at which $$MC=MR$$. Shouldn't the yellow region represent a positive economic profit in this case? I can understand that at any output level above the one at which $$ATC=MR$$, the firm would start making losses. But isn't the quote above incorrect, or am I missing something? Sure, you might be making a suboptimal profit at output $$= Q’$$, but you’re certainly not making an outright loss.
In perfect competition, $$MR=P$$.
When $$MC>P$$, marginally you're generating losses but you may still be profitable overall. This is represented by your (positive) yellow area, which is smaller than the blue area (the optimal one).
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2019-09-16 20:15:12
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https://math.stackexchange.com/questions/1653494/doesnt-this-lp-norm-estimate-for-all-p-give-me-an-l-infty-bound
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# Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?
Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and $n$.
Can I not just take $n \to \infty$ in this inequality to conclude that $v \in L^\infty(\Omega)$? The reason I ask is, that the author of this paper (see page 300, where my $v$ is his $u^+$) proves after proving the above quoted statement that $v \in L^\infty(\Omega)$ by a proof by contradiction.
• prove by contradiction that you can :-) – reuns Feb 13 '16 at 18:23
• – Svetoslav Feb 13 '16 at 18:36
You totally can, since $C$ is independent of $n$. Sometimes not doing so is illustratively useful, but you can.
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2020-01-22 11:38:05
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https://apalache.informal.systems/docs/adr/011adr-smt-arrays.html
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# ADR-011: alternative SMT encoding using arrays
authorrevision
Rodrigo Otoni1.6
This ADR describes an alternative encoding of the KerA+ fragment of TLA+ into SMT. Compound data structures, e.g. sets, are currently encoded using the core theory of SMT, with the goal being to encode them using arrays with extensionality instead. The hypothesis is that this will lead to increased solver performance and more compact SMT instances. We target the Z3 solver and will use the SMT-LIB Standard (Version 2.6) in conjunction with Z3-specific operators, e.g. constant arrays.
For an overview of the current encoding check the TLA+ Model Checking Made Symbolic paper, presented at OOPSLA'19. In the remainder of the document the use of the new encoding and the treatment of different TLA+ operators are described.
## 1. CLI encoding option
The encoding using arrays is to be an alternative, not a replacement, to the already existing encoding. Given this, a new option is to be added to the check command of the CLI. The default encoding will be the existing one. The option description is shown below. The envvar SMT_ENCODING can also be used to set the encoding, see the model checking parameters for details.
--smt-encoding : the SMT encoding: oopsla19, arrays (experimental), default: oopsla19 (overrides envvar SMT_ENCODING)
### Code changes
The following changes will be made to implement the new CLI option:
• Add new string variable to class CheckCmd to enable the new option.
• Add new smtEncoding field to SolverConfig.
• Add new class SymbStateRewriterImplWithArrays, which extends class SymbStateRewriterImpl.
• Use the new option to set the SolverConfig encoding field and select between different SymbStateRewriter implementations in classes BoundedCheckerPassImpl and SymbStateRewriterAuto.
## 2. Testing the new encoding
The new encoding should provide the same results as the existing one, the available test suite will thus be used to test the new encoding. To achieve this, the unit tests needs to be made parametric w.r.t. the SolverConfig encoding field and the implementations of SymbStateRewriter, and the integration tests need to be tagged to run with the new encoding.
### Code changes
The following changes will be made to implement the tests for the new encoding:
• Refactor the classes in tla-bmcmt/src/test to enable unit testing with different configurations of SolverConfig and implementations of SymbStateRewriter.
• Add unit tests for the new encoding, which should be similar to existing tests, but use a different solver configuration and SymbStateRewriterImplWithArrays instead of SymbStateRewriterImpl.
• Add integration tests for the new encoding by tagging existing tests with array-encoding, which will be run by the CI with envvar SMT_ENCODING set to arrays.
## 3. Encoding sets
Sets are currently encoded in an indirect way. Consider a sort some_sort and distinct elements elem1, elem2, and elem3 of type someSort, as defined below.
(declare-sort some_sort 0)
(declare-const elem1 some_sort)
(declare-const elem2 some_sort)
(declare-const elem3 some_sort)
(assert (distinct elem1 elem2 elem3))
A set set1 containing elem1, elem2, and elem3 is currently represented by a constant of type set_of_some_Sort and three membership predicates, as shown below.
(declare-sort set_of_some_Sort 0)
(declare-const set1 set_of_some_Sort)
(declare-const elem1_in_set1 Bool)
(declare-const elem2_in_set1 Bool)
(declare-const elem3_in_set1 Bool)
(assert elem1_in_set1)
(assert elem3_in_set1)
(assert elem2_in_set1)
The new encoding has each set encoded directly as an array whose domain and range equal the set's sort and the Boolean sort, respectively. SMT arrays can be thought of as a functions, as this is exactly how they are represented internally in Z3. Set membership of an element elem is thus attained by simply setting the array at index elem to true.
One important point in the new encoding is the handling of set declarations, since declaring an empty set requires the setting of all array indexes to false. This can be easily achieved for finite sets by explicitly setting each index, but falls outside the quantifier-free fragment of first-order logic in the case of infinite sets, e.g. the set of integers. To handle declarations of infinite sets we rely on Z3's constant arrays, which map all indexes to a fixed value. Below is an example using the new encoding.
(declare-const set2_0 (Array some_sort Bool))
(declare-const set2_1 (Array some_sort Bool))
(declare-const set2_2 (Array some_sort Bool))
(declare-const set2_3 (Array some_sort Bool))
(assert (= set2_0 ((as const (Array some_sort Bool)) false)))
(assert (= set2_1 (store set2_0 elem1 true)))
(assert (= set2_2 (store set2_1 elem2 true)))
(assert (= set2_3 (store set2_2 elem3 true)))
The store operator handles array updates and receives the array to be updated, the index, and the new value, returning the updated array. For array access, the select operator can be used, which receives an array and an index and returns the value at the given index, as shown below.
(assert (= (select set2_2 elem1) true)) ; SAT
(assert (= (select set2_2 elem2) true)) ; SAT
(assert (= (select set2_2 elem3) true)) ; UNSAT
(assert (= (select set2_3 elem1) true)) ; SAT
(assert (= (select set2_3 elem2) true)) ; SAT
(assert (= (select set2_3 elem3) true)) ; SAT
For consistency, the new encoding uses constant arrays to declare both finite and infinite arrays.
### Code changes
The following changes will be made to implement the new encoding of sets:
• Add alternative rewriting rules for sets when appropriate, by extending the existing rules.
• All alternative rules will be suffixed with WithArrays.
• The new rules will not rely on LazyEquality and will aim to use SMT equality directly.
• Only the generation of SMT constraints will be modified by the new rules, the other Arena elements will remain unchanged.
• In class SymbStateRewriterImplWithArrays, add the new rules to ruleLookupTable by overriding the entries to their older versions.
• Add four new Apalache IR operators in ApalacheOper, Builder, ConstSimplifierForSmt, and PreproSolverContext, to represent the array operations.
• The selectInSet IR operator represents the SMT select.
• The storeInSet IR operator represents the SMT store.
• The unchangedSet IR operator represents an equality between the current and new SSA array representations. This is required to constraint the array representation as it evolves. It is important to note that this operator assumes that all arrays are initially empty, so an element not explicitly added is assumed to not be in the array. To check absence of an element, selectInSet should be used with negation.
• The smtMap IR operator represents the use of SMT map.
• In class Z3SolverContext, add/change appropriate methods to handle SMT constraints over arrays.
• The main changes will de done in declareCell and the new mkSelect, mkStore, and mkUnchangedSet methods, as these methods are directly responsible for creating the SMT constraints representing sets and set membership.
• With the new IR operators, the "in-relation" concept, which underpins declareInPredIfNeeded and getInPred, will not be applied to the new encoding. Cases for the new IR operators will be added to toExpr, which will default to TlaSetOper.in and TlaSetOper.notin for the existing encoding.
• The smtMap IR operator will be used to encode the TLA+ set filter operation. It constructs a temporary array that contains the evaluation of the filter's predicate for each set element and uses SMT map to compute the intersection of the set being filtered and the set represented by the temporary array constructed.
• Cases for FinSetT and PowSetT will be added to getOrMkCellSort, as these types are no longer represented by uninterpreted constants.
• cellCache will be changed to contain a list of cells, in order to handle the effects of push and pop in the SSA assignment of sets. The following examples illustrates this need.
(assert (= set_0 ((as const (Array Int Bool)) false)))
(assert (= set_1 (store set_0 5 true)))
(push)
(assert (= set_2 (store set_1 6 true)))
(push)
(assert (= set_3 (store set_2 7 true)))
(assert (= (select set_3 7) true))
(pop 2)
(assert (= (select set_1 7) false)) ; Without the list we would query set_3 here
## 4. Encoding functions and sequences
Functions are currently encoded as sets of pairs, with each pair representing a mapping present in the function. The first element of a pair is a tuple containing some function arguments and the second element is the return value given by such arguments. The handling of functions is thus given by operations over sets and tuples. Sequences of type T are currently encoded as tuples of form ⟨start,end,fun⟩, where start and end are integers and fun is a function from integers to T. The new encoding of functions will thus encompass sequences, as their tuple representations is intended to be kept.
The new encoding will, like the current one, also map tuples of arguments to return values, but will do so natively instead of simply relying on sets. A function will be represented by two SMT arrays. The first array will store the domain of the function and will be encoded as a standard TLA+ set. The second array will store the mappings, having sort <S1,...,Sn> as its domain, with Si being the sort of argument i, and the sort of the function's codomain as its range. The sorts of the array domain and range can be infinite, but the domain of the function itself, and by implication the number of mappings tuples, will always be finite.
To encode the TLA+ function finSucc = [x \in {1,2,3} |-> x + 1 ], which computes the successors of integers from 1 to 3, we first have to declare its domain, as shown below; tuples are represented here as per the OOPSLA'19 encoding.
(declare-sort Tuple_Int 0) ; Sort of <Int>
(declare-const tuple_with_1 Tuple_Int) ; <1>
(declare-const tuple_with_2 Tuple_Int) ; <2>
(declare-const tuple_with_3 Tuple_Int) ; <3>
(declare-const finSucc_domain_0 (Array Tuple_Int Bool))
(declare-const finSucc_domain_1 (Array Tuple_Int Bool))
(declare-const finSucc_domain_2 (Array Tuple_Int Bool))
(declare-const finSucc_domain_3 (Array Tuple_Int Bool))
(assert (= finSucc_domain_0 ((as const (Array Tuple_Int Bool)) false))) ; {}
(assert (= finSucc_domain_1 (store finSucc_domain_0 tuple_with_1 true))) ; {<1>}
(assert (= finSucc_domain_2 (store finSucc_domain_1 tuple_with_2 true))) ; {<1>,<2>}
(assert (= finSucc_domain_3 (store finSucc_domain_2 tuple_with_3 true))) ; {<1>,<2>,<3>}
The array storing the function's domain is used to guard the definition of the array storing the function's mappings, since mappings should only be present for values in the domain. The array storing the mappings of finSucc is shown below.
(declare-const finSucc_0 (Array Tuple_Int Int))
(declare-const finSucc_1 (Array Tuple_Int Int))
(declare-const finSucc_2 (Array Tuple_Int Int))
(declare-const finSucc_3 (Array Tuple_Int Int))
(assert (ite (select finSucc_domain_3 tuple_with_1)
(= finSucc_1 (store finSucc_0 tuple_with_1 2))
(= finSucc_1 finSucc_0)))
(assert (ite (select finSucc_domain_3 tuple_with_2)
(= finSucc_2 (store finSucc_1 tuple_with_2 3))
(= finSucc_2 finSucc_1)))
(assert (ite (select finSucc_domain_3 tuple_with_3)
(= finSucc_3 (store finSucc_2 tuple_with_3 4))
(= finSucc_3 finSucc_2)))
Note that, unlike with the new encoding for sets, we do not use constant arrays. The reason is that the function's domain cannot be altered, so the array has to constrain only the values in said domain. Function application can be done by simply accessing the array at the index of the passed arguments. A function application with arguments outside the function's domain leads to an unspecified result in TLA+, which is perfectly captured by unconstrained entries in the SMT array. Below are some examples of function application.
(assert (= (select finSucc_3 tuple_with_1) 2)) ; SAT
(assert (= (select finSucc_3 tuple_with_2) 3)) ; SAT
(assert (= (select finSucc_3 tuple_with_3) 4)) ; SAT
(declare-const tuple_with_4 Tuple_Int) ; <4>
(assert (= (select finSucc_3 tuple_with_4) 16)) ; SAT
Although a function's domain cannot be altered, its image can be changed via the TLA+ function update operator. The update will be encoded as a guarded array update, as illustrated below; attempting to update an entry outside the function's domain will lead to no change happening.
(declare-const finSucc_4 (Array Tuple_Int Int))
(declare-const finSucc_5 (Array Tuple_Int Int))
(assert (ite (select finSucc_domain_3 tuple_with_1) ; [finSucc EXCEPT ![1] = 9]
(= finSucc_4 (store finSucc_3 tuple_with_1 9))
(= finSucc_4 finSucc_3)))
(assert (ite (select finSucc_domain_3 tuple_with_4) ; [finSucc EXCEPT ![4] = 25]
(= finSucc_5 (store finSucc_4 tuple_with_4 25))
(= finSucc_5 finSucc_4)))
(assert (= (select finSucc_5 tuple_with_1) 2)) ; UNSAT
(assert (= (select finSucc_5 tuple_with_1) 9)) ; SAT
(assert (= (select finSucc_5 tuple_with_4) 16)) ; SAT
In contrast to the current encoding, which produces a number of constraints that is linear in the size of the set approximating the function when encoding both function application and update, the new encoding will produce a single constraint for each operation. This will potentially lead to a significant increase in solving performance.
### Code changes
The following changes will be made to implement the new encoding of functions:
• Add alternative rewriting rules for functions when appropriate, by extending the existing rules. The same caveats stated for the rewriting rules for sets will apply here.
• The sets of pairs used in the current encoding are the basis for the counter-example generation in SymbStateDecoder. In order to continue having counter-examples, these sets will keep being produced, but will not be present in the SMT constraints. They will be carried only as metadata in the Arena.
• Update class SymbStateRewriterImplWithArrays with the rules for functions.
• Update the storeInSet IR operator to also store function updates. It will have the value resulting from the update as an optional argument.
• Since functions will be encoded as SMT arrays, the selectInSet, storeInSet, and unchangedSet IR operators will be used when handling them. A future refactoring may rename these operators.
• Update class Z3SolverContext to handle the new SMT constraints over arrays.
• A case for FunT will be added to getOrMkCellSort.
• In declareCell, functions will be declared as arrays, but will be left unconstrained.
• The mkStore method will be updated to also handle functions. It will have an additional optional argument containing the value to be stored in the range of the array. The new argument's default value is true, for the handling of sets.
• The mkNestedSelect method is added to support set membership in function sets, i.e., f \in [S -> T]. The nesting has first funAppRes = f[s \in S], followed by funAppRes \in T.
TODO
TODO
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2022-05-29 07:33:06
|
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|
https://tex.stackexchange.com/questions/202853/moderncv-class-prevent-indentation-in-address-section-after-making-it-left-just
|
# moderncv class: prevent indentation in address section after making it left-justified
I am using the moderncv class (http://www.ctan.org/pkg/moderncv) to write my CV. In order to modify the address section, I modified moderncvstyleclassic.sty.
I want to get the text of the address section left-justified (which worked), but now I have the problem that the first line of the address (addressstreet) is indented.
My question is: How do I prevent this indentation of the first line within this information box
So far I did not find any solutions for this problem.
Code in moderncvstyleclassic.sty with my change commented:
%% code above omitted
% optional detailed information box
\newbox{\makecvtitledetailsbox}%
\savebox{\makecvtitledetailsbox}{%
\begin{tabular}[b]{@{}l@{}}%% MY MODIFICATION <<-- l for left-justified (instead of "r")
\ifthenelse{\isundefined{\@countryinfo}}{}{\makenewline\@countryinfo}%%##moved up to use as country
\ifthenelse{\isundefined{\@mobile}}{}{\makenewline\mobilesymbol\@mobile}%
\ifthenelse{\isundefined{\@phone}}{}{\makenewline\phonesymbol\@phone}%
\ifthenelse{\isundefined{\@fax}}{}{\makenewline\faxsymbol\@fax}%
\ifthenelse{\isundefined{\@homepage}}{}{%
\ifthenelse{\equal{\@homepagetitle}{}}% \homepagetitle could be empty
\ifthenelse{\isundefined{\@extrainfo}}{}{\makenewline\@extrainfo}%##moved up to use as country
\end{tabular}
}%
%% code below omitted
Using the \patchcmd macro:
\makeatletter
\patchcmd{\makecvtitle}{@{}r@{}}{@{}l@{}}{}{}
\makeatother
after the style declaration.
Before:
After:
Or if you want to manually edit the .sty file (discouraged) just change from:
\makenewline\addresssymbol\@addressstreet
to:
\addresssymbol\@addressstreet\makenewline
For also putting the extrainfo below city use the following set of commands instead of the previous:
\makeatletter
\patchcmd{\makecvtitle}{@{}r@{}}{@{}l@{}}{}{}
\patchcmd{\makecvtitle}{\makenewline\@extrainfo}{}{}{}
\makeatother
result:
the country field is optional, if you set the address like this:
\address{street and number}{postcode city}{country}
you'll have a country. Otherwise, like this:
\address{street and number}{postcode city}
you'll have no country (as in my last picture).
## Complete MWE
\documentclass[11pt,a4paper,sans]{moderncv}
\moderncvstyle{classic}
\makeatletter
\patchcmd{\makecvtitle}{@{}r@{}}{@{}l@{}}{}{}
\patchcmd{\makecvtitle}{\makenewline\@extrainfo}{}{}{}
\makeatother
\moderncvcolor{blue}
\usepackage[scale=0.75]{geometry}
% personal data
\name{John}{Doe}
\title{Resumé title}
\phone[mobile]{+1~(234)~567~890}
\phone[fixed]{+2~(345)~678~901}
\phone[fax]{+3~(456)~789~012}
\email{john@doe.org}
\homepage{www.johndoe.com}
\social[github]{jdoe}
\photo[64pt][0.4pt]{picture}
\quote{Some quote}
\begin{document}
\makecvtitle
\section{Education}
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2020-09-28 19:48:04
|
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https://mathematica.stackexchange.com/questions/124083/highlighting-repeated-matches-with-stringcases
|
# Highlighting repeated matches with StringCases
This is a two-part question:
First, given
txt = "the quick brown fox jumps over the lazy dog"
Why does this repeated pattern match both instances:
txt // StringCases[(t : "the") .. :> t]
{"the", "the"}
Yet this matches only the 2nd instance?
txt // StringCases[(pre___ ~~ t : "the" ~~ post___) .. :>
Row[{pre, Style[t, Red], post}]]
{"the quick brown fox jumps over (the) lazy dog"}
(* note output above uses: pre <> "(" <> t <> ")" <> post *)
Second, provided the first part is solved, what's the best way to output a single string with all instances highlighted, rather than a list of matches?
• I once asked something similar. – wxffles Aug 16 '16 at 21:22
String patterns will always match the longest string possible, since pre___ and post___ match anything the second pattern will swallow the whole string. It will not look for further matches inside already matched parts of the string. This is related to the Overlaps option, but there is no way to get what we want with Overlaps and this pattern.
With Shortest we can modify this behavior, perhaps not to achieve what we want though:
StringCases[
txt,
Shortest[pre___ ~~ t : "the" ~~ post___ ..] :> Row[{pre, Style[t, Red], post}]
]
The best way to achieve highlighting is using StringReplace with StyleBox:
StringReplace[
txt,
"the" -> "\*StyleBox[\"the\", FontColor->RGBColor[1, 0, 0]]"
]
Example highlighting several different words:
StringReplace[
txt,
(t : "the" | "jumps") :> "\*StyleBox[\"" <> t <> "\", FontColor->RGBColor[1, 0, 0]]"
]
• Although I didn't mention it in the Q, I need to be able to match Alternatives, eg ("the" | "jumps") - hence was gravitating toward StringCases. Can you make your without writing a helper function? – alancalvitti Aug 16 '16 at 21:24
• @alancalvitti Yes, see my update. – C. E. Aug 16 '16 at 21:26
• I had this as a separate answer but it's just like yours: Row@Apply[List]@ StringReplace[x : "the" | "jumps" :> Highlighted[x]]@txt, producing this – Jason B. Aug 16 '16 at 21:37
• @JasonB Had never seen Highlighted before, so thanks for posting :) – C. E. Aug 16 '16 at 21:40
This is just like C.E.'s answer, using StringReplace, but with a different highlighting method
highlightText[words_List] := ReplaceAll[
StringReplace[x : (Alternatives @@ words) :> Highlighted[x]]@#,
StringExpression[a__] :> Row[{a}]] &
This will work on strings with and without the keywords but, unlike C.E.'s answer, when the keywords are present your result is not a String, but a Row. For display in a notebook, the difference is minimal, but it is worth taking note of.
highlightText[{"the", "jumps"}]@"the quick brown fox jumps over the lazy dog"
highlightText[{"the", "jumps"}]@"That lazy dog went to sleep"
• Jason, very nice, ready to accept. However, if there's no match, then the output is Row[string]. Can you fix? – alancalvitti Sep 17 '16 at 23:24
• @alancalvitti Ja, but it might be until tomorrow before I can open MMA – Jason B. Sep 18 '16 at 0:47
|
2019-11-18 00:40:17
|
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|
https://chemistry.stackexchange.com/questions/69452/measuring-or-calculating-the-ph-of-4m-naoh-i-prepared-the-solution-in-milli-q-w
|
# Measuring or calculating the pH of 4M NaOH? I prepared the solution in Milli Q water
I prepared $4\ \mathrm{M}$ solution of $\ce{NaOH}$, and the glass electrode I use cannot measure it the $\mathrm{pH}$ correctly. What are the other options do I have to exactly know the solution's $\mathrm{pH}$?
• This link explains the problem. all-about-ph.com/acid-and-alkaline.html The following link is for a high pH electrode but I couldn't find any specifications for it. :-( awedirect.co.uk/ph/ph-electrode/… – MaxW Feb 28 '17 at 17:39
• Do you really need to know pH or just that solution is 4.00 molar NaOH? Can you assume that the solution is "pure" NaOH? No NaCl, no KOH. I'm wondering about a sodium electrode, or conductivity. Carbonate from the air will be a problem. – MaxW Feb 28 '17 at 17:58
• Did you use tap water or distilled water? – Bob Mar 1 '17 at 0:45
• thanks ringo, I had calculated the same way, and I had made it in Miili q water, pH 5.7, and using AR grade NaOH. – angie Mar 3 '17 at 17:20
For $\mathrm{pH}>12$ and $\ce{Li+}$ or $\ce{Na+}$ concentrations greater than $0.1\ \mathrm{M}$, glass electrodes experience alkali error. Because the concentration of $\ce{H3O+}$ ions in solution is so low, interference from alkali metal ions (like $\ce{Na+}$) becomes noticeable and causes the electrode to read lower than the true $\mathrm{pH}$ of the solution. $\ce{K+}$ typically cause less error than $\ce{Li+}$ or $\ce{Na+}$ due do its larger size.
Since sodium hydroxide is a strong base and dissociates completely into $\ce{Na+}$ and $\ce{OH-}$ ions, though, it's actually quite simple to just calculate the $\mathrm{pH}$.
$$K_\mathrm{w}=\ce{[H3O+][OH-]} \Rightarrow$$ $$1 \times 10^{-14} \ \mathrm{M^2}=\ce{[H3O+] \cdot 4\ \mathrm{M}} \Rightarrow$$ $$\ce{[H3O+]}=2.5 \times 10^{-15}\ \mathrm{M}$$ $$\mathrm{pH}=–\log(2.5 \times 10^{-15}\ \mathrm{M})=14.60$$
If you are worried about whether or not you truly have a $4\ \mathrm{M}$ solution of $\ce{NaOH}$, you should try titrating it with a known concentration of acid (in the past I always used potassium hydrogen phthalate). This process is called standardization in acid-base parlance.
|
2020-07-12 17:30:03
|
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https://web.unican.es/portal-investigador/publicaciones/detalle-publicacion?p=ART4205
|
# Researchers portal
Buscar
Estamos realizando la búsqueda. Por favor, espere...
## Cross section measurement of t-channel single top quark production in pp collisions at s=13TeV
Abstract: The cross section for the production of single top quarks in the t channel is measured in proton?proton collisions at 13TeV with the CMS detector at the LHC. The analyzed data correspond to an integrated luminosity of 2.2fb?1. The event selection requires one muon and two jets where one of the jets is identified as originating from a bottom quark. Several kinematic variables are then combined into a multivariate discriminator to distinguish signal from background events. A fit to the distribution of the discriminating variable yields a total cross section of 238±13(stat)±29(syst) pb and a ratio of top quark and top antiquark production of Rt-ch.=1.81±0.18(stat)±0.15(syst). From the total cross section the absolute value of the CKM matrix element Vtb is calculated to be 1.05±0.07(exp)±0.02(theo). All results are in agreement with the standard model predictions.
Otras publicaciones de la misma revista o congreso con autores/as de la Universidad de Cantabria
Fuente: Physics Letters B 772 (2017) 752-776
Editorial: Elsevier
Fecha de publicación: 01/09/2017
Nº de páginas: 25
Tipo de publicación: Artículo de Revista
ISSN: 0370-2693,1873-2445
Url de la publicación: https://doi.org/10.1016/j.physletb.2017.07.047
### Autores/as
SIRUNYAN, A. M.
JUAN RAMON CASTIÑEIRAS DE SAA
MARCOS FERNANDEZ GARCIA
JUAN GARCIA FERRERO
MARIA AMPARO LOPEZ VIRTO
|
2023-03-22 20:21:50
|
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https://rationalwiki.org/wiki/RationalWiki:Saloon_bar/Archive70
|
# RationalWiki:Saloon bar/Archive70
This is an archive page, last updated 4 January 2012. Please do not make edits to this page.
## Does Andy ever visit/read this website?
I always wondered if Andy ever visited Rationalwiki.........?Waronstupidity (talk) 00:07, 28 July 2010 (UTC)
Some of his sysops definitely do, but Andy reading RW would depend on whether or not he's a total liar or as deluded as he appears to be. Concernedresident omg!!! ponies!!! 00:13, 28 July 2010 (UTC)
Eh, his responses to what's here would probably just be like what's already in the assquote template. --GastonRabbit (talk) 00:50, 28 July 2010 (UTC)
I doubt that he reads RW. The man doesn't have much interest in reading, let alone consider or think about, anything that doesn't promote conservatism. Hell, he sometimes doesn't even read what does promote his views. Have you seen him summarize some of the news articles he links to on the mainpage of Conservapedia? ~SuperHamster Talk 01:10, 28 July 2010 (UTC)
"Waronstupidity" (specialty not in breast cancer), you're embarking on a fool's errand. I looked at many of your edits before reverting them and found propaganda supporting tax-payer funded abortion and higher taxes, so admit it, you're committed to censoring the Bible. Denying this calls into question all of your credibility on the fact that homeschooling prevents genocide. Got news for you: it's a myth that were atheists in ancient times. There may not even be true atheists today. Help yourself by opening your mind, before it's too late. The truth shall set you free.--Aschlafly 23:58, 22 May 2022 (UTC)
Human is Andy. The whole Conservapedia-RationalWiki dynamic is a figment of his and your imaginations. The rest of us are not real. Sterile caterer 02:07, 28 July 2010 (UTC)
I certainly believe that Andy has visited RW as he has made requests to have certain items removed, so he must have checked beforehand, but I doubt that he is a regular reader in the way that TK, Rob, Ken & Karajou are. Lily Inspirate me. 08:29, 28 July 2010 (UTC)
Andy who? P-Foster (talk) 02:38, 28 July 2010 (UTC)
You have to have at least a little knowledge for hunger to seek out the opinions of your detractors. why would Andy have any knowledge hunger, he already knows everything right? --Opcn (talk) 09:56, 28 July 2010 (UTC)
Hunger for knowledge, surely? But I suppose that's a fair assessment. If you think that "I believe that..." is synonymous with "it's clearly true that..." you're not going to change your mind. And not ever wanting to change your mind, and thus avoiding painful cognitive dissonance, means avoiding looking for evidence that could prove you wrong - overcompensating will then mean avoiding looking at evidence at all, which we observe when he posts likes to news articles that clearly don't support his interpretation, as if he's just linking to them as a superficial attempt at what we might call "cargo cult citation". Of course, there'll be a different post facto explanation for this in his head, specifically in this case "they're scum", "they're irrelevant", "they're hateful" - and these tinted mental glasses are going to apply even if he was to read this site, it's certainly true of the ones we know read it and have observed reacting to it - and this makes a bit of a protective filter from cognitive dissonance, letting through the few things that support you while turning those things that you don't agree with into "lies" or "misrepresentations", or perhaps going as far as turning them on their head as proof some some sort of conspiracy (I call it a dissonance filter, I'm not sure if there's a proper term for it). If you're under the impression that it's all lies then even the blatantly factual detractions, say, LArron pointing out issues with his "exponential growth" thing, becomes lies; they're no longer true because the source is not trustworthy, and therefore "I'm still right regardless" (does the same apply at the more liberal RationalWiki? Yes and no, but that's a different matter). It's possible that Andy's extreme conservative rhetoric about hateful liberals (everything falling just short of seriously accusing them of baby eating), might be this dissonance filter triggering, protecting him from dissenting opinions in order to fully preserve his own. d hominem 13:07, 28 July 2010 (UTC)
## Jon Stewart
said Loya Jirga! ħuman 03:25, 28 July 2010 (UTC)
Dare I hypothesize that perchance he was not referencing ours? DickTurpis (talk) 03:30, 28 July 2010 (UTC)
Surely he meant ours!!! I will be on show next week with Toast and Ace! ħuman 05:15, 28 July 2010 (UTC)
I'd be more convinced if he said Lovely Jerboas. Lily Inspirate me. 09:41, 28 July 2010 (UTC)
Stewart asked but I am a busy man. AceX-102 10:34, 28 July 2010 (UTC)
Ooooh... now that would be interesting... d hominem 11:49, 28 July 2010 (UTC)
what, seriously? what will you be talking about -- my name is slugboy 11:57, 28 July 2010 (UTC)
The merits of aural gin intake. MARCVS ANTONIVS 13:01, 28 July 2010 (UTC)
## Recent incidents which have made you think of RW
Mine: Here, or perhaps here, about 30 seconds in. Megaten (talk) 11:14, 28 July 2010 (UTC)
## Mental health and other things
Please forgive the stream of consciousness writing...
Apparently a range of "new" disorders could mean we're all a bit mentally unstable. But while that interests me, the related article on why patients want to be pipolar is even better. I know a few people who are genuinely bipolar and a know a few people who claim to be pipolar (similarly, I know people who are autistic and people who claim to be autistic). On the one hand, I agree with the conclusions in the first article about people liking to have a name for their symptoms - I have first/second hand experience of this - because it focuses the mind on something to cure. You can't fight it if it doesn't have a name, you can't rally a battlecry against it, basically - it's part of the therapy. However, I think there's also a buck-passing aspect; you see it with many addictions, such as sex addiction and internet addiction, that have been shown to be non-clinical (at least in 95%+ of cases). You don't want to blame yourself, you want to blame your disease, it's not your fault and you don't like believing it is - again, part of the therapy, I suppose.
This brings me back to people wanting to have a mental illness and claiming that they do so. My rule-of-thumb for the difference between people claiming to have depression/autism/pipolar/etc and people who actually have those is that the genuine sufferers tend to keep quiet while the ones who claim (I'm not going to say that they're "faking" but I'm going to say that they very clearly exaggerate their symptoms, and apply a "disorder" status to what are perfectly normative personality traits) happily shout it from the rooftops. I think a lack of stigmatisation caused by people like Stephen Fry "coming out" isn't enough to make people want a disorder; it's a trend, and a compensation for a perceived lack of personality. The people who claim to have disorders, in my experience, tend to be alternative kids who are a bit insecure - although I agree you could reverse the cause and effect here, I'm not totally convinced that's the primary factor. These people seem to think that their disorder makes them more interesting; they bring it up in conversation about themselves, it's like "oh shit, I'm not interesting enough for people to accept me, let's tell them I'm autistic!!" (maybe this is autism speaking, I don't know). There's one girl I know who has perfectly normative mood swings, is usually fine but can be a bit bitchy - nothing out of the ordinary, she gets a bit drunk, she shouts and makes a scene, hands up who hasn't done that at some point and secretly enjoyed the attention. But instead of calming down and just either saying sorry or "well, I was a bit of a bitch there" it's just "it's my bipolar talking!!". While this is just attempting an excuse to cover up and save face there is an element of it being trendy and more interesting to have such an excuse to hand.
The thing is, I think this applies to a lot of irrational thought too; face it, the universe is boring - "what you see is what you get" (I'm not going to address the issue of quantum mechanics or scientific discoveries, the vast majority of the population aren't scientists, or don't find the world's true intricacies particularly fascinating). No one likes this mundane existence; they want the action of the movies, the lives of the movie stars, the rocks tars, and the people who inexplicitly get on the cover of OK! magazine. They want this escapism not just to exist at a distance, but to apply to them because they're afraid of admitting that reality, for lack of a better term, sucks. I think it's ultimately an insecurity; you're life is interesting but because it doesn't appear on a magazine cover you don't think it's interesting enough (I'm not literally blaming celebrity obsessed magazines, this is more metaphorical). Gods, miracles, magic medicines, ghosts, demonic possessions, urban legends and increasingly equally mental disorders - they're all tools just to escape how mundane reality is.
All these things seem to be governed by the same set or principles of making the world a more "interesting" place, rather than adapting your own views of what "interesting" means to find the universe fascinating. d hominem 12:53, 28 July 2010 (UTC)
The dl;dr version: life sucks and everyone is just in denial of it. d hominem 12:54, 28 July 2010 (UTC)
I'm afraid I have a severe case of paragraphedolia: I am incapable of reading large amounts of text unless they are appropriately paragraphed. MARCVS ANTONIVS 12:59, 28 July 2010 (UTC)
That would be a by-product of it being far longer than intended. It has been rearranged. d hominem 13:09, 28 July 2010 (UTC)
Nicely put sir, nicely put. I know that, in my youth, I liked to think that having "issues" made me more interesting - in reality it made me a pretentious burke. Now, having watched my son have some very real issues - I'm glad he made it through adolescence without being one of the rising number of teenage male suicides, it was a close run thing at times and the NHS mental health services are next to useless - I know that it has far more to do with a very real pain and he would do anything to be clear of it. Jack Hughes (talk) 16:12, 28 July 2010 (UTC)
That's another good point, I might incorporate it into my "rule of thumb". The people I know with the real problems have tried to do something about it, the others tend not to. Or they go for medication and little else - and again making a very big show and dance about taking their magic pills (which they are as it seems a lot ant-depressents are just placebos). d hominem 16:43, 28 July 2010 (UTC)
I basically agree that we are all a bit mentally unstable, & it's healthy to be aware of our own "issues", since the people who are convinced that they're 100% well balanced tend to be less self-aware about their own insecurities & mental/emotional habits. As for actually being diagnosed with a named condition, it's a double-edged sword. Obviously clear definable conditions such as bipolar or schizophrenia should be diagnosed & treated, but overdiagnosis of vaguer conditions such as personality disorders is risky, especially with children & adolescents.
There's something a little bit like political correctness at work - whereas we might simply regard somebody as bad-tempered, for example, or easily embarrassed or sulky or socially awkward, these behaviours are increasingly viewed in the context of some form of mild disorder. This isn't necessarily always wrong, but it does divide people's behaviour into categories & can give a convenient excuse for negative behaviour as Armondikov said.
As someone who's struggled with a lot of this stuff myself, I can certainly understand the desire to be diagnosed - it's appealing to look for an easy answer that makes sense of the things you don't like about yourself or that others may not like about you. But the crux of it, going back to my comment about us all being a little mentally unstable, is that we all have our own insecurities and mental habits and (sometimes negative) thought patterns and emotional issues, & sometimes these can lead us to do or say things that we probably shouldn't, but we do ultimately have responsibility for acting on those impulses or otherwise.
Labelling people as just bad-tempered, awkward, etc. can be too negative as it just blames the individual without really acknowledging the psychological context for that behaviour, but on the other hand labelling the individual as having a condition goes to the other extreme by placing the blame on the condition rather than the individual. It's hard to find a middle ground, but the best attitude is just to be aware that we're all a bit mental in our own ways, and better off for knowing it & dealing with it, without necessarily needing to find a name for it. WëäŝëïöïďMethinks it is a Weasel 18:09, 28 July 2010 (UTC)
"...genuine sufferers tend to keep quiet while the ones who claim [to have them] ...happily shout it from the rooftops." —Armondikov Pardon me for turning your "is" into an "ought"—I believe that it's impossible to express any personal observation without letting some amount of personal judgment seep in — but that's a bit old-school of you, isn't it? Keeping it quiet just feeds into the old-fashioned idea that mental conditions are something to be ashamed of — i.e. "it is best for people who are different to keep quiet about it".
Another thing—don't forget to blame the parents'! Many parents with "problem children" (regardless of whether said children are "problems" because of the parents' own behavior, the child's social environment, or because of an inherent condition) are desperate for some sort of help—any help. Giving the problem a name — even a misdiagnosed one — provides something for them to latch onto so they can "solve" the problem. Radioactive afikomen Please ignore all my awful pre-2014 comments. 19:51, 28 July 2010 (UTC)
It's not an "old fashioned" opinion, it's an observation based on people I know. There are people who have gone into some quite intense therapy, but they rarely talk about it. It's not that people are ashamed of it or want to keep it to themselves - after all, they are getting help. They just don't brag about it. And the ones that are bragging don't tend to be the ones going to get any help. d hominem 23:38, 28 July 2010 (UTC)
## Spain is losing its Machismo!
Adios to bullfighting in Catalonia? Olé?......olé?......olé????? DickTurpis (talk) 13:05, 28 July 2010 (UTC)
Already been WIGOW'ed CS Miller (talk) 13:30, 28 July 2010 (UTC)
D'OH! DickTurpis (talk) 13:34, 28 July 2010 (UTC)
## Kevin Trudeau's Secret Societies
Dunno if this is old news but there was a series of YouTube videos thrown up on 10JUL10 of a radio interview in Los Angeles where Kevin claims to have been a member of "The Brotherhood", a secret society that controls everything and that they can use devices to detect people with DNA that vibrates at certain frequencies.
For an initiation fee and annual dues, he has moles planted in ALL of the SEEKRIT SOCIETIES (CIA, former KGB, Freemasons, Bildergerg, Illuminati, Skull and Bones, etc.) and can "get you access" to their SEEKRIT INFORMASHUNS to make YOU a billionaire.
But he's not in it for the money. He just wants to help you. I guess that's what the fees are for.
The Foxhole Atheist (talk) 15:51, 28 July 2010 (UTC)
I forgot to mention... IN ADDITION, he's willing to teach YOU the techniques to make your OWN DNA vibrate at the SAME FREQUENCIES as those in THE HIGHEST LEVELS of these very secret societies! For FREE! Just pay your initiation and dues and it's all yours! The Foxhole Atheist (talk) 15:55, 28 July 2010 (UTC)
I've heard of the DNA tracking before. Is this the same guy who caused a minor uproar in the rationalist community when it was offered to help track down Madeline McCann? d hominem 16:01, 28 July 2010 (UTC)
No. It's Kevin Trudeau, the same shyster that brought the world Mega-Memory and Natural Cures, among other atrocities. The Foxhole Atheist (talk) 22:27, 28 July 2010 (UTC)
## Greatest. Painting. Ever.
What more do you need?
MDB (talk) 16:10, 28 July 2010 (UTC)
Moar rule 34? --JeevesMkII The gentleman's gentleman at the other site 21:46, 28 July 2010 (UTC)
Moar electric guitars (interesting to note that many of the copies of this crop out Leia...) d hominem
Indiana Jones fighting Mecha-Godzilla and with Death Star II blocking the sun. Now that would be an epic picture --Thanatos (talk) 01:53, 29 July 2010 (UTC)
## Comparing new governments
New UK government and new(ish) US government. While Cameron and his cobbled together coalition of the largely willing have seemingly done an awful lot in their few weeks in power, Obama looks dead in the water.
Is that right? Just prompted by Heffer's piece in today's Telegraph. Or is it just an outsider's perception because we don't experience the detail? Ajkgordon (talk) 10:37, 28 July 2010 (UTC)
I'm not familiar with the actions the new UK government has taken so far, but this is an apples-to-oranges comparison, and a pretty biased one at that. The UK has a parliamentary system that ensures the government will always have a legislative majority backing it and seeing its programme through. Obama, on the other hand, was elected indpendently of Congress, and although he has Democratic majorities in both chambers to work with, the supermajority requirements against filibusters and the small-state bias in the Senate make it incredibly difficult to push controversial laws against a determined opposition. What is he supposed to do if, for example, not even a watered-down cap-and-trade bill gets the unanimous support of his own caucus and the few sensible Republicans that are left? He did score some victories, the health care overhaul was a significant progress, if not exactly what his base was hoping for, and recently, he got his financial regulation bill approved, but given the likely outcome of the midterm elections, that'll probably be it for the foreseeable future. As for bias, it seems like the author has bought into that whole supply-side-economics bullshit, and his criticism about Obama's performance here couldn't be more incoherent - "There's a huge deficit, but we mustn't let the Bush tax cuts expire! We've got huge unemployment, but stimulus is evil!". Just the same old nonsense that the right has been peddling for decades, and the talk about Newt Gingrich suggests that the author would like nothing better than having Obama replaced by another Republican ideologue.
If there's one thing that I'd criticize Obama for, it's his refusal to formulate a coherent liberal philosophy and defend it against the raging loons on the right (that's assuming that he actually is a liberal, which might not be the case). I agree with that article that the Sherrod episode is a perfect illustration of this administration's overly sensitive approach to completely unfounded allegations and criticism from Fox News and the like - they're going to attack him no matter what he does, and there's no use in placating them. This indecisiveness will put him in danger of facing an energized opposition, while his own base is dissatisfied and won't give him the necessary support come 2012. Röstigraben (talk) 11:19, 28 July 2010 (UTC)
Yes, it's difficult to compare the two. I think sometimes that the US system is geared up to resist change because there is such a disparity between the different branches of government that all need to work together independently to bring something into force. I think we've yet to feel the force of what Cameron's government has done, though. It'll take a while to realise how naive it is to think you can slash public spending that much and not be heavily affected, there is a bit of waste terms but they're not going to save that much (what is it, up to 25% or more in some places?) by buying cheaper pencils and cancelling things like "sensitivity training" or whatever. We're going to see roads fall apart, schools slowly crumble and see the bureaucracy - as much as people hate the word, they don't understand how important some of it it - disintegrate leaving us fucked if we want to do something as simple as get a passport. I'm not saying a deficit can be ignored, but it's too much and too soon and it's too much to assume that the slack will be picked up by wilful volunteers and the private sector. d hominem 11:48, 28 July 2010 (UTC)
Any of the columnists in the Telegraph can be safely regarded as space-fillers, loons or both. They are a waste of electrons to read. Heffer in particular. Even if he was right about something, he'd still be wrong - David Gerard (talk) 20:53, 28 July 2010 (UTC)
@ Armond, yes, the US system is intentionally designed to make it difficult to get anything done, especially big changes. ħuman 22:32, 28 July 2010 (UTC)
Yeah, like riders. WHAT IN THE FUCK IS THAT SHIT ABOUT?!?!!? d hominem 16:00, 29 July 2010 (UTC)
## Atheist Campaign - NZ
Across the road from my office
Downtown
Both these billboards have lasted 1 month, neither of which have been vandalized. Wouldn't last long in the states though I bet. AceX-102 02:16, 29 July 2010 (UTC)
Would that be because the choice of green on white ensured that no one could read them? --JeevesMkII The gentleman's gentleman at the other site 04:35, 29 July 2010 (UTC)
Add them to our Atheist bus campaign article. Lily Inspirate me. 06:59, 29 July 2010 (UTC)
this crap is ridiculous.Waronstupidity (talk) 15:58, 29 July 2010 (UTC)
You'll have to clarify that, as most of these are reactions to larger, more prominent, and better funded campaigns by churches that have a tendency to be very fearmongering. I suggest reading Ariane Sherine's early articles on the subject. d hominem 16:02, 29 July 2010 (UTC)
How so? EddyP (talk) 16:12, 29 July 2010 (UTC)
I want to move to NZ now. Pie, Alps, beer and atheists. Oh, Maoris and rugby too. Uhm, and Ace, I s'pose too. There are indeed many estimable things about your interestingly far-away nation, Dr.McWicked, Sir. DogPMarmite Patrol 16:26, 29 July 2010 (UTC)
Don't forget the orcs! Radioactive afikomen Please ignore all my awful pre-2014 comments. 20:12, 29 July 2010 (UTC)
Don't forget the low crime, high standard of living, more broadband access than anywhere else in the world, cheap cannabis and Wellington has more cafe's, bars and restaurants than New York (per capitia that is). AceX-102 20:44, 29 July 2010 (UTC)
Must admit that it's one of the places that I've always wanted to visit. It does seem to have a lot of things packed into a small area. (Ok, many of those things are sheep - but still...)--BobSpring is sprung! 20:49, 29 July 2010 (UTC)
Population wise though it is quite sparse. Particularly inland as 90% of NZ's population live within 30km of the coast. AceX-102 20:55, 29 July 2010 (UTC)
Any R.Wikians that want to visit I'd be more tan happy to show them around and offer a bed for a few nights. AceX-102 20:56, 29 July 2010 (UTC)
I'd really love to take you up on that - but unfortunately it ain't going to happen. :-( --BobSpring is sprung! 20:59, 29 July 2010 (UTC)
A few friends I know are moving to NZ for jobs. Apparently it rocks. £1000 minimum for flights, apparently, so no. d hominem 00:23, 30 July 2010 (UTC)
Oh well. Any takers, just let me know. AceX-102 00:30, 30 July 2010 (UTC)
I'll be moving there in March for grad school. I have been watching Flight of the Conchords to learn how to say "eggs." Eiiiigs.--talk 00:49, 30 July 2010 (UTC)
Careful, Ace, if my life continues on this aimless path I may take you up on that eventually. DickTurpis (talk) 00:52, 30 July 2010 (UTC)
AD, look forward to seeing you in the flesh. Dick - bring it on. AceX-102 01:32, 30 July 2010 (UTC)
## Nomination
Inspired by GoonieSpunk2012, I would like to nominate Tom Lehrer as our Poet Laureate and Official Songperson Mascot. ħuman 05:00, 27 July 2010 (UTC)
Seconded. Lord of the Goons The official spikey-haired skeptical punk 05:04, 27 July 2010 (UTC)
Thirded, I've been a Lehrer fan for yonks, despite the fact that he predates me by several decades. One has to curse Henry Kissinger for putting an end to his writing career. Lily Inspirate me. 08:51, 27 July 2010 (UTC)
P.S. Following the YouTube links I came across an interview with John Cleese - "why people, who haven't commited any punishable offence, listen to country and western music is absolutely beyond me." Lily Inspirate me. 08:53, 27 July 2010 (UTC)
You might like Roy Zimmerman
Then it is settled. Tom Lehrer is officially RationalWiki's Poet Laureate and songwriter. Lord Goonie Hooray! I'm helping! 06:55, 30 July 2010 (UTC)
## Friends?
Who wants to be friends? Please sign below using asterisks. ħuman 07:06, 27 July 2010 (UTC)
## Girl sues Uni to hate gays
We have another Chase Harper here--Thanatos (talk) 02:41, 28 July 2010 (UTC)
Oh dear, she's training to be a counsellor. I'm afraid bigotry of any sort is not a good asset for that. Lily Inspirate me. 08:40, 28 July 2010 (UTC)
Someone needs to sit that girl down and explain to her that she is practicing her right to hate gays at the same time as she is practicing her right not to get an advanced degree in counseling, and that she should think hard about the difference between having the right to do something and having the ability to avoid the consequences. --Opcn (talk) 10:25, 28 July 2010 (UTC)
A right to do something and a right to avoid the consequences... I like that. d hominem 11:50, 28 July 2010 (UTC)
Cenk Uygur is right, as usual. People would be far less sympathetic if she said she was intolerant of black people, but it amounts to the same. I'd go further, what it amounts to is the same as someone in medical school saying "it's my right to not believe in Germ Theory so I can skip these classes". It doesn't work like that. People who practice phrenology will not pass the requirements to become psychiatrists. ID advocates won't pass the requirements to be a biologist. Racists and homophobes won't pass the requirements to be a counsellor. People have a right to not believe in steering wheels, but they won't get a drivers license with that in their heads. d hominem 12:01, 28 July 2010 (UTC)
Step 1: Make your sadistic and control urges part of a religion.
Step 2: Complain about "freedom of religion"
Step 3: ????
Step 4: Profit! Sen (talk) 14:45, 28 July 2010 (UTC)
Well, I think she is a c*nt and general scum of the earth... but I do think I might agree with her on this. It's a pathetic belief she holds, but it's still her freedom to have it. And in this case it wasn't so much that her misguided belief was making her fail part of the required eduction, the school was making her follow courses to change her mind. Armondikov's analogy is incorrect in this case, it's not that someone is failing medical school because they don't believe in germ theory, it's that someone is passing the tests in medical school even though they don't believe in germ theory, but the school is giving them extra assignments because of their retarded beliefs.. And that's kind of unfair. If these kinds of beliefs were a requirement for the degree, it should have been part of the curriculum in the first place. --GTac (talk) 15:07, 28 July 2010 (UTC)
She is training to be a counsellor, a job which basically requires that you do not have prejudices against the people you are counselling. Would you really be happy if a racist or sexist was allowed to do the same? MARCVS ANTONIVS 18:35, 28 July 2010 (UTC)
Why make everyone "learn" things they already know? She doesn't meet the University's standards for a counselor. I'm doubtful any amount of counseling is going to change that, but I could be wrong. To me the issue is whether we should be taxed to fund institutions we disagree with. Because it is a State school, she has the right to determine the curriculum to some degree. Lumenos (talk) 16:11, 28 July 2010 (UTC)
what the fuck wikipedia? What the fuck? --Opcn (talk) 19:02, 28 July 2010 (UTC)
Yes Marcus, I'd be ecstatic when a racist or sexist would become a counsellor, you saw right through me! But the thing is, there is no such prerequisite at all to become a counsellor. If she hates the gays and becomes a career counsellor, then it shouldn't matter a damn thing. The only thing that matters is that you have to be professional enough to keep your own personal opinions out of your work. By your logic, I should be forced to take extra lessons if I would follow a career in counselling, because I loathe people wearing fedoras.. --GTac (talk) 11:19, 29 July 2010 (UTC)
Do you trust someone to keep their opinions out of a professional context, especially considering the context of this occupation we're talking about? A non-judgemental attitude is a requirement, just as a grasp of mathematics is a requirement for most science degrees. If you don't grasp maths well, you can get put on an additional course to "change your mind" about how well you grasp maths. A tolerant attitude to all comers is something for a counsellor, so you get put on a course to educate you further in the world of The Gays. The line between what constitutes a mere personal opinion and what constitutes professional knowledge gets quite blurred in many occupations, so it becomes nothing more than special pleading when you want to say what is opinion and what is knowledge for the sake of what is allowable. d hominem 16:17, 29 July 2010 (UTC)
IF that were so, it should obviously have been reflected in the curriculum. If she were able to pass everything while being a homophobic, then apparently she has the requirements. Otherwise, they should adjust the curriculum and apply it to EVERYBODY, it's nonsense to suddenly give higher demands for a single student. Yes, it's easy to hate bigoted cunts like that, but that doesn't mean she doesn't have the same rights as others. --GTac (talk) 17:17, 29 July 2010 (UTC)
You're assuming that "being able to meet all the requirements" and "being a homophobe" are compatible. Part of being certified as a counselor is being able to treat anyone who shows up at your door. If you turn away gays, then the university can lose it's accreditation. Or at least, that will, I assume, be their argument. Quaru (talk) 20:28, 29 July 2010 (UTC)
There's a problem with adjusting a curriculum to fit people rather than adjusting (the politically correct would call it "teaching") the people to fit the curriculum; and that is standards. The point of a qualification is to raise yourself above others and prove it through testing an examination. Quaru mentioned accreditation, which is where larger organisations gather together to define these standards and say what to teach and what to accept for a qualification - and it return people who pass the standards get the clout of this massive collective of institutions saying "this person is competent". If you cut standards to fit everybody then there is little point in having them because they become merely the lowest common denominator; not only can anyone be a doctor, but everyone does become a doctor because the standards have been lowered to apply to everyone. Now, regarding singling people out, if a single person doesn't fit a certain standard you fight to bring them up to it, not fight to lower your standards to their level - that's unfair on others. The reason you don't see this specific case "applied to everyone" is because everyone else undoubtedly fits that standard already, and thus educating them further in that subject is pointless. You don't waste time giving extra maths lessons to a science student who can grasp it perfectly well any more than you lower the standard of qualification to accept people who can't do it. In this case, most of the students won't be openly homophobic so there's no need to make a fuss in that respect, they've already got the right attitude that is needed to reach the standard. d hominem 07:26, 30 July 2010 (UTC)
Put another way (harken back to the math for science example), the other students "test out" of the "sensitivity training" by not needing it. Like is you want to study physics but they discover that not only haven't you been exposed to calculus, but you haven't taken any trig yet. Pile on the catch-up courses! ħuman 16:29, 30 July 2010 (UTC)
### same thing happening in Michigan
...a trend? P-Foster (talk) 17:51, 28 July 2010 (UTC)
## IRC 107 <pretty big news if you ask me>
I just stumbled upon this case, and am surprised it is not getting a little more attentoin. The IRC 107 is the internal revenue code that makes housing and other support for ministers of churches not part of their income. It is this code that lets the mega churches pay for the multi-million dollar mansions and gold toilet seats of the seriously corrupt, but is common practice for many smaller churches as well. The Freedom from Religion Foundation is challenging it on establishment clause grounds. tmtoulouse 16:15, 28 July 2010 (UTC)
Interesting. I notice that the wording is "a minister of the gospel". Does this exclusion only apply specifically to Christian clergy? If so, that would seem to be a grossly unconstitutional discrimination & I'm surprised it's been allowed to stand unchallenged for so long. WëäŝëïöïďMethinks it is a Weasel 17:29, 28 July 2010 (UTC)
Very interesting. A further bit of Googling shows that the relevant section states:
TITLE 26 - INTERNAL REVENUE CODE
Subtitle A - Income Taxes
CHAPTER 1 - NORMAL TAXES AND SURTAXES
Subchapter B - Computation of Taxable Income
PART III - ITEMS SPECIFICALLY EXCLUDED FROM GROSS INCOMESec.
107. Rental value of parsonages
In the case of a minister of the gospel, gross income does not include -
(1) the rental value of a home furnished to him as part of his compensation; or
(2) the rental allowance paid to him as part of his compensation, to the extent used by him to rent or provide a home.
So it appears to be specifically Christian. And in IRC 256 it appears they can also claim tax relief for the purchase of another home at the same time,
PART IX - ITEMS NOT DEDUCTIBLE
(6) Section not to apply with respect to parsonage and military housing allowances
No deduction shall be denied under this section for interest on a mortgage on, or real property taxes on, the home of the
taxpayer by reason of the receipt of an amount as -
(A) a military housing allowance, or
(B) a parsonage allowance excludable from gross income under section 107.
There is an article from Journal of Accountacy (2002) which deals with the constitutionality issue and notes that Bush renewed the act in 2002. Lily Inspirate me. 14:48, 30 July 2010 (UTC)
I very seriously doubt it only applies to Christian ministers -- there certainly would have been a legal challenge by now if it did. MDB (talk)
## How Cruel!
I can't believe this. People are eating clever cuddly creatures like squirrels. Don't they understand that we should only eat ugly stupid creatures like pigs and cows?--BobSpring is sprung! 18:41, 29 July 2010 (UTC)
If one needed any indication that the recession is far from over... Vulpius (talk) 19:17, 29 July 2010 (UTC)
Are you posting this to bait resident vegetarian Tom Moore? Radioactive afikomen Please ignore all my awful pre-2014 comments. 20:13, 29 July 2010 (UTC)
Not at all. Actually I'm sympathetic to many of Tom's views. I'm just amused by the outrage that people feel when squirrels get eaten rather than composted.--BobSpring is sprung! 20:32, 29 July 2010 (UTC)
I love the reaction from Viva (which I always thought was one of those "music" channels). MARCVS ANTONIVS 20:29, 29 July 2010 (UTC)
I am ok with this, although I'd like to know the opinion of some microbiologist/zoonose experts. There are a ton of illnesses/parasites that can be transmitted from animals to humans (because we are animals too, imagine that) and he is basically talking about wild squirrels, which are basically rodents. Our other non-cuddly self-replicating meat factories might be "common" but they are also the ones we have learned how to keep relatively healthy/clean and healthy from. Sen (talk) 20:47, 29 July 2010 (UTC)
We eat deer, wild boar and a whole lot of wild birds. Doesn't seem to cause a big problem.--BobSpring is sprung! 20:57, 29 July 2010 (UTC)
I think that a lot of commenters are missing the point: the complaints quoted here are not about "cute" animals - they're cynical attempts at manipulation by the powerful, covert vegetarian / vegan lobby. Jenny Seagrove, despite her fluff pretending that this is just about "wildlife", is a known militant vegetarian activist, and, like most vegetarians, a bona fide nutcase who also advocates for deregulation of the quack "herbal remedy" industry [1]
We all remember Sea Kittens, right? d hominem 20:49, 29 July 2010 (UTC)
I do! In particular, I am fondly remembering the sea kitten sticks I ate with tartar sauce for dinner last night and an absolutely to-die-for Cajun spiced cat-sea kitten sandwich I had at the Louisville RV show last weekend... Mmmm... Cayenne and hot sauce... *drool* The Foxhole Atheist (talk) 22:14, 29 July 2010 (UTC)
I have "The Joy of Cooking" from Bobbs-Merril (apparently the Thirty-third printing). It has a drawing of how to skin a squirrel on page 515. It's hilarious to look at, as it appears fairly comical. Also, on the opposing page 514, there is depiction of how to skin a rabbit, equally humorous. Picture not included, because I have no authorization to replicate it. --Eira OMTG! The Goat be Praised. 23:41, 29 July 2010 (UTC)
Skinning them is a lot like peeling a grape. Off with the head, out with the entrails, and then peeeeeel away the hide.--talk 00:43, 30 July 2010 (UTC)
It's not cool to eat Tree Kittens. Secret Squirrel (talk) 01:08, 30 July 2010 (UTC)
Do Red Tree Kittens taste different than Grey Tree Kittens? I have plenty of Red Tree Kittens within range, might be worth picking up a .22 rifle to augment the chili. For Teh Greys I'd have to patrol the woods where I growed up, I think. Simpler to Kill Teh Tree Kittehs I bait with my compost buckits... Yum! ħuman 06:12, 30 July 2010 (UTC)
OMG, now you have me craving squirrel chili... --Eira OMTG! The Goat be Praised. 06:23, 30 July 2010 (UTC)
Cannibals. Secret Squirrel (talk) 00:47, 31 July 2010 (UTC)
According to Joy of Cooking, you neither disembowel it nor decapitate it prior to the skinning. Pic would be related if it were ok. --Eira OMTG! The Goat be Praised. 06:21, 30 July 2010 (UTC)
JoC is so 20th Century. Nowadays, you pop a cap in its ass, bite off its head, rip out its guts, peel off the skin, season to taste and eat it raw. ħuman 06:41, 30 July 2010 (UTC)
I've been told that you should only eat baby grey squirrels as the adults are extremely tough and require a lot of cooking. We are fortunate to have some red squirrels near by at Formby. Personally I am all for grey squirrel culls along (nasty American imports) as well mink and other aliens which are destroying our native species. There was a mink farm not far from where some friends lived and the stupid animal rights people came and let them all out. Needless to say it has a terrible effect on the other wildlife, both mammals and fish. Lily Inspirate me. 07:22, 30 July 2010 (UTC)
A red squirrel is as big as a small dog, I'll be damned if I'm eating its slippery sack of guts when I stew it. Although I guess I'm out of the Scouts and a vegetarian now, so it might be a bit moot.--talk 14:40, 30 July 2010 (UTC)
You must have verrry big red squirrels in USA or mighty small pooches. European reds don't get over 12 oz. including bones & fur. Lily Inspirate me. 15:12, 30 July 2010 (UTC)
American dogs are evolutionarily adapted to fit in the tiny handbags of C list celebrities. It's a sterling example of symbiosis with fashion designers. --JeevesMkII The gentleman's gentleman at the other site 16:37, 30 July 2010 (UTC)
Oh, also. A decade or so ago, there was a weird little bubble fad for more exotic meat in the UK, with stuff like ostrich and kangaroo meat appearing briefly in supermarkets. It's kind of shame that went away about as quickly as it came. I'd be nice to have Elvis brand microwaveable squirrel burgers and such like. --JeevesMkII The gentleman's gentleman at the other site 16:44, 30 July 2010 (UTC)
Kangaroo exotic, pfeh! It's damn fine done right. Mind you, that wasn't until about 15 years ago, up to then it was regarded as dog meat - David Gerard (talk) 16:52, 30 July 2010 (UTC)
## Lymphatic Drainage Techniques
Amusing topic? Sounds perfect for Leechology, but I suspect that's not what they "do". ħuman 06:55, 30 July 2010 (UTC)
Doesn't seem that amusing. Macmillan are pretty reliable and woo free. Jack Hughes (talk) 12:48, 30 July 2010 (UTC)
Hmmmm, thanks! ħuman 16:10, 30 July 2010 (UTC)
## I seriously Hope God Exist.
For I prefer that to the Nihilistic Atheist view that life is meaningless and end dramatically with oblivion.
Seriously how you people live knowing that (your belief) there is nothing after life and that all the beautiful thing you saw was useless. Also how did you prove that there nothing after such death? Did you ever die?
I must admit that i am Obsessed with death and made lot of research and personal opinion tell me there must be something, but i know you guy will call me a Moron for not believing that after life there is Abyss and Oblivion.
But seriously how did you know (since that what you are sure it is truth) that there nothing after death? what are your proof? I can't prove the existence of God but can you disprove the Existence of the Afterlife?Waronstupidity (talk) 23:26, 27 July 2010 (UTC)
i would prefer Hell over Oblivion. Waronstupidity (talk) 23:27, 27 July 2010 (UTC)
I hope God exists, too. I just doubt it very much, and see no particular reason to think it true. I can't embrace something I don't think is true just because the lie is comforting.--talk 23:29, 27 July 2010 (UTC)
If there nothing after death it mean life is meaningless, life is just here to make us suffer for nothing. why make us intelligent then?
why!? seriously its driving me crazy..... i can't stop thinking about it.Waronstupidity (talk) 23:32, 27 July 2010 (UTC)
I would find a life with God more purposeless than one without. What would be the purpose of life if afterwards you knew you'd have an eternity more? Without an afterlife everything in your life has importance for it could all be your final act - so make it count. AceX-102 23:33, 27 July 2010 (UTC)
i am obsessed over this and its driving me nut and that one of the reason i stopped coming here, you guy scare the shit out of me with your Nihilistic comment, i feel a void each time i read an Atheist message, you guy scare me.Waronstupidity (talk) 23:35, 27 July 2010 (UTC)
Atheism =/= Nihilism. AceX-102 23:36, 27 July 2010 (UTC)
I hope god exist and i would burn in hell if i could choose between hell and Oblivion.Waronstupidity (talk) 23:38, 27 July 2010 (UTC)
there many sub proof that afterlife may exist but scientific seem to want it to not exist........... if i were a scientist i would persue my study to prove it then prove the existence of god over disproving it. Scientific don't want god to exist Waronstupidity (talk) 23:40, 27 July 2010 (UTC)
Are you OK Waron? Last time we got a little worried about you. AceX-102 23:41, 27 July 2010 (UTC)
Sorry we're scary. But really, think of atheism as just removing something you don't need. There's been a big dingy couch in the corner of the room, left there by well-meaning parents. It's stained with disgusting events of the past and is silly in its own right. We're moving the couch out. Now you get to decide what you want to put there instead. You can study philosophical Buddhism, and use zazen to strive for greater intellectual enlightenment. Or you can build a secular philosophy of kindness. Or you can find meaning in art and literature. The couch is gone: now you can decorate how you want, with something that's actually beautiful and useful to you.--talk 23:41, 27 July 2010 (UTC)
I personally believe in God and feel it ignorant to think otherwise but I know most will differ here. Ex-Troll Cheerleader (talk) 23:43, 27 July 2010 (UTC)
technically i am not okay...... my new med make me stress more........ its calm my paranoia ( i suffer from schizophrenia) and my other symptom but it make my anxiety worst by like 5x more.......... and i can't stop thinking about death.......... its driving me mad...... i stopped comming here cause this site make me stress more.........Waronstupidity (talk) 23:44, 27 July 2010 (UTC)
I went to church some week ago and i felt so good..... but this site and many other make me lose faith... i know you guy must think i am weak.....Waronstupidity (talk) 23:47, 27 July 2010 (UTC)
Perhaps you shouldn't stop by here if it upsets you? AceX-102 23:48, 27 July 2010 (UTC)
And you should really tell someone about your meds dude. AceX-102 23:49, 27 July 2010 (UTC)
I believe in God and dont tell me Im weak or Ill have to block you, lol. Ex-Troll Cheerleader (talk) 23:50, 27 July 2010 (UTC)
Thanks to Canada health care system i can only see my psychiatrist once per month so if my med are bad i must wait it out.......Waronstupidity (talk) 23:51, 27 July 2010 (UTC)
If there is a god then let's hope it's not the one described in the Bible. That god has serious issues, and I reckon I'd rather spend eternity with nothing to read but Gene Ray. Concernedresident omg!!! ponies!!! 23:57, 27 July 2010 (UTC)
just curious what is the % of Atheist/theist on this wiki? Waronstupidity (talk) 23:58, 27 July 2010 (UTC)
That's a good question. I assume atheists/agnostics in in a very high majority, but that's about as much as I know. Shame. It's nice to have intelligent theists around for their views and help in the theological side of things. Concernedresident omg!!! ponies!!! 00:00, 28 July 2010 (UTC)
i consider myself a mad(you know what i mean) Open minded Theist :-) hahahehe Waronstupidity (talk) 00:01, 28 July 2010 (UTC)
You're not a firebrand preacher, and you don't appear to have explosives strapped to you, so you're very welcome here. Just maybe best to avoid the religious stuff on RW if it's getting you down at the moment. Concernedresident omg!!! ponies!!! 00:03, 28 July 2010 (UTC)
you're right, but i like it here (beside the blatent Atheist strict view) Waronstupidity (talk) 00:06, 28 July 2010 (UTC)
You must not think highly of this life if you think it has no meaning without an afterlife. I think that kind of thinking is rather sad, because it's implying that you think life is meaningless and an afterlife is all that matters (I'm agnostic and personally think a lack of afterlife means that people should make the most of this life, and derive meaning from what's in it and not something outside it, especially if there's no evidence for said thing outside it anyway). I also agree with what Concernedresident said about the Biblical God. --GastonRabbit (talk) 00:45, 28 July 2010 (UTC)
I would say do whatever makes you feel comfortable. If going to church helps you, then go; there's lots of worse things you could be doing, and no-one here will think the worse of you for it. I personally intend to start believing in some sort of afterlife; it probably makes life a lot more comforting. EddyP (talk) 08:31, 28 July 2010 (UTC)
I'm going to start believing I'm rich and famous for the same reason.--BobSpring is sprung! 05:48, 29 July 2010 (UTC)
And I'm going to believe that Lumenos is an alien from another planet in order to bring some comfort to me. Lord of the Goons The official spikey-haired skeptical punk 05:52, 29 July 2010 (UTC)
I got your "meaning of life" right here, buddy. Prepare to spend 6 1/2 minutes in heaven. ħuman 02:17, 28 July 2010 (UTC)
Waron - you might try reading an article of ours which covers many of these points.--BobSpring is sprung! 05:47, 28 July 2010 (UTC)
Although the relevant answer on that FAQ could be trebled or quadrupled in size if you want to make it thorough. d hominem 11:54, 28 July 2010 (UTC)
Whether there is an afterlife or not is immaterial to me. I either fade into oblivion or I get some sort of come-uppance. If the latter happens then I expect to be judged by my actions and motives which happen not to be based on any arbitrary religious edicts. Can I really respect an omnipotent being who demands that I prostrate myself and worship him at least once a week like some sort of petty despot, or says that I must discriminate other people for their sexuality, or that I may not appear in my natural state on a beach, but then overlooks whether I respect other people or creatures, or try not to leave the planet in a worse state than I found it? To me, religion often promotes the petty over the principal. Personally I think that you can be more moral by not believing in a god where your actions may be driven by the threat of punishment or promise of reward. Lily Inspirate me. 15:28, 28 July 2010 (UTC)
"Whether there is an afterlife or not is immaterial..." Haha, I just noticed that. Very funny. ħuman 16:31, 30 July 2010 (UTC)
Creative writing can often seem like an uphill struggle so it's nice when people appreciate carefully crafted phrasing. Lily Inspirate me. 14:10, 31 July 2010 (UTC)
### Pantheism
If life is "meaningless" because it is temporary, an eternal life is all the more meaningless. There could be something that is somewhat god-like and somewhat impersonal. The Eastern pantheistic God, is eminent instead of transcendent. This God is with us, we are a part of it and it is us. Life has meaning because we can experience things like pleasure, pain, love, etc. Meaning is obviously everywhere. Atheism (meaning the disbelief in a personal god) says nothing about afterlives. (But since we are persons and we are a part of God, this God is personal.) To understand the afterlife, don't listen to the promises of magicians and charlatans, but instead try to understand our essence and self-identity. Is there life after this moment? Or have you died and been born again? What connects our past self to our future self? What part of "you" do you want to live and what part do you want to die? These are questions that get us closer to understanding the meaning of "life" and the meaning of "afterlife". ~ Lumenos (talk) (other talk pages: LI1, LI2, WP) 00:09, 28 July 2010 (UTC)
Wow, that was fairly coherent. Nice job. ħuman 02:11, 28 July 2010 (UTC)
Thanks. I made my biggest fopaux, right before I left for my hike and now that I got some sunlight in my eyes and blood to my brain I have successful lumenation! I have the music video accompaniment also, with a psychedelic Hinduish theme. This is Carmensita by Devendra Banhart with his girlfriend Natalie Portman and a blue woman.
Fopaux? Lily Inspirate me. 08:25, 28 July 2010 (UTC)
I suspect Lumenos meant faux pas. CS Miller (talk) 11:51, 28 July 2010 (UTC)
Yes, sorry. I guess the Urban dictionary had a neologism, although I did mean it was a mistake and not merely unnormal. ~ Lumenos (talk) (other talk pages: LI1, LI2, WP) 17:41, 28 July 2010 (UTC)
## Any Mac Users Ever Have This Problem?
ahhh, the YSOD!
This morning, while I was playing a game of Civilization IV, the display on my iMac suddenly went all yellow. I couldn't do anything on the system, not even shut it down by holding the power button. I unplugged it, and plugged it back in.
When I restarted it, the screen stayed dark, and I got the start-up chime repeatedly, with it fading each time it repeated till it was almost silent, then the "sound cycle" repeated.
I've got an appointment at the Apple Store this afternoon, but I'm curious if anyone else ever experienced that, and what the problem was.
A little googling indicated it might be a power supply problem. I'm hoping for that, since it's apprently rather cheap to fix. MDB (talk) 16:34, 29 July 2010 (UTC)
I had a similar problem when I accidently set all permissions on my disk improperly. The solution was to revert to the last Time Machine archive. You're using Time Machine to backup your disk, aren't you? It will take a few hours or more depending on the size of your disk image, but it's certainly worth trying. Just pop in your install disk to see if you can boot to DVD. If so, see if you can access your HDD and poke around. That will rule out your power supply hypothesis. ÑR/Señor Admin/¡hablen ustedes! 16:49, 29 July 2010 (UTC)
I do have a time machine backup, yes. I hooked it up to my old MacBook and the last backup was less than an hour since it failed. So, all I'll lose is my crushing of the Dutch in Civilization. MDB (talk) 16:54, 29 July 2010 (UTC)
Ha! If you try to crush the Dutch, we crush your computer's puny HDD! You should welcome your orange overlords! --GTac (talk) 17:25, 29 July 2010 (UTC)
Sure, what are you going to do, force me to wear wooden shoes with splinters? MDB (talk) 17:50, 29 July 2010 (UTC)
Either that or kick you in the chest with aluminum cleats... ScientificRigor (talk) 18:31, 29 July 2010 (UTC)
I think your problem is that you were playing a game on a Mac. Macs are only designed for posing with your laptop on your knee and typing one handed (because your other hand is stroking your well kept ironic beard) while you make videos and use media applications in a coffee house in North London. d hominem 21:35, 29 July 2010 (UTC)
What Armondikov is of course true, but what's actually happened is your graphics card has snuffed it. The yellow screen is a classic symptom. Probably heat damage, and it'll very typically finally die when you're playing a game. The motherboard will probably need to be replaced. This will learn you for buying anything from Apple. --JeevesMkII The gentleman's gentleman at the other site 03:21, 30 July 2010 (UTC)
The wierd startup sound is called the "chimes of death", see WP:Macintosh startup. I had a Mac back when I was a little girl that had this problem and it had a frown on the monitor. Ex-Troll Cheerleader (talk) 05:28, 30 July 2010 (UTC)
Way-ull, I tried to boot it again when I got home from work yesterday. It did boot... and I got video corruption again shortly after I logged into my account. So, I shut it down and hauled it to the Apple Store.
## Help with html
I have a blog at blogspot. As you may know, I can put a html script in the sidebar of my blog. It would be so nice to have some pictures and text there, and I can do that, but is it somehow possible to make them appear in random order, just like a random article is featured on RW's main page? You know, one second this picture and text, the other second another picture and text. Is there some html script that allows to do that? Can I find it anywhere? --Idiot number 59 (talk) 18:47, 1 August 2010 (UTC)
Randomising requires either JavaScript or something server-side. You can't do it with just HTML (which is a markup, not scripting, language). MARCVS ANTONIVS 08:14, 2 August 2010 (UTC)
Hmm I found this:
<script language="JavaScript"><!--
images = new Array(2); //init array -- change the number here to the number of images you have
//Creates the first image -- note: MUST start from 0
images[0] = "<a h ref = ><img src= alt= border='0'>";
//Creates the second image -- duplicate or remove where necessary
images[1] = "<a h ref =
><img src= alt= border='0'></a>";
images[2] = "<a h ref = ><img src= alt= border='0'></a>";
index = Math.floor(Math.random() * images.length); //select from the array
document.write(images[index]); //output
--></script>
And it works fine expect that it allows only pictures. Is it possible to change it so that I could add some text as well? --Idiot number 59 (talk) 08:33, 2 August 2010 (UTC)
All the code does is build an array of <img> tags, and then write them into the document in a random order (allowing duplicates). You could replace them with image[0]=" Witty comment 1 " etc. CS Miller (talk) 10:17, 2 August 2010 (UTC)
And if you want both, just use the html to image[0]="<a h ref = ''><img src='' alt='' border='0'></a> Witty comment 1 " or some such.. Quaru (talk) 17:50, 2 August 2010 (UTC)
I never realized "h ref" could be broken in two EDIT NOTE, oh, I see, it's blacklisted. What might work better, if you want it to be caption-like, is image[0]="<a h ref = ''><img src='' alt='' border='0'></a><br/><center>Witty comment 1</center>" For long witty comments, drop the "center" tags. You might also want to use "size" parameters inside the "img" tag - "width=x height=y". ħuman 03:16, 3 August 2010 (UTC)
Also you can drop the "a" tags if you don't want to link the image to anything. ħuman 03:16, 3 August 2010 (UTC)
Thank you all very much. Strange thing is that it works on my webpage and it works on my OTHER blogs but it actually doesn't work on the blog on which I wanted to put it. That blog has all other sort of crazy html and java stuff, can it be the reason? Or what else? --Idiot number 57 (talk) 13:49, 3 August 2010 (UTC)
## Harry Houdini
We should have an article. [5] ħuman 07:53, 2 August 2010 (UTC)
We did, but it escaped. Lily Inspirate me. 08:53, 2 August 2010 (UTC)
Damn, he's a sneaky bastid! ħuman 09:15, 2 August 2010 (UTC)
Is that two drums and a cymbal I can hear falling off a cliff... oh, oh... wait... BA-DUM TISH!!! There it is. d hominem 13:27, 2 August 2010 (UTC)
I stubbed my toe on it. Moar help? Please? ħuman 06:31, 3 August 2010 (UTC)
I was going to look at this last night, but forgot. I've got a couple of books on Houdini - I'll have a flick through later and add what seems on-mission. Worm(t | c) 09:21, 3 August 2010 (UTC)
Houdini's debunking was the inspiration behind Penn & Teller: Bullshit!. They make a few references to him directly throughout the series, though mostly in the first few episodes. d hominem 09:33, 3 August 2010 (UTC)
## Glenn Beck
Living on the eastern side of the pond I have only a vague idea of who Glenn Beck is from what people write here and the occasional mention on Radio 4's Americana. However, I came across an article in the LA Times which appears to insuate that Beck is ratcheting up people's fear and encouraging them to buy gold while at the same time being sponsored on Faux News by a company which sells gold with comparatively high mark-ups. Our Glenn Beck page doesn't mention him saying "buy gold now", perhaps someone who watches him could comment and maybe even add this to the article. Lily Inspirate me. 10:15, 2 August 2010 (UTC)
Mother Jones has an article on that very topic. MDB (talk) 11:32, 2 August 2010 (UTC)
I made a brief addition to the Glenn Beck article, referencing the MoJo article. MDB (talk) 11:41, 2 August 2010 (UTC)
Goldline - David Gerard (talk) 12:03, 2 August 2010 (UTC)
What I truly love about this gold hysteria right now is how half the commercials on TV are telling you that prices are high and you should sell your gold for cash. Then you get another group telling you how great an investment gold is, and how it's the perfect time to use your cash to buy gold. Maybe I should do both, then I'm sure to come out ahead! DickTurpis (talk) 15:21, 2 August 2010 (UTC)
One of the reasons I'll be glad when the economy picks up is we'll finally quit seeing those damn "sell your gold" commercials. MDB (talk) 16:18, 2 August 2010 (UTC)
But then I won't be able to start my business "Crisp Ca$h", where people trade in their old, wrinkly dollar bills for new, crisp ones. (For a small fee, of course.) Radioactive afikomen Please ignore all my awful pre-2014 comments. 23:37, 2 August 2010 (UTC) You, my friend, just might be a fucking genius. Don't let on that banks do it for free is all. DickTurpis (talk) 23:45, 2 August 2010 (UTC) I think Glenn Beck is the personification of the Mormon oath of vengeance HE is destroying America--Thanatos (talk) 02:12, 3 August 2010 (UTC) I don't know if it's goldline, but most of my liberal radio hosts are sponsored by someone selling gold, and they all do the voicing for the ads. Randi Rhodes' ad scared the shit out of me until I realized it was an ad, not "editorial" content. ħuman 03:07, 3 August 2010 (UTC) I couldn't find it in the on-line version of the Mother Jones article, but I think the print version did acknowledge that some of the leftie hosts sold gold, too. (It was only a couple of paragraphs in the article -- it is Mother Jones, after all.) MDB (talk) 12:01, 3 August 2010 (UTC) ## Richard Dawkins on BBC World Service Richard Dawkins is presenting a 4-part series on the Human Genome, on the BBC World Service. It's on at 10:30BST Wednesdays and repeated on 14:32BST Saturday (in Europe at least). The first episode has been broadcast. For those of you in the UK, it's on iPlayer, here http://www.bbc.co.uk/iplayer/episode/p008r4l4/Discovery_Age_Of_The_Genome_Episode_1/ . Otherwise, tune in. CS Miller (talk) 17:58, 2 August 2010 (UTC) Alternative link http://www.bbc.co.uk/programmes/p008r4l4 , with other broadcast times. CS Miller (talk) 18:03, 2 August 2010 (UTC) I think that I already mentioned this when it was on Radio 4 about a month ago. The home page is at http://www.bbc.co.uk/programmes/b00ssmcp so you can listen to all episodes without waiting for the rest of the World Service series. I saved them all as mp3s. Lily Inspirate me. 18:11, 2 August 2010 (UTC) Now that I think about, I vaguely remember someone (probably you) mentioning it before. CS Miller (talk) 18:52, 3 August 2010 (UTC) ## WHo's got better sense of humor Or who is funnier.... me or MC? Honestly.... --Idiot number 59 (talk) 05:56, 3 August 2010 (UTC) Lumenous, and RobS, both have you beat. To a cinder. ħuman 06:28, 3 August 2010 (UTC) Ace. Definitely Ace. His quotes are top notch. d hominem 09:25, 3 August 2010 (UTC) Personally, I can detect no difference between #59 and MC. But having a sense of humour is not the same as being funny. Lily Inspirate me. 09:52, 3 August 2010 (UTC) No difference between me and MC? Seriously? MC is an ass, but I am wonderful, humorous, exciting, good-looking, have lot of class and good pottery writing skills that you might have noticed. Lily, if you want me to write a poem for you, you just need to ask. --Idiot number 59 (talk) 10:30, 3 August 2010 (UTC) Idiot number Spilling his food on the floor Autumn leaves perish AceX-102 10:47, 3 August 2010 (UTC) Well, this is simply boring, but I appreciate your attempts to compete with me, not that you have any chances whatsoever. --Idiot number 59 (talk) 10:50, 3 August 2010 (UTC) vomitous mutant a pestilence embodied where are my trousers? AceX-102 10:53, 3 August 2010 (UTC) And what makes you think that we are interested in a trouserless Ace? --Idiot number 57 (talk) 11:52, 3 August 2010 (UTC) Well, his fiance certainly is : ) Radioactive afikomen Please ignore all my awful pre-2014 comments. 16:07, 3 August 2010 (UTC) He has a fiance? Really? I thought that everyone here are either gays or nerds. (Except me, obviously). Well, Ace, congratulations! And please upload one trouserless picture of yourself and I can tell if your fiance really wants you or maybe she has some other motives that makes her stay with you. --Idiot number 57 (talk) 16:27, 3 August 2010 (UTC) I would be lying if I said Im not a nerd, but then again, so is my boyfriend. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 02:33, 4 August 2010 (UTC) Right? I catch my wife editing wookiepedia all the time.. I'm waiting for her to make an account here, I've noticed her lurking recently. And on topic, neither of you are funny. Quaru (talk) 03:28, 4 August 2010 (UTC) Of course the resident cheerleader has the awesomeist sense of humor. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 23:10, 3 August 2010 (UTC) Ace = hilarity. AceX-102 23:15, 3 August 2010 (UTC) I'd like you to write me a pottery, Idiot number 57. ħuman 02:11, 4 August 2010 (UTC) Do you want love, hate or joke pottery? --Idiot number 59 (talk) 03:39, 4 August 2010 (UTC) ## Evangelical kids I found this in a footnote to Chapter 1 (The “evolutionizing” of a culture) from War of the Worldviews on AIG. "It is estimated by the Barna Institute that in this generation two-thirds of the children from evangelical homes will leave the church after they leave home. For more information, see State of the Church: 2002 by George Barna." Now I wonder why that may be the case? Do they just find out that all they've been taught is crap after all? Lily Inspirate me. 12:06, 3 August 2010 (UTC) I was hearing of some study recently about kids that go off to university and their religious beliefs. Ones doing STEM subjects are likely to have lost any faith long before getting there. On the other hand those doing humanities tend to start off believers but a staggering 90% (I forget the precise figure, but it was staggering) lose religious belief in college/university. This is postulated to be due to exposure to alternative ideas, which invariably will happen if you leave home to live on your own, either in the "real world" or in college. Without the constant barrage of demands from a close family unit, specific faith dies pretty quickly under exposure to other faiths - how many people figured religion was bullshit after discovering that there was actually more than one? d hominem 12:11, 3 August 2010 (UTC) Bob Altemeyer documents this with numbers in The Authoritarians - David Gerard (talk) 12:14, 3 August 2010 (UTC) When they talk about "leaving the church", do they mean that the kids become atheists, or does that include those who simply switch to a mainline denomination? I've got no idea how they even arrived at those numbers, but if it's the former, that would be a pretty radical break for a strongly religious group in a country that is still 80% Christian. Röstigraben (talk) 13:46, 3 August 2010 (UTC) When these people say "the Church" they usually mean "our Church". So yes, they probably do become more mainstream or liberal in their religious views. d hominem 13:52, 3 August 2010 (UTC) I'd be willing to bet that many "left" the church long before going to college, but just didn't feel free or comfortable "coming out" about it until they had some (even slight) independence of living situation. And I used the phrase "coming out" intentionally, since it is a similar story. ħuman 20:23, 3 August 2010 (UTC) ## More questions than Answers Staggering my way round the AIG site I came across their Answer books which are available online. Just to show how well researched they are I noticed that in the section Real Scientists of Can Creationists Be Real Scientists they state that Nicolas Steno was Dutch, when everyone knows that in fact he was Danish. Lily Inspirate me. 12:28, 3 August 2010 (UTC) "Hollywood has often portrayed Christians as ignorant, mean-spirited Bible-thumpers, while skeptics are depicted as reasonable, intelligent thinkers." Since when?? In most things I see the skeptics are nihilistic killjoys who disbelieve everything at first sight while the people with faith end up proved right and convert them to the religion/conpiracy theory/paranormal explanation/whatever. d hominem 12:30, 3 August 2010 (UTC) I love how their list of famous creationists falls over a cliff in the later half of the 19th century. We go from famous scientists to nothings, failures and self-promoters. It's almost like something happened in the 19th century that made the scientific community get some new ideas on origins. Funny that. --JeevesMkII The gentleman's gentleman at the other site 13:29, 3 August 2010 (UTC) That was going to be one of my main refutations if I can ever be bothered. It's a bit like Andy appropriating historical figures as homeschooled before there was wide-spread public schooling. Lily Inspirate me. 13:33, 3 August 2010 (UTC) Well, that's one of the main reasons that an "argument from authority" is fallacious. When you're an expert, not only are you an expert just in your own field, but also in your own time. Good scientists change their mind as they age and do research; something that you can't do when you're dead pretty much by definition. So we see this march of progress, evolutionary biologists are very unlikely to cite Origin in a paper published today, just as much as physicists aren't going to cite Principia. Equally, these modern scientists are unlikely to get referenced 150 years from now. Invariably, dead people are going to have outdated beliefs and theories, it's stupidly obvious. d hominem 13:57, 3 August 2010 (UTC) The above comment shows an Extreme prejudice against Dead People.--Tolerance (talk) 19:06, 3 August 2010 (UTC) ## Help! Hopefully one of the techies here can help lil' ol' me. Just loaded Skype on my laptop and now for some reason, altho calls play through headset, all other sounds come out through the speakers, even when headset plugged in. Any ideas? Running Win7 with HP IDT High Def codec? At the moment, the playback tab under sound, shows speakers as default device and headphones as default comm device, but there seems to be no way to set headphone to default device. Any ideas? Ta muchly in advance. --PsyGremlinPrata! 12:13, 3 August 2010 (UTC) It's nothing stupid like the headset just being plugged into the microphone slot and not headphones jack? Usually the sound only shuts off automatically when you put something into a headphone jack - so it probably won't if it's just a microphone or going through USB. See if you can get it through the headphones when you've shut off the sound via the laptop's mute button (most have them on the same row as the keys for monitor brightness and volume). d hominem 12:16, 3 August 2010 (UTC) Would that it were so simple. With sound muted, nothing plays on speakers or headphones, but when I dial Skype Sound test - perfect sound through headphones. Very odd. seems to be a common problem but the Skype forum doesn't answer the question. --PsyGremlinPraat! 12:22, 3 August 2010 (UTC) Well, I thought it'd be more helpful than "have you tried turning it off and on again". Anyway, the mute button tends to be overall mute, hence why Spotify can pick it up and inconveniently pause adverts when you hit it, I wasn't sure if Skype would be affected the same way. Other ideas would be to check your volume mixers to see if the headphones are a separate channel to the speakers, and making sure all your drivers for that headset are up to date and working. Is the sound test only through the headphones? I'd just graze around the Skype options to see if anything looks like it would isolate headphones and speakers. d hominem 12:27, 3 August 2010 (UTC) Thanks AK, much appreciated (and I saw what you did with that IT Crowd line :) ) I'll keep fiddling and hopefully something will happen.--PsyGremlinParla! 12:37, 3 August 2010 (UTC) Hahaha. Foolish mortal. You've just fallen victim to DRM. Because your headset isn't certified as a trusted audio path, there is no way to make it play all the audio from your computer. If they let you plug in just any old device, you might be naughty and copy music through the analogue hole. You filthy pirate. Enjoy the computer you "own." --JeevesMkII The gentleman's gentleman at the other site 15:21, 3 August 2010 (UTC) That is very strange indeed. I presume it used to work properly, ie, plugging in headphones killed the speakers in the flappie? Mine works fine, though I think I am plugged into line out, not 'phones. Heck, it ought to not even be a "computer" thing, it ought to just be a mechanical switch in the jack. ħuman 20:18, 3 August 2010 (UTC) ## Jeremy Clarkson On "Yesterday" "Inventions that changed the world": Florida has more gunshot wounds per annum than the rest of the world together. Most Floridan gunshot wounds are caused by spouses and lovers. - I know it's only anecdotal but WTF!.Him (talk) 12:56, 3 August 2010 (UTC) This WikiAnswers site give conflicting information. From that the vast majority of gunshot deaths are suicides. WP:Gun_violence_in_the_United_States seems to back this up. Jack Hughes (talk) 13:42, 3 August 2010 (UTC) He might have said that - I wasn't actually watching the odious twerp it was just on in the background. Him (talk) 13:55, 3 August 2010 (UTC) Zimbabwe and several East European and South American countries actually have higher murder rates per capita than the US. Colombia much higher. As for Somalia and Afghanistan I'm not sure the statistics are very reliable. I would be inclined to treat this as seriously as when Jeremy Clarkson described a car's horsepower in terms of how many inches deep it can cover Belgium. Remember he is a comedian. WSC (talk) 15:12, 3 August 2010 (UTC) Note that the original comment suggested gunshot wounds rather than deaths. Hard to find stats though. Worm(t | c) 15:43, 3 August 2010 (UTC) >EC) "Comedian" - that's a strange way to spell "idiot". Him (talk) 15:44, 3 August 2010 (UTC) ## Order a poem about yourself! If you want a poem about yourself, just tell me I write mainly love and hate poems, but also some ironic/mocking stuff. NB - you must be an interesting person, because there is nothing to write about boring persons. Take a look at my other works: --Idiot number 57 (talk) 16:37, 3 August 2010 (UTC) Even if you aren't charging for the service, I'll still say it's not a very good deal. OH SNAP! Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:06, 3 August 2010 (UTC) Psh, you and your silly poems. I'd love a poem about myself, Idiot number 57. Though something tells me that it will be quite the opposite of a love poem. ~SuperHamster Talk 17:52, 3 August 2010 (UTC) You are the most boring person on this wiki. What on Earth should I write about you? --Idiot number 59 (talk) 19:08, 3 August 2010 (UTC) You could always write a poem about how boring I am. ~SuperHamster Talk 21:51, 3 August 2010 (UTC) Oh Super Hamster How boring you really are makes my anus weep AceX-102 21:54, 3 August 2010 (UTC) ## What should I do about the mouse my cat just caught? My cat just caught a mouse, which is on its back, twitching every now and then. Do you think it's playing dead, or actually dead? If it's still alive I'll toss it back into the field, but if it's already deceased I'll let my cat have it. Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:00, 3 August 2010 (UTC) Eat the mouse. Or if you don't like the taste of mice, let your cat have it. That's what I would do. --signed by an Oniontalk edits 17:15, 3 August 2010 (UTC) Did he bring it to you or did you catch him with it? If he brought it to you, you should take it and reward him. 207.67.17.45 (talk) 17:18, 3 August 2010 (UTC) I caught her with it. Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:21, 3 August 2010 (UTC) I am not a veterinarian (though I play a zoo guard on teh Internets), but the twitching sounds like damage to the spinal cord. I suggest throwing it back in the field - if it recovers, good for it, if it doesn't... well, birds of pray need to eat, too.--ZooGuard (talk) 17:24, 3 August 2010 (UTC) Birds of prey, I mean. Freudian slip? --ZooGuard (talk) 17:28, 3 August 2010 (UTC) I think it only counts as a Freudian slip if you say "birds of penis" : ) Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:30, 3 August 2010 (UTC) I would just let the cat have it anyway. Put it out of its misery. --signed by an Oniontalk edits 17:27, 3 August 2010 (UTC) There was an early Victorian recipe on Radio 4 to make sugar mice. Basically you boil the mouse for about 5 minutes, remove the skin then steep in a concentrated sugar solution for about two weeks until fully preserved. They don't tend to use real mice now just use pink sugar and gelatine with string for tails. Lily Inspirate me. 17:28, 3 August 2010 (UTC) Thanks everyone! I let her have it—I've waited a while and it appears quite dead. Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:30, 3 August 2010 (UTC) Then you followed the advice I was going to give- "Let God decide".--BobSpring is sprung! 17:55, 3 August 2010 (UTC) As an aside, mice don't "play dead". They try like hell to get away while the cat "plays" with them until they stop being "fun". Some domestic cats will actually eat them, but since they don't smell or look like cat food, most just leave them on their pet human's pillow. ħuman 19:34, 3 August 2010 (UTC) ## FBI goes after Wikipedia for use of logo http://www.bbc.co.uk/news/technology-10851394. Guess who else reproduces it. I think the Federal Bureau of Investigations should be spending their time, you know, actually investigating stuff. Just a thought. --signed by an Oniontalk edits 17:18, 3 August 2010 (UTC) Mike Godwin responds, I love that guy. tmtoulouse 17:22, 3 August 2010 (UTC) What gives? Godwin doesn't compare Larson to a Nazi once. DickTurpis (talk) 17:24, 3 August 2010 (UTC) Having now read the response above I've got to admit it's pretty good. He gives Lenski a run for his money in terms of put downs.--BobSpring is sprung! 18:40, 3 August 2010 (UTC) Deep in the festering bowels of Wikimedia as I crawl, I've seen some really quite marvellous letters from Mr Godwin. Mostly of course he spends his time quietly sorting out problems in a mutually beneficial manner. But I think the word "Godwinated" should mean something along the lines of "Lenskied". Perhaps in thirty years they can be packaged and reprinted - David Gerard (talk) 19:38, 3 August 2010 (UTC) Especially investigating crimes against Conservapedia. Radioactive afikomen Please ignore all my awful pre-2014 comments. 17:24, 3 August 2010 (UTC) Actually CP has a nice copy of the logo too. How come our FBI article doesn't have one? Should we add one in solidarity with WP, or leave it out because the fools at CP have one?--BobSpring is sprung! 18:28, 3 August 2010 (UTC) I think we should add it. The FBI are seemingly just being douches about it. QuaruSuzuki - You can't explain that! 03:49, 4 August 2010 (UTC) HEIL THE FBI AGENT ADOLF SHITLER! Ex-Troll Cheerleader (talk) 19:40, 3 August 2010 (UTC) Probably the agent that went after them is a butt-hurt vandal. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 23:08, 3 August 2010 (UTC) ## TK strays from party line TK just posted this on mainpageright, showing how RINO McCain is 20 points ahead of the Democrat, but primary challenger and Teatard Hayworth (Andy's favorite) is 5 points behind. This isn't the story you should be highlighting, Teacake. DickTurpis (talk) 17:22, 3 August 2010 (UTC) Are you sure you're in the right room? Lily Inspirate me. 17:28, 3 August 2010 (UTC) No. DickTurpis (talk) 17:57, 3 August 2010 (UTC) ## SEX tape The Eva Mendes sex tape. Thanks to PZ Myers Him (talk) 22:25, 3 August 2010 (UTC) And that is despite it currently competing with Omid Djalili! d hominem 23:27, 3 August 2010 (UTC) LOL. ħuman 03:49, 4 August 2010 (UTC) ## Apropos of very little, does anyone besides me read Homestuck? I mention this because it just occurred to me that TK's compulsive scheming reminds me of one of the characters--arachnidsGrip. Except I'm pretty sure TK doesn't have a giant spider for a mother. --Gulik (talk) 22:42, 3 August 2010 (UTC) ## Wire in the Blood Highly recommended guys. The episodes where the motivation is religious are definitely very poignant. These are probably less frequent than the BDSM related episodes, but really, they're very good, especially the one that's just been on tonight (so should be available on ITV player if you can get it). I reckon I identify with Tony Hill very much in the "compassionate atheist" routine. But yeah, definitely recommended. d hominem 23:12, 3 August 2010 (UTC) ## The Ground Zero Mosque I'm curious... what do you good people think about the proposed mosque near Ground Zero? My initial thought was that the opposition was a bunch of anti-Muslim crackpots. But then I heard that the Anti-Defamation League was opposed to it, and, while they're hardly likely to be especially sympathetic to Muslims, they aren't the standard garden variety bigots either. On the other paw, I've heard it said that it's only a "mosque" in the broadest sense; it's more of as Islamic Community Center with a prayer space. (There won't be any minarets, for instance.) The impression that I got was that, were it not for the signage, you wouldn't think it was any different from any other Manhattan building. On yet another paw, I've seen charges from at least some reasonable sources that the group building it does have some ties to extremist groups. I'm not sure how deep those ties go, though. And while I'm not going to join in with the "ZOMG! Muslims!" crowd, I do think that there's merit to the argument that this is rubbing salt into the wound of 9/11. Even if the Muslim group building the center has no intention of that sort, it will still hurt people directly affected by the attacks. And switching sides again, America shouldn't be denying building permits based on the religion of the occupants. So, I'm conflicted. MDB (talk) 12:41, 3 August 2010 (UTC) I'm not sure there's mosque-building efforts anywhere in any western country that can't be claimed to have "ties" to "extremist" groups. It's worse than conspiracy theorists using the term "linked to", because people might take it seriously. What you're describing above contains huge amounts of fog - David Gerard (talk) 12:53, 3 August 2010 (UTC) You can read the ADL's statement here. The most telling part is the last paragraph: "Proponents of the Islamic Center may have every right to build at this site, and may even have chosen the site to send a positive message about Islam. The bigotry some have expressed in attacking them is unfair, and wrong. But ultimately this is not a question of rights, but a question of what is right. In our judgment, building an Islamic Center in the shadow of the World Trade Center will cause some victims more pain – unnecessarily – and that is not right." This is simply bullshit. Of course it's about rights, and there is no question in the juridical sense. As long as they properly applied for the relevant building permits and obtained them on purely technical terms, Muslims have the right to build mosques anywhere they please. You either uphold this right for everyone, or you dismantle it. You can't affirm a universal constitutional right to religious freedom and then turn around and say, "But not in this case! Some people don't like what they're planning!". Just imagine where society would be headed if we started chastising people for - or even prohibit them from - doing things that piss some others off? The freedom of speech and artistic expression would be gone in an instant, and we'd have to accomodate those people with the most restrictive opinions about morality. I used to have a small amount of respect for the ADL because they're usually very firm proponents of the separation of church and state, but this statement is completely opposed to the philosophy they claim to uphold. Röstigraben (talk) 13:33, 3 August 2010 (UTC) That's a valid point -- "linked to" is a weasel word. Well, a weasel phrase. MDB (talk) 13:28, 3 August 2010 (UTC) It's also a question of motive and intent, which you can't easily show or prove one way or another. Specifically, do people want to build a mosque just to piss off 9/11 victims and their relatives? If yes, then they're being assholes and in the wrong; if no, then the objectors are assholes and in the wrong. As you can't prove this either way (very easily, at least) you're torn between upholding religious freedom for all, or play a fairly dangerous game of a special exception for an exceptional event. I wouldn't like to call that one as there's a fine line between standing up for universally accepted principles against the wishes of a majority and completely destroying the idea of democratic freedom to decide what does and does not happen. d hominem 13:43, 3 August 2010 (UTC) The point of constitutional law and human rights is to establish a set of principles that must not be violated, no matter how large the majority opposing them should get, and religious freedom is one of those. I find it hard to imagine that this building would have been designed for the exact purpose of pissing off 9/11 victims, but as you said, we can't prove it either way. In such cases, presumption of innocence applies. Röstigraben (talk) 13:52, 3 August 2010 (UTC) THe ADL's statement isn't quite clear, but I'm taking them as not calling for the NYC gov't to stop the building; they're asking the Islamic group "please find another location". There is a difference. MDB (talk) 13:52, 3 August 2010 (UTC) The point is that constitutions get amended, moral zeitgeists evolve, views and tolerances change. One mechanism for this sort of change are major, unpredictable and catastrophic Black Swan type events. That's why it's a difficult call, because when the majority wants something badly, you have to change, and you can't easily pick and choose when you're going to listen and when you're not going to listen. d hominem 14:01, 3 August 2010 (UTC) This also draws you into some interesting line-drawing arguments. At what distance from the WTC would a mosque be allowed? A mile? Two miles? What? And what kind of Muslims? Shia? Sunni? Would athiests who saw Muslims and Christians as equally wrong be allowed to build a community?--BobSpring is sprung! 14:36, 3 August 2010 (UTC) "building an Islamic Center in the shadow of the World Trade Center" - ignoring the very sick joke implicit in that (well, and I guess there are still four buildings that are part of the WTC?), what time of day is this shadow measured, does the community center simply have to be hidden behind another building... etc. etc. Reminds me of the NRA convention soon after Columbine - tacky, legal, makes sense (and is legal) to protest, does not make sense (and is illegal?) to outlaw. ħuman 19:58, 3 August 2010 (UTC) (EC)I'm not advocating a pick-and choose approach at all, quite the opposite. Freedom of religion is a universal human right that is enshrined in the constitutions of all modern democracies, and Americans in particular should be proud that their country broke new ground in establishing it. It's a right that all humans enjoy from birth simply by virtue of being human, it's not granted by the state, and can't be taken away by the state. That means that no state and no majority of its constituents can legitimately infringe upon it. It's obviously not universally respected around the world, and in a majority of Muslim countries, for example, there might not even be a popular demand for it. But those governments and societies which violate it are rightfully regarded as backwards and criticized for it. This concept did take quite some time to develop (at least in its modern universalist form), but it undoubtedly represents a monumental progress that can't be overturned. No matter what happens and how many people want to abrogate these rights, they just can't do it as long as the rule of law prevails in a country, because domestic constitutional and international human rights courts would stop them from doing so. Röstigraben (talk) 14:45, 3 August 2010 (UTC) I don't think you're going to find many people on this site that will argue that the community center should be stopped legally. There is, though, at least something to be said for the argument that even if they should have the right to build it there, they still shouldn't build it there. The head of the ADL compared it to the effort some years back to build a Carmelite convent just outside of Auschwitz. I don't think anyone thought a bunch of nuns were intentionally trying to offend Holocaust survivors, it was a question of sensitivity. I realize that there are definitely differences, but there are parallels, too. MDB (talk) 14:54, 3 August 2010 (UTC) It can't be stopped legally by anyone, there's just no case against it. As for sensitivity, there's nothing wrong at all with stating your concerns and asking them nicely to voluntarily stop this project, even though I don't think they should have to. If they still want to go ahead with it, it must be accepted. But that's not what the opponents of the mosque/community center have been doing, they've been crying bloody murder for months now and show no sign of relenting. Some right-wing lunatic even compared this project to erecting a kamikaze memorial at Pearl Harbor, if I remember correctly. Ideologues have taken over and are now trying to turn this into a national wedge issue. This would've been an excellent opportunity for the ADL to affirm their commitment to universal religious freedom, but instead, they chose to lend credibility to people who are fundamentally opposed to these rights and try to exploit the issue for political gains. Röstigraben (talk) 15:10, 3 August 2010 (UTC) Well, usually, by default I am against nay kind of church building getting build, anywhere. On the other hand it makes the west/secular societies look good and will/can be used as a talking point by "freedom of thought/beliefs" groups inside various islamic countries. And I think that in the end, that is more important. I belong in those who think that you can't win against religion by "banning" it, rather by slowly making it laughable and impotent. Having the moral high ground so that other groups can easily push for more multireligious (or a-religious) environments inside islamic countries is worth it more, since I believe that having multiple "street shops" for various religions, creates an effect of making them look silly. Freedom of religion, does eventually actually lead to deterioration of a religion's "dominance" aura and "unquestionable" status in an area. So to sum up. It's because I want to start eroding islam in islamic countries, that I want the mosque to be built, so that rights group can then all might and high say "see, we/they even allow mosques to be built on ground zero!" and on occasion an islamic politician to be saying "oh well, whatever" and give permission for churches and synanogues, which in turn will lead to more islamic high streets where different preachers will be saying "we are the truth", "no we are!", and kids will be able to more easily start suspecting that all is bullshit. Finally there is a rather important "freedom" aspect, regarding property rights. I mean, they bought the land. It's theirs. If conservatives/the mob can now start removing property rights by simply "not liking them" then this can set a dangerous precedent. Frankly, I say fuck it, they should have the right to erect a big, fat neon sign there saying "Islam pisses over all", and in turn of course conservatives can erect signs all around saying that "Mohhamet sucks pigs". Which in turn, as they look silly, gives me the freedom to erect billboards saying "all religions are stupid" and "hey, look at this cool gadget. Isn't science great? Has religion ever done anything as cool as that?". And it is my personal theory again, that in such freedom respecting societies "we" again have the advantage, because true egoists don't care what other people say about them anyway, while the religious can't stand to have their Gods/Symbols despoiled without being able to respond. It makes them look powereless. Finally, the opposition never had an actual moral point in the first place, because the entire thing is driven by the idea that all muslims are terrorists. That's the beginning of ideas that start dehumanizing people without evidence, rather than focusing down on the individual level, and should not be pandered at. And the muslims in question rightfully don't have any obligation to be "sensitive", because 9/11 it is not something they are supposed to recognise as "theirs" in the first place. It's like saying that a new church is a monument to pedophilles, because lots of christian priests are that. Or that I am supposed to care if someone shot a school, simply because he wore the same brand of shoes as me. Its one thing to call the religious stupid, its another to start making people responsible for stuff they didn't do. Collective guilt of people is an extremly dangerous meme, only a tiny bit away from collective punishment as well. And a conservative who is ready to dehumanize someone and grab someone else's property rights because "he is terrorist, cuz muslim, thus justified", is only a tiny bit away from starting violation other rights as well "cuz muslim". So to re-sum up. I "don't oppose" the construction of the mosque even if I want to see it empty & bankrupt. If you start banning those things it creates a tangled moral & legal mess to navigate through when you can go for freedom(tm) and be done with it. Excellent trolling too. Sen (talk) 18:15, 3 August 2010 (UTC) There's also the sort-of issue of whether there are any other appropriate sites for the structure in the general area. I suspect decent real estate is hard to come by in Manhattan. ħuman 20:13, 3 August 2010 (UTC) Bloomberg lays the smack down. Sums up why freedom of religion is an absolutely vital freedom - David Gerard (talk) 10:10, 4 August 2010 (UTC) ## The paediatrician story I vaguely remember seeing a retelling of the "paediatrician mistaken for paedophile" story somewhere here, but I can't find it at the moment. Here's a link to a BBC article that tracks down the original incident - a female paediatrician returned one day from work to find "PAEDO" spray-painted over her front door. --ZooGuard (talk) 17:10, 3 August 2010 (UTC) Stories like that show how brainless people are. Both words begin with the letters p-a-e-d, so they must be the same! --signed by an Oniontalk edits 17:29, 3 August 2010 (UTC) That's fair enough. In this case paedo is obviously short for paediatrician. I have yet to return home and find electro scrawled on my wall though. DeltaStarSenior SysopSpeciationspeed! 23:59, 4 August 2010 (UTC) ## Yea sayeth the LORD, I containeth not the empty set Marvel at this pile of mathematical nonsense I found on the interwebs: Genius. I'm not that hot of a mathematician, but isn't A (THE FATHER!) exactly the same set as E (THE IMPERFECT UNIVERSE!)? --JeevesMkII The gentleman's gentleman at the other site 17:54, 3 August 2010 (UTC) Not necessarily; E = B, but B is a proper subset of A, so they're not equal. Whether the whole mess is inconsistent, I can't say. MARCVS ANTONIVS 18:03, 3 August 2010 (UTC) Is B actually guaranteed to be a proper subset of A? Seems to me that only happens if A contains the empty set. I guess there's nothing in the definition of A that actually prevents that happening. I can't for the life of me figure out what the fuck it's all supposed to mean though. --JeevesMkII The gentleman's gentleman at the other site 18:13, 3 August 2010 (UTC) I have no idea what it could possibly mean, and I suspect the writers haven't either, given how they use the word "perfect". MARCVS ANTONIVS 18:17, 3 August 2010 (UTC) -scratches head- So does God contain the imperfect universe or not? You would have though that to be a True or False question. Sen (talk) 18:20, 3 August 2010 (UTC) I really don't know. I have no idea why they keep going on about Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): B \cup \emptyset either, since union with the empty set is the original set; it's like adding 0. Also, if they define "perfect" as "not containing the empty set", it doesn't follow that A is perfect. MARCVS ANTONIVS 18:24, 3 August 2010 (UTC) It's probably quantum. Sen (talk) 18:26, 3 August 2010 (UTC) I suppose you could take the statements that "foo is perfect" as declaring that the empty set is not a member of foo. In which case, the intersection of P and A really would just be A. --JeevesMkII The gentleman's gentleman at the other site 18:29, 3 August 2010 (UTC) The intersection of P and A is not A, since $\emptyset \in A, \emptyset \notin P$. MARCVS ANTONIVS 18:35, 3 August 2010 (UTC) As far as I can tell, P, B, C, D and E are all perfect, and the only imperfect set is A (still assuming "perfect" means "does not contain the empty set"). We can therefore conclude that the Father is the only imperfect thing in existence, and therefore Christianity is false. MARCVS ANTONIVS 18:40, 3 August 2010 (UTC) QED. I'm glad we cleared that one up, then. I wonder what the pope will do for a job now? --JeevesMkII The gentleman's gentleman at the other site 18:47, 3 August 2010 (UTC) Where'd you find this, by the way? MARCVS ANTONIVS 14:04, 4 August 2010 (UTC) Atheistic version. Now that was simpler, wasn't it? Sen (talk) 18:50, 3 August 2010 (UTC) ## Im a terrible person Should I reveal my true identity to everybody on Wikipedia where I am an admin and a bureaucrat? I dont deserve it as I am a bad, bad person. Someone slap me with a trout for being such a bad bad person. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 23:41, 3 August 2010 (UTC) YES! I'd like it if you do, for I'm dying to know who you are on there - right now, I'm in a doubt as to whether or not you are what you really say you are. And you're not a bad person - you're an ex-bad person. ~SuperHamster Talk I have something called split personality syndome. Im afraid to tell people my white-hat user account at WP after some of the things Ive done as a black-hat. You wouldnt believe me if I told you what my white hat account was, but youll believe that I was LBHS Cheerleader and Grawp among other things. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 23:52, 3 August 2010 (UTC) Oohhh, you're pretty notorious. So you like 4chan and enjoy email bombing, huh? ~SuperHamster Talk 23:56, 3 August 2010 (UTC) Aahhh, and you're also part of the whole "Hagger" thing - I've heard of that before. You've got a lot of time on your hands to not only make a ton of accounts to violate Wikipedia's rules, but to work your way up to a bureaucrat (which I am still in doubt that you are one). ~SuperHamster Talk 00:00, 4 August 2010 (UTC) I wasnt the only "Grawp" only one of them. There were many. Im also part of the "Phone Losers of America" (Google it). Ill email you my name on Wikipedia not that youll believe me. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 00:08, 4 August 2010 (UTC) Wouldn't Checkuser reveal your real account? Or did you use some fancy-schmancy way of avoiding it? And nah, I probably won't believe you. ~SuperHamster Talk 00:21, 4 August 2010 (UTC) Stop messing around, Godwin - David Gerard (talk) 00:06, 4 August 2010 (UTC) Checkuser would not work. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 00:25, 4 August 2010 (UTC) I don't want to be needlessly rude, but this sort of repeated thing just sounds like boasting to me. Yes, you're very clever for fooling everyone, but questions like these seem more like advertising yourself than ethics or conversation.--talk 02:41, 4 August 2010 (UTC) If I discuss anything else people get mad. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 03:01, 4 August 2010 (UTC) I hereby offer to solve your dilemma: Simply email your pal C®ackeЯ with your username and password and I will carry on in the same vein (and style if needs be) so that you are free in mind and body to werk your dark woerks of nastiness. I might even promise NOT to share the username/pw with a certain TK person. 03:06, 4 August 2010 (UTC) C®ackeЯ Going back a bit: I'd normally agree with you AD, but this actually led to a nice email conversation between me and ETC (ooh, fancy little acronym there). So it's not all boasting, per se. ~SuperHamster Talk 03:10, 4 August 2010 (UTC) All claims concerning Wikipedia in this section are best treated as being of literary value only - David Gerard (talk) 07:39, 4 August 2010 (UTC) I care about this question just enough to write this comment about how little I care.--BobSpring is sprung! 08:54, 4 August 2010 (UTC) Cheerleader; send me you evil alter ego and I will troll wikipedia for you. 86.40.103.195 (talk) 09:09, 4 August 2010 (UTC) I've a lot of experience there. If you think Gerard is a moose, wait till you see some of the feckless asperger's ridden morons I've destroyed over there. 86.40.103.195 (talk) 09:10, 4 August 2010 (UTC) Marcus gets more and more like Ken with his silly "Essays", persona of many disguises and imagined vanquishing of his enemies like some modern Don Qixote. I can well imagine that both of them are still living with their mother. Lily Inspirate me. 10:14, 4 August 2010 (UTC) Actually, I would LOVE ♥ to see you write that youre a cheerleader at lemon bay high and how much pchs sucks and be sure mention what whores the pchs cheerleaders are. Oh and dont forget to write all about how "PCHS NJROTC" fucked the port shithole cheer captain and gave her crabs. Do it all at wp:Talk:Port Charlotte High School. And do it from your infamous 86 range. While youre at it entertain TK over at CP with tales about how he licks young girls cunts for shits and giggles. Do that from you 86 range too. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 19:26, 4 August 2010 (UTC) ## Anne Rice and christianity Anne Rice has quit the christian faith: "In the name of Christ, I refuse to be anti-gay. I refuse to be anti-feminist. I refuse to be anti-artificial birth control. In the name of Christ, I quit Christianity and being Christian," and predictably gets lambasted by christians for her trouble. http://www.facebook.com/annericefanpage FreeThought (talk) 04:36, 4 August 2010 (UTC) I think this has been mentioned elsewhere. AceX-102 04:38, 4 August 2010 (UTC) Sorry, I couldn't find it anywhere except a brief mention on the conservapedia talk page. FreeThought (talk) 04:43, 4 August 2010 (UTC) WIGO somewhere. I can't find myself responding with more than "meh" though. I mean, it's the woman who's fundamentally responsible for the trend of teenage girls being obsessed with homoerotic vampires just stating the freaking obvious about religious fundamentalists. d hominem 15:52, 4 August 2010 (UTC) ## Damn we need rain We've only had 1.5 inches, and that fell in 15 minutes, since like late May. The trees are wilting. ħuman 06:52, 4 August 2010 (UTC) June was really dry here but July was quite wet (especially in Wales), and it's raining again this morning. There's still a shortage of water in the reservoirs of the North West though. I heard on the radio that globally 14 weather records have been broken so far this year. Lily Inspirate me. 07:11, 4 August 2010 (UTC) Good thing there's no global warming. QuaruSuzuki - You can't explain that! 16:10, 4 August 2010 (UTC) Coincidence: it started absolutely stair-rodding down at 17:10 BST here. Him (talk) 16:16, 4 August 2010 (UTC) ## Help for Karajou? http://www.cracked.com/article_18680_political-cartoons-lowest-form-communication.html - π 13:01, 4 August 2010 (UTC) Enjoyable article. I just automatically sniggered when I spotted "After Eden". d hominem 13:33, 4 August 2010 (UTC) I always laugh at After Eden, but probably not for the reasons the illustrator intended. - π 13:44, 4 August 2010 (UTC) I don't think After Eden would be funny even if I accepted its basic premise. It reminds me of Mallard Fillmore in that it's little more than a soapbox for the artist. MDB (talk) 15:27, 4 August 2010 (UTC) Don't forget their earlier article about (web)comics! http://www.cracked.com/article_17607_the-5-circles-baffling-web-comic-hell.html Radioactive afikomen Please ignore all my awful pre-2014 comments. 03:06, 5 August 2010 (UTC) ## Wherein I get my political news from CP, probably regret it. I ought to know better than to place any stock in what CP has to say, but this week they seem to be gloating about how many republican incumbents are losing their primaries to more "CP friendly" candidates. Now I have no idea who either the incumbents or their challengers are, but I can only assume if CP is happy then the winners are all fringe whack jobs who want guns, theocracy and to bake the poor in to soylent green. Can someone in the know tell me if this is going to backfire hilariously on the reps when the non-crazy portion of the electorate find out just who they're being asked to choose between? I have my fingers crossed. --JeevesMkII The gentleman's gentleman at the other site 15:37, 4 August 2010 (UTC) I voted in the republican primaries yesterday, as I had to vote against a lot of douches. It worked. The guy I voted for will be on the ticket. So I know in my state, it's already backfired. I mostly find it funny, because the tea party is trying really hard to get their people on the ticket, and mostly they have no chance of winning against a democrat if they do, as they simply do not represent anything but the extreme right, and no one in the middle will vote for them. And if you have noticed anything of the gubernatorial races, Rick Snyder, "One Tough Nerd" made the ticket in Michigan. (I saw his ad on CNN when they were discussing the races, so I assume it's gotten around, maybe not..) And the guy I voted against, Mike Cox. Literally his ad was along the lines of, "I stood against Obamacare. I fought for lower taxes. I stand with Arizona in their immigration law" There was another tea party talking point, maybe abortion. Yeah. Fuck that guy. I honestly don't know who's running on the democratic ticket. But there's my "news from the street" or some shit. QuaruSuzuki - You can't explain that! 15:54, 4 August 2010 (UTC) It reminds me of growing up in a small, Free State town (the Free State is our big sky/cow/backward hick country for the uninformed). At election time, the choice was between the Konservatieve Party (Conservative Party) and the Herstigte Nasionale Party (Reformed National Party), both ultra right-wingers, who broke away from the National party, because they felt the latter wasn't enforcing apartheid strongly enough. Hopefully the US electorate will wise up to the lesser of two evils.--PsyGremlinPrata! 15:59, 4 August 2010 (UTC) It seems to me that the smartest move for the GOP this midterm would be to run centrists against Democratic incumbents. Instead, the teabagger types are running extremists against their own incumbents. Very strange strategy. ħuman 19:38, 4 August 2010 (UTC) And I thought it was just a ruse to throw the election so the Democrats would have to deal with the recession. But since they haven't shouted "BAZINGA!" yet, I assume they must be serious... d hominem 22:13, 4 August 2010 (UTC) The primary results were mixed, with some establishment and some tea party candidates each getting in the running. There is no one particularly extreme right now, so there probably won't be a backlash. Not very exciting and not exactly a sweeping indictment of the establishment. However the big story of tea party v. GOP establishment is the Nevada Senate race. CP is mostly parroting that one, but basically Reid (Democratic Senate majority leader) is like twenty points down versus a generic Republican, but because the GOP candidate is tea partier Sharon Angle - and she turns out to be crazy and stupid in equal measure - Reid now has a comfortable lead. It's early yet, so a lot will depend on the behavior of the new candidates - it's too hard to guess how extreme some of the tea partiers are right now.--talk 01:42, 5 August 2010 (UTC) ## Palin daughter breaks off engagement - again Here. Is mummy meddling, do you think? EddyP (talk) 08:36, 4 August 2010 (UTC) Heaven forfend! - David Gerard (talk) 10:12, 4 August 2010 (UTC) Every time I hear a news story about Bristol and Levi (America's Sweethearts), I'm reminded of the time I was watching the local news while visiting my parents. Some sordid news story was on, and I remarked, "Feh. Trailer park trash." Mom replied, "son, they may be all they can afford." I responded, "Mother. It's a story about a twenty eight year old woman who got her eighteen year old boyfriend to go into her trailer and shoot her estranged husband with a shotgun." Mom thought for a second, and said, "okay, they're trailer park trash." MDB (talk) 13:14, 4 August 2010 (UTC) There's some Jeremy Kyle material right there! d hominem 15:59, 4 August 2010 (UTC) Is that good? MDB (talk) 16:09, 4 August 2010 (UTC) No. Absolutely not. The Jeremy Kyle show features people that make trailer park trash appear like educated royalty (if that's not an oxymoron). DeltaStarSenior SysopSpeciationspeed! 23:32, 4 August 2010 (UTC) I always think "The Palins are filmed in front of a live studio audience"--Thanatos (talk) 00:47, 5 August 2010 (UTC) Ah, so, to put it in analogy terms, Jeremy Kyle:Great Britain::Jerry Springer:United States. MDB (talk) 10:24, 5 August 2010 (UTC) ## Evil socialist National Health Service Had a nasty fall whilst playing footy and landed very hard on my left hand. It swelled up and was pretty sore. I've broken my scaphoid in my right hand before, and I was concerned that if I'd busted anything in my left and it healed incorrectly I'd be fucked, so off I went to the local hozzy. Walked in, seen by the triage nurse, examined, x-rayed, nothing broken just soft tissue damage. In and out in less than an hour, the staff were extremely professional and impeccably polite. I do hope the communist Kenyan doesn't force such horrors on the US. DeltaStarSenior SysopSpeciationspeed! 15:10, 4 August 2010 (UTC) Me too. That shit is scary. And you weren't just put down like a horse? Lucky you. QuaruSuzuki - You can't explain that! 15:21, 4 August 2010 (UTC) Fuuuuck. You escaped the death panels? Lucky - David Gerard (talk) 15:25, 4 August 2010 (UTC) He's just been added to their list for consideration later. It'll take the Evil Socialist Bureaucracy (tm) a few years to get around to him. MDB (talk) 15:28, 4 August 2010 (UTC) How long before the Cameron Clegg mafia fuck it up? Him (talk) 15:30, 4 August 2010 (UTC) Until the new government gets its rules sorted out, I believe there is a firm eighteen-week limit on the waiting list for the death panels - David Gerard (talk) 15:31, 4 August 2010 (UTC) I believe the death panels are one of the quangos that are going to be abolished and replaced by local referenda. Lily Inspirate me. 15:43, 4 August 2010 (UTC) So you didn't come out infected by Marxism? d hominem 15:46, 4 August 2010 (UTC) Marxism is sexually transmitted, you can only get it by sleeping with a Cultural Studies major. --The Emperor Kneel before Zod! 03:09, 5 August 2010 (UTC) So how exactly does the NHS work? Arent hospitals owned by companies just like in the US but just the government dictates the insurance? A friend of mine was admitted to the London Bridge Hospital while on vacation in England and I heard its owned by an American company. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 13:04, 5 August 2010 (UTC) No. The NHS is part of the state, and is funded by taxes. MARCVS ANTONIVS 13:57, 5 August 2010 (UTC) ## Yes Men I'm not sure if you guys know about the Yes Men. Saw their "Change the World" feature yeserday. I must admit, I'm in two minds. I love them taking big business to the cleaners, but some of the tactics made me feel uncomfortable - like the Bhopal hoax. --PsyGremlinHable! 15:45, 4 August 2010 (UTC) Bhopal Hoax? QuaruSuzuki - You can't explain that! 15:48, 4 August 2010 (UTC) Never mind, about the cash? I was worried you were going to say something about people not believing it happened.. I know people at Dow, I know it happened.. :p QuaruSuzuki - You can't explain that! 15:52, 4 August 2010 (UTC) That would be a pretty shockingly stupid conspiracy if they thought the Bhopal disaster was a hoax... d hominem 15:54, 4 August 2010 (UTC) No, no. What they did is pose as a spokesman for Dow Chemical and were interviewed on BBC, and said they'd be paying$12 billion to Bhopal survivors. a) that shaved \$3-billion off their market cap that day, b) forced them to admit there was no Bhopal relief, c) sadly, raised hopes of the sufferers in Bhopal for a few hours. That part I didn't like. --PsyGremlinSpeak! 16:00, 4 August 2010 (UTC)
Absolutely fucking brilliant. I have never heard of them before, but they are now my new heroes. Thankyou Psygremlin. DeltaStarSenior SysopSpeciationspeed! 22:52, 4 August 2010 (UTC)
That was pretty awesome. They've done some amazing stuff, like suggesting an 'acceptable loss of life vs profit' model. The "Change the World" film is very good, and very scary when you see what big business is prepared to believe.--PsyGremlin講話 10:32, 5 August 2010 (UTC)
Whoops, my bad. It's "Fix the World". handy dl linky here. --PsyGremlinSnakk! 10:38, 5 August 2010 (UTC)
## ¿Español?
¿Estàn otros hispanohablantes en RW? Porque hablo inglès... pero no es mi idioma nativo. ¡Tengo curiosidad! Ricardo 18:03, 4 August 2010 (UTC)
¿Que? QuaruSuzuki - You can't explain that! 18:13, 4 August 2010 (UTC)
Puedo hablar Español, pero cuando tengo que escribir entonces todas mis faltas son muy obvios.--BobSpring is sprung! 18:19, 4 August 2010 (UTC)
This wiki is also available in Serbo-Croat. (Not!) Lily Inspirate me. 18:35, 4 August 2010 (UTC)
I look forward to having to move all our previous work to en.rationalwiki.org... ħuman 22:15, 4 August 2010 (UTC)
Translations wouldn't be a bad idea, but fuck me, the workload... let's just leave Google Translator to do the hard work for us, right? d hominem 00:37, 5 August 2010 (UTC)
¡Hola Ricardo! Mis padres vienen de Nicaragua, pero vivo en Canadá. Donde estoy casi nadie habla español, entonces no tengo mucho oportunidad de practicar. --Night Jaguar (talk) 01:14, 5 August 2010 (UTC)
Bonjourno! Jeg sprechen ikkje de Deutch. DeltaStarSenior SysopSpeciationspeed! 01:36, 5 August 2010 (UTC)
Spanish speaker rising hand & containing urge to correct details in comments above --Xyr (talk) 02:04, 5 August 2010 (UTC)
Creo que sería tan difícil traducir artículos existentes que sería sin punta intentarlo... crear contento nuevo sería mejor que traducir. Yo participaría en eso. Lyra § talk 06:39, 5 August 2010 (UTC)
Ehkä sekin energia olisi paremmin käytetty tämänhetkisen sisällön parantamisessa. Timppeli 07:08, 5 August 2010 (UTC)
if I vaporized this page just for lulz? Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 00:51, 5 August 2010 (UTC)
I dont think anyone would be mad, more they'd think you were stupid and deliberately inciting drama. AceX-102 00:54, 5 August 2010 (UTC)
People already think that. Im really not as stupid as I act here but its just become habitual to act stupid here since that seems (no offense) to be the in thing here. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 00:58, 5 August 2010 (UTC)
I don't think you'd get any laughs for vaping MC's userpage. AceX-102 01:14, 5 August 2010 (UTC)
No. As much as he deserves it, no. Do not delete his page, do not say his name three times. He will be back, I know, but why give him another thing to bitch about?--Thanatos (talk) 01:16, 5 August 2010 (UTC)
Although I agree with you all, that it'd be a bad thing, I do admit that I've often been very, very temped to vape it. Or just alter it up. But I kinda' like that he calls me out on it. It makes me laugh. So that's kept me from it so far. He's such a useless tool. Like a broken shovel, that just won't stay in the garbage.. QuaruSuzuki - You can't explain that! 02:13, 5 August 2010 (UTC)
Im with Quaru on this. I know what to expect from MC and I could give a shit less if MC wrote about me in chicken coop, called me the devil, or pissed all over himself, but I dont want to piss off everybody else over it. But I still think itd be funny to troll the troll. Ex-Troll CheerleaderI'm a teenage girl; get ALLstate to protect yourself from Mayhem like me 02:33, 5 August 2010 (UTC)
itd be funny to troll the troll we did that a year ago - wasn't funny. AceX-102 02:34, 5 August 2010 (UTC)
We wouldn't have let anyone get away with it when *you* were actively trolling the site (i.e. a few weeks/months back), so I would count on the same amount of respect from you. P-Foster (talk) 03:10, 5 August 2010 (UTC)
Christ on an Easter stick, kiddies, who remembers the first RW "troll" who people thought was every troll on sight? ħuman 05:17, 5 August 2010 (UTC)
Who could forget everybody's favorite bogeyman TK? Radioactive afikomen Please ignore all my awful pre-2014 comments. 08:58, 5 August 2010 (UTC)
Heh I thought perhaps you meant Bohdan, grand master of sockpuppetry, if I recall correctly. (Or whatever he calls himself now.) Lyra § talk 09:49, 5 August 2010 (UTC)
You do realise that I am a sock of Bohdan? I came out as a sock many moons ago. Nobody believed me. 86.40.106.235 (talk) 12:01, 5 August 2010 (UTC)
No Marcus, you came in your sock. Lily Inspirate me. 17:17, 5 August 2010 (UTC)
Nah, MC, I knew Bohdan, I worked with Bohdan, you, sir, are no Bohdan. ħuman 18:33, 5 August 2010 (UTC)
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2022-05-22 23:58:31
|
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https://financetrain.com/calculate-and-interpret-covariance-and-correlations
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# Calculate and Interpret Covariance and Correlations
Covariance defined
In probability theory and statistics, covariance measures the comovement between two variables i.e. the amount by which the two random variables show movement or change together.
If the two variables are dependent then the covariance can be measured using the following formula:
For two independent variables the joint densities are separated and the equation becomes:
There is a difference between the covariance of two random variables which is linked to the joint probability distribution and the sample covariance which is an estimated value of the parameter.
Covariance interpreted
In financial markets covariance is positive when the variables show similar behaviour i.e. larger values of one variable correspond to larger values of another variable and the same holds true for smaller values. When the covariance is negative it means the exact opposite i.e. larger values of one variable correspond to smaller values of another variable.
The strength of the linear relationship however cannot be easily interpreted by the magnitude of the calculated value. In order to interpret the strength a related measure called correlation is used.
Correlation defined
The covariance measure is scaled to a unitless number called the correlation coefficient which in probability is a measure of dependence between two variables. Dependence broadly refers to any statistical relationship between two variables or two sets of data.
The formula for correlation between two variables is as follows:
The covarince is scaled by the product of the two standard devations of the variables. This measure is called the Pearson correlation which holds true only when the relationship between two variables is linear in nature. When the relationship is non-linear in nature Spearman correlation or rank correlation is used in order to account for the deviation from linearity.
Correlation interpreted
Pearson: The correlation number would always be in the range of -1 to +1. A value of 1 means that the variables always move in the same direction and a value of -1 means the two always move in the opposite direction. In the case where the variables are independent the covariance is zero which means the correlation is also zero. In other words the two variables do not exhibit any movement relative to each other. Any number in between indicates that the one number moves less positively or negatively in relation to changes in another number.
Spearman: This measure is useful when there might be errors in the data and is less sensitive to outliers and more robust.
Refer to the spreadsheet Covariance-Correlation.xlsx for detailed calculations.
# R Programming Bundle: 25% OFF
Get our R Programming - Data Science for Finance Bundle for just $29$39.
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2022-12-08 05:59:22
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http://wiki.ece.cmu.edu/ddl/index.php/Convex_function
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# Convex function
A convex function is a function such that
$\alpha f(\mathbf{x}) + (1-\alpha)f(\mathbf{y}) \geq f(\alpha \mathbf{x} + (1-\alpha)f(\mathbf{y})$
$\forall \mathbf{x,y}, 0 \leq \alpha \leq 1$
Conceptually (and quite roughly) a convex function can be thought of as being shaped like a "U", where a straight line drawn between any two points on the function always lies on or above the function.
In optimization, convexity of the objective function and constraints is an important property that enables proof of global optimality and allows the use of efficient numerical methods.
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2013-05-23 18:03:48
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http://openstudy.com/updates/558b8666e4b0c2c049014164
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1. sdfgsdfgs
2. kmullis6
@danish071996 @sammixboo @Whitemonsterbunny17
3. kmullis6
Can any of you guys help me?
4. anonymous
@ganeshie8
5. anonymous
@perl
6. anonymous
@wio
7. kmullis6
Ok.
8. perl
x is the number of pages
9. kmullis6
So that's for part A?
10. perl
I am using the basic rate equation cost per page * number of pages = total cost
11. perl
cost per page of office scanner * no. pages > flat rate to rent scanner + cost per page of rented scanner
12. perl
$\Large .09x >70 +.02x$
13. kmullis6
Ok. Sorry, I'm confused. So what you had before was a mistake, and what is above is for A now?
14. perl
yes I am revising as I think about the problem
15. kmullis6
Oh ok
16. perl
THe english inequality should explain how I got the math part. Want to go over it one more time. just to be sure
17. kmullis6
18. kmullis6
I think I get it. Thank you
19. perl
We want $\Large \rm rented ~ scanner ~ cost < office ~scanner ~ cost$$\Large \rm rental ~cost + \frac{cost}{page }\cdot \# pages < \frac{cost}{ page} \cdot \# pages$$\Large \rm 70 +\frac{0.02}{1 page } \cdot x ~pages <\frac{0.09}{1 page} \cdot x~pages \\~$ Finally we have $$\Large 70 + .02x < .09 x$$
20. perl
for part B you can solve the inequality for x
21. kmullis6
1,000
22. perl
correct :)
23. perl
Edit If she scans more than a 1000 pages in a year, she will save money using the rented scanner. if she scans less than a 1000 pages a year she will save money not renting and using the office equipment.
24. kmullis6
Thank you so much for all of your help! (: @perl
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2017-01-23 21:40:20
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http://www.evanlray.com/stat344ne_s2020/schedule.html
|
Click on the text like “Week 1: Jan 20 – 24” to expand or collapse the items we covered in that week.
I will fill in more detail and provide links to lecture notes and labs as we go along. Items for future dates are tentative and subject to change.
#### Wed, Jan 22
• In class, we will work on:
• Register for GitHub here if you haven’t already; I will ask you to provide your GitHub user name in the questionairre below.
• Fill out a brief questionnairre (if you are taking two classes with me, you only need to fill out this questionairre once)
• Fill out this brief poll about when my office hours should be held (if you are taking two classes with me, you only need to fill out this poll once)
• Chollet:
• Skim sections 1.1 – 1.3. This is pretty fluffy and we’ll mostly either skip it or talk about it in much more depth later, but it might be nice to see now for a little context.
• Goodfellow et al.:
• Read the intro to section 5.5 (but don’t worry about KL divergence), read section 5.5.1, lightly skim section 6.1, read the first 3 paragraphs of section 6.2.1.1, and skim sections 6.2.2.1, 6.2.2.2, and 6.2.2.3. We will talk about this over the next day or two.
• Videos
• I moved the videos that were here to later days.
• Homework 1
• Written part due 5pm Wed, Jan 29
• Coding part due 5 pm Fri, Jan 31
#### Fri, Jan 24
• In class, we will work on:
• Maximum likelihood and output activations for binary classification. I don’t have any lecture notes for this.
• Matrix formulation of calculations for logistic regression across multiple observations. I don’t have any lecture notes for this, but it’s written up in Lab 01.
• Highlights of NumPy: https://github.com/mhc-stat344ne-s2020/Python_NumPy_foundations/blob/master/Python.ipynb
• Continue/finish readings listed for Wed, Jan 22.
• Take a look at the NumPy document listed above. You can’t run it directly on GitHub, but if you want you could sign into colab.research.google.com and try out some of the code there. Also cross-reference this with the Numpy section in Chollet.
• Videos: Here are some videos of Andrew Ng talking about logistic regression and stuff we did today; you don’t need to watch these, but feel free if you want a review:
• Logistic regression set up: youtube
• Loss function just thrown out there without justification: youtube
• Discussing set up for loss function via maximum likelihood: youtube
• Start at thinking about “vectorization”, i.e. writing things in terms of matrix operations: youtube
• More on vectorization, but I think this video is more complicated than necessary and you might skip it: youtube
• Vectorizing logistic regression. Note that Andrew does this in the variant where your observations are in columns of the X matrix. I want us to understand that you can also just take the transpose of that and get a just-as-valid way of doing the computations, just sideways. This is worth understanding because different sources and different software packages will organize things different ways and you want mental flexibility. We talked about this in class but I don’t know of a video that explains things with the other orientation. youtube
• Broadcasting in NumPy – This is among my least favorite examples, apologies!
• Homework 1
• Written part due 5pm Wed, Jan 29
• Coding part due 5 pm Fri, Jan 31
#### Mon, Jan 27
• In class, we will work on:
• Maximum likelihood and output activations for regression. Lecture notes: pdf
• Lab 1 about numpy calculations relevant to logistic regression – complete and turn in by Friday, Jan 31.
• Homework 1
• Written part due 5pm Wed, Jan 29
• Coding part due 5 pm Fri, Jan 31
• Lab 1
• Due 5pm Fri, Jan 31
#### Wed, Jan 29
• In class, we will work on:
• Overview/details of maximum likelihood for logistic regression. Demo visualization here.
• Overview/details of maximum likelihood for linear regression. Demo visualization here.
• Maximum likelihood and output activations for multi-class classification. No lecture notes, but see the video linked below.
• Summary of models, activation functions, and losses so far: pdf
• Videos
• Some of you may find these videos helpful:
• Softmax regression is another word for multinomial regression, which is the equivalent to logistic regression for multi-class classification.
• Homework 1
• Written part due 5pm today Wed, Jan 29
• Coding part due 5 pm Fri, Jan 31
• Lab 1
• Due 5pm Fri, Jan 31
#### Fri, Jan 31
• In class, we will work on:
• Quiz on logistic regression
• Highlights from last class (see also first page of lecture notes below)
• More concrete example of calculations for multinomial logistic regression
• Lecture notes: pdf
• We also wrote out that for $$m$$ observations, \begin{align*} z &= \begin{bmatrix} z^{(1)} & \cdots & z^{(m)} \end{bmatrix} = \begin{bmatrix} z^{(1)}_1 & \cdots & z^{(m)}_1 \\ z^{(1)}_2 & \cdots & z^{(m)}_2 \\ \vdots & \ddots & \vdots \\ z^{(1)}_K & \cdots & z^{(m)}_K \end{bmatrix} \\ &= \begin{bmatrix} b_1 + w_1^T x^{(1)} & \cdots & b_1 + w_1^T x^{(m)} \\ b_2 + w_2^T x^{(1)} & \cdots & b_2 + w_2^T x^{(m)} \\ \vdots & \ddots & \vdots \\ b_K + w_K^T x^{(1)} & \cdots & b_K + w_K^T x^{(m)} \end{bmatrix} \\ &= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_K \end{bmatrix} + \begin{bmatrix} w_1^T \\ w_2^T \\ \vdots \\ w_K^T \end{bmatrix} \begin{bmatrix} x^{(1)} & x^{(2)} & \cdots & x^{(m)} \end{bmatrix} \end{align*} where the last equals sign uses Python broadcasting to pull out the vector $$b$$.
• Hand out doing the calculations: pdf
• General Python stuff (also posted on resources page)
• Plotting with matplotlib pdf
• Basic use of Keras pdf
• Homework 1
• Coding part due 5 pm today Fri, Jan 31
• Lab 1
• Due 5pm Fri, Jan 31
• Homework 2
• Coding part due 5pm Fri, Feb 7
• Videos
• Some of you may find this sequence of three videos introducing Keras helpful (note that I will follow the code in Chollet, which differs slightly from code used in these videos):
#### Mon, Feb 03
• In class, we will work on:
• Hidden layers and forward propagation. Lecture notes: pdf. Note there is an error on the bottom of page 1: for multinomial regression, you use a softmax activation, not a sigmoid activation.
• Lab 02
#### Fri, Feb 07
• In class, we will work on:
• Loss functions, activation functions, and their derivatives. pdf
• Gradient checking. Lecture notes: pdf
• Lab 03
#### Wed, Feb 12
• In class, we will work on:
• Continue on backpropagation with hidden layers – notes posted Monday
• Start lab on backpropagation
#### Mon, Feb 17
• In class, we will work on:
• Overfitting and regularization.
• Lecture notes: pdf
• Python notebook: GitHub
• Homework 3 due 5pm Fri, Feb. 21
• Videos:
• Chapter 4 of Chollet
• Section 7.1.1 of Goodfellow et al up through Equation 7.5.
#### Fri, Feb 21
• In class, we will work on:
• Start on convolutional neural networks: pdf
• Homework 3 due 5pm today Fri, Feb. 21
#### Mon, Feb 24
• In class, we will work on:
• Point of problem 1 on last HW – linear activations in hidden layers don’t achieve anything: pdf
• Miscellaneous stuff about CNNs: pdf
• First example of a full CNN: pdf
• Homework 4 due 5pm Fri, Feb. 28
#### Wed, Feb 26
• In class, we will work on:
• Generators in Python, Data Augmentation for image data: pdf
• Lab on CNNs
• Homework 4 due 5pm Fri, Feb. 28
#### Fri, Feb 28
• In class, we will work on:
• Overview of common architectures, transfer learning
• Homework 4 due 5pm Fri, Feb. 28
#### Mon, Mar 02
• In class, we will work on:
#### Wed, Mar 04
• In class, we will work on:
• Maybe we’ll do Midterm 1 covers material up through Wed, Feb 19
#### Fri, Mar 06
• In class, we will work on:
• Or else possibly we’ll do Midterm 1 covers material up through Wed, Feb 19
#### Mon, Mar 09
• In class, we will work on:
#### Wed, Mar 11
• In class, we will work on:
#### Fri, Mar 13
• In class, we will work on:
#### Mon, Mar 16
• No Class: Midsemester Break. Safe travels!
#### Wed, Mar 18
• No Class: Midsemester Break. Safe travels!
#### Fri, Mar 20
• No Class: Midsemester Break. Safe travels!
#### Mon, Mar 23
• In class, we will work on:
#### Wed, Mar 25
• In class, we will work on:
#### Fri, Mar 27
• In class, we will work on:
#### Mon, Mar 30
• In class, we will work on:
#### Wed, Apr 01
• In class, we will work on:
#### Fri, Apr 03
• In class, we will work on:
#### Mon, Apr 06
• In class, we will work on:
#### Wed, Apr 08
• In class, we will work on:
#### Fri, Apr 10
• In class, we will work on:
#### Mon, Apr 13
• In class, we will work on:
#### Wed, Apr 15
• In class, we will work on:
#### Fri, Apr 17
• In class, we will work on:
#### Mon, Apr 20
• In class, we will work on:
#### Wed, Apr 22
• In class, we will work on:
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2020-02-26 20:37:25
|
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|
https://web2.0calc.com/questions/help_73747
|
+0
# help
+1
84
1
Let $$z = 3[\cos(85^{\circ})+i\sin(85^{\circ})]$$. Find the smallest positive value of $$r$$ such that $$r[\cos(45^{\circ})+i\sin(45^{\circ})] = z^n$$ for some positive integer $$n$$.
Dec 20, 2019
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2020-04-06 07:46:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9980412125587463, "perplexity": 797.6083010628977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371620338.63/warc/CC-MAIN-20200406070848-20200406101348-00009.warc.gz"}
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https://stacks.math.columbia.edu/tag/01NZ
|
Lemma 27.16.6. In Situation 27.15.1. The scheme $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ constructed in Lemma 27.15.4 and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$.
Proof. Let $X$ be the scheme representing the functor $F$. Note that $X$ is a scheme over $S$ since the functor $F$ comes equipped with a natural transformation $F \to h_ S$. Write $Y = \underline{\text{Proj}}_ S(\mathcal{A})$. We have to show that $X \cong Y$ as $S$-schemes. We give two arguments.
The first argument uses the construction of $X$ as the union of the schemes $U_ d$ representing $F_ d$ in the proof of Lemma 27.16.5. Over each affine open of $S$ we can identify $X$ with the homogeneous spectrum of the sections of $\mathcal{A}$ over that open, since this was true for the opens $U_ d$. Moreover, these identifications are compatible with further restrictions to smaller affine opens. On the other hand, $Y$ was constructed by glueing these homogeneous spectra. Hence we can glue these isomorphisms to an isomorphism between $X$ and $\underline{\text{Proj}}_ S(\mathcal{A})$ as desired. Details omitted.
Here is the second argument. Lemma 27.15.5 shows that there exists a morphism of graded algebras
$\psi : \pi ^*\mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \mathcal{O}_ Y(n)$
over $Y$ which on sections over affine opens of $S$ agrees with (27.10.1.3). Hence for every $y \in Y$ there exists an open neighbourhood $V \subset Y$ of $y$ and an integer $d \geq 1$ such that for $d | n$ the sheaf $\mathcal{O}_ Y(n)|_ V$ is invertible and the multiplication maps $\mathcal{O}_ Y(n)|_ V \otimes _{\mathcal{O}_ V} \mathcal{O}_ Y(m)|_ V \to \mathcal{O}_ Y(n + m)|_ V$ are isomorphisms. Thus $\psi$ restricted to the sheaf $\pi ^*\mathcal{A}^{(d)}|_ V$ gives an element of $F_ d(V)$. Since the opens $V$ cover $Y$ we see “$\psi$” gives rise to an element of $F(Y)$. Hence a canonical morphism $Y \to X$ over $S$. Because this construction is completely canonical to see that it is an isomorphism we may work locally on $S$. Hence we reduce to the case $S$ affine where the result is clear. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2021-10-20 11:08:55
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https://www.merry.io/algebraic-topology/18-attaching-spaces-and-cell-complexes/
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In this lecture we introduce a particular cool class of topological spaces, called cell complexes.
The importance of cell complexes (also called CW complexes) in algebraic topology cannot be overstated. This semester we will only scratch the surface: we'll show that many common spaces carry the structure of a cell complex, and in a couple of lectures time we'll define cellular homology, which is a superior version of singular homology tailor-made for cell complexes.
We won't study the main reason they are important until next semester. Roughly speaking, we'll see in Algebraic Topology II that from the point of view of any homology/homotopy functor, every space is a cell complex! More precisely, we'll show that one can "approximate" an arbitrary topological space $X$ by a cell complex $Y$ in such a way that a homology/homotopy functor $H_n$ or $\pi_n$ cannot tell the difference.
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2019-10-21 11:00:18
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https://theculture.sg/2018/04/solutions-to-review-1/
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Question 1
(i)
$y = f(x) = \frac{x^2 + 14x + 50}{3(x+7)}$
$3y(x+7) = x^2 + 14x + 50$
$x^2 + (14-3y)x + 50 - 21 y = 0$
$\text{discriminant} \ge 0$
$(14-3y)^2 - 4(1)(50-21y) \ge 0$
$196 - 84y + 9y^2 - 200 + 84y \ge 0$
$9y^2 - 4 \ge 0$
$(3y - 2)(3y + 2) \ge 0$
$y \le - \frac{2}{3} \text{~or~} y \ge \frac{2}{3}$
(ii)
Using long division, we find that
$y = \frac{x^2 + 14x + 50}{3(x+7)} = \frac{x}{3} + \frac{7}{3} + \frac{1}{3(x+7)}$
So the asymptotes are $y = \frac{x}{3} + \frac{7}{3}$ and $x = -7$
Question 2
(i)
$x^2 - 9y^2 + 18y = 18$
$x^2 - 9(y^2 - 2y) = 18$
$x^2 - 9[(y-1)^2 - 1^2] = 18$
$x^2 - 9(y-1)^2 + 9 = 18$
$x^2 - 9(y-1)^2 = 9$
$\frac{x^2}{9} - (y-1)^2 = 1$
This is a hyperbola with centre $(0, 1)$, asymptotes are $y = \pm \frac{x}{3} + 1$, and vertices $(3, 1)$ and $(-3, 1)$.
$y = \frac{1}{x^2} + 1$ is a graph with asymptotes $x = 0$ and $y=1$.
Use GC to plot.
(ii)
$\frac{x^2}{9} - (y-1)^2 = 1$—(1)
$y = \frac{1}{x^2} + 1$ —(2)
Subst (2) to (1),
$\frac{x^2}{9} - (\frac{1}{x^2} + 1 - 1)^2 = 1$
$\frac{x^2}{9} - (\frac{1}{x^2})^2 = 1$
$x^2 - \frac{9}{x^4} = 9$
$x^6 - 9 = 9x^4$
$x^6 - 9x^4 - 9 = 0$
(iii)
From graph, we observe two intersections. Thus, two roots.
Question 3
(ai)
$\sum_{r=1}^n (r+1)(3r-1)$
$= \sum_{r=1}^n (3r^2 + 2r -1)$
$= \sum_{r=1}^n 3r^2 + \sum_{r=1}^n 2r - \sum_{r=1}^n 1$
$= 3 \sum_{r=1}^n r^2 + 2 \sum_{r=1}^n r - \sum_{r=1}^n 1$
$= 3 \frac{n}{6}(n+1)(2n+1) + 2 \frac{n}{2}(1 + n) - n$
$= \frac{n}{2}(n+1)(2n+1) + n(1+n) - n$
$= \frac{n}{2}(n+1)(2n+1) + n^2$
(aii)
$2 \times 4 + 3 \times 10 + 4 \times 16 + ... + 21 \times 118$
$= 2 [2 \times 2 + 3 \times 5 + 4 \times 8 + ... + 21 \times 59]$
$= 2 [(1+1) \times (3 \cdot 1 - 1) + (2+1) \times (3 \cdot 2 -1) + (3+1) \times (3 \cdot 3 -1) + ... + (20+1) \times (3 \cdot 20 -1) ]$
$= 2 \sum_{r=1}^{20} (r+1)(3r-1)$
$= 2 [\frac{n}{2}(n+1)(2n+1) + n^2 ]$
$= n(n+1)(2n+1) + n^2$
$= n(2n^2 + 3n + 1) + n^2$
$= 2n^3 + 4n^2 + n$
(bi)
$\frac{2}{(r-1)(r+1)} = \frac{A}{r-1} - \frac{B}{r+1}$
$2 = A(r+1) - B(r-1)$
Let $r = -1$
$2 = - B(-2) \Rightarrow B = 1$
Let $r = 1$
$2 = A(2) \Rightarrow A = 1$
$\therefore \frac{2}{(r-1)(r+1)} = \frac{1}{r-1} - \frac{1}{r+1}$
(bii)
$\sum_{r=2}^n \frac{1}{(r-1)(r+1)}$
$= \frac{1}{2} \sum_{r=2}^n \frac{2}{(r-1)(r+1)}$
$= \frac{1}{2} \sum_{r=2}^n (\frac{1}{r-1} - \frac{1}{r+1})$
$= \frac{1}{2} [ 1 - \frac{1}{3}$
$+ \frac{1}{2} - \frac{1}{4}$
$+ \frac{1}{3} - \frac{1}{5}$
$...$
$+ \frac{1}{n-3} - \frac{1}{n-1}$
$+ \frac{1}{n-2} - \frac{1}{n}$
$+ \frac{1}{n-1} - \frac{1}{n+1}]$
$= \frac{1}{2} [1 + \frac{1}{2} - \frac{1}{n} - \frac{1}{n+1}]$
$= \frac{1}{2} (\frac{3}{2} - \frac{n+1+n}{n(n+1)})$
$= \frac{3}{4} - \frac{2n+1}{2n(n+1)}$
(biii)
As $n \to \infty$, $\frac{1}{n} \to 0$ and $\frac{1}{n+1} \to 0$, the sum of series tends to $\frac{3}{4}$, a constant. Thus, series is convergent.
(biv)
$\sum_{r=5}^{n+3} \frac{1}{(r-3)(r-1)}$
Replace $r$ by $r + 2$. Then we have
$\sum_{r=3}^{n+1} \frac{1}{(r-1)(r+1)}$
$= \sum_{r=2}^{n+1} \frac{1}{(r-1)(r+1)} - \frac{1}{(2-1)(2+1)}$
$= \frac{3}{4} - \frac{2(n+1)+1}{2(n+1)[(n+1)+1]} - \frac{1}{3}$
$= \frac{5}{12} - \frac{2n+3}{2(n+1)(n+2)}$
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2018-06-25 13:29:25
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https://publications.hse.ru/en/articles/317469613
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• A
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Regular version of the site
High extremes of Gaussian chaos processes: a discrete time approximation approach
Theory Probability and its Applications. 2018. Vol. 63. No. 1. P. 1-21.
A. I. Zhdanov, V. I. Piterbarg.
Let $\mathbf{\boldsymbol{\xi}}(t)=(\xi_{1}(t),\ldots,\xi_{d}(t))$ be a Gaussian zero mean stationary a.s. continuous vector process. Let $g\colon{\mathbb{R}}^{d}\to {\mathbb{R}}$ be a homogeneous function of positive degree. We study probabilities of high extrema of the Gaussian chaos process $g(\mathbf{\boldsymbol{\xi}}(t))$. Important examples are products of Gaussian processes, $\prod_{i=1}^{d}\xi_{i}(t)$, and quadratic forms $\sum_{i,j=1}^{d}a_{ij}\xi_{i}(t)\xi_{j}(t)$. Methods of our studies include the Laplace saddle point asymptotic approximation and the double sum asymptotic method for probabilities of high excursions of Gaussian processes. For the first time, using the double sum method, we apply the discrete time approximation with refining grid.
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2020-08-08 17:58:18
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|
https://unix.stackexchange.com/questions/272872/concatenate-pdfs-but-extend-pdfs-to-be-even-number-of-pages
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# Concatenate PDFs but extend pdf's to be even number of pages
I want to concatenate a bunch of PDFs but for printing purposes I would prefer that empty pages are added to each document that have an odd number of pages. Can I do this with PDFTK?
• Can you use LaTeX? – terdon Mar 29 '16 at 17:07
• Sure I can @terdon – adamse Mar 29 '16 at 17:38
Here's a simple little script that will iterate over all pdf files in the current directory and concatenate them into a single PDF, using LaTeX. PDFs with an odd number of pages will have an extra blank page added after them:
#!/bin/bash
cat<<EoF > all.tex
\documentclass{article}
\usepackage{pdfpages}
\begin{document}
EoF
## rename the PDFs to something safe
c=0;
for f in *pdf
do
## Link the PDF with a safe name
ln -s "$f" "$c".pdf
## Include the PDF in the tex file
printf '\includepdf[pages=-]{%s.pdf}\n' "$c" >> all.tex; ## Get the number of pages pages=$(pdfinfo "$c".pdf | grep -oP '^Pages:\s*\K\d+') ## Add an empty page if they are odd [[$(expr "$pages" % 2) != 0 ]] && printf '%s\n' "\newpage\null" >> all.tex ((c++)); done printf '\\end{document}' >> all.tex; pdflatex all.tex Since this is LaTeX, you can do all sorts of extra stuff. For example, you can have each PDF in its own section, with a clickable table of contents: #!/bin/bash cat<<EoF > all.tex \documentclass{article} \usepackage{pdfpages} \usepackage[colorlinks=true,linkcolor=blue,linktoc=page]{hyperref} \begin{document} \tableofcontents \newpage EoF ## rename the PDFs to something safe c=0; for f in *pdf do ## Link the PDF with a safe name ln -s "$f" "$c".pdf ## Include the PDF in the tex file cat<<EoF >> all.tex \section{${f//.pdf}}
\includepdf[pages=-]{$c.pdf} EoF ## Get the number of pages pages=$(pdfinfo "$c".pdf | grep -oP '^Pages:\s*\K\d+') ## This time, because each PDF has its own section title ## we want to add a blank page to the even numbered ones [[$(expr "$pages" % 2) = 0 ]] && printf '%s\n' "\newpage\null\newpage" >> all.tex ((c++)); done printf '\\end{document}' >> all.tex; ## Need to run it twice to build the ToC pdflatex all.tex; pdflatex all.tex; • may be \cleardoublepage ? – JJoao Mar 31 '17 at 10:13 Certainly. Just create a blank page with (for example) echo "" | ps2pdf -sPAPERSIZE=a4 - blank.pdf and add blank.pdf to every document that has an odd number of pages. E.g. pdftk \ BLANK=blank.pdf \ A=foo1.pdf \ B=foo2.pdf \ C=foo3.pdf \ cat A BLANK B BLANK C \ output bar.pdf • Thanks, but I'm looking for a more automated way since I have many PDFs. – adamse Mar 29 '16 at 14:45 • Ah, An automated way should also be possible, you'd have to ask pdfinfo for the size of the PDFs and proceed accordingly. You should make clear in your question that you are looking for an automated method, and also perhaps include an example of what you have in mind. – Faheem Mitha Mar 29 '16 at 15:21 • Interesting! I'll try to make something. – adamse Mar 29 '16 at 15:22 For future wanderers like myself. I wrote a script combining the answer from @Faheem Mitha and the answer here #!/bin/sh # USAGE: ./concat-pdf.sh *.pdf output-file.pdf ALL_FILES="" BLANK_FILE="/tmp/blank.pdf" for OUTPUT_FILE; do true; done if test -f "$OUTPUT_FILE"; then
echo "$OUTPUT_FILE already exists" exit 1 fi echo "" | ps2pdf -sPAPERSIZE=a4 -$BLANK_FILE
for PDF_FILE in $* ;do if [$PDF_FILE != "$OUTPUT_FILE" ]; then pages=$(strings < $PDF_FILE | sed -n 's|.*Count -\{0,1\}$$[0-9]\{1,\}$$.*|\1|p' | sort -rn | head -n 1) ALL_FILES="$ALL_FILES $PDF_FILE" [$((pages%2)) -ne 0 ] && ALL_FILES="$ALL_FILES$BLANK_FILE"
fi
done
echo "pdftk $ALL_FILES cat output output.pdf" pdftk$ALL_FILES cat output output.pdf
rm -f \$BLANK_FILE
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2019-11-18 04:35:11
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https://www.physicsforums.com/threads/sine-waves-and-4th-derivative-and-4-dimensional-space-time-relationship.428866/
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# Sine waves and 4th derivative and 4 dimensional space-time relationship?
1. Sep 14, 2010
### ThePhysicsGuy
The 4th derivative of a sine wave is itself. This, of course, not only is a fact, but can be used to define the sine wave. My question is, are there any theories relating the abundance of sine waves in QM to the fact that the largest dimensions of space-time are together 4 dimensional? For example, I have often heard string theory explained as vibrating branes. Vibration tends to imply sine wave, so does string theory explain the causes of this vibration in terms of a 4th derivative being equal to the undifferentiated function?
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2018-03-23 19:46:36
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https://nxttime.wordpress.com/category/filter/nonlinear-complementary-filter/
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Archives for category: Nonlinear complementary filter
This time I’ll show you the first results of incorporating a 3-axis magnetometer (or compass) into my IMU filter.
A quick round-up first. I have a IMU sensor that houses a 3-axis accelerometer and a 3-axis gyro sensor. Well, actually I got two, one that I built myself and one from Dexter Industries. To use this type of sensor I have written a non-linear complementary filter (NLC-filter) who’s job it is to fuse the output of the two sensors to a drift free low noise attitude signal. The filter tells me the tilt over the x-axis and tilt over the y-axis. It also tells me the heading. The heading signal is not drift-free. The reason for this is that the accelerometer can provide tilt data but it cannot provide heading data. For this you need a compass.
A compass sensor should really be called an magnetometer because it doesn’t give you heading like a sailors compass does. Instead it gives you the force of the magnetic field over three axis. From this one can calculate heading.
It was technically not very difficult to incorporate the magnetometer data into the filter. Although it took me a long time to figure out how to do this. The main thing was to tune the PI-controller for the compass. I’m not finished with this but I can show you some first results of the filter anyway. The measurements were taken with the setup that you see in the picture. The NXT is mounted on a platform some 20 cm above the ground. This is to minimize effects of the steel enforced concrete floor. It can rotate endlessly as it is mounted on a turn table and driven by a NXT motor and a worm wheel. The worm wheel in combination with a regulated motor gives a very stable rotation speed. The sensors are mounted 10 cm from the NXT to prevent disturbances from the NXT. Measurements are sent over bluetooth to the PC in real time.
this first graph was taken while the NXT was stationary. It shows how stable the filter signal is. Note that the scale of the vertical ax is 0.1 degree, the vertical axis ranging from -4 to 0 degrees. The roll and pitch signals are very stable. During the 15 minutes of the test these signals stayed within a bandwidth of 0.4 degrees. This is all that is left of the gyro drift. The yaw signal (that is controlled by the compass) is less stable, it floats between -0.8 and -2.0 degrees. But also in the yaw signal the gyro drift is eliminated to a large extend. I am very, very pleased with these noise levels as the bandwidth of noise from an accelerometer is around 5 degrees.
The second graph compares the output signal from the compass with that of the filter. This time the NXT was spinning around the Z-axis. The graph shows the results of a full spin. . You basically see two descending lines, indicating a constant movement of the sensors. The signal from the compass, in blue, is not exactly over the signal from the filter. There is a slight difference in the heading that these two report. The reason for this is that the setup of the NXT is not exactly level. The compass signal suffers from this. Its line is not really straight but it is like an S-curve due to this. The filter signal on the other hand is tilt compensated and almost straight. The smoother red line also indicates a lower noise level.
This last image shows the effect of a magnetical disturbance on the compass sensor and on the filter. While taking this graph I slowly moved a head phone along side of the sensor.
The compass suffers from this disturbance, at the peak the signal is over 10 degrees of. The filter on the other hand is not affected by the disturbance. It detects the disturbance and ignores the compass signal while it is disturbed. During this time the filter relies solely on the gyro to keep track of heading.
I think the low noise levels of the filter are nice. But the true advantages are in the tilt compensation and the robustness to magnetic disturbances. The filter is equally robust to disturbances in acceleration I think. This however I cannot show you using this setup.
I have included a compass into my INS filter. It now is drift-free and accurate over all three axis. It also is very robust to magnetic disturbances. The compass itself is not. It gives me great pleasure to see the first results.
Tuning the filter is difficult though. I have used the NXJChartingLogger that will be part of Lejos 0.9.1 a lot. The filter provides all kinds of output that is displayed real time on the pc. Different parameters are logically grouped into different graphs. The graphs provide information on the state of the filter and thus help in tuning.
I have also included a method to introduce errors in the sensor data. This enables me to examine the performance of the filter.
As a last thing I have made the filter calculate the initial state of the filter. With this the filter settles accurately (<.5degrees) within half a second.
The performance dropped to 97 Hertz with three sensors attached. But I haven't optimized the compass code yet.
In later posts I'll demonstrate the performance of the filter.
By request I will publish the Lejos software for the dIMU sensor.
The software contains of drivers for the Dexter Industries IMU sensor, programs to calibrate the sensor and the IMU filter that I wrote about on this blog before. The filter can be used for other IMU sensors as well. Also included is the sample program that utilizes this filter. It is the very same program I used in the video’s published in a previous post. There is no formal support and no warranty. If you are going to use it you will have to rely on your own wit, the javadoc and on this blog.
You can download the software from this link. After downloading you will have to extract it keeping the directory structure intact. Make sure yor compiler can find the code.
# Using the IMU drivers
The dIMU consists of two sensors, a gyroscope and a accelerometer, each has it’s own driver. The gyroscope driver is called L3G4200D and the driver for the accelerometer is MMA7455L. (I use the technical names of the sensors a driver names.) On top of these drivers I made user interfaces that allowes you to examine, configure and calibrate the sensors. The user interfaces are called L3G4200D_E and MMA7455L_E respectively. You can use these programs as driver as well. It gives your programs access to the additional functionality but it needs the NXT screen and might result in larger programs. There is a sample program included that gives access to the user interfaces, it is called testDIMU.
This is how you enable the drivers in your code,
SensorPort.S4.i2cEnable(I2CPort.HIGH_SPEED);
MMA7455L accel = new MMA7455L(SensorPort.S4);
L3G4200D gyro = new L3G4200D(SensorPort.S4);
The first line instructs Lejos to use high speed I2C. The sensors support this.
This is how you get data (rate, acceleration and tilt) from the sensors.
float[] rate = new float[3];
gyro.fetchAllRate(rate);
float[] accel = new float[3];
accel.fetchAllAccel(accel);
float[] tilt = new float[3];
accel.fetchAllTilt(tilt);
As you can see the drivers return data from all three axis is one call. If you just need data from one axis you can get it from the arrays. The order is X, Y, Z. The gyro returns degrees per second by default. The accelerometer returns Milli-G for acceleration and Degrees for tilt, also by default. If you want to use other units in your programs you can change the default values like this.
gyro.setRateUnit(RateUnits.RPS);
accel.setAccelUnit(AccelUnits.MS2);
This changes the default units to radians per second, meter per second^2 and radians respectively. Other units are available, check the javaDoc. The default unit can be overridden per data request by specifying the desired unit as second parameter in the FetchAll… methods.
# Configuring the sensors
The gyro can be configured for dynamic range and sample rate using the setRange and setSampleRate methods. As a rule one should select the lowest value that your application allows for. This gives data of best possible quality.
The accelerometer cannot be configured. I found this of little use.
# Calibrating the sensors
The gyro of my dIMU doesn’t really need calibrating. however there is the possibility to do so. Calibration is started b calling gyro.calculateOffset(). During calibration the sensor should remain motionless. Calibration settings of the gyro are not stored, so they are lost when your program terminates. (Storing calibration settings of gyro sensors is of no use as the calibration values depend on environmental factors and are not stable over time.)
The accelerometer needs calibration. The user interface driver provides functionality to calibrate the sensor and to store the calibration settings. The (base) driver looks for stored calibration settings upon initialization and loads these automatically when they are available. Calibration settings of the accelerometer are stable over time so you’ll need to do this just once. Each of the three axes has to be calibrated separately. To calibrate an axis one has to point it straight up first and hold still while the calibration routine collects data samples. Then the axis has to be pointed straight down and held still for some time. Follow the on screen instructions and do not forget to save the settings after calibration.
# Using the IMU filter
The IMU filter can be used with any three-axis gyro and any three-axis accelerometer as long as their drivers implement the RataData and TiltData interfaces. This is how you initialise the filter
NLCFilter attitude = new NLCFilter(gyro, accel);
attitude.start();
The two parameters to the filter constructor are the gyro driver and accelerometer driver. One could leave out the accelerometer, the filter will work but values will drift over time. The second line of code starts the filter. The filter needs 2 to 5 seconds to settle at start up, therefore you need to keep the sensor motionless and more or less level for a few seconds. You can find out if the filter is ready settling with the Initializing() method.
The IMU filter keeps track of the attitude, or pose, of your sensor/robot. You can query the roll, pitch and yaw angles this way
attitude.setTiltUnit(TiltUnits.RADIANS);
float roll=attitude.getRoll();
float pitch=attitude.getPitch();
float yaw=attitude.getYaw();
or
float[] tilt = new float[3];
attitude.fetchAllTilt(tilt);
By default these methods return angles in radians. You can change this by setting a different unit with the setTiltUnit() method.
You can also use the filter to transform data from a body frame to a world frame. This is useful if another sensor returns data that needs to be corrected for the robots current attitude. the next example transforms a distance returned by a UltraSonic sensor to world coordinates. The example assumes the us and IMU sensors are aligned and that the US sensor measures along the X-axis.
Matrix usBody=new Matrix(1,3);
Matrix usWorld=null;
us = new UltrasonicSensor(SensorPort.S1);
usBody.set(0,0,us.getDistance());
usWorld=attitude.transformToWorld(usBody);
The matrix usWorld now contains the distance from sensor to the detected object over the X, Y and Z axis.
# Configuring and tuning the IMU filter
By default the IMU filter updates as often as possible. It’s update frequency is over 100 Hertz. To free computer resources one can lower the update frequency by using the setFrequency() method. The filter will try to match the specified frequency. A parameter value of 0 will run the filter at maximum speed.
The filter can be tuned to increase the quality of its output. I advise you not to tune the filter until you are familiar with it and understand its inner workings. Tuning consists of altering the P and I values of it’s internal PI controller. The I value takes care of gyro drift cancellation and the P value controls how fast attitude is corrected by use of tilt data from the accelerometer. The values can be modified by using the setKp() and setKi() methods.
There are two ways the filter can assist you in tuning. It keeps track of the integral of absolute errors, or absI. This is a measure of the total error of the filter over time. The smaller the error (given a fixed period) the better the filter performs. /* The filter also allows you to send data over bluetooth to the pc for examination. For this one has to use the NXTChartingLogger on the pc that is part of the Lejos distribution. You instruct the filter to send its state to the pc by supplying a NXTDataLogger object with the setDataLogger() method. */
# Running the demo program
The demo program is called testIMU. At startup of this program the sensor must be horizontal and motionless. The sensor is assumed to be aligned ith the NXT brick with the sensor plug facing to the same direction as the sensor ports. Once you see the wire frame you can start moving the sensor.The demo has four display modes:
• Wire frame. Here it shows a wire frame of the sensor on the NXT screen
• Rotation matrix. The screen will show the content of the rotation matrix. In this matrix the current attitude is stored. The matrix is also used to convert body coordinates to world coordinates by matrix multiplication..
• Roll, Pitch, Yaw. The screen shows the Roll, Pitch, Yaw angles of the sensor.
• Update speed. The screen shows the actual update speed of the filter.
You can browse through the display modes by using the arrow keys. The enter key resets the filter. The clip below shows the demo program in action in wire frame mode.
Some time ago I wrote about transformation matrices. This time I want to show you how to use it. You’ll see how much you can benefit from it.
I wrote a small class that can draw a compass rose on the NXT screen. The class contains a collection of points that are the start and end points of the lines that make the compass rose. It has a method to draw these lines. But before drawing the lines they are transformed using the current attitude of the IMU sensor. The transformation is a matrix multiplication of the compass rose definition and the attitude matrix that the IMU filter maintains. This is done in line 4 of the code example below. R is the transformation matrix, rose is the compass definition. The function toScreen on the same line centers the image on the screen.
public void draw(Matrix R) {
Matrix ss;
Graphics g=new Graphics();
ss=toScreen(R.times(rose));
for (int i=0;i<elements;i++) {
g.drawLine((int)ss.get(0,i*2),(int)ss.get(1,i*2),(int)ss.get(0,i*2+1),(int)ss.get(1,i*2+1));
}
}
As you can see the transformation matrix allows you to perform very complex functionality, three rotations, using a very simple function, times().
Below is a video of the application in action. You might want to play this video in HD as it is quite hard to see the details on the NXT screen.
This application just shows a compass rose. But the same technique can be used in a wide variety of applications. One such application could be to transform data from range sensors to world coordinates. This will ease the process of mapping the environment. As the matrix maintained by the IMU filter is 3 dimensional you can even make a 3D map.
Transformation matrices are also very useful for robots with holonomic wheels. Here they can be used to transform robot speed into wheel speed.
Here is another video. This time it features the dIMU from Dexter Industries.
In a previous post I showed how the Filter for the IMU sensor combines both gyro and accelerometer data to get drift-free low noise information about the robots orientation in space. The filter performs a second function, it translates sensor output to a world frame of reference. In this post I’ll explain what I mean by that and how the filters works in this respect.
# Body frame of reference versus the world frame of reference
Suppose you are standing upright facing north when somebody next to you asks you to turn right for 180 degrees. After you turned so you’ll be facing south. Clear enough. Now suppose you are lying on the ground face down and head pointing south. If you are asked again to turn right for 180 degrees by the same person you will have to consider two options regarding the meaning of the word right. The first option is that right should be in reference to your body. If you turn like this you’ll end up face up while still pointing north. The second option is to treat right in reference to the person that asked the question. If you turn like that you’ll end up still facing down but with the head pointing south.
The same goes with robots and the sensors they carry. Your robot might have a gyro that indicates that the robot turns clockwise, but if the robot is upside down you’ll see itt turning counter clockwise. This is because the robot and the gyro attached to it have a different reference than you have.
To distinguish between the two references we call the first one the body frame of reference and the second the world frame of reference. The world frame of reference is supposed to be inert, a global frame where everything can relate to. Words like up, down, north and south are related to the world frame of reference. The body frame of reference belongs to the robot and the sensors. When the robot moves its body frame of reference moves along with it. Robot commands like drive forward are always in respect to the body frame of reference. Likewise, data from the sensors are also expressed in the body frame of reference.
In robotics, you’ll frequently have to translate information from one frame of reference into the other. For example, when you calculate the position of the robot (world frame of reference) from data from the tachometer (body frame of reference). The process to translate information from one frame to another is called transformation.
# Transformations
A transformation requires the use of trigonometry. Suppose again that you want to calculate the position of the robot (world frame of reference) from data from the tachometer (body frame of reference). Let us assume that the tachometers indicate the the robot has travelled straight forward for 30 cm. In terms of body frame it is said that it moved over the x-axis with 30 cm and over the y-axis with 0 cm. Suppose the robot was facing east when it travelled and you have called the east to west axis the y-axis as is customary. It is clear then that in the world frame of reference the position of the robot increases by 30 cm along the y-axis and maintained its position along the x-axis. In mathematical terms the transformation from body frame to world frame is:
Xw = Xb * cos(a) - Yb * sin(a)
Yw = Xb * sin(a) + Yb * cos(a)
Where the parameter a stands for the angle between the robot (or better the body frame of reference) and the world. In the above example a=90, making cos(a)=0 and sin(a)=1. as a result the x and y values are swapped in the transformation.
Transforming sensor output to world frame of reference is the second function of the IMU filter. However, It does so in a 3D space and not in a 2D space as the example.
The preferred way to model a transformation mathematically is by using matrices and vectors.
This is the preferred way as it offers a convenient and short way to describe both the transformation and sensor output. Without going in too much detail, the transformation is stored in a transformation matrix of 3 by 3 elements. The sensor output is stored in a vector of 3 elements. To transform a vector from body frame of reference to world frame of reference one has to execute a matrix multiplication on the body vector using the transformation matrix. In mathematical terms:
Vw = Vb X T
Where T stands for transformation matrix. To go from world to body the formula is:
Vb = Vw X Tinv
Where Tinv represents the inverse matrix of T.
Internally the filter maintains a transformation matrix. The robots current attitude can be calculated from the transformation matrix.
Here a quick post of some graphs that show the effect of the filter I use for my IMU.
You might have seen a previous posts where I show you that my sensor is capable of detecting a grain of sugar under the NXT. If you haven’t seen it, check the post and video here. Few months ago I changed from robotC to Lejos. And now I have rewritten the sensor drivers and the filter in Java.
The Lejos developers are currently working on a very nice logging feature that allows users to display data from the NXT real time on the pc. I got hold of the beta code and tested it without any problems. I used the NXTDataLogger (this is the name of the Java class) to made the intrinsics of the IMU filter visible.
The IMU sensor combines a 3-axis gyroscope and a 3-axis accelerometer. The filter reads the output from both sensors, combines there signals in a smart way and provides stable information about the orientation (yaw, pitch and roll) of the robot.
An accelerometer can provide tilt data (pitch and roll) as can be seen in the first graph. This sensor however has two serious drawbacks. First it is quite noisy, that’s why the graph lines seems so hairy. Second, it does not distinguish tilt from acceleration. So, the output from this sensor can have different causes. This is why you cannot make a balancing robot with just an accelerometer.
A gyroscope is far less noisy, that is why the second graph shows smooth lines. It is also not affected by acceleration. However also this sensor has its particularities. First, it doesn’t measure tilt, but it measures rate of turn. This can only be translated into tilt if the starting condition is known. To get the tilt at any time you have to integrate all the readings and the starting condition. In the process small errors are introduced, in the long run they add up to la large error. The green line in the graph shows this effect. At about 42 seconds the sensor rotates faster than its maximum range (>2000 degrees/second), it then gives faulty readings, these are integrated in the calculated tilt. As a result the line ends higher than it started, even though the sensor was turned back to its original position. The second effect that makes a gyro hard to use in long runs is called drift. Due to temperature changes or voltage drops the base level of a gyroscope changes. This means that after some time the gyro seems to be turning slowly when it is still being stationary. This effect is well visible in the blue line. This goes up slowly. This effect can not be eliminated by calibration, unless you are able to calibrate the sensor along the run.
The filter is like a PI-controller. It uses the data from the gyroscope as the basis of it’s output. The lines in the third graph, that shows the output of the filter, are smooth because of this. It uses the data from the accelerometer to correct the output from any errors that have been build up. But it does so slowly in order to keep the noise out of the output signal. The P-value of the PI controller corrects for errors in the gyro signal. It also makes the filter usefull when starting conditions are unknown. The filter also sums past errors to get an I-value, this value is used to correct for drift in the gyro data. The result is a graph with smooth drift-free lines.
Two final notes. First, the green line (the yaw-line) is not drift-free. This is because an accelerometer cannot provide information of orientation in the XY-plane. You need a compass sensor to provide this information. Second, the range is expressed in cosine of the angle, so 1 corresponds to zero degrees, 0 corresponds to 90 degrees.
Here is a short update of my works.
I’m doing two projects at the same time.
First, there is Sidbot. It waits till I master Java threads.
Second, I’m building a holonomic or Killough pilot in Lejos. This project hangs on me not knowing how far the Robot has travelled. This info is needed for the pilot to work with a navigator.
Actually, my main project now is learning java. This goes well but there us a lot to it.
Few weeks ago I ordered two more BlinkM’s. These are I2C driven all color led’s. I wrote a driver to address these led’s. The led’s are now mounted on each corner of Sidbot. So Sidbot can shine all colors to all Sides of it’s triangular body. Each led is driven individually so each one can have it’s own color. I made a nice flashlight show to test the driver.
I finished drivers for my IMU too. The filter to fuse the signals is also finished but it can’t run in a separate thread yet.
I also have been on holidays and busy with work. But these matters don’t concern this blog.
I totally forgot to post the video I made of the IMU sensor.
This clips shows how accurate the IMU sensor can be. This IMU sensor combines a 3 axis gyro and a 3 axis accelerometer. The signals from both sensors are combined using a nonlinear complementary filter to get an output signal that has low noise and is drift free. (Well actually it is only drift free over 2 axis. It still drifts over the Z-axis.) This home made sensor will show a blue light when it is horizontal using a 0.5 degrees margin. Otherwise the light will turn red. The full color LED is a BlinkM.
You’ll find more about this sensor in my previous posts.
Here is the code for the filter that calculates attitude from gyro and accelerometer. The functional bits of the code are well commented in the code itself. The main structure of the code will be described in this post. A practical description of the filter can be found in this article. The theory is in this bunch of articles.
The filter loops, giving frequent updates of an estimation of the robots attitude. Therefore the code is in one endless loop. The faster the filter loops the better. But the execution speed is limited due to hardware limitation and other processes that need CPU time. To preserve time for other processes the iteration speed of the program is controlled. At the start of the program the sensors and data structures need to be initialized. Hardware initialization is done in a separate function called init (not included in the code below). This concentrates all hardware depended code into one function and keeps the main function easily reusable, even if you have different hardware.
As stated before the loop contains the filter. Aside from the code that implements the filter there is code to supply information to the user. These parts of the code are all commented away. If they are useful to you, you can uncomment them. Even then you can switch these parts of the code on or of by changing a true to false in the if statement surrounding this code. There are three ways to supply information to the end user. Data can be logged to excel, displayed on the NXT screen or a full color led (BLINKM) can be triggered if certain conditions are met.
If you find the code useful or informative please press the “like” button. This might encourage me to make more posts like this.
#pragma systemfile
// this defines the iteration time of the filter in msec
#define IMU_speed 25
#include "math.c"
#include "matrixDirectives.c"
#include "I2C.c"
#include "IMU_acclMatrix.c"
#include "IMU_gyroMatrix.c"
/*
// Optional code to log the filter state in an excel fie
#define NXT2excelMaxData 8
#include "nxt2excel.c"
*/
// this struct will hold attitude, position (Todo) and speed (Todo) of the robot
typedef struct
{
matrix C;
bool initialized;
} tPosition;
tPosition position;
void init();
void normalizeM(matrix &in);
{
float duration;
matrix U, T;
vector I, P, Rate;
long lastTime, start, wait;
bool home, oldhome;
// PI controller parameters, these are used to tune the filter
float kp=0.0005,ki=0.005, id=0.99;
// Initialize the sensors
init();
lastTime=nPgmTime;
eraseDisplay();
// initialize cosine matrix, the cosine matrix holds the attitude of the robot
position.C[0][0]=1;
position.C[0][1]=0;
position.C[0][2]=0;
position.C[1][0]=0;
position.C[1][1]=1;
position.C[1][2]=0;
position.C[2][0]=0;
position.C[2][1]=0;
position.C[2][2]=1;
// iterate forever, but halt if one of the sensorsreturns an error
while (I2C_gyro.error==NO_ERR && I2C_accl.error==NO_ERR)
{
start=nPgmTime;
alive();
// get sensor readings,
// gyro readings (in rad/sec) are stored in a vector (array of 3 elements) called gyro.rate
// acceleration readings (in g) are stored in a vector called accl.g
Gyro_getGyroBiased();
Accl_getAcclBiased();
// for better filter output the exact time for each iteration is measured
duration=(nPgmTime-lastTime)/1000.0;
lastTime=nPgmTime;
// nPgmTime wraps around when reaching maxint, correct for this
if (duration<0) duration=IMU_speed/1000.0;
/*
// Optional code to log the filter state in an excel fie
if (false)
{
for (int Rcol=0;Rcol<3;Rcol++)
{
for (int ii=0;ii<3;ii++)
NXT2excelSendMessage();
}
}
*/
// scale the turning rate (rad/sec) to a change in attitude (rad)
VscaleVS(gyro.rate,duration,Rate);
// corract the gyro signal for offset (I) and accumulated errors (P), but do it slowly (ki, kp)
// at the same time, convert it to a skew symmetric matrix
// for speed not matrix functions are used here
U[2][1]=gyro.rate[0]*duration-I[0]*ki-P[0]*kp;
U[1][2]=-U[2][1];
U[0][2]=gyro.rate[1]*duration-I[1]*ki-P[1]*kp;
U[2][0]=-U[0][2];
U[1][0]=gyro.rate[2]*duration-I[2]*ki-P[2]*kp;
U[0][1]=-U[1][0];
// transform the change in attitude in respect to robot axis into change in attitude in respect to global axis
// for speed not matrix functions are used here
// matrix equivalent: MproductMM(position.C,U,T);
T[0][0]=position.C[0][1]*U[1][0]+position.C[0][2]*U[2][0];
T[0][1]=position.C[0][0]*U[0][1]+position.C[0][2]*U[2][1];
T[0][2]=position.C[0][0]*U[0][2]+position.C[0][1]*U[1][2];
T[1][0]=position.C[1][1]*U[1][0]+position.C[1][2]*U[2][0];
T[1][1]=position.C[1][0]*U[0][1]+position.C[1][2]*U[2][1];
T[1][2]=position.C[1][0]*U[0][2]+position.C[1][1]*U[1][2];
T[2][0]=position.C[2][1]*U[1][0]+position.C[2][2]*U[2][0];
T[2][1]=position.C[2][0]*U[0][1]+position.C[2][2]*U[2][1];
T[2][2]=position.C[2][0]*U[0][2]+position.C[2][1]*U[1][2];
// add the cange in attitude to the attitude to get an updated attitude
MsumMM(position.C,T,position.C);
// Renormalize matrix
// due to small errors the matrix gets corrupted over time (its axis are no longer perpendicular
// this function corrects this
normalizeM(position.C);
// correct the attitude using tilt info from the accelerometer
// but only if there is no or little acceleration, otherwise titl info can not be trusted
if (abs(sqrt(accl.g[0]*accl.g[0]+accl.g[1]*accl.g[1]+accl.g[2]*accl.g[2])-1.0)<.2)
{
// Calculate difference (error) between culculated attitude and accelerometer
// and use this in a proportional feedback (P)
P[0] = (position.C[2][1]*accl.g[2]) - (position.C[2][2]*accl.g[1]);
P[1] = (position.C[2][2]*accl.g[0]) - (position.C[2][0]*accl.g[2]);
P[2] = (position.C[2][0]*accl.g[1]) - (position.C[2][1]*accl.g[0]);
// Add the error to the accumulated error and use this in a integral feedback (I)
I[0]=I[0]*id+P[0]*duration;
I[1]=I[1]*id+P[1]*duration;
I[2]=I[2]*id+P[2]*duration;
}
else
{
P[0]=0;
P[1]=0;
P[2]=0;
}
/*
// optional code to drive the LED in the sensor
if (true)
{
// test if the robot is in it's starting attitude
if ( abs(position.C[2][0])<.009 && abs(position.C[2][1])<.009)
home=true;
else
home=false;
// change the color of the LED if needed
if (home != oldhome)
{
if (home)
blinkM_setColour(0 , 0, 128);
else
blinkM_setColour(64 , 0, 0);
}
oldhome=home;
}
*/
// optional code to display sensor and filter data
if (true && time1[T1]>200)
{
eraseDisplay();
nxtDisplayTextLine(7, "%5d",wait);
//displayM(position.C);
//displayV(accl.g,1);
//displayV(gyro.rate,0);
ClearTimer(T1);
}
//while (nNxtButtonPressed==-1);
//while (nNxtButtonPressed!=-1);
// wait for next iteration
// the program does not assume a constant execution speed,
// instead it measures every iteration, and waits the exact amount of time to guarantue constant iteration speed
wait=IMU_speed-(nPgmTime-start);
if (wait<0)
{
wait=0;
//PlaySound(soundException);
}
wait1Msec(wait);
}
}
void init()
{
// configure port;
SensorType[S3]=sensorI2CCustom;
nI2CRetries=3;
position.initialized=false;
accl.initialized=false;
gyro.initialized=false;
blinkM_setColour(0 , 4, 0);
// configure accl;
I2C_accl.port=S3;
Accl_init();
Accl_getBias();
accl.initialized=true;
// configure gyro;
I2C_gyro.port=S3;
Gyro_init();
Gyro_getBias();
gyro.initialized=true;
if (I2C_gyro.error || I2C_accl.error)
{
nxtScrollText("IMU: %3d, %3d",I2C_gyro.error , I2C_accl.error);
blinkM_setColour(63 , 0, 0);
PlaySound(soundException);
wait1Msec(2000);
}
blinkM_setColour(0 , 0 , 0);
position.initialized=true;
}
//normalizes a matrix (takes 2.87 msec)
void normalizeM(matrix &in)
{
float error, renorm;
vector vx,vy, xor, yor, zor;
VgetRowMS(in, 0,vx);
VgetRowMS(in, 1,vy);
dotProductVVS(vx, vy,error); // eq.18
if (abs(error)>0.05)
{
VscaleVS(vy,error/-2.0, xor);
VsumVV(vx, xor, xor); //eq.19
VscaleVS(vx,error/-2.0, yor);
VsumVV(vy, yor, yor); //eq.19
VcrossProductVV(xor,yor, zor); //eq.20
dotProductVVS(xor, xor,error);
renorm = .5 *(3 - error); //eq.21
for ( int Rcol=0; Rcol<Msize;Rcol++)
in[0][Rcol]=renorm*xor[Rcol];
dotProductVVS(yor, yor,error);
renorm = .5 *(3 - error); //eq.21
for ( int Rcol=0; Rcol<Msize;Rcol++)
in[1][Rcol]=renorm*yor[Rcol];
dotProductVVS(zor, zor,error);
renorm = .5 *(3 - error); //eq.21
for ( int Rcol=0; Rcol<Msize;Rcol++)
in[2][Rcol]=renorm*zor[Rcol];
}
}
|
2018-09-25 03:12:39
|
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|
https://socratic.org/questions/prove-that-p-auubuuc-p-a-p-b-p-c-p-annb-p-bnnc-p-annc-p-annbnnc#646119
|
# Prove that? : P(AuuBuuC)=P(A)+P(B)+P(C)-P(AnnB)-P(BnnC)-P(AnnC)+P(AnnBnnC)
## $P \left(A \cup B \cup C\right) = P \left(A\right) + P \left(B\right) + P \left(C\right) - P \left(A \cap B\right) - P \left(B \cap C\right) - P \left(A \cap C\right) + P \left(A \cap B \cap C\right)$ I can show that with diagram, but how to prove it with formulas?
Aug 7, 2018
#### Explanation:
$\text{Prerequisite : } P \left(A \cup B\right) = P \left(A\right) + P \left(B\right) - P \left(A \cap B\right) \ldots . \left(\star\right)$.
$P \left(A \cup B \cup C\right) = P \left(A \cup D\right) , \text{ where, } D = B \cup C$,
$= P \left(A\right) + P \left(D\right) - P \left(A \cap D\right) \ldots \ldots \ldots . \left[\because , \left(\star\right)\right]$,
$= P \left(A\right) + \textcolor{red}{P \left(B \cup C\right)} - \textcolor{b l u e}{P \left[A \cap \left(B \cup C\right)\right]}$,
$= P \left(A\right) + \textcolor{red}{P \left(B\right) + P \left(C\right) - P \left(B \cap C\right)} - \textcolor{b l u e}{P \left(A \cap B\right) \cup \left(A \cap C\right)} ,$
=P(A)+P(B)+P(C)-P(BnnC)-color(blue){[P(AnnB)+P(AnnC)-P((AnnB)nn(AnnC)],
$= P \left(A\right) + P \left(B\right) + P \left(C\right) - P \left(A \cap B\right) - P \left(B \cap C\right) - P \left(A \cap C\right) + P \left(A \cap B \cap C\right) ,$
as desired!
|
2022-01-27 14:54:16
|
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|
https://www.jobilize.com/online/course/3-3-projectile-motion-two-dimensional-kinematics-by-openstax?qcr=www.quizover.com&page=6
|
3.3 Projectile motion (Page 7/16)
Page 7 / 16
Conceptual questions
Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither $\text{0º}$ nor $\text{90º}$ ): (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at $t=0$ ? (d) Can the speed ever be the same as the initial speed at a time other than at $t=0$ ?
Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither $\text{0º}$ nor $\text{90º}$ ): (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?
For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?
During a lecture demonstration, a professor places two coins on the edge of a table. She then flicks one of the coins horizontally off the table, simultaneously nudging the other over the edge. Describe the subsequent motion of the two coins, in particular discussing whether they hit the floor at the same time.
Problems&Exercises
A projectile is launched at ground level with an initial speed of 50.0 m/s at an angle of $30.0º$ above the horizontal. It strikes a target above the ground 3.00 seconds later. What are the $x$ and $y$ distances from where the projectile was launched to where it lands?
$\begin{array}{lll}x& =& \text{1.30 m}×{10}^{2}\\ y& =& \text{30}\text{.9 m.}\end{array}$
A ball is kicked with an initial velocity of 16 m/s in the horizontal direction and 12 m/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air? (c)What maximum height is attained by the ball?
A ball is thrown horizontally from the top of a 60.0-m building and lands 100.0 m from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground?
(a) 3.50 s
(b) 28.6 m/s (c) 34.3 m/s
(d) 44.7 m/s, $50.2º$ below horizontal
(a) A daredevil is attempting to jump his motorcycle over a line of buses parked end to end by driving up a $\text{32º}$ ramp at a speed of . How many buses can he clear if the top of the takeoff ramp is at the same height as the bus tops and the buses are 20.0 m long? (b) Discuss what your answer implies about the margin of error in this act—that is, consider how much greater the range is than the horizontal distance he must travel to miss the end of the last bus. (Neglect air resistance.)
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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|
2020-02-22 13:35:48
|
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|
https://www.physicsforums.com/threads/how-to-offset-a-polynomial.674318/
|
# How to Offset a polynomial
1. Feb 25, 2013
### zzinfinity
How to "Offset" a polynomial
Suppose I have a function for a curve, for example y=x2. I want to find a function to "offsets" it by 2 units. That is, I want a larger parabola that is exactly 2 units away from my original parabola. What I have in mind is the offset command in AutoCAD. Is there a simple transformation that can be done to my function to do this? And if its not possible, is it possible to do with functions other than polynomials? I've been thinking about this for a while, and I feel like its not possible, but I was wondering if anyone had ever encountered an algebraic way to do this.
2. Feb 25, 2013
### pwsnafu
Re: How to "Offset" a polynomial
What do you mean? What does "exactly 2 units away" mean? What do you mean by a "a larger parabola"?
3. Feb 25, 2013
### micromass
Staff Emeritus
Re: How to "Offset" a polynomial
Maybe you should include a drawing or picture of what you want to do?
4. Feb 25, 2013
### Simon Bridge
Re: How to "Offset" a polynomial
I think he means something like this:
For $y=x^2$ ... the outer curve would have roots $x = \pm 2$ and intersect the y axis at $y=-2$. With those three points, a parabola can be determined.
Maybe more like: the outer boundary of the set of all points within 2 units of the curve.
5. Feb 25, 2013
### JJacquelin
6. Feb 25, 2013
### dodo
Re: How to "Offset" a polynomial
These formulas are producing offsets in directions normal to the curve. The reference to CAD software makes me think that the OP simply wants to translate the parabola. (Guesswork here.)
In which case, a direction of translation should be specified. For example, to translate along the X-axis, you would add/subtract some constant "c" to "x":$$y = (x \mp c)^2$$
To translate along the Y-axis, you add/subtract to "y":$$y = x^2 \pm c$$
To translate in an arbitrary (straight-line) direction, you would do a combination of the above. If you want to translate "c" units in a direction determined by an angle "theta" (with respect to the positive X-axis), then the horizontal translation is $c \cos \theta$ and the vertical translation is $c \sin \theta$, leaving the formula as$$y = (x - c \cos \theta)^2 + c \sin \theta$$
But yes, I agree with micromass than a picture would help to avoid guessing.
7. Feb 25, 2013
### fortissimo
Re: How to "Offset" a polynomial
If you want to have your parabola precisely two units away from your orignal one, at every point, then you will not obtain a parabola. This would be equivalent to having a ball (with a given radius) rolling around at the perimeter of the parabola, noting the curve it traces out. For instance, for y1 = x^2 and y2 = x^2 + c the Euclidian distance between these two curves approaches zero as x tends to infinity, while the distance at x=0 is c. Scaling and translating the parabola will not produce the desired result either, since the parabolas will move farther apart as x tends to infinity.
Last edited: Feb 25, 2013
8. Feb 25, 2013
### zzinfinity
9. Aug 27, 2014
### mathmut
Hello,
mine is a similar question.
Suppose we make a co-ordinate system with y and x axes.locate a point 5 units above the vertex which is at the ORIGIN.this point is the focus.now, we make a dashed line 5 units below parallel to y=0 and this is our directrix.considering this origin we sketch a parabola according to the equation y2=4fx where f is the focus.lets call this parabola 1
NOW,the question is.i intend to OFFSET the parabola 1 curve 2 units BELOW.such that the new curve(i am keeping it un named and un-defined and ambigious by calling it a curve) i get is 2 units exactly below the parabola 1 with the same curvature(similar to offseting a line or a semi-circle in a CAD modelling software such as pro-e or SW).WILL THIS NEW CURVE BE A PARABOLA (as per its definition of equal distances from directrix and focus)???
10. Aug 27, 2014
### Simon Bridge
Welcome to PF;
Do you mean that each point on the second curve has a y value that is 2 units less than the y value of the first, for the same value of x? (Sketch what you mean and check.)
11. Aug 27, 2014
### mathmut
Dear Simon,
i actually want to know if this NEW curve is a parabola or not? will this NEW parabola take on a NEW EQUATION? or will it have the SAME equation as the parabola 1 ? because we have just offsetted the whole parabola 1 2 units downwards.
12. Aug 28, 2014
### Simon Bridge
Don't know for sure but it sounds like the new curve will also be a parabola with a different equation. To be sure, I need you to answer my questions.
If you follow suggestions, you should be able to work it out for yourself anyway.
Note - shifting a curve given by f(y)=g(x) x0 a units to the right and b units up gives a new equation: f(y-b)=g(x-a), but it is the same curve in a new position.
|
2017-12-11 23:13:39
|
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|
http://homerreid.github.io/scuff-em-documentation/examples/NanostructureCasimirPolder/NanostructureCasimirPolder/
|
# Casimir-Polder potentials in dielectric nanostructures
In this example, we exploit scuff-em's support for periodic geometries to compute Casimir-Polder potentials for atoms inside dielectric waveguides. Our basic test example will be the 1D photonic crystal studied in this paper:
We will illustrate the use of scuff-em's Casimir-Polder module scuff-caspol by reproducing the results of Hung et al. for a 1D lattice, then extend the calculation to the case of a 2D square lattice. The files for this example may be found in the share/scuff-em/examples/NanobeamCasimirPolder subdirectory of your scuff-em installation.
## gmsh geometry and surface mesh for nanobeam unit cell
The gmsh geometry file UnitCell.geo describes just the portion of the nanobeam surface that lies within the unit cell, i.e. the cell that is infinitely periodically replicated to yield the full geometry. To produce a discretized surface-mesh representation of this geometry, we run it through gmsh:
% gmsh -2 UnitCell.geo
This produces a file named UnitCell.msh, which I rename to NanoBeamUnitCell_1006_.msh because (a) I will be using it as the unit cell of a beam geometry, in contrast to a different use I will find for the same unit cell below; and (b) the mesh has 1006 interior edges (this is the number that defines the memory and computation time requirements for the scuff-em calculation; it may be found by running scuff-analyze --mesh UnitCell.msh). You can open the .msh file in gmsh to visualize the unit-cell mesh:
% gmsh NanoBeamUnitCell_1006.msh
Note the following:
• For 1D periodic geometries in scuff-em, the direction of infinite extent must be the x direction.
• The sidewalls normal to the $y$ and $z$ directions are meshed, but the sidewalls normal to the $x$ direction are not meshed for this structure, because those surfaces are not interfaces between different dielectrics.
• For surfaces that straddle the unit-cell boundaries (as is the case here), each triangle edge that lies on the unit-cell boundary must have an identical image edge on the opposite side of the unit cell. An easy way to achieve this is to use extrusions in gmsh, as in the .geo file above.
• In this case the unit cell is 0.367 $\mu$m long. This and other geometric parameters can be modified by editing the file UnitCell.geo or directly on the gmsh commmand line using the -setnumber option.
## scuff-em geometry file for dielectric nanobeam
A scuff-em [geometry file]scuffEMGeometries describing an extended nanobeam consisting of infinitely many repetitions of the above unit cell filled with a dielectric material of constant relative permittivity $\epsilon=4$, is NanoBeam_1006.scuffgeo. The file reads, in its entirety,
LATTICE
VECTOR 0.367 0.0
ENDLATTICE
OBJECT Nanobeam
MESHFILE NanoBeamUnitCell_1006.msh
MATERIAL CONST_EPS_4
ENDOBJECT
We can use scuff-analyze to visualize the geometry described by this .scuffgeo file:
% scuff-analyze --geometry NanoBeam_1006.scuffgeo --WriteGMSHFiles --Neighbors 3
[The option --Neighbors 3 requests that, in addition to the unit-cell geometry, the first 3 periodic images of the unit cell in both the positive and negative directions (for a total of 5 copies of the unit cell) be plotted as well. This helps to convey a slightly better sense of the actual infinite-length structure being simulated.] This produces the file NanoBeam_1006.pp, which you can view in gmsh:
% gmsh NanoBeam_1006.pp
Note that the visualization file produced by scuff-analyze includes extra triangles (visible at the left end of the structure) that are not present in the unit-cell geometry. These are called straddlers, and they are added automatically by scuff-em to account for surface currents that flow across the unit-cell boundaries in periodic geometries.
## gmsh geometry and surface mesh for nanoarray unit cell
It's easy to generalize all of this to a geometry with two-dimensional periodicity. The only modification required to the unit-cell mesh is that we have to remove the sidewalls normal to the $y$ direction. This can be done using the same gmsh geometry we used above for the nanobeam unit cell, but with the extra command-line option -setnumber LDim 2 on the gmsh command line. (Here LDim stands for "lattice dimension".)
% gmsh -2 -setnumber LDim 2 UnitCell.geo
This produces a file named UnitCell.msh, which I rename to NanoArrayUnitCell_800_.msh. It looks like this
## scuff-em geometry file for dielectric nanoarray
A scuff-em geometry file describing a dielectric surface extended in two dimensions and perforated with a square lattice of the holes pictured above is NanoArray_800.scuffgeo. The file reads
LATTICE
VECTOR 0.367 0.0
VECTOR 0.000 0.845
ENDLATTICE
OBJECT NanoArray
MESHFILE NanoArrayUnitCell_800.msh
MATERIAL CONST_EPS_4
ENDOBJECT
Again we use scuff-analyze to visualize the geometry described by this .scuffgeo file:
% scuff-analyze --geometry NanoArray_800.scuffgeo --WriteGMSHFiles --Neighbors 3
## Evaluation points for Casimir-Polder potentials
For a Casimir-Polder calculation we will want to evaluate the Casimir-Polder potential at range of points. We put the Cartesian coordinates of these points into a text file named EPFile, which looks like this:
0.1835 0.4225 -0.500
0.1835 0.4225 -0.480
0.1835 0.4225 -0.460
...
0.1835 0.4225 0.480
0.1835 0.4225 0.500
and defines a line of points running through the middle of the hole in the beam unit cell.
To double-check that the evaluation points we specify are actually where we expect them to be vis-a-vis the meshed surfaces in our problem, we can ask scuff-analyze to plot the evaluation points together with the unit-cell geometry:
% scuff-analyze --geometry NanoBeam_1006.scuffgeo --EPFile EPFile --WriteGMSHFiles
This confirms that our EPFile describes a line of evaluation points running through the middle of the hole in the beam structure, as desired.
## Setting up the Casimir-Polder calculation
To compute the Casimir-Polder potential on, say, a rubidium atom at the above evaluation points for the 1D and 2D extended structures, we now say simply
% scuff-caspol --geometry NanoBeam_1006.scuffgeo --EPFile EPFile --Atom Rubidium
and/or
% scuff-caspol --geometry NanoArray_800.scuffgeo --EPFile EPFile --Atom Rubidium
These calculations will produce files named NanoBeam_1006.out and NanoArray_800.out tabulating the Casimir-Polder potential experience by the rubidium atom at each of the specified evaluation points.
For more information on scuff-caspol, see the old scuff-caspol documentation, which is thorough and up-to-date (though it does not cover CP calculations in extended geometries) despite having not yet been ported from its previous format.
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2017-06-24 03:25:45
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https://www.gamedev.net/forums/topic/508809-pvs--portal-culling-and-recursion/
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# PVS : Portal Culling and Recursion
This topic is 3372 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hello, i've been working on portal culling ; very standard method : I draw the room where the player is. For each drawed rooms, i check all the portals this room has and if a portal is visible across a 2D rectangle, i render the room on the other side of this portal passing the passed rectangle clipped with the portal's rectangle. The rectangle is initially the entire screen. The first problem is obvious : we have to check for loops. Thus we introduce a flag, telling if a room has already been considered, if it is the case, we skip it. The real problem is described by this simple example : As you can see there are some loops possible. Here is what happens with the above method :
Player is in A -> A is drawed
A's portals : [ 1, 2 ]
portal 1 is visible, so let's draw B
B's portals : [ 1, 3 ]
portal 1 is visible, so let's draw A
A has already been seen, skip it
portal 3 is visible, let's draw C
C's portals : [ 2, 3, 4 ]
portal 2 is not visible through 3, skip
portal 3 is visible, let's draw B
B has already been seen, skip
portal 4 is not visible through 3, skip
portal 2 is visible, let's draw C
C has already been seen, skip
the end
Fact is that we missed the draw of room D wich is visible !!! I'm looking for a very simple solution, if there's one. This is realtime culling... How does (for example DOOM3) handle this ? Thanks for your time ! [Edited by - jfdegbo on September 19, 2008 7:06:15 AM]
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Instead flagging a room, you could flag a portal. Rooms can be visible through multiple portals, eventually via multiple sectors as well.
Loop through your portals and flag them. Now it can happen that you add a sector twice or more. Therefore you use another flag to tell if the sector already has been added. If you encounter a room that has already been inserted, just proceed like normal, except don't add it twice.
< check all portals in room A >
2.- Portal (1)AB has not been checked yet, look into room B
3.- room B has not been added yet, add room B to the PVS
4.- Portal (3)BC has not been checked yet, look into room C
5.- room C has not been added yet, add room C to the PVS
6.- You can't see anymore portals via portal AB > BC in room C
>> Back to room A, next portal (2) AC
all the portals there yet, so proceed.
8.- Portal (4) CD has not been checked yet, look into room D
10.- No more portals in room D, back to room C
11.- Don't know exactly how you check if a portal is inside your frustum, but it could be possible that portal (3) CB will be checked. This one has already been flagged before in step 4 though, so you can skip it.
greetings,
Rick
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It's been ages since I did portal stuff, but IIRC I built a list of visible rooms and their transformation and stopped when duplicates are found. You do your usual traversal of the portals (probably depth-first since that's easiest), and find the cumulative transformation applied by the current path of portals. Then you check to see if that particular room/transformation pair is already in your "to draw" list, and if it's not then add it. If it's already present then you've done with this room and return to the parent.
You need to store the transformation/room pair as otherwise you can't do neat rendering tricks like recursive rooms and other "impossible" room layouts.
Edit: I missed the bit where you're doing proper clipping of portals to each other. In that case I think you should always keep traversing the graph until you find no portals are currently visible. In your example that means you'll visit C twice, and the second time you'll correctly add D.
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Thank you Rick, this works... in a way.
OrangyTang : i'm not sure i understand what you mean by "transformation"...
But i have another problem.
When considering a portal p2 through a portal p1, i do the following : (very standard stuff, too :))
* project p1 vertices according to camera
* create the bounding rectangle of the projected 2D vertices
* do the same for p2
* clip p2's 2D rectangle inside p1's 2D rectangle
if the resulting clip is not a line or a point, then p2 is visible through p1.
Now, here's the problem :
If room A and B have two adjacent portals 1 & 2 leading to each other, ie A -1-> B -2-> A and B has a single portal 3 to room C
A -1- -2- B -3- C
If we are in A, we could have the following situation scenario :
add A portal 1 not checked add B portal 1 checked, skip portal 2 can be seen through 1 (because 2d overlap), not checked room A already seen, all portals checked, skip portal 3 not visible from 1 (could be like that) portal 2 checked, skipthe end
we missed portal 3 that was possible to be seen through 2
Is it very wrong to do 2D clipping for portals ? I am missing anything ?
Thanks again !
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I believe my solution won't have that problem, as you'll just visit B twice, and C will be added correctly. Then you trim your list of duplicates so you don't draw B twice.
When I say transformations, it's usual to attach a matrix transform to each portal so you can do spatial tricks like infinite corridors or mirrors. If you're not up to that yet then you can ignore it (for a regular map like your examples all transformations would be the identity matrix).
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Well, I think you are right.
I've tested this and it works. I finally did some sort of (very optimized) stack of traversal with state restoration, and this seems to work perfectly.
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2017-12-13 15:22:11
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https://www.quantumstudy.com/let-f-be-the-force-acting-on-a-particle-having-position-vector-r-and-t-be-the-torque-of-this-force-about-the-origin-then/
|
# Let F be the force acting on a particle having position vector r and T be the torque of this force about the origin. Then
Q: Let $\vec{F}$ be the force acting on a particle having position vector $\vec{r}$ and $\vec{T}$ be the torque of this force about the origin. Then
(a) $\vec{r} . \vec{T} = 0$ and $\vec{F} . \vec{T} = 0$
(b) $\vec{r} . \vec{T} = 0$ and $\vec{F} . \vec{T} \ne 0$
(c) $\vec{r} . \vec{T} \ne 0$ and $\vec{F} . \vec{T} = 0$
(d) $\vec{r} . \vec{T} \ne 0$ and $\vec{F} . \vec{T} \ne 0$
Ans: (a)
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2022-07-04 06:33:01
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https://www.nextgurukul.in/examCorner/Board-Papers/CBSE/X/Maths/2013/Delhi.htm
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clear
## Change Papers
Back
CBSE X
Delhi
MATHS PAPER 2013
Time allowed: 180 minutes; Maximum Marks: 90
General Instructions: 1) All questions are compulsory. 2) The question paper consists of thirty questions divided into 4 sections A, B, C and D. Section A comprises of ten questions of 01 mark each, Section B comprises of five questions of 02 marks each, Section C comprises ten questions of 03 marks each and Section D comprises of five questions of 06 marks each. 3) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4) There is no overall choice. However, internal choice has been provided in one question of 02 marks each, three questions of 03 marks each and two questions of 06 marks each. You have to attempt only one of the alternatives in all such questions. 5) In question on construction, drawing should be near and exactly as per the given measurements. 6) Use of calculators is not permitted.
SECTION A
### Question 1
1. The common difference of the A.P. $\frac{1}{p}$, $\frac{1−p}{p}$, $\frac{1−2p}{p}$..................is:
### Question 2
2. In Fig. 1, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA ⊥ PB, then the length of each tangent is:
### Question 3
3. In Fig.2, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively, If AB = 29 cm, AD = 23 cm, ∠B = 90° and DS = 5 cm, then the radius of the circle (in cm.) is:
### Question 4
4. The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30°. The distance of the car from the base of the tower (in m.) is:
### Question 5
5. The probability of getting an even number, when a die is thrown once, is :
### Question 6
6. A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime-number less than 23, is :
### Question 7
7. In Fig. 3, Find the area of triangle ABC (in sq. units) is:
### Question 8
8. If the difference between the circumference and the radius of a circle is 37 cm, then using π = $\frac{22}{7}$, the circumference (in cm) of the circle is:
(A) 154
(B) 44
(C) 14
(D) 7
SECTION B
### Question 9
9. Solve the following quadratic equation for x:
4$\sqrt{3}{x}^{2}+5x−2\sqrt{3}=0$
### Question 10
10. How many three−digit natural numbers are divisible by 7?
### Question 11
11. In Fig. 4, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, BE and CF.
### Question 12
12. Prove that the parallelogram circumscribing a circle is a rhombus.
### Question 13
13. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a king nor a queen.
### Question 14
14. Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions 14 cm × 7 cm. Find the area of the remaining card board. [ Use Ï€ = $\frac{22}{7}$]
SECTION C
### Question 15
15. For what value of k, are the roots of the quadratic equation kx (x − 2) + 6 = 0 equal?
### Question 16
16. Find the number of terms of the A.P. 18, 15$\frac{1}{2}$, 13, …...., - 49$\frac{1}{2}$ and find the sum of all its terms.
### Question 17
17. Construct a triangle with sides 5 cm, 4 cm and 6 cm. Then construct another triangle whose sides are $\frac{2}{3}$ times the corresponding sides of first triangle.
### Question 18
18. The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole. [ $\sqrt{3}$= 1.732 ]
### Question 19
19. Prove that the points (7, 10), (−2, 5) and (3, −4) are the vertices of an isosceles right triangle.
### Question 20
20. Find the ratio in which the y-axis divides the line segment joining the points (−4, − 6) and (10, 12). Also find the coordinates of the point of division.
### Question 21
21. In Fig.5, AB and CD are two diameters of a circle with centre O, which are perpendicular to each other. OB is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.[ use π = $\frac{22}{7}$]
### Question 22
22. A vessel is in the form of hemispherical bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14 cm and the total height of the vessel is 13 cm. Find the total surface area of the vessel.[Ï€ = $\frac{22}{7}$]
### Question 23
23. A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy. [Ï€ = $\frac{22}{7}$]
### Question 24
24. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc. [ Use π = $\frac{22}{7}$]
SECTION D
### Question 25
25. Solve the following for x:
$\frac{1}{2a+b+2x}=\frac{1}{2a}$ + $\frac{1}{b}$ + $\frac{1}{2x}$
### Question 26
26. Sum of the areas of two squares is 400 ${\mathrm{cm}}^{2}$. If the difference of their perimeters is 16 cm, find the sides of the two squares.
### Question 27
27. If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first n terms.
### Question 28
28. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
### Question 29
29. In fig. 6, l and m are two parallel tangents to a circle with centre O, touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that ∠DOE = 90°
### Question 30
30. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building.
### Question 31
31. A group consists of 12 persons, of which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest. Which of the above values you prefer more?
### Question 32
32. The three vertices of a parallelogram ABCD are A(3, −4), B(−1, −3) and C(−6, 2). Find the coordinates of vertex D and find the area of ABCD.
### Question 33
33. Water is flowing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour.
### Question 34
34. A bucket open at the top, and made up of a metal sheet is in the form of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs 10 per 100${\mathrm{cm}}^{2}$. [Use π = 3.14]
|
2017-08-23 21:32:17
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http://www.tedbeaudry.net/arcadia-bluffs-mqhso/6725a7-which-of-the-following-has-negligible-mass%3F
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## which of the following has negligible mass?
The Spring Is Initially Compressed A Distance X0x0. and the tabletop. This mass i s : Afterward, which of the following is true? After the collision, in which no energy is lost find: a) the rotational speed of … A rock with a mass of 10 kg has a weight of 16 N on the moon. If the blocks are released from rest with the connecting cord taut, what is their total … D) The mass of an individual gas molecule is negligible. This preview shows page 2 - 3 out of 3 pages.. The spring is initially compressed a distance x0x0. If the collar is released from rest at A and travels along the smooth guide, determine its speed just before it strikes B. Get 1:1 … (iii) The tension on the string. A particular carbon isotope has an atomic number of 6 and an atomic mass of 14. d. A proton has a positive charge and a negligible mass. Discussion TA: 2 In the figure, the pulley has negligible mass and is frictionless. Electron. Calculate: (i) The angular velocity. Determine the minimum force P needed to move the post. All Rights Reserved. As shown in the figure below, two masses m1 = 5.00 kg and m2 which has a mass 50.0 Percent that of m1, are attached to a cord of negligible mass which passes over a frictionless pulley also of negligible mass. The puller is frictionless and may be treated as a uniform solid . Block B acquires a speed of 1.60 m/s. Correct answers: 3 question: A block on a rough, horizontal surface is attached to a horizontal spring of negligible mass. A string of negligible mass has a bucket tied at the end. d. Which of the following is unique for any given element? (c) An electron has negligible mass compared to the mass of a neutron. Answers. MEDIUM . (b) You place the spring vertically with one end on the floor. Block A in the figure has mass 1.00 kg and block B has mass 3.00 kg. It is supported at one end by a pin and at the other end by a post having a mass of 50 kg and negligible thickness. You then drop a 1.20-kg book onto it from a height of 0.80 m above the top of the spring. By definition. The string is 60cm long and the buckets has a mass of 45g. The amount of interest you'll get on your savings is negligible, so you might as well spend your money. The spring constant of the spring is kk. The linkage ABD is formed by connecting two 4-kg bars and a collar of negligible mass. This preview shows page 2 - 4 out of 7 pages. A 15.0 kg mass and a 10.0 kg mass are suspended by a pulley that has a radius of 10.0 cm and a mass of 3.0 kg. Glider A has a mass of 0.90 kg and glider B has a mass of 0.60 kg. When did organ music become associated with baseball? A) A neutron has no charge and has a mass of 1 amu. molecule electron proton element neutron 9. 1 Answer to A diver of weight 580 N stands at the end of a diving board of length L = 4.5 m and negligible mass (Fig. Click hereto get an answer to your question ️ A cord of negligible mass is wound around the rim of a flywheel (disc) of mass 20 kg and radius 20 cm . Radioactive isotopes are useful in scientific research because _____. Puck A, of mass m/2 comes in and hits B. Who was the lady with the trophy in roll bounce movie? They then simultaneously push each other horizontally. Select all that apply. A negatively charged subatomic particle that has negligible mass and moves around in a cloud around the nucleus. The cord has negligible mass and causes the pulley to rotate without slipping. A) 22 B) 102 C) 122 D) 202 О А Puck A, of mass m/2 comes in and hits B. The graph shows the variation with time t of the speed v of a ball of mass 0.50 kg, that has been released from rest above the Earth’s surface.. In contrast, the electron has a negligible mass of .0005 amu. Assume the pulley to be friction-less and of negligible mass; K1= 20 N/m K2= 40 N/m m=100 kg c.Neutrons have no charge and negligible mass. After the collision, in which no energy is lost find: a) the rotational speed of … It accounts for less than 1/1000 th of the total mass. For simplicity, we will use the amu unit for the three subatomics. (a) How far must the spring be compressed for 3.20 J of potential energy to be stored in it? Objects of negligible mass have a mass that can be neglected. A block of mass of 10 kg travels in the positive direction along a surface with negligible friction. The platform P (fig.) But no real gas strictly obeys the gas equation at all temperatures and pressures. Two masses 2 kg and 3 kg are attached to a rod of negligible mass. b. C) An electron is negatively charged and has a mass of 1 amu. When passing through the vertical, the string slips a bit from the pivot so that its length increases by a small amount d(d L) in negligible … neutrons, protons, and electrons that this carbon isotope has is _____. 10. Block A has a mass of 2.5 kg, and block B 4.0 kg. Initially, glider A moves toward glider B, which is at rest. Which of the following has negligible mass a Electron b Neutron c Atom d. Which of the following has negligible mass? Using preliminary test we have to decide whether the above series is di... question_answer. The mass of an electron is only about 1/2000 the mass of a proton or neutron, so electrons contribute virtually nothing to the total mass of an atom Step 2: Statements (a) Protons have a charge of - 1. An uncharged atom of nitrogen (atomic number = 7) has _____. I imagine myself hitting a ping-pong ball very strong, and the fact that the ping pong ball has very little mass doesn't imply that the force I exerted is small, but that the aceleration is very large. The block collides with an object of an unknown mass that is at rest. The Moon, whose center is 3.84 × 10 8 m from the Earth’s center, has mass 7.35 × 10 22 kg. An electron, with relatively negligible mass meaning the mass of an electron is so insubstantial that it isn't counted towards the total weight of the atom. Starting from rest, the speed of the two masses is 3.80 m/s at the end of 3.16 s. At that time, the kinetic energy of the system is 77.0 J and each mass has moved a distance of 6.00 m. Determine the lighter mass. Students also viewed these Mechanics questions. A 2.5 kg block sits on an inclined plane with a 30 degree inclination. The collar of negligible size has a mass of 0.25 kg and is attached to a spring having an unstretched length of 100 mm. (d) Number of proton equals to number of electrons on a stable atomAStatements b, c and dBStatements a, b and What is the final speed of block A? How do you put grass into a personification? An electron has a negative charge and a mass of approximately 1 amu. The board is fixed to two pedestals separated by distance d = 1.5 m. Of the forces acting on the board, what are the (a) magnitude and (b) direction (Up or … Both neutrons and protons are assigned as having masses of 1 amu each. The rod is cooled at $0^\circ C$, but prevented from contracting by attaching a mass at the lower end. a) atomic mass b) number of electrons c) atomic number d) number of protons e) None of the listed responses is correct. What is the best way to fold a fitted sheet? Q: A football is kicked 60.0 meters. 7. Phosphorus-32 (radioactive) has _____ than phosphorus-35 (normal). A 15.0 kg mass and a 10.0 kg mass are suspended by a pulley that has a radius of 10.0 cm and a mass of 3.0 kg. Block A has a mass of 1.3 kg, block B has a mass of 2.7 kg, and angle è is 26 °. 17. 8. The pulley has negligible mass and spins with negligible friction about its axle. glass does not become a true liquid until temperatures are greater than or equal to 515515 °c. The spring is compressed such that the block is located at position X. (a) Protons have a charge of - 1. A ball of mass M is attached to a string of length R and negligible mass. Block A has a mass of 1.0 kg, block B has a mass of 2.0 kg, and angle θ is 30o. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by m, the time period becomes $\text{5T/3,}$ then the ratio of $\frac{m}{M}$ is Physics. A 2.5 kg block sits on an inclined plane with a 30 degree inclination. 14. 9. How much money does The Great American Ball Park make during one game? (ii) The centripetal acceleration. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Two Identical pucks B and C, each of mass m, are connected by a rod of length l and negligible mass that is free to rotate about its center. Which of the following aspects of kinetic-molecular theory explain why a plasma-based light doesn’t get unbearably hot? If your impeached can you run for president again? Q. Some time after the block is released and travels in the direction shown in the figure, the spring compression is xfxf. Is green skull in the pirate bay is good? What disturbs me is the assertion that every total force over an object of negligible mass is negligible. The block has an initial momentum 18 %. However, electrons … The block has a mas of 2.5 kg. Two people of unequal mass are initially standing still on the ice with negligible friction. 14. An Atwood's machine consists of masses m1 and m2, and a pulley of negligible mass and friction. The beam AB has a negligible mass and thickness and is subjected to a triangular distributed loading. (A) 400 K and 200 K (B) 200 K and 400 K (C) 400 K and 400 K (D) 800 K and 400 K 14. ( Concept 2.2) neutron element atom electron proton 8. Answer. Which of the following descriptions of a subatomic particle is correct? What is the analysis of the poem song by nvm gonzalez? Which one of the following subatomic particles has appreciable mass and lacks a charge? If m1 and m2 start from rest, after they have each traveled a distance h = 2.70 m, use energy content to determine the following. A student must determine the change in momentum of the cart as it is pulled across the horizontal surface from the moment the cart is released from rest to the moment immediately before the cart collides with the pulley. for example, lemonade flows more easily than fudge syrup. The block collides with an object of an unknown mass that is at rest. E) Dipole-dipole attraction Nov 30,2020 - A pendulum is made of a massless string of length L and a small bob of negligible size and mass m. It is released making an angle q0 (1 rad) from the vertical. Also, when bonds form, some mass is converted into energy, but this is negligible when compared to the mass … The subatomic particle which carries a unit negative charge and has negligible mass - electron. When the ball is at point P, the string is horizontal. 2001BI. Students also viewed these Mechanics questions. B) London force. d.An electron has far less mass than either a proton or a neutron. There are a number of particles of particularly low mass. The coefficient of sliding friction is: 0.2. When something is meaningless or insignificant because it is so little, it's negligible. Q.2) A beam AB has a negligible mass and thickness, and supports the 200-kg uniform block as shown in Figure 2 below. (p. 29) proton: neutron: electron: element: molecule: 10. 12-27). 1.6 m/s^2. The Spring Constant Of The Spring Is Kk. Reasoning Statements: Nearly equal to.Because block is almost in free fall or, because block has negligible mass and the tension in the string is nearly zero. If the blocks are released from rest with the connecting cord taut, what is their total … What is the value of the acceleration of gravity (g) on the moon? a. the number of neutrons b. the number of protons c. the number of electrons d. the mass number. Expert Answer 100% (1 rating) Previous question Next question Get more help from Chegg. Question: A Block Of Mass Mm Is Launched By A Spring Of Negligible Mass Along A Horizontal Surface Of Negligible Friction. B) A proton is positively charged and has a mass of 1 amu. Which of the following has negligible mass? b. Which of the following. Its got 10 electrons and 9 neutrons. Air resistance is negligible. 2) Which of the following is NOT an intermolecular force? Course Hero is not sponsored or endorsed by any college or university. The spring is compressed such that the block is located at position X. Find the location of the point on this rod where the rotational … A children’s toy consists of a cart whose very light wheels are attached to … The block is released from rest. Isotopes of an element will always differ in _____. Compared to a single proton, as in Hydrogen-1. Review - CR Chapter 1, 2, 3, and 4 Spring 2014, Florida State College at Jacksonville • BSC 2010, Lone Star College System, North Harris • BIO 1408. As shown in the diagram above, one end of a light string of mass m and length L is connected to an oscillator and the other is connected to a hanging mass M so that standing waves can be set up in the string. If a horizontal force F is applied at the midpoint of the rod, find (a) the tension in the wire, and (b) the horizontal and (c) the If an element has an atomic number of 9 and a mass number of 19, then, which of the following statements is INCORRECT? The angle of the incline is 9 = 35°. negligible. angle of the incline is 9 = 35°. are even more negligible in mass. See more. (A) 10 39 N (B) 10 29 N (C) 10 19 N (D) 10 9 N. 36. D) An electron is negatively charged and has negligible mass А OB ОС OD Question 4 Mercury-202 has how many neutrons in its nucleus? See more. C) Hydrogen bond. a. Kavungya answered the question on June 1, 2019 at 08:04. The other end of the spring is attached to a wall. The Block Is Released From Rest. A student must determine the change in momentum of the cart as it is pulled across the horizontal surface from the moment the cart is released from rest to the moment immediately before the cart collides with the pulley. The figures above show two cases in which masses are suspended from the ends of the rod. 3.Particles are small. The number of protons in an uncharged atom _____. The spring which has negligible mass is not fastened to either of the blocks and drops to the surface after it has expanded. An element has 8 protons, 9 neutrons, and 8 electrons. Which of the following subatomic particles has appreciable mass and lacks a charge? The cord has negligible mass and causes the pulley to rotate without slipping. The ball moves clockwise In a vertical circle, as shown above. D) Van der waals force. 2.All particles are in constant random motion. A spring of negligible mass has force constant k = 1600 N/m. Which of the following is the best estimate of the gravitational force of the Earth on the Moon? Masses for the three subatomic particles can be expressed in amu (atomic mass units) or grams. find the fahrenheit temperatures for which glass is a liquid. Does harry styles have a private Instagram account? Who is the longest reigning WWE Champion of all time? The bucket is swung horizontally making 6 revolutions per second. If the collar is released from rest at A and travels along the smooth guide, determine its speed just before it strikes B. Two Identical pucks B and C, each of mass m, are connected by a rod of length l and negligible mass that is free to rotate about its center. What is the rhythm tempo of the song sa ugoy ng duyan? A steady pull of 25 N is applied on the cord as shown in the figure. AIIMS 2000: A rod of length 1.4 m and negligible mass has two masses of 0.3 kg and 0.7 kg tied to its two ends. What is the denotative and connotative meaning of clouds? How many, 11. 35. In each case the unknown mass m is balanced by a known mass… The other end of the spring is attached to a wall. A rod of length 1.4 m and negligible mass has two masses of 0.3 kg and 0.7 kg tied to its two ends. Electrons have much less mass than either protons or neutrons, and The block has an initial momentum 18 %. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. A thin rod of negligible mass and area of cross-section $4\times10^{-6}m^{2}$, suspended vertically from one end has a length of 0.5 m at $100^\circ C$. Correct answers: 3 question: A block on a rough, horizontal surface is attached to a horizontal spring of negligible mass. A metal other than iron and copper which shows variable valency - mercury. Block A has a mass of 1.3 kg, block B has a mass of 2.7 kg, and angle è is 26 °. Q. The collar of negligible size has a mass of 0.25 kg and is attached to a spring having an unstretched length of 100 mm. The force exerted on the block as a function of time is shown on the graph. 4.Except during collisions, interactions between particles are negligible. Angular acceleration of flywheel is : c. A neutron has no charge and its mass is negligible. The molecules have negligible mass. There are 10 neutrons in its nucleus. 13. The force exerted on the block as a function of time is shown on the graph. A force of 5 N is required to slide a block at a constant speed over the dry surface of a table. A particular carbon isotope has an atomic number of 6 and an atomic mass of 14. 1.Particles exchange energy through elastic collisions. A rod of negligible mass is pivoted at a point that is off center, so that length l 1 is different from length l2. (p. Which of the following statements is/are correct? Physics. Negligible definition, so small, trifling, or unimportant that it may safely be neglected or disregarded: The extra expenses were negligible. The respective number of. electrons are still much heavier than neutrinos or photons, which Find the location of the point on this The pulley has negligible friction in its axle and negligible mass. A gas which obeys the gas laws and the gas equation P V = n R T strictly at all temperatures and pressures is said to be an ideal gas.The molecules of ideal gases are assumed to be volume less points with no attractive forces between one another. In Figure the pulley has negligible mass, and both it and the inclined plane are frictionless. A block of mass mm is launched by a spring of negligible mass along a horizontal surface of negligible friction. Which one of the following has negligible mass? The coefficients of static friction at B … so are negligible in calculating the mass of an atom. The electron may be considered as having negligible mass, particularly in heavier elements. How long will the footprints on the moon last? A rigid, vertical rod of negligible mass is connected to the floor by a bolt through its lower end, as shown in. The most common form of calcium has 20 protons, 20 neutrons, and 20 electrons. A) Covalent bond. When a beta particle is released, the atom's mass remains approximately the same because it loses an electron, which has such a small mass in comparison to the whole atom that it is negligible. However, How old was Ralph macchio in the first Karate Kid? In the Figure the pulley has negligible mass, and both it and the inclined plane are frictionless. The puller is frictionless and may be treated as a uniform solid . The number of protons in an uncharged atom _____. Which of the following subatomic particles has appreciable mass and lacks a charge? Two blocks of steel, the first of mass 1 kg and the second of mass 2 kg, are in thermal equilibrium with a third block of aluminum of mass 2 kg that has a temperature of 400 K. What are the respective temperatures of the first and second steel blocks? The respective number of neutrons, protons, and electrons that this carbon isotope has is _____. What I understand for negligible is very small mass compared to forces. The number of protons in one atom of an element is that element's ___. The pulley has negligible friction in its axle and negligible mass. Negligible definition, so small, trifling, or unimportant that it may safely be neglected or disregarded: The extra expenses were negligible. 16. (viscosity is the property of a fluid that resists internal flow. The force of air resistance is not negligible. (b) Neutrons have no charge. has negligible mass and is tried down so that the 0.4 m long cords keep a 1 m long spring compressed to 0.6 m. When nothing is on the platform. Consider a hypothetical atom with an atomic number of 4 and a net electronic charge of +1. A mass M is suspended from a spring of negligible mass. Determine the natural frequency of the system shown. A block of mass of 10 kg travels in the positive direction along a surface with negligible friction. See the answer. fudge syrup has a higher viscosity than lemonade.) The rod also has a wire connected between its top end and the floor. (p. 29) proton: neutron: electron: atom: element: 9. 10. (iv) The linear velocity. 15. Start studying Exam 3 Test. Part B Select a point value to view scoring criteria, solutions, and/or examples and to score the response. What floral parts are represented by eyes of pineapple? In the Figure the pulley has negligible mass, and both it and the inclined plane are frictionless. There are a number of particles of particularly low mass. By definition. One may also ask, can you be negligible? Its atomic number and atomic mass, respectively, are. Copyright © 2021 Multiply Media, LLC. Which of the following is the moment of inertia I of the rod rotating anticlockwise about point X? This problem has been solved! This is false. 12. This question is in two parts.Part 1 is about mechanics and thermal physics.Part 2 is about nuclear physics.. Part 1 Mechanics and thermal physics. Electrons have much less mass than either protons or neutrons, and so are negligible in calculating the mass of an atom. Why don't libraries smell like bookstores? How did Rizal overcome frustration in his romance? To put it another way, if you have decided to not bother calculating in the mass of some object, because you have decided for whatever reason that it's small enough that it won't significantly affect the result of some calculation, then you would compute that that object travels along a geodesic. The center of gravity (CG) is located at the mid-point of the beam AB. Point Q is at the bottom of the circle and point Z is at the top of the circle. A proton has a positive charge and a mass of approximately 1 amu. If the blocks are released from rest with the connecting cord taut, what is their total kinetic energy when block B has fallen 25 cm? The number of protons in an uncharged atom _____. 13. This textbook can be purchased at www.amazon.com. Protons have a charge of + 1 (b) Neutrons have no charge. Negligible friction about its axle ) you place the spring is attached to a triangular distributed loading puller is and. 26 ° the longest reigning WWE Champion of all time speed over the dry surface of a particle! Examples and to score the response a higher viscosity than lemonade. collision, in which masses suspended... Assume the pulley to rotate without slipping a bolt through its lower end scientific research because _____ 20! Launched by a known mass… 2001BI scoring criteria, solutions, and/or examples and score... Using preliminary Test we have to decide whether the above series is di... question_answer viscosity. A particular carbon isotope has an atomic mass, and both it and the inclined plane are frictionless of kg... If your impeached can you run for president again how much money does the American. Iron and copper which shows variable valency - mercury bucket is swung horizontally making 6 revolutions per.! Disturbs me is the best way to fold a fitted sheet all?! Is green skull in the first Karate Kid constant k = 1600 N/m the angle of the is. Through its lower end, as shown in Figure 2 below find: a of. And angle è is 26 ° how much money does the Great American ball Park make during game... Incline is 9 = 35° of interest you 'll get on your is., 9 neutrons, and block B has a negligible mass sits on which of the following has negligible mass? inclined are! 0.25 kg and block B has a negligible mass - electron 9 neutrons, protons, 20. A hypothetical atom with an object of negligible mass along a horizontal which of the following has negligible mass?! And 20 electrons the moon degree inclination than 1/1000 th of the total mass lemonade. and are! Both neutrons and protons are assigned as having masses of 1 amu bucket tied at the lower,! 0.90 kg and 0.7 kg tied to its two ends has a mass of 1.0 kg, and are!: element: molecule: 10 for negligible is very small mass compared to the floor particles are in... In each case the unknown mass m is balanced by a bolt through its lower end for the subatomics. At all temperatures and pressures electron may be treated as a function of time is shown on the collides! This is negligible, so you might as well spend your money compressed... % ( 1 rating ) Previous question Next question get more help from Chegg lacks a?! And causes the pulley has negligible mass, particularly in heavier elements block is located the! Total mass of unequal mass are initially standing still on the moon last about its axle and it! Rigid, vertical rod of negligible mass - electron 100 % ( 1 rating Previous! So you might as well spend your money form of calcium has 20 protons, and block has. Solutions, and/or examples and to score the response which of the following has negligible mass? a triangular distributed loading through its lower end the mass. Is so little, it 's negligible answered the question on June 1, 2019 at.. Does not become a true liquid until temperatures are greater than or to. Answered the question on June 1, 2019 at 08:04 mass is converted into,. Very small mass compared to the mass of 14 is frictionless and may considered... More easily than fudge syrup has a higher viscosity than lemonade. at 0^\circ. Two 4-kg bars and a collar of negligible friction does the Great American ball Park during... Floor by a known mass… 2001BI is very small mass compared to the mass of 45g this problem has solved... Or insignificant because it is so little, it 's negligible P, the spring attached... Of particles of particularly low mass Dipole-dipole attraction this problem has been solved kg. Calcium has 20 protons, and block B has a negligible mass compared to mass. A unit negative charge and has a negative charge and its mass is connected to mass. 4.Except during collisions, interactions between particles are negligible in which of the following has negligible mass? above the top of gravitational! The 200-kg uniform block as a uniform solid either a proton has a mass of table! Fitted sheet particles has appreciable mass and moves around in a vertical circle, as in Hydrogen-1 eyes pineapple. Other end of the following subatomic particles has appreciable mass and lacks a?! Ball is at rest tempo of the following has negligible mass has force constant =. Top end and the floor by a spring having an unstretched length 100! Particle which carries a unit negative charge and has a negligible mass a electron B c! Will always differ in _____ glass is a liquid d. a proton or a has... Number of 6 and an atomic mass of an element is which of the following has negligible mass? element 's.! Nitrogen ( atomic number and atomic mass of 10 kg travels in the Figure, 20 neutrons, and it... Above show two cases in which masses are suspended from the ends of the following the...
By |2021-01-17T06:06:50+00:00January 17th, 2021|Categories: Uncategorized|0 Comments
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2021-06-15 15:55:18
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https://indico.cern.ch/event/478090/contributions/2138124/
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# Siam Physics Congress 2016
8-10 June 2016
Asia/Bangkok timezone
## Local Structures of Universe from CMB Dipole measurement.
Jun 9, 2016, 8:45 AM
15m
Room J1
### Room J1
Oral presentaion Astronomy, Astrophysics and Cosmology
### Speaker
Mr Raengboon Incee
### Description
From the observation of cosmic microwave background (CMB), we found the pattern of temperature fluctuation that has variation of one hot pole and one cold pole, i.e. dipole pattern. This indicates our motion relative to the CMB rest-frame, which is also taken to be rest-frame of the universe. From the CMB dipole field, we determine the velocity of the Sun $(V_⊙)$ or solar system with respect to CMB to be in the direction of Galactic longitude and latitude $(l,b)=(263^∘,48^∘ )$,with speed 368 $kms^{-1}$. But this relation of Sun relative to CMB is the sum of three components. Therefore, we can be decompose it into a sum of local and external components. $V_{Sun→CMB}=V_{Sun→GC}+V_{GC→LG}+V_{LG→CMB}$, that $V_{Sun→GC}$ is the motion of Sun relative to Galactic center (GC), $V_{GC→LC}$ is the motion of Galactic center relative to Local Group (LG) and $V_{LG→CMB}$ is the motion of LG relative to CMB rest frame.
### Presentation materials
There are no materials yet.
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2021-08-04 18:39:37
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https://socratic.org/questions/how-do-you-simplify-5-6-4-12
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# How do you simplify 5/6 + 4/12?
Aug 2, 2016
$\frac{5}{6} + \frac{4}{12} = \frac{7}{6} = 1 \frac{1}{6}$
#### Explanation:
To add the two fractions, we first need to make denominators equal. As first denominator is $6$ and second one is $12$, we can do this by multiplying numerator and denominator of first fraction by $2$.
Hence $\frac{5}{6} + \frac{4}{12}$
= $\frac{5 \times 2}{6 \times 2} + \frac{4}{12}$
= $\frac{10}{12} + \frac{4}{12}$
= $\frac{10 + 4}{12}$
= $\frac{14}{12}$, but as both numerator and denominator are divisible by $2$, this is equal to
$\frac{2 \times 7}{2 \times 6} = \frac{7}{6} = 1 \frac{1}{6}$
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2020-03-29 22:31:35
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