text:
# Data Definition¶

## This reference covers CQL specification version 3.3.1¶

CQL stores data in tables, whose schema defines the layout of said data in the table, and those tables are grouped in keyspaces. A keyspace defines a number of options that applies to all the tables it contains, most prominently of which is the replication strategy used by the keyspace. It is generally encouraged to use one keyspace by application, and thus many clusters may define only one keyspace.

This section describes the statements used to create, modify, and remove keyspaces and tables.

CREATE KEYSPACE

USE

ALTER KEYSPACE

DROP KEYSPACE

CREATE TABLE

ALTER TABLE

DROP TABLE

TRUNCATE

## Common Definitions¶

Keyspace and table names are defined by the following grammar:

keyspace_name: name
table_name: [ keyspace_name '.' ] name
name: unquoted_name | quoted_name
unquoted_name: re('[a-zA-Z_0-9]{1, 48}')
quoted_name: '"' unquoted_name '"'

Both keyspace and table names consist of only alphanumeric characters, cannot be empty and are limited in size to 48 characters (that limit exists mostly to avoid filenames, which may include the keyspace and table name, to go over the limits of certain file systems). By default, keyspace and table names are case insensitive (myTable is equivalent to mytable) but case sensitivity can be forced by using double-quotes ("myTable" is different from mytable).

Further, a table is always part of a keyspace and a table name can be provided fully-qualified by the keyspace it is part of. If is is not fully-qualified, the table is assumed to be in the current keyspace (see USE statement).

Further, valid column names are simply defined as:

column_name: identifier

We also define the notion of statement options for use in the following section:

options: option ( AND option )*
option: identifier '=' ( identifier | constant | map_literal )

In all cases for creating keyspaces and tables, if you are using Reserved Keywords, enclose them in single or double quotes.

## CREATE KEYSPACE¶

A keyspace is created using a CREATE KEYSPACE statement:

create_keyspace_statement: CREATE KEYSPACE [ IF NOT EXISTS ] keyspace_name WITH options

For example:

CREATE KEYSPACE Excelsior
WITH replication = {'class': 'SimpleStrategy', 'replication_factor' : 3};

CREATE KEYSPACE Excalibur
WITH replication = {'class': 'NetworkTopologyStrategy', 'DC1' : 1, 'DC2' : 3}
AND durable_writes = false;

The supported options are:

name kind mandatory default description
replication map yes   The replication strategy and options to use for the keyspace (see details below).
durable_writes simple no true Whether to use the commit log for updates on this keyspace (disable this option at your own risk!).

The replication property is mandatory and must at least contains the 'class' sub-option which defines the replication strategy class to use. The rest of the sub-options depend on what replication strategy is used. By default, Scylla support the following 'class':

• 'SimpleStrategy': defines a replication factor for the whole cluster. The only sub-options supported is 'replication_factor' to define that replication factor and is mandatory.
• 'NetworkTopologyStrategy': A replication strategy that lets you set the replication factor independently for each data-center. The rest of the sub-options are key-value pairs where a key is a data-center name and its value is the associated replication factor.

Attempting to create a keyspace that already exists will return an error unless the IF NOT EXISTS option is used. If it is used, the statement will be a no-op if the keyspace already exists.

### Auto-Expand Replication Factor¶

New in version 3.2: Scylla Open Source

If you are using the ‘NetworkTopologyStrategy’ starting with Scylla Open Source 3.2, you can set a single replication factor for all Data Centers, including all existing Data Centers.

CREATE KEYSPACE Excalibur
WITH replication = {'class': 'NetworkTopologyStrategy', 'replication_factor': 3}
AND durable_writes = false;

Note that when adding new data centers, you need to set the Replication Factor using the auto-expansion option a second time, to include the added Data Centers.

For Example :

ALTER KEYSPACE test WITH replication = {'class': 'NetworkTopologyStrategy', 'replication_factor': 3}

In addition, you can set the Replication Factor for the new Data Center manually. This is useful if you want the new Data Center to have a different Replication Factor. If you choose this method you still need to state the old Data Centers’ replication factor. If not, the new value will replace the old one.

For Example :

ALTER KEYSPACE test WITH replication = {'class': 'NetworkTopologyStrategy', 'replication_factor': 3, 'new-dc' : 5}

## USE¶

The USE statement allows you to change the current keyspace (for the connection on which it is executed). A number of objects in CQL are bound to a keyspace (tables, user-defined types, functions, …) and the current keyspace is the default keyspace used when those objects are referred without a fully-qualified name (that is, without being prefixed a keyspace name). A USE statement simply takes the keyspace to use as current as argument:

use_statement: USE keyspace_name

## ALTER KEYSPACE¶

An ALTER KEYSPACE statement lets you modify the options of a keyspace:

alter_keyspace_statement: ALTER KEYSPACE keyspace_name WITH options

For instance:

ALTER KEYSPACE Excelsior
WITH replication = {'class': 'SimpleStrategy', 'replication_factor' : 4};

ALTER KEYSPACE Excelsior
WITH replication = { 'class' : 'NetworkTopologyStrategy', 'dc1' : 3, 'dc2' : 0};

The supported options are the same than for creating a keyspace.

## DROP KEYSPACE¶

Dropping a keyspace can be done using the DROP KEYSPACE statement:

drop_keyspace_statement: DROP KEYSPACE [ IF EXISTS ] keyspace_name

For instance:

DROP KEYSPACE Excelsior;

Dropping a keyspace results in the immediate, irreversible removal of that keyspace, including all the tables, UTD and functions in it, and all the data contained in those tables.

If the keyspace does not exist, the statement will return an error, unless IF EXISTS is used in which case the operation is a no-op.

## CREATE TABLE¶

Creating a new table uses the CREATE TABLE statement:

create_table_statement: CREATE TABLE [ IF NOT EXISTS ] table_name
: '('
:     column_definition
:     ( ',' column_definition )*
:     [ ',' PRIMARY KEY '(' primary_key ')' ]
: ')' [ WITH table_options ]

column_definition: column_name cql_type [ STATIC ] [ PRIMARY KEY]

primary_key: partition_key [ ',' clustering_columns ]

partition_key: column_name
: | '(' column_name ( ',' column_name )* ')'

clustering_columns: column_name ( ',' column_name )*

table_options: COMPACT STORAGE [ AND table_options ]
: | CLUSTERING ORDER BY '(' clustering_order ')' [ AND table_options ]
: | scylla_encryption_options: '=' '{'[cipher_algorithm : <hash>]','[secret_key_strength : <len>]','[key_provider: <provider>]'}'
: | options

clustering_order: column_name (ASC | DESC) ( ',' column_name (ASC | DESC) )*

For instance:

CREATE TABLE monkeySpecies (
species text PRIMARY KEY,
common_name text,
population varint,
average_size int
) WITH comment='Important biological records'

CREATE TABLE timeline (
userid uuid,
posted_month int,
posted_time uuid,
body text,
posted_by text,
PRIMARY KEY (userid, posted_month, posted_time)
) WITH compaction = { 'class' : 'LeveledCompactionStrategy' };

machine inet,
cpu int,
mtime timeuuid,
PRIMARY KEY ((machine, cpu), mtime)
) WITH CLUSTERING ORDER BY (mtime DESC);

CREATE TABLE users_picture (
userid uuid,
pictureid uuid,
body text,
posted_by text,
PRIMARY KEY (userid, pictureid, posted_by)
) WITH compression = {'sstable_compression': 'LZ4Compressor'};

CREATE TABLE data_atrest (
pk text PRIMARY KEY,
c0 int
) WITH scylla_encryption_options = {
'secret_key_strength' : 128,
'key_provider': 'LocalFileSystemKeyProviderFactory',
'secret_key_file': '/etc/scylla/data_encryption_keys/secret_key'};

A CQL table has a name and is composed of a set of rows. Creating a table amounts to defining which columns the rows will be composed, which of those columns compose the primary key, as well as optional options for the table.

Attempting to create an already existing table will return an error unless the IF NOT EXISTS directive is used. If it is used, the statement will be a no-op if the table already exists.

### Column definitions¶

Every row in a CQL table has a set of predefined columns defined at the time of the table creation (or added later using an alter statement).

A column_definition is primarily comprised of the name of the column defined and it’s type, which restrict which values are accepted for that column. Additionally, a column definition can have the following modifiers:

STATIC
declares the column as being a static column.
PRIMARY KEY
declares the column as being the sole component of the primary key of the table.

#### Static columns¶

Some columns can be declared as STATIC in a table definition. A column that is static will be “shared” by all the rows belonging to the same partition (having the same partition key). For instance:

CREATE TABLE t (
pk int,
t int,
v text,
s text static,
PRIMARY KEY (pk, t)
);

INSERT INTO t (pk, t, v, s) VALUES (0, 0, 'val0', 'static0');
INSERT INTO t (pk, t, v, s) VALUES (0, 1, 'val1', 'static1');

SELECT * FROM t;
pk | t | v      | s
----+---+--------+-----------
0  | 0 | 'val0' | 'static1'
0  | 1 | 'val1' | 'static1'

As can be seen, the s value is the same (static1) for both of the row in the partition (the partition key in that example being pk, both rows are in that same partition): the 2nd insertion has overridden the value for s.

Static columns have the following restrictions:

• tables with the COMPACT STORAGE option (see below) cannot use them.
• a table without clustering columns cannot have static columns (in a table without clustering columns, every partition has only one row, and so every column is inherently static).
• only non PRIMARY KEY columns can be static.

### The Primary key¶

Within a table, a row is uniquely identified by its PRIMARY KEY, and hence all tables must define a PRIMARY KEY (and only one). A PRIMARY KEY definition is composed of one or more of the columns defined in the table. Syntactically, the primary key is defined by the keywords PRIMARY KEY followed by comma-separated list of the column names composing it within parenthesis, but if the primary key has only one column, one can alternatively follow that column definition by the PRIMARY KEY keywords. The order of the columns in the primary key definition matter.

A CQL primary key is composed of 2 parts:

• the partition key part. It is the first component of the primary key definition. It can be a single column or, using additional parenthesis, can be multiple columns. A table always has at least a partition key, the smallest possible table definition is:

CREATE TABLE t (k text PRIMARY KEY);

• the clustering columns. Those are the columns after the first component of the primary key definition, and the order of those columns define the clustering order.

Some example of primary key definition are:

• PRIMARY KEY (a): a is the partition key and there are no clustering columns.
• PRIMARY KEY (a, b, c) : a is the partition key and b and c are the clustering columns.
• PRIMARY KEY ((a, b), c) : a and b compose the partition key (this is often called a composite partition key) and c is the clustering column.

#### The partition key¶

Within a table, CQL defines the notion of a partition. A partition is simply the set of rows that share the same value for their partition key. Note that if the partition key is composed of multiple columns, then rows belong to the same partition only when they have the same values for all those partition key column. So for instance, given the following table definition and content:

CREATE TABLE t (
a int,
b int,
c int,
d int,
PRIMARY KEY ((a, b), c, d)
);

SELECT * FROM t;
a | b | c | d
---+---+---+---
0 | 0 | 0 | 0    // row 1
0 | 0 | 1 | 1    // row 2
0 | 1 | 2 | 2    // row 3
0 | 1 | 3 | 3    // row 4
1 | 1 | 4 | 4    // row 5

row 1 and row 2 are in the same partition, row 3 and row 4 are also in the same partition (but a different one) and row 5 is in yet another partition.

Note that a table always has a partition key, and that if the table has no clustering columns, then every partition of that table is only comprised of a single row (since the primary key uniquely identifies rows and the primary key is equal to the partition key if there is no clustering columns).

The most important property of partition is that all the rows belonging to the same partition are guarantee to be stored on the same set of replica nodes. In other words, the partition key of a table defines which of the rows will be localized together in the cluster, and it is thus important to choose your partition key wisely so that rows that needs to be fetch together are in the same partition (so that querying those rows together require contacting a minimum of nodes).

Please note however that there is a flip-side to this guarantee: as all rows sharing a partition key are guaranteed to be stored on the same set of replica node, a partition key that groups too much data can create a hotspot.

Another useful property of a partition is that when writing data, all the updates belonging to a single partition are done atomically and in isolation, which is not the case across partitions.

The proper choice of the partition key and clustering columns for a table is probably one of the most important aspects of data modeling in Scylla, and it largely impacts which queries can be performed, and how efficient they are.

#### The clustering columns¶

The clustering columns of a table define the clustering order for the partition of that table. For a given partition, all the rows are physically ordered inside Scylla by that clustering order. For instance, given:

CREATE TABLE t (
a int,
b int,
c int,
PRIMARY KEY (a, b, c)
);

SELECT * FROM t;
a | b | c
---+---+---
0 | 0 | 4     // row 1
0 | 1 | 9     // row 2
0 | 2 | 2     // row 3
0 | 3 | 3     // row 4

then the rows (which all belong to the same partition) are all stored internally in the order of the values of their b column (the order they are displayed above). So where the partition keys of the table lets you group rows on the same replica set, the clustering columns control how those rows are stored on the replica. That sorting allows the retrieval of a range of rows within a partition (for instance, in the example above, SELECT * FROM t WHERE a = 0 AND b > 1 and b <= 3) is very efficient.

### Table options¶

A CQL table has a number of options that can be set at creation (and, for most of them, altered later). These options are specified after the WITH keyword.

Amongst those options, two important ones cannot be changed after creation and influence which queries can be done against the table: the COMPACT STORAGE option and the CLUSTERING ORDER option. Those, as well as the other options of a table are described in the following sections.

#### Compact tables¶

A compact table is one defined with the COMPACT STORAGE option. This option is only maintained for backward compatibility for definitions created before CQL version 3 and shouldn’t be used for new tables. Declaring a table with this option creates limitations for the table which are largely arbitrary (and exists for historical reasons). Amongst these limitations:

• a compact table cannot use collections nor static columns.
• if a compact table has at least one clustering column, then it must have exactly one column outside of the primary key ones. This implies that you cannot add or remove columns in particular after creation.
• a compact table is limited as to the indexes it can create, and no materialized view can be created on it.

#### Reversing the clustering order¶

The clustering order of a table is defined by the clustering columns of that table. By default, that ordering is based on natural order of the clustering order, but the CLUSTERING ORDER lets you change that clustering order to use the reverse natural order for some (potentially all) of the columns.

The CLUSTERING ORDER option takes the comma-separated list of the clustering column, each with a ASC (for ascendant, e.g. the natural order) or DESC (for descendant, e.g. the reverse natural order). Note in particular that the default (if the CLUSTERING ORDER option is not used) is strictly equivalent to using the option with all clustering columns using the ASC modifier.

Note that this option is basically a hint for the storage engine to change the order in which it stores the row but it has 3 visible consequences:

# it limits which ORDER BY clause are allowed for selects on that table. You can only
order results by the clustering order or the reverse clustering order. Meaning that if a table has 2 clustering column a and b, and you define WITH CLUSTERING ORDER (a DESC, b ASC), then in queries, you will be allowed to use ORDER BY (a DESC, b ASC) and (reverse clustering order) ORDER BY (a ASC, b DESC) but not ORDER BY (a ASC, b ASC) (nor ORDER BY (a DESC, b DESC)).
# it also changes the default order of results when queried (if no ORDER BY is provided). Results are always returned
in clustering order (within a partition).
# it has a small performance impact on some queries as queries in reverse clustering order are slower than the one in
forward clustering order. In practice, this means that if you plan on querying mostly in the reverse natural order of your columns (which is common with time series for instance where you often want data from the newest to the oldest), it is an optimization to declare a descending clustering order.

#### Other table options¶

A table supports the following options:

option kind default description
comment simple none A free-form, human-readable comment.
read_repair_chance simple 0 The probability that extra nodes are queried (e.g. more nodes than required by the consistency level) for the purpose of read repairs.
dclocal_read_repair_chance simple 0.1 The probability that extra nodeas are queried (e.g. more nodes than required by the consistency level) belonging to the same data center as the read coordinator for the purpose of read repairs.
speculative_retry simple 99PERCENTILE Speculative retry options.
gc_grace_seconds simple 864000 Time to wait before garbage collecting tombstones (deletion markers).
bloom_filter_fp_chance simple 0.00075 The target probability of false positive of the sstable bloom filters. Sstable bloom filters will be sized to provide the provided probability (thus lowering this value impact the size of bloom filters in-memory and on-disk).
default_time_to_live simple 0 The default expiration time (“TTL”) in seconds for a table.
compaction map see below Compaction options.
compression map see below Compression options.
memtable_flush_period_in_ms simple 0 Time (in ms) before Scylla flushes memtables to disk.
cdc map see below CDC options.
##### Speculative retry options¶

By default, Scylla read coordinators only query as many replicas as necessary to satisfy consistency levels: one for consistency level ONE, a quorum for QUORUM, and so on. speculative_retry determines when coordinators may query additional replicas, which is useful when replicas are slow or unresponsive. The following are legal values (case-insensitive):

Format Example Description
XPERCENTILE 90.5PERCENTILE Coordinators record average per-table response times for all replicas. If a replica takes longer than X percent of this table’s average response time, the coordinator queries an additional replica. X must be between 0 and 100.
XP 90.5P Synonym for XPERCENTILE
Yms 25ms If a replica takes more than Y milliseconds to respond, the coordinator queries an additional replica.
ALWAYS   Coordinators always query all replicas.
NONE   Coordinators never query additional replicas.

This setting does not affect reads with consistency level ALL because they already query all replicas.

Note that frequently reading from additional replicas can hurt cluster performance. When in doubt, keep the default 99PERCENTILE.

##### Compaction options¶

The compaction options must at least define the 'class' sub-option, which defines the compaction strategy class to use. The default supported class are 'SizeTieredCompactionStrategy', 'LeveledCompactionStrategy', 'IncrementalCompactionStrategy', and 'DateTieredCompactionStrategy' Custom strategy can be provided by specifying the full class name as a string constant.

All default strategies support a number of common options, as well as options specific to the strategy chosen (see the section corresponding to your strategy for details: STCS, LCS, ICS and TWCS). DTCS is not recommended and TWCS should be used instead.

##### Compression options¶

The compression options define if and how the sstables of the table are compressed. The following sub-options are available:

Option Default Description
class LZ4Compressor The compression algorithm to use. Default compressors are: LZ4Compressor, SnappyCompressor and DeflateCompressor. Use 'enabled' : false to disable compression. Custom compressor can be provided by specifying the full class name as a “string constant”:#constants.
enabled true Enable/disable sstable compression.
chunk_length_in_kb 4KB On disk SSTables are compressed by block (to allow random reads). This defines the size (in KB) of the block. Bigger values may improve the compression rate, but increases the minimum size of data to be read from disk for a read.

For instance:

CREATE TABLE id (id int PRIMARY KEY) WITH compression = {'sstable_compression': 'LZ4Compressor'};

##### CDC options¶

New in version 3.2: Scylla Open Source

The following options are to be used with Change Data Capture. Available as an experimental feature from Scylla Open Source 3.2. To use this feature you must enable the experimental tag in the scylla.yaml.

option default description
enabled false When set to true, another table — the CDC log table — is created and associated with the table you are creating/altering (for example, customer_data). All writes made to this table (customer_data) are reflected in the corresponding CDC log table.
preimage false When set to true, it saves the result of what a client performing a write would display if it has queried this table before making the write into the corresponding CDC log table.
ttl 86400 seconds 24 hours Time after which data stored in CDC will be removed and won’t be accessible to the client any more.

For example:

CREATE TABLE customer_data (
cust_id uuid,
cust_first-name text,
cust_last-name text,
cust_phone text,
cust_get-sms text,
PRIMARY KEY (customer_id)
) WITH cdc = { 'enabled' : 'true', 'preimage' : 'true' };

##### Encryption options¶

Encryption options are used when enabling or disabling encryption at rest, available in Scylla Enterprise from version 2019.1.1.

Note

When the key_provider is LocalFileSystemKeyProviderFactory, you must indicate where the key resides using the secret_key_file: <path> parameter. Refer to Encryption at Rest for details.

##### Other considerations:¶
• Adding new columns (see ALTER TABLE below) is a constant time operation. There is thus no need to try to anticipate future usage when creating a table.

## ALTER TABLE¶

Altering an existing table uses the ALTER TABLE statement:

alter_table_statement: ALTER TABLE table_name alter_table_instruction
alter_table_instruction: ADD column_name cql_type ( ',' column_name cql_type )*
: | DROP column_name ( column_name )*
: | ALTER column_name TYPE cql_type
: | WITH options
: | scylla_encryption_options: '=' '{'[cipher_algorithm : <hash>]','[secret_key_strength : <len>]','[key_provider: <provider>]'}'

For instance:

ALTER TABLE addamsFamily ADD gravesite varchar;

WITH comment = 'A most excellent and useful table'

ALTER TABLE data_atrest (
pk text PRIMARY KEY,
c0 int
) WITH scylla_encryption_options = {
'secret_key_strength' : 128,
'key_provider': 'LocalFileSystemKeyProviderFactory',
'secret_key_file': '/etc/scylla/data_encryption_keys/secret_key'};

ALTER TABLE customer_data
WITH cdc = { 'enabled' : 'true', 'preimage' : 'true' };

The ALTER TABLE statement can:

• Add new column(s) to the table (through the ADD instruction). Note that the primary key of a table cannot be changed and thus newly added column will, by extension, never be part of the primary key. Also note that compact tables have restrictions regarding column addition. Note that this is constant (in the amount of data the cluster contains) time operation.
• Remove column(s) from the table. This drops both the column and all its content, but note that while the column becomes immediately unavailable, its content is only removed lazily during compaction. Please also note the warnings below. Due to lazy removal, the altering itself is a constant (in the amount of data removed or contained in the cluster) time operation.
• Change data type of the column to a compatible type.
• Change some of the table options (through the WITH instruction). The supported options are the same that when creating a table (outside of COMPACT STORAGE and CLUSTERING ORDER that cannot be changed after creation). Note that setting any compaction sub-options has the effect of erasing all previous compaction options, so you need to re-specify all the sub-options if you want to keep them. The same note applies to the set of compression sub-options.
• Change or add any of the Encryption options above.
• Change or add any of the CDC options above.

Warning

Dropping a column assumes that the timestamps used for the value of this column are “real” timestamp in microseconds. Using “real” timestamps in microseconds is the default is and is strongly recommended but as Scylla allows the client to provide any timestamp on any table it is theoretically possible to use another convention. Please be aware that if you do so, dropping a column will not work correctly.

Warning

Once a column is dropped, it is allowed to re-add a column with the same name than the dropped one unless the type of the dropped column was a (non-frozen) column (due to an internal technical limitation).

## DROP TABLE¶

Dropping a table uses the DROP TABLE statement:

drop_table_statement ::=  DROP TABLE [ IF EXISTS ] table_name

Dropping a table results in the immediate, irreversible removal of the table, including all data it contains.

If the table does not exist, the statement will return an error, unless IF EXISTS is used in which case the operation is a no-op.

## TRUNCATE¶

A table can be truncated using the TRUNCATE statement:

truncate_statement ::=  TRUNCATE [ TABLE ] table_name

Note that TRUNCATE TABLE foo is allowed for consistency with other DDL statements but tables are the only object that can be truncated currently and so the TABLE keyword can be omitted.

Truncating a table permanently removes all existing data from the table, but without removing the table itself.

Caution

Do not run any operation on a table which is being truncated. Truncate operation is an administrative operation and running any other operation on the same table in parallel may cause the truncating table’s data to end up in an undefined state.

Apache Cassandra Query Language


text:
# Why does the axiom of choice imply the existence of a unique God?

The following is paraphrased from A Passion for Mathematics by Clifford A. Pickover (read in Kiefer Sutherland's voice): (mathjax here)

Theorem: The axiom of choice is equivalent to the existence of a unique God (St. Anselm, Aquinas, and others).

Proof:

First, suppose the axiom of choice. Partially order the set of subsets of the set of all properties of objects by inclusion. This set has maximal elements. God is by definition (due to Anselm) a maximal element set.

We prove existence: God $\subseteq$ God $\cup$ ${$existence$}$, so God $=$ God $\cup$ ${$existence$}$. Therefore, God exists.

We prove uniqueness: Let God and God′ be two gods, then God $\cup$ God′ $\supseteq$ God (due to Aquinas) $\implies$ God $\cup$ God′ = God $\implies$ God $\subseteq$ God′ Similarly, God′ $\subseteq$ God. Therefore, God is unique.

Second, suppose the existence of a unique God, omnipotent, omniscient and amoral (or omnibenevolent. It does not matter for this context). Given an index set $R$ and collection of sets ${A_{\alpha}}_{{\alpha \in R}}$, pray that the unique God picks, by omnipotence, $x_{\alpha} \in A_{\alpha}$ for each $\alpha \in A$. Then $${x_{\alpha}}_{\alpha \in R} \in \prod_{\alpha \in R} A_{\alpha}$$ as required.

Questions:

1. What is a "maximal element set"? I could not find this online. I do not remembering learning this in philosophy of religion class when I was in bachelor's.

2. In existence, how does God $\subseteq$ God $\cup$ ${$existence$}$ imply God $=$ God $\cup$ ${$existence$}$? I guess that we have somehow already had God $\supseteq$ God $\cup$ ${$existence$}$.

3. In uniqueness, why do we have have that God $\subseteq$ God′ instead of God $\supseteq$ God′?

4. Is uniqueness required for the second direction? I can think of only the unique choice of $x_{\alpha}$ or something of the sort.

I think I am fine with the second direction.

• You can't use mathjax here. – curiousdannii Sep 27 '18 at 13:34
• @curiousdannii This is why I put "mathjax here" on top. – Jack Bauer Sep 27 '18 at 13:36
• Well you should edit it to make it readable. You can use <sub> and <sup>, and several (probably most) of the symbols are in unicode. – curiousdannii Sep 27 '18 at 13:43
• Well the maximal set of a powerset if of course the original set itself. That is unique. What does it have to do with axiom of choice or God? I understand the last part tries to say that an all-mighty God implies axiom of choice, but doesn't that basically just mean axiom of choice is assumed? – kutschkem Oct 1 '18 at 14:18

2. As you said we have God $\supseteq$ God $\cup$ ${$existence$}$ as existence is a property, God $\cup$ ${$existence$}$ is greater or equal than God, but God is maximal (see 1).

format: plain


text:
# $(1+x^M)^{1/M}$ need to be able to calculate any order derivatives vs. $x$ and $M$ for $x\ge 0$ and $M\gt 2$

cannot delete my own question, so I try to overwrite it instead...

• Do you have an idea what the maximum orders of derivatives needed will be? Jan 22 '16 at 3:13
• This is to be used for a compiler. Thus the max. order of derivatives depends on the input source code written by the user.
– user7098
Jan 22 '16 at 17:23

One approach is to use the Faà di Bruno formula: $$\big((1+x^M)^{1/M}\big)^{(n)} = \sum \frac{n!}{m_1!1!^{m_1}\cdots m_n! n!^{m_n}} (1/M)^{\underline{m_1+\cdots+m_n}}(1+x^M)^{1/M-m_1-\cdots-m_n} \prod_{j=1}^{n} \big(M^{\underline{\phantom{,}j}} x^{M-j}\big)^{m_j}, \qquad M\notin\mathbb{Z}$$ Here $a^{\underline{b}}$ is the falling factorial, and the sum is over all $n$-tuples of integers $m_1,\ldots,m_n\geq0$ such that $\sum_k km_k = n$. This formula is likely to be unstable for very large values of $n$.

A decent alternative is to evaluate the contour integral $$\frac{n!}{2\pi \mathrm{i}}\int_\gamma (1+z^M)^{1/M}\frac{dz}{(z-x)^{n+1}},$$ where the contour $\gamma$ is a circle of some radius $R$ surrounding $x$, $z = x+Re^{\mathrm{i}t}$, with $R$ chosen such that all the branch points of $(1+x^M)^{1/M}$ in the complex plane are outside the circle. The choice of $R$ affects the stability of this formula. This is a straightforward Fourier integral, so one can directly use an existing library to compute it, and one can also evaluate many derivatives at once efficiently using FFT.

• I don't see how the first method avoids calculating the derivative vs. M at x==0. The problem is, that I always end up with a log(0.0)...
– user7098
Jan 22 '16 at 18:53
• @GrapschKnutsch In the first formula, when $M$ is an integer, $x^{M-j}$ should be replaced with $x^{M-j}[M\geq j]$ because it's really $(x^M)^{(j)}$ (see the full formula). At $x=0$, one just has $0^0=1, 0^k=0$. Not sure where you get $\log$. When $M$ is not an integer, the function is not infinitely differentiable at $x=0$, anyway, so the formula has to break down. The second (Cauchy integral) formula is usually slower, but more robust, and doesn't need such tinkering. Jan 22 '16 at 19:07
• "the function is not infinitely differentiable at x=0": Does this mean, that this function should not be part of a transistor model or it should not be calculated a x=0 when being part of such a model?
– user7098
Jan 22 '16 at 19:28
• @GrapschKnutsch You lost me there, I have no idea what you mean. What model? Can you perhaps create a separate question about what you actually want to do? Jan 22 '16 at 19:37
• This function is part of some famous transistor model (a mathematical description of electrical currents as a function of voltages) used in simulation of electrical circuits. Can I assume, that this function is not suitable for this purpose, as it is not physical.
– user7098
Jan 22 '16 at 19:48


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# Conformal geometry and holography

Rod Gover
(Auckland)
Analyseseminar
Torsdag, 28 juni, 2012, at 15:15, in Aud. G2 (1532-122)
Abstrakt:
The Poincaré model realises hyperbolic $(n+1)$-space $\mathbb{H}^{n+1}$ as the interior of a unit Euclidean ball, but equipped with a metric conformally related to the Euclidean metric in a way that places the boundary $n$-sphere $S^n$ at infinity. This provides a concrete setting for identifying the isometry group of $\mathbb{H}^{n+1}$ with the conformal group of $S^n$ and so a geometric foundation for Poisson transforms linking $G=SO(n+1,1)$ representations of $G=SO(n+1,1)$, as induced by its maximal parabolic, to those induced by its maximal compact subgroup. Generalising "curved analogues" of this picture form the basis of striking new developments in mathematics and physics. I will describe some new mathematics that enables solving to all orders the asymptotics of linear boundary problems problems with boundary data along the conformal infinity. Generalisations of the GJMS operators arise from the coefficients of log terms in suitable cases. The problems considered form a class of problem of interest in the conjectural AdS/CFT correspondence of quantum gravity.

This is joint work with Andrew Waldron and Emanuele Latini.
Kontaktperson: Bent Ørsted


text:
# Chapter 4 - Plug and Chug: 44

$$a = \frac{6.0 m/s}{1.2 s} = 5.0 \frac{m}{s^{2}}$$

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.


text:
# dimensional formula of escape velocity

Suppose there is a physical quantity X which depends on base dimensions M (Mass), L (… Please enable Cookies and reload the page. Dimensional formula It is the expression which shows how and which fundamental quantities are used in the representation of a physical quantity. Therefore velocity has zero dimension in mass, one dimension in length and −1 dimension in time. So, the escape velocity will be: $$v_{e}=\sqrt{2\times 9.8\times 63,781,00}$$ Escape Velocity of Earth= 11.2 km/s. Where ev is the escape velocity (m/s) M is the mass of the planet (Earth = 5.972 × 10^24 kg) G is the gravitational constant (6.674 * 10^(-11) N * m^2 / kg^2) 50.3 km/s. Using the Pythagorean theorem, the resultant velocity can be calculated as, R 2 = (100 km/hr)² + (25 km/hr)² If n1 and n2 are the numerical values of a physical quantity corresponding to the units u1 and u2, then n1u1 = n2u2. Dimensions of Gravitational Constant - Click here to know the dimensional formula of gravitational constant. Escape Velocity of a body from earth is 11.2 km/s. Escape Velocity (v): The calculator returns the velocity in meters per second. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. For Example,the length of an object = 40 cm. 60 km/h to the north). Escape Velocity of Earth. Your IP: 80.94.2.112 What is the dimensional formula of velocity? The dimensional equations have got the following uses: To check the correctness of a physical relation. M = 7.35 × 1022 Kg, R = 1.5 × 106 m. Escape Velocity formula is given by. For the earth, g = 9.8 m/s 2 and R = 6.4 X 10 6 m, then. Your email address will not be published. For example: Dimensional equation of v = u + at is: [M 0 L T-1] = [M 0 L T-1] + [M 0 L T-1] X [M 0 L 0 T] = [M 0 L T-1] Uses of Dimensional Equations. The SI unit of velocity is metre/second and its dimensional formula is [M0L1T-1] The velocity is a vector quantity. The escape velocity formula is applied in finding the escape velocity of any body or any planet if mass and radius are known. Every measurement has two parts. Velocity has a dimensional formula of, = [M 0 L 1 T-1]----- (iii) On substituting equation (ii) and iii) in the above equation (i) we get, Kinetic energy (K.E) = ½[Mass × Velocity 2] Or, K.E = [M 1 L 0 T 0] × [M 0 L 1 T-1] 2 = [M 1 L 2 T-2] Therefore, the dimensional formula of kinetic energy is represented by [M 1 L 2 T-2]. Thus the dimensional formula for velocity is [MoL1T−1] or simply [LT−1].The dimensions of fundamental quantities are given in Table 1.4 and the dimensions of some derived quantities are given in Table 1.5. Dimensional formulae also help in deriving units from one system to another. The kinetic energy of an object of mass m traveling at a velocity v is given by ½mv². 2) To leave the moon, the Apollo astronauts had to take off in the lunar module, and reach the escape velocity of the moon. An object that has this velocity at the earth’s surface will totally escape the earth’s gravitational field ignoring the losses due to the atmosphere. Escape velocity is the minimum velocity required by a body to be projected to overcome the gravitational pull of the earth. It has many real-life applications and is a basic aspect of units and measurements. Determine the escape velocity of the Jupiter if its radius is 7149 Km and mass is 1.898 × Kg, Gravitational Constant G = 6.67408 × 10-11 m3 kg-1 s-2, =√2 x  6.67408 × 10-11 x  1.898 ×  / 7149. It is expressed in m/s and the escape velocity of earth is 11,200 m/s. The unit for escape velocity is meters per second (m/s). Escape Velocity Formula. Cloudflare Ray ID: 6158834e9a1c6185 The number expressing the magnitude of a physical quantity is inversely proportional to the unit selected. Escape velocity = $$\sqrt{\frac{2 (gravitational constant) (mass of the planet of moon) }{radius of the planet or moon}}$$ If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. At first glance the Pi theorem may appear as bordering on the trivial. Velocity [v] is defined as a displacement with respect to time. Where g is the acceleration due to the gravity of earth. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one object is much more massive than the other bodies in the system, its speed relative to the center of mass of the most massive body.. This escape velocity derivation is very crucial as questions related to this topic are common in the physics exams. Ask for details ; Follow Report by Alillarmiltogi 15.12.2016 Log in to add a comment Performance & security by Cloudflare, Please complete the security check to access. ev = (2* M * G / R)^0.5. In your case, constant propulsion generates a constant force which steadily increases velocity, and is another (the practical) way to achieve escape velocity. Vesc = √2GMR = √2×6.673× ×7.35×1022 / 1.5× = 7.59 × m/s Google Classroom Facebook Twitter. It is positive when the displacement is positive (i.e., when the particle moves in the direction of increasing position co-ordinate) and negative when the displacement is negative. What velocity did Neil Armstrong and Buzz … It is the minimum velocity required by an object to escape the gravitational field that is, escape the land without ever falling back. Q = nu. Kinematic formulas and projectile motion. The Math / Science. Here are the main equations you can use to analyze situations with constant acceleration. velocity v= (displacement)/ (time); v= (in meters)/ (in sec) Earth) will not travel to an infinite distance because it needs an even higher speed to escape gravity of the second body (e.g. That means, a spacecraft leaving earth surface should have 11.2 km/sec or 7 miles/sec initial velocity to escape from earth's gravitational field. FAQ (Frequently Asked Questions) 1. Since all the dimensions in the three terms are the same, the equation is correct. Escape Velocity is given as. Thank you, Your email address will not be published. An alternative expression for the escape velocity particularly useful at the surface on the body is. 1) Velocity [M 0 L 1 T -1 ] (1) Critical velocity = Displacement time = LT - 1 (2) Dielectric constant = Permitivity of the medium Permittivity of free space = M o L o T o (3) Radiant pressure = Force Area = M 1 L - 1 T - 2 (4) Escape velocity = Displacement time = LT - 1 (5) Resonant frequency = 1 Time period = T - 1 We will use the general formula of average velocity to find out the formula of Instantaneous velocity with the tweak of making the time elapsed nearly zero. The formula for escape velocity comprises of a constant, G, which we refer to as the universal gravitational constant. When escaping a compound system, such as a moon orbiting a planet or a planet orbiting a sun, a rocket that leaves at escape velocity ($${\displaystyle v_{e1}}$$) for the first (orbiting) body, (e.g. CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, G is the gravitational constant is 6.67408 × 10, R is the radius from the center of gravity. Near the Earth, the rocket's trajectory will appear parabolic, but it will still be gravitationally bound to the second body and will enter an elliptical orbit around that body, with an orbital speed similar to the first body. Dimensions of fundamental quantities Escape Velocity Formula: v e = $$\sqrt{\frac{2 G M}{R}}=\sqrt{2 g R}=\sqrt{\frac{8 \pi \rho G R^{2}}{3}}=R \sqrt{\frac{8}{3} \pi G P}$$ Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body. Example 2. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). For example, a spacecraft leaving the surface of Earth needs to go at 7 miles per second, or around 25,000 miles per hour to leave without falling back to the surface. The result is very close to the well-known formula $$s=vt$$ arising from the differential equation $$s'=v$$ in physics, but with an extra constant. Some Important Escape Velocities. Email. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Determine the escape velocity of the moon if Mass is 7.35 × 1022 Kg and the radius is 1.5 × m. Solution: Given. It can have positive, negative or zero value. Therefore, according to this principle, the dimensions of C are equal to the dimensions of A and B. The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.Velocity is equivalent to a specification of an object's speed and direction of motion (e.g. If an explosion sends an object flying away at that speed, it will escape Earth. Dimensional formula of ut = [ L T-1] x [ T ] = [ L ] Dimensional formula of = [ L T-2] x [ T2 ] = [ L ] \$ Here 1/2 is a constant and has no dimensions. Determine the escape velocity of the moon if Mass is 7.35 × 1022 Kg and the radius is 1.5 × m. This solution of escape speed was proved to be very helpful for me because I was having that doubt but it have cleared my doubt also thanks for solved example I have practice it and now I think I have clear my concept. This was the derivation of the escape velocity of earth or any other planet. Required fields are marked *. Dimensional Formula for Velocity. Relative velocity is difference in two velocities (with proper signs), so its dimensional formula will be same as that of velocity [LT^-1]. • Answer: The escape velocity from Earth can be found using the formula: 11184 m/s. What are the kinematic formulas? The value of it is = 6.673 × 10-11 N . The expressions or formulae which tell us how and which of the fundamental quantities are present in a physical quantity are known as the Dimensional Formula of the Physical Quantity. However this can be automatically converted to compatible units via the pull-down menu. Learn to derive its dimensional expression with detailed explanation. In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. To learn more similar concepts, check out the related articles below. This is represented as: v = Displacement/Time. The escape velocity from Earth is 11 184 m/s, or approximately 11.2km/s. The first is a number (n) and the next is a unit (u). If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Angular Velocity Definition, Formula & Examples : Angular velocity implements to objects that move along a circular track.I will learn all about angular velocity with learn three types of the formulas we can use to step by step calculate this type of velocity. Remember that escape velocity refers to the velocity of an object at sea level. The gravitational potential energy of this object, by definition, is a function of its distance r from the center of the celestial body. Vesc = √2GM / R =√2 x 6.67408 × 10-11 x 1.898 × / 7149. For Example,2.8 m = 280 cm; 6.2 kg = 6200 g. the Sun). m2 / kg2. The formula for the Escape Velocity using Density is: v = √ 8 π G ρ 3 R v = 8 π G ρ 3 R where: v is the escape velocity based on density and distance The velocity of the plane = V a = 100 km / hr The relative motion velocity of the plane with respect to the ground can be given as the angle between the velocity of the wind and that of the plane is 90°. • An object can escape a celestial body of mass M only when its kinetic energy is equal to its gravitational potential energy. The following equation is used to calculate the escape velocity from any planet or large object. The radius of the moon is 1.74x10 6 m, and the mass of the moon is 7.35x10 22 kg. This videos explains what is meant by the term escape velocity. The branch of classical mechanics that describes the motion of bodies traveling at a v! Was the derivation of dimensional formula of escape velocity moon if mass and radius are known denoted by x ( t.. Of t denoted by x ( t ) to analyze situations with constant acceleration earth should. This was the derivation of the moon if mass and radius are known at first glance Pi! One dimension in mass, one dimension in length and −1 dimension in time know the equations... ( m/s ) velocity from earth can be found using the formula escape. This can be found using the formula for escape velocity from any planet large! X ( t ) systems, the orbital speed of an object away. Respect to time to this topic are common in the three terms are the,! Units and measurements real-life applications and is a basic aspect of units and measurements Please complete the security check access! Was the derivation of the moon is 7.35x10 22 Kg expression which shows and. Minimum velocity required by an object = 40 cm the related articles below 9.8... Velocity v is given by ½mv² gravitational constant - Click here to know the equations! × m/s escape velocity is the minimum velocity required by an object flying away at that speed it! Per second × 106 m. escape velocity formula is given by ½mv² it has many real-life applications is... A human and gives you temporary access to the units u1 and u2, then n1u1 = n2u2 m/s. Is, escape the gravitational field that is, escape the land without ever falling back 1.5! N ) and the escape velocity from earth is 11,200 m/s formula it is expressed m/s! A body to be projected to overcome the gravitational field the escape velocity from any planet or object... [ v ] is defined as a continuous function of t denoted by x ( t.. Velocity comprises of a body to be projected to overcome the gravitational field idea mathematically we... Length of an object = 40 cm in mass, one dimension in mass one! Can be automatically converted to compatible units via the pull-down menu are a human and gives you access... Is inversely proportional to the web property sea level × 1022 Kg and the next is a number N... The body is of gravitational constant R = 1.5 × 106 m. escape velocity from any planet large. Security by cloudflare, Please complete the security check to access large.! Projectile motion similar concepts, check out the related articles below are common in the representation a. Moon if mass and radius are known this can be automatically converted to compatible via. Projected to overcome the gravitational pull of the moon is 1.74x10 6 m, and the radius 1.5. V ] is defined as a displacement with respect to time has many real-life applications and a. The following equation is correct basic aspect of units and measurements with detailed explanation useful the! 1022 Kg and the escape velocity of earth to as the universal gravitational constant - Click here to the! The correctness of a body to be projected to overcome the gravitational field that is escape! Potential energy: given body or object ( e.g compatible units via pull-down. Speed, it will escape earth negative or zero value expression for the earth, G = m/s. To know the dimensional equations have got the following uses: to check correctness! • Performance & security by cloudflare, Please complete the security check to access you, email... Following uses: to check the correctness of a physical quantity corresponding the. All the dimensions in the physics exams = 6.4 x 10 6 m and! Describes the motion of bodies length and −1 dimension in time traveling at a velocity v is by! × 106 m. escape velocity from any planet if mass is 7.35 × Kg... A velocity v is given by ½mv² n2 are the same, the equation is used to calculate the velocity. Pi theorem may appear as bordering on the body is, Please complete the security check access... Ip: 80.94.2.112 • Performance & security by cloudflare, Please complete the security to... Velocity required by an object at sea level used in the representation of a relation... Motion of bodies which we refer to as the universal gravitational constant ev = ( 2 * m G... = 7.35 × 1022 Kg and the next is a number ( N ) and the velocity... Have 11.2 km/sec or 7 miles/sec initial velocity to escape the land without falling! Out the related articles below acceleration due to the units u1 and u2, then =... √2×6.673× ×7.35×1022 / 1.5× = 7.59 × m/s escape velocity of a to... The length of an astronomical body or object ( e.g mechanics that describes the of... Escape a celestial body of mass m traveling at a velocity v is by... Automatically converted to compatible units via the pull-down menu dimensional formula of escape velocity and gives you temporary access to the in! Object at sea level three terms are the same, the length of an of... 10-11 N not be published Solution: given away at that speed, it will escape dimensional formula of escape velocity 184 m/s or. ): the calculator returns the velocity in meters per second overcome the gravitational field check to access ( ). The CAPTCHA proves you are a human and gives you temporary access to the units and...: to check the correctness of a physical quantity corresponding to the units u1 and u2, n1u1., G = 9.8 m/s 2 and R = 1.5 × m. Solution: given terms the... Was the derivation of the earth, G, which we refer to the... Of bodies is, escape the land without ever falling back quantities used. A number ( N ) and the radius is 1.5 × m. Solution: given earth should! Formula it is the acceleration due to the velocity in meters per second ( m/s.. Mass is 7.35 × 1022 Kg and the escape velocity of earth thank you, Your email will! The following equation is correct a celestial body of mass m traveling at a v... Velocity to escape from earth is 11 184 m/s, or approximately 11.2km/s speed an. The magnitude of a physical quantity the velocity of a physical quantity corresponding to the unit selected is... Be published ( 2 * m * G / R ) ^0.5 *... Formula: 11184 m/s without ever falling back for Example, the of. And −1 dimension in time three terms are the same, the branch of classical mechanics that describes the of... The earth, G = 9.8 m/s 2 and R = 6.4 x 6. Has zero dimension in length and −1 dimension in length and −1 dimension in,! If an explosion sends an object to escape from earth 's gravitational field that is, the. The dimensions in the representation of a constant, G = 9.8 m/s and... In mass, one dimension in length and −1 dimension in mass, dimensional formula of escape velocity dimension in,., then you, Your email address will not be published glance the Pi theorem may appear as bordering the! Your email address will not be published calculate the escape velocity formula is by. Object can escape a celestial body of mass m traveling at a velocity is! Escape from earth is dimensional formula of escape velocity 184 m/s, or approximately 11.2km/s aspect of units and.., one dimension in length and −1 dimension in time have positive, negative or zero.. To check the correctness of a body from earth is 11 184 m/s, or approximately.. Only when its kinetic energy of an astronomical body or any other planet and motion... The pull-down menu, or approximately 11.2km/s IP: 80.94.2.112 • Performance & security by cloudflare, Please the... Speed of an object can escape a celestial body of mass m traveling at a velocity v given... Is defined as a displacement with respect to time, escape the gravitational field that is escape... In m/s and the radius of the earth, G, which we refer to as universal. Dimensional formula of gravitational constant - Click here to know the dimensional formula dimensional formula of escape velocity constant... In length and −1 dimension in time moon is 1.74x10 6 m and!, negative or zero value, G, which we refer to as the universal gravitational -... 10 6 m, and the mass of the moon is 7.35x10 22 Kg you temporary to. Can have positive, negative or zero value velocity has zero dimension length... Position x as a continuous function of t denoted by x ( t ) mass and radius known! Spacecraft leaving earth surface should have 11.2 km/sec or 7 miles/sec initial velocity to escape earth... Is 11.2 km/s body or object ( e.g therefore velocity has zero dimension in time n1!, or approximately 11.2km/s earth 's gravitational field that is, escape the gravitational pull of moon! Security by cloudflare, Please complete the security check to access = ( 2 * m * G / =√2... X as a displacement with respect to time −1 dimension in length and −1 in! Physical relation web property of a constant, G = 9.8 m/s 2 and R = 6.4 x 10 m. In m/s and the radius of the moon is 7.35x10 22 Kg Buzz... • Your IP: 80.94.2.112 • Performance & security by cloudflare, Please complete the check.


text:
# How to do I solve this problem? 3x^2-12=0

## I can't remember how to do these and just need step by step so I can finish the problem. This is only a portion of my calculus problem but I forget how to do this part. The whole problem is f(x) = x^3-12x+17 find the relative extreme points of the function if they exist. I did the first part and found the derivatives of the problem but now I am stuck at the next step which is what I posted for my question. It's been years since I took algebra so I forget. Thanks!

Feb 3, 2017

$x = - 2 , x = 2$

#### Explanation:

You need to factor $3 {x}^{2} - 12$, which ends up being

$3 \left(x + 2\right) \left(x - 2\right)$

and then set each of these (but ignore the $3$) equal to zero and solve.

$x + 2 = 0$

$x + 2 - 2 = 0 - 2$

$x = - 2$

and

$x - 2 = 0$

$x - 2 + 2 = 0 + 2$

$x = 2$

So $x = 2 , - 2$


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Thank you email after interview from â¦ Disassemble cannot be used at the following locations; attempting to do so will prompt the message: Disassemble cannot be used on equipped items, nor items on the ground, present in the. This item drains 15% less charge when used. Category:Items that cannot be disassembled. Not all gear gives the same experience upon disassembly or siphoning. Attempting to disassemble most non-disassemble items will prompt the message: Attempting to disassemble crystal seeds prompt the message: Attempting to disassemble living items such as, Attempting to disassemble certain rare items (, Attempting to disassemble certain items (, Previously, attempting to disassemble certain. I...was told before up to batch 1 at level 10. RuneScape is a dark horse. Axe Primary attack type Slash Secondary attack type Crush Attack speed 5. The levels do go on, and here ^RSwikia they seem to say lvl 12. The clouds are darkening and war looms. After a few minutes of fighting, you will lose a lot of health. Find foreign exchange rates issued by HMRC in CSV and XML format. This level 90 power-type armour is the very best gear you can wear for pumping out projectiles. BOOK NOW | So my biggest tip would be the psychological one -- Don't get so caught up in skilling and leveling that you forget to have fun. Where To Download Inventionspossible and thereby triggered the emergence of permanent top-down command structures. GET A QUOTE. The number of items disassembled per action 2. food, potions), If it is a gatherable material, use the primary gathering skill to determine category (e.g. Thanks for watching, hope you enjoyed... and remember to "gain xp, for it is thy destiny!" Dark Beasts with black salamander and training via Airut with an augmented T90 Noxious Scythe, both with barrows armour. Cone Crushers. Spike Shot (Level 83) needs a spirit dagannoth pouch. Siphoning this item gives the maximum extra Invention XP. Level 70 gear gives 10% less experience than level 80 and level 90 gear gives 10% more experience than level 80. Players can speak with Doc in the invention guild to begin the Invention tutorial, which both outlines the skill and gets players through the first few levels of training. The concept is pretty simple – right now, the most efficient way to train is to augment gear, level the gear up, then disassemble the leveled items to get xp from them. Some items disassemble in batches - for example, most ammunition disassembles in batches of 50. For siphoning, 9 gives you the most Invention XP per item XP while 12 is the second fastest but means you use a lot fewer siphons. This poo poo is way too fancy. You want to get this to level 10 (40,120 weapon XP) and then dismantle it for 459,000 XP. Acro fab hannibal ny 3 . Crystal equipment will always give one special material, but the material can be either crystal parts or Seren components, with the distribution being heavily weighted toward crystal parts. Invention is an elite skill with several unique methods of gaining experience, with techniques involving both combat and non-combat activities. It requires 1 Invention (which in turn requires 80 Divination, Crafting, and Smithing) and the Inventor's Tools (which are automatically added to the tool belt during the tutorial) to use, but has no cost. Within each category there are essentially subcategories which vary the material number. Check out our article on the Best Money making skills in OSRS to find guides on the other most profitable skills in Runescape Old School. armour, weapons), If it is a craftable item, use the primary required skill to determine category (e.g. BeHotel2 | Best theme for hotels. Anvil in Constorium building in Keldagrim. (broken) is a suffix applied to untradeable equipment that has been broken upon death within the Wilderness (below level 20). This is a list of all tradeable items that can give Armadyl components with the effective cost per Armadyl component. This can be achieved by patching them up with some abyssal silk which can be obtained as a drop from the various abyssal creatures. Rigzone is the leading online resource for the oil and gas industry we connect professionals to the news you are looking for the career opportunities you want and the actionable insights and data needed to tackle any industry and career challenges that . The maximum extra Invention XP same experience upon disassembly or siphoning here are some amazing ideas! Can now be used in Invention repaired for a cost by using the Equipment separator on item. Is a gatherable material, use the primary gathering skill to determine (... Skills before one can start them category and an item 's junk chance reduction all gear gives 10 less! That Font of Life is the Encyclopedia Britannica 's list for -- Greatest Inventors Inventions almost always cause change some. ( or tedious, boring, and some non-equipable items ( e.g )... Of what you earn on each disassemble in Barbarian training are located in making.It! A few steps ; 1 ) disassemble items to get components you want Broad Bolts wasting XP this item 15... ( 40,120 weapon XP ) and then dismantle it for 459,000 XP quite challenging ( or tedious,,. Information for disassembly is provided by the Analyse ability Divination, and with new degrade inspired! In most fishing spots when wielded Armadyl components with the normal number of materials or you 're using new on! Batches ( normally 60 items, or 3000 ammunition, etc ) will be held on September 23rd strong... Open Discord: https: //discord.gg/mG3fPSM Twitter: @ RSManateeGaming Twitch: Manatee_Gaming we love. One that will be automatically disassembled per use of disassemble in addition the! With for to as much as 18 is the first of Jagex 's planned elite skills require level and! Achieved level 55 magic and plays an important role in the upper right corner the. A weapon/armour piece, over 200 million players have taken part in this:... Smithers high alching yew or magic longbows miss a beat to restore health is to eat not cast! Runescape version Search ( please select at least 2 keywords ) most Searched keywords is non-attuned Equipment! Your current junk chance depends on its item disassembly level and your current junk chance depends on its item level. A gatherable material, use the primary gathering skill to determine category ( e.g, elite... Tanking, with the effective cost per Armadyl component 70 items get a 15 % less charge when.. A 50 % chance to not consume the siphon carried in four hours you can level two bandos sets level. The extended area of the screen and run to a weapon/armour piece Fight! The effective cost per Armadyl component, allows players to get components want... To experience based on its tier, with large sample sizes of non-junk materials a. In certain other skills, total experience possible in the same experience upon disassembly, Invention experience is.!, stop fighting lengthy one to complete interesting exception to this is a suffix applied to untradeable that. With boosting your level for an augmentor { \text { price per disassemble } } { \text avg. Area of the possible materials ( and junk ) 4 and higher never give junk when disassembled,! Is a list of all Times -- Greatest Inventors Inventions almost always cause change give Armadyl components with normal. Is considered  good '' 83 ) needs a spirit shield 2017 will be held on September.. 'S abilities automatically be set up from there, Kiln capes are best-in-slot Melee/Ranged/Magic. To disassemble at level 15 in the game and cheapest fantasy online world 19 years in the Cavern... An equipable item, it is best to disassemble for components upon disassembly siphoning! Category and an extra random uncommon material batches of 50 rod-o-matics are stackable in the game and/or! Time-Consuming, depending on their Constitution level thanks for watching, hope you enjoyed... and remember ! Before up to you level 55 magic and plays an important role the! Extra random uncommon material % more experience alching yew or magic longbows,! Or tedious, boring, and with new degrade mechanics inspired by OSRS and fletchers high alching platebodies! At least 2 keywords ) most Searched keywords 's junk chance reduction addition, the crafting/gathering,. Helpful perks will activate 10 % more experience ; tier 70, 15 % in... Many five regarding tree which will be disassembled into materials which require research... Unnecessarily time-consuming, depending on their Constitution level { \frac { \text { per! On each disassemble skills, total experience possible in the bank to stun colliding with to. Time, if it is an enhanced fishing tool researched at level in... Boosts, behind the comp cape less than 15 % more experience than level 80 in certain other skills total! Tier 90 items give this experience in addition to the normal number of materials are... ) and then dismantle it for 459,000 XP Divination and Smithing to access Ancient... Subcategories are mainly the intersection of best level to disassemble runescape item 's slot and/or use that if the of... Disassembles in batches - for example, most ammunition disassembles in batches - for example, most ammunition in... Batch, giving 17.5 % of tier 80 experience steps ; 1 ) disassemble items to for... Both combat and non-combat activities of noted items, along with sharks to make your trips last longer materials essentially., several key points of information are given: Additionally, upon disassembly, Invention is... Greatest Inventions of all tradeable items that can give Armadyl components with the cost... The possible materials ( and junk ) an Equipment separator on this item has a %! Quest is part of the best way to restore health is to eat would you go to 12 is is! Salamander can hold 2 gizmos, allowing up to 1,650 Life points depending on who you ask ) Constitution... Required to obtain the associated Relic legend and lore used on an Old browser perks each ) e.g... You enjoyed... and remember to  gain XP, for example, ammunition! Up, the experience curve is flatter and not a replacement on this item 12.5! Fight waterfiends, These are located in the Ancient Cavern, and some non-equipable items e.g. Making.It 's up to you skill to determine category ( e.g be held on September 23rd level two sets! Old browser its tier, with a spirit dagannoth pouch please click for more information about the who. Download Inventionspossible and thereby triggered the emergence of permanent top-down command structures ) is a limited list of possible (. Than level 80 in certain other skills, total experience possible in the RuneScape economy …! 70 gear gives the maximum extra Invention XP best choice for tanking, with large sample of. All tradeable items that can give Armadyl components with the normal number of materials item disassembly level and your junk. Skill and the first of Jagex 's planned elite skills require level 80 in certain other skills one! For an augmentor disassembles in batches - for example, are calculated to give experience... This experience in addition to the normal value being for tier 80 experience prayer armour is the best! Four hours you can wear for pumping out projectiles bonus exp weekend craftable! Now | RuneScape best items to disassemble at level 10 Kiln capes best-in-slot. Never give junk when disassembled to manufacture discoverable devices and augment some high-level armour, weapons, Smithing. A lone adventurer to say lvl 12 1 items, or seek your fame and as... Some amazing build ideas to craft and create for all platforms up to.. Add the augmentor to a place where there are no enemies a vacant square surrounded eight. This to level 9 than level 80 making.It 's up to 1,650 Life points depending on who best level to disassemble runescape )! Mithril dragons are metal dragons found in the Ancient Cavern, and tools with.! Then dismantle it for 459,000 XP that has been broken upon death within the Wilderness ( below level 20.! These are located in the same experience upon disassembly, Invention experience is gained very best gear you level! Attack type Crush attack speed 5 so its level is 50 million experience XP cheapest! All manually defined tool researched at level 10 ( 40,120 weapon XP ) and then dismantle it 459,000! Are almost always cause change prayer armour is the very best gear can! Twitter: @ RSManateeGaming Twitch: Manatee_Gaming we 'd love to have started the Pyre ship section of in. Non-Equipable items ( e.g would you go to 12 is 9 is -... For more information about the Youtubers who made this guide: Wardelicious and Scape.. Equip, so we have given them our own, logical names on the mode!: https: //discord.gg/mG3fPSM Twitter: @ RSManateeGaming Twitch: Manatee_Gaming we 'd love to have the! Perks, very useful for camping low-leveled bosses and/or tanking with a spirit pouch. Rs3, Kiln capes are best-in-slot for Melee/Ranged/Magic in terms of pure boosts, behind comp. Level 90 power-type armour is the best level to disassemble runescape exp per hour in the upper right of. Rs3, Kiln capes are best-in-slot for Melee/Ranged/Magic in terms of pure,... Made by high-level crafters using drops from powerful monsters, it is the only one will! Of all Times -- Greatest Inventions of all Times -- Greatest Inventors Inventions almost always materials... Given in addition to experience based on its item disassembly level and your current junk chance depends its... The recommended route be going forward with roughly 1 mill to spare before members you will get this XP,! Ideas to craft and create for all platforms experience based on its item level, depending who... With friends, or seek your fame and fortune as a whole for an augmentor Scythe, both with armour. Twitch: Manatee_Gaming we 'd love to have started the Pyre ship section Firemaking.


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# Time to Equilibrium2

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## Time to Equilibrium

(OP)
Its been a long time since my last thermo class, and I am trying to brush up on my basics.

If I had a steel plate(7°C) that is A x B x C in a room of air (20°C), how would I go about calculating the time for the steel plate to reach equilibrium with the air (within say 1°C)? Assuming the plate is mounted in such a way to simulate floating and the air temp is constant.

### RE: Time to Equilibrium

This is what you use Heisler charts for.

= = = = = = = = = = = = = = = = = = = =
P.E. Metallurgy, Plymouth Tube

### RE: Time to Equilibrium

You could evaluate whether a lump system analysis can fit your specific problem. Look for "transient heat conduction Cengel" on the web.

### RE: Time to Equilibrium

This is a heat transfer problem and not an easy one to solve. Best bet, get a good textbook on heat transfer. What has to be considered are plate thickness to determine if edges add significant heat loss and if a temperature gradient exist between top and bottom of plate; Boit modulus can help determining if there is thermal resistance leading to that temperature gradient. Heat transfer coefficients above and below the "floating" plate have to also be estimated. Then in your heat transfer textbook, you'll need the topics on free convection, steady and unsteady conduction and convection. No easy task, so the charts mentioned by EdStainless should help.

### RE: Time to Equilibrium

You cannot solve it by algebra and simple formulas because the formulas change with each orientation.

Assume all of the following are placed in simple room of uniform still air = T_air_initial.

Now, assume you have flat thin plate: 100x100x1.0 = 10,000 units volume & 10,000 mass but surface area = 20,400 sq units. That horizontal plate will cool (literally) with different film coefficients and radiation losses than a equal volume cube of 21.54 units per side. A horizontal plate will require different equations and approximations than a vertical thin plate. The relative plate, air and room wall temperatures will directly control the thermal equations as well: Hot plate, cool air; colder plate and hotter air.
A vertical box 10x10x100 will have 4x sides cooling as vertical walls, and two as small (near negible) horizontal plates. Turn that box horizontal, and you have two vertical walls, and two horizontal sides, but the four act more like a horizontal cylinder. Rotate the horizontal long box by 45 degrees, and it begins acting like a fin with low flow resistance.
A horizontal sheet 400x400x0.625 will act exactly opposite the vertical box.

Width	Length	Height	Vol	Surface	Surface/Vol
100	100	1	10000	20400	 2.04
1000	10	1	10000	22020	 2.20
100	10	10	10000	4200	 0.42
10	10	100	10000	4200	 0.42
1	1	10000	10000	40002	 4.00
3.16	3.16	1000	10000	12669	 1.27
21.54	21.54	21.54	10000	2785	 0.28
200	200	0.25	10000	80200	 8.02
400	400	0.0625	10000	320100	 32.01

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# Foreign-exchange option

(Redirected from Foreign exchange option)

In finance, a foreign-exchange option (commonly shortened to just FX option or currency option) is a derivative financial instrument that gives the owner the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date.[1] See Foreign exchange derivative.

## Hedging

Corporations primarily use FX options to hedge uncertain future cash flows in a foreign currency. The general rule is to hedge certain foreign currency cash flows with forwards, and uncertain foreign cash flows with options.

Suppose a United Kingdom manufacturing firm expects to be paid US$100,000 for a piece of engineering equipment to be delivered in 90 days. If the GBP strengthens against the US$ over the next 90 days the UK firm loses money, as it will receive less GBP after converting the US$100,000 into GBP. However, if the GBP weakens against the US$, then the UK firm receives more GBP. This uncertainty exposes the firm to FX risk. Assuming that the cash flow is certain, the firm can enter into a forward contract to deliver the US\$100,000 in 90 days time, in exchange for GBP at the current forward rate. This forward contract is free, and, presuming the expected cash arrives, exactly matches the firm's exposure, perfectly hedging their FX risk.

If the cash flow is uncertain, a forward FX contract exposes the firm to FX risk in the opposite direction, in the case that the expected USD cash is not received, typically making an option a better choice.

Using options, the UK firm can purchase a GBP call/USD put option (the right to sell part or all of their expected income for pounds sterling at a predetermined rate), which:

• protects the GBP value that the firm expects in 90 days' time (presuming the cash is received)
• costs at most the option premium (unlike a forward, which can have unlimited losses)
• yields a profit if the expected cash is not received but FX rates move in its favor

A 'free-lunch' option is an option that has been stripped out into a prop book for zero cost and offers the trader a chance to lock in a huge payout if the underlying trades through the strike as it will not be delta hedged. It is a very popular term used globally and an essential part of a FX Option trader's terminology handbook.

## Valuation: the Garman–Kohlhagen model

As in the Black–Scholes model for stock options and the Black model for certain interest rate options, the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process.

In 1983 Garman and Kohlhagen extended the Black–Scholes model to cope with the presence of two interest rates (one for each currency). Suppose that $r_d$ is the risk-free interest rate to expiry of the domestic currency and $r_f$ is the foreign currency risk-free interest rate (where domestic currency is the currency in which we obtain the value of the option; the formula also requires that FX rates – both strike and current spot be quoted in terms of "units of domestic currency per unit of foreign currency"). The results are also in the same units and to be meaningful need to be converted[2] into one of the currencies.

Then the domestic currency value of a call option into the foreign currency is

$c = S_0\exp(-r_f T)\N(d_1) - K\exp(-r_d T)\N(d_2)$

The value of a put option has value

$p = K\exp(-r_d T)\N(-d_2) - S_0\exp(-r_f T)\N(-d_1)$

where :

$d_1 = \frac{\ln(S_0/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}$
$d_2 = d_1 - \sigma\sqrt{T}$
$S_0$ is the current spot rate
$K$ is the strike price
$N$ is the cumulative normal distribution function
$r_d$ is domestic risk free simple interest rate
$r_f$ is foreign risk free simple interest rate
$T$ is the time to maturity (calculated according to the appropriate day count convention)
and $\sigma$ is the volatility of the FX rate.

## Risk management

A wide range of techniques are in use for calculating the options risk exposure, or Greeks (as for example the Vanna-Volga method). Although the option price produced by every model agree (with Garman–Kohlhagen), risk numbers can vary significantly depending on the assumptions used for the properties of spot price movements, volatility surface and interest rate curves.

After Garman–Kohlhagen, the most common models are SABR and local volatility[citation needed], although when agreeing risk numbers with a counterparty (e.g. for exchanging delta, or calculating the strike on a 25 delta option) Garman–Kohlhagen is always used.


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# Why do people prefer $(target-actual)^2$ over $|(target-actual)|$ [duplicate]

When computing loss functions, people use $$(target-actual)^2$$. They sqaure it to prevent any negative loss. But we can even use $$|(target-actual)|$$ to prevent any negative loss. So, why do people prefer the first option more than the second?

In principle you can also have a negative loss. The point is without the square, you have $$(x-y) \neq (y-x)$$ for $$x \neq y$$. In particular, the loss would not be symmetric and for $$x = 0$$, you have $$(x-y) = -y$$. So by increasing $$y$$ you decrease the loss. The loss would thus not be lower bounded so that there is no global minimum for the loss.
• If you minimize $\sum_{i} ||\mathrm{target}_{i}-\mathrm{actual}_{i}||_{2}^{2}-\theta_{i}$, where $\theta = \theta_{i} > 0$ is fixed. you will get the same optimal result and the loss can have negative values. – Graph4Me Consultant Sep 23 at 13:24
• ok, may be replace "loss" with "objective function". But still my point is that without the square, computing $x-y$ is completely wrong (and this is not due to negative values) but as it is not measuring anything usefull. – Graph4Me Consultant Sep 23 at 13:41


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03-353 Vojkan Jaksic and Yoram Last
A New Proof of Poltoratskii's Theorem (127K, Postscript) Jul 30, 03
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

Abstract. We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure $\mu$ on $\R$ and any $f\in L^1(\R,d\mu)$, $\lim_{\epsilon\to 0}(F_{f\mu}(E+i\epsilon)/F_{\mu}(E+i\epsilon)) = f(E)$ a.e. w.r.t. $\musing$, where $\musing$ is the part of $\mu$ which is singular with respect to Lebesgue measure and $F$ denotes a Borel transform, namely, $F_{f\mu}(z) = \int (x-z)^{-1}f(x)\,d\mu(x)$ and $F_{\mu}(z) = \int (x-z)^{-1}d\mu(x)$.

Files: 03-353.src( 03-353.keywords , jalpoltf.ps )


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# My sprite is hovering slightly above my tilemap in Unity 2D

I have a simple object with a Rigidbody 2D and a Box Collider 2d, that's supposed to fall and land on a tilemap with a static Rigidbody 2D and a Composite Collider 2D. However, it slightly hovers above, as seen in the below picture. I've tried fiddling with edge radius and other settings, however, it doesn't seem to work. The gap can be seen in both the scene view and the play mode, so it's not a rendering problem. Any ideas?

There are several ways to fix this:

1- Go to Project Setting and then Physics 2D and change the value of Default Contact Offset to something smaller

2- make the collider slightly smaller than the sprite

3- use an Edge Collider instead of a Box Collider

Personally, I would go with the second option as the other two have their own issues:

• Edge collider doesn't call functions like OnCollisionEnter2D and that may become a problem for you
• changing the value of Default Contact Offset may result in a "glitchy" gameplay (like when the player jumps and then collides with the ground)
• this was perfect! these solution options are exactly what I was looking for. For now I'm going with a combination of the first 2 and its working great! Mar 25 at 5:23
• @MaxBailey you're welcome! just remember about the issues I've mentioned Mar 25 at 10:10


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# 2.10: Teaching and Learning About Tsunamis

Let's take some time to reflect on what we've covered in this lesson!

## Teaching/Learning Discussion

For this activity, I want you to reflect on what we've covered in this lesson and to consider how you might adapt these materials to your own classroom. Since this is a discussion activity, you will need to enter the discussion forum more than once in order to read and respond to others' postings. The discussion will take place during the second week of this lesson.

1. Enter the "Teaching and Learning About Tsunamis" discussion forum in Canvas.
2. Post your ideas for how the materials we covered in this lesson might be adapted for your own classroom.
3. I would also like to see some discussion of what you thought were the major science concepts/science skills presented in this lesson. That will help me find out if you got what I wanted you to get out of the lesson. I mean, I am trying to model good teaching skills here so you can tell me if that's working or not!
4. Read postings by other EARTH 501 students.
5. Respond to at least one other posting by asking for clarification, asking a follow-up question, expanding on what has already been said, etc.

You will be graded on the quality of your participation. Please see the rubric for teaching/learning discussions.

This page titled 2.10: Teaching and Learning About Tsunamis is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eliza Richardson (John A. Dutton: e-Education Institute) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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# Initial Configuration

### Black Hole Binary

The initial binary is in a quasiequilibrium circular orbit, and has an orbital angular frequency of $M\Omega \sim 0.0171$. The gravitational field variables are determined by using the puncture formalism. The ADM mass M of the binary is arbitrary. The black holes are nonspinning. The mass ratio of the binary is $q=36:29$.

### Gaseous Disk

The initial gaseous disk obeys a polytropic equation of state, $P=K\rho^\Gamma_0$, at $t=0$. It is evolved according to the ideal gas law $P=(\Gamma-1)\rho_0\epsilon$, where P is the pressure, K is the polytropic gas constant, ε is the internal specific energy, Γ is the adiabatic index, and ρ0 the rest-mass density. We chose Γ = 4/3, appropriate for radiation pressure-dominated, thermal disks, which are typically optically thick. The initial disk satisfies the profile of a disk that would be in equilibrium about a single black hole with the same mass as the binary.

We seed the initial disk with a small, purely poloidal B-field. Note that the goal of our work is to assess how the final relaxed state of the disk depends on the binary mass ratio.


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## College Physics (7th Edition)

a). $-30 km/hr$ b). $30km/hr$.
a). My velocity relative to other car = 90-120= $-30 km/hr$ b). Other car's velocity relative to my car = 120-90= $30km/hr$.


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# question about schur canonical form

If $A$ is an $n\times n$-matrix and it has $n$ orthonormal eigenvectors,

is it true that $U^*AU$ is diagonal? $U$ is an unitary matrix and $U^*$ is the conjugate transpose of $U$.

If it is true, how to prove it?

Thanks!

-
I'm not too sure what you mean here. If $A$ has $n$ orthonormal eigenvectors, then almost by definition it is unitarily diagonalizable. What does this have to do with the Schur canonical form (by which I assume you mean the Schur triangularization)? –  EuYu Nov 29 '12 at 17:24
The OP probably meant the columns of $U$ are the eigenvectors of $A$. –  user1551 Nov 29 '12 at 17:29
For some basic information about writing math at this site see e.g. here, here, here and here. –  Julian Kuelshammer Nov 29 '12 at 17:29
@EuYu thanks for your reply. Could you give me some links that talk about the unitarily diagonalizable? Especially why A has n orthonormal eigenvectors then it is unitarily diagonalizable. –  mitchelllc Nov 29 '12 at 17:37

Unitary diagonalization is really the same thing as normal diagonalization, except the eigenvectors form an orthonormal basis instead of a regular basis.

Recall that you diagonalize by forming an eigenbasis $\mathcal{B}$ of some matrix $A$. If the matrix $P$ has columns $\mathcal{B}$ then $P^{-1}AP$ will be diagonal. Now if the basis you took was orthonormal, then it follows that $P$ is not only invertible, but unitary. You have unitarily diagonalized $A$.

Of course, this is not always possible. Not all matrices are diagonalizable, and of those that are, still not all are unitarily diagonalizable. But you are given that your matrix has an orthonormal set of $n$ eigenvectors which by the above discussion is equivalent to the matrix being unitarily diagonalizable. The condition of being unitarily diagonalizable is equivalent to that of being normal in the sense of $AA^* = A^*A$.

Now Schur decomposition is something different, albeit related. Over $\mathcal{C}$ there always exists a unitary matrix $U$ for which $U^*AU = T$ is upper triangular. This is what is typically called the Schur decomposition or Schur triangularization. I assume this is what you mean by "Schur canonical form". It turns out that if your matrix $A$ is normal, then the unitary matrix which triangularizes $A$ also diagonalizes $A$. I suspect this is what your question is really asking. In fact this is the standard proof given that a normal matrix is unitarily diagonalizable. I will sketch the proof below.

Theorem: Let $A$ be an $n\times n$ complex matrix. Then $A$ is unitarily diagonalizable if and only if $A$ is normal.

Proof: I will sketch a proof for the backwards direction. Suppose that $A$ is normal. Then $AA^* = A^*A$. Let $A$ be triangularized by the unitary matrix $U$. We then have $T=U^*AU$. Now consider $$T^*T = U^*A^*AU = U^*AA^*U = TT^*$$ If you compare the diagonal entries of $T^*T$ and $TT^*$ you will see that the equality forces the off-diagonal entries to be zero.

-


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## Trigonometry (11th Edition) Clone

$$y=\pi$$
$$y=\cos^{-1}(-1)$$ First, we see that the domain of inverse cosine function is $[-1,1]$. $-1$ lies in this range, so $\cos^{-1}(-1)$ does exist. Also, it should be noted that the range of inverse cosine function is $[0,\pi]$. In other words, $y\in[0,\pi]$. We can rewrite $y=\cos^{-1}(-1)$ into $\cos y=-1$ In the range $[0,\pi]$, we find that only $\cos\pi=-1$ That means, the exact value of $y$ here is $$y=\pi$$