Library example_scottish_club
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This example is taken from
Basic Predicate Calculus (1.3.4)
A private club has the following rules:
Now, we show that these rules are so
strict that no one can be accepted.
Scottish Club Example
A private club has the following rules:
- Every non-Scottish member wears red socks.
- Every member wears a kilt or does not wear red socks.
- The married members do not go out on Sunday.
- A member goes out on Sunday if and only if he is Scottish.
- Every member who wears a kilt is Scottish and married.
- Every Scottish member wears a kilt.
Section club.
Variables Scottish RedSocks WearKilt Married GoOutSunday : Prop.
Hypothesis rule1 : ~Scottish -> RedSocks.
Hypothesis rule2 : WearKilt \/ ~RedSocks.
Hypothesis rule3 : Married -> ~GoOutSunday.
Hypothesis rule4 : GoOutSunday <-> Scottish.
Hypothesis rule5 : WearKilt -> Scottish /\ Married.
Hypothesis rule6 : Scottish -> WearKilt.
One-line "proof" of a 5-variable tautology
~(rule1 /\ rule2 /\ rule3 /\ rule4 /\ rule5 /\ rule6)
Or, equivalently, this states that all the rules together make a contradiction. So, this is also a tautology:
(rule1 /\ rule2 /\ rule3 /\ rule4 /\ rule5 /\ rule6) -> False
Lemma NoMember : False.
Proof.
tauto.
Qed.
End club.
Print NoMember.
Explicit proof from the rules
- Assume a club member wears a kilt.
- Then he is both Scottish and Married (rule5).
- Then he does NOT go out on Sundays (rule3).
- Then he cannot be Scottish (contrapositive from rule4: ~GoOutSunday -> ~Scottish).
- On the other side, assume a club member does not wear a kilt.
- Then he cannot be Scottish (contrapositive from rule6: ~WearKilt -> ~Scottish).
- Then he must wear red socks (rule1).
- Then ~(WearKilt \/ ~ RedSocks) is true (negation of rule2).
The Law of Excluded Middle shows that this is impossible, since each member either wears a kilt or he does not.
Section clubExplicit.
Variables Scottish RedSocks WearKilt Married GoOutSunday : Prop.
Hypothesis rule1 : ~Scottish -> RedSocks.
Hypothesis rule2 : WearKilt \/ ~RedSocks.
Hypothesis rule3 : Married -> ~GoOutSunday.
Hypothesis rule4 : GoOutSunday <-> Scottish.
Hypothesis rule5 : WearKilt -> Scottish /\ Married.
Hypothesis rule6 : Scottish -> WearKilt.
These two axioms are often needed (and they are not provable
in intuitionist mathematics). They are usually imported
from Coq libraries:
Require Import Classical_Prop.
In this brief example we do not import this library with axioms, we state these two basic axioms locally.
Require Import Classical_Prop.
In this brief example we do not import this library with axioms, we state these two basic axioms locally.
Axiom ExcludedMiddle: forall A:Prop, A \/ ~ A.
Axiom ContraPos : forall A B: Prop, (A -> B) <-> (~B -> ~A).
Lemma NoMemberExplicit : False.
Proof.
elim (ExcludedMiddle WearKilt).
intros KiltHypothesis.
pose (rule5 KiltHypothesis) as H1.
destruct H1 as [H2 H3].
pose (rule3 H3) as H4.
destruct rule4 as [rule4A rule4B].
assert ((Scottish -> GoOutSunday) -> (~GoOutSunday -> ~Scottish)).
apply ContraPos with (A:=Scottish) (B:=GoOutSunday).
pose (H rule4B) as H6.
pose (H6 H4) as H7.
pose (H7 H2) as CONTRA.
contradiction CONTRA.
Now assume that there is no kilt
intros NoKiltHypothesis.
assert ((Scottish -> WearKilt) -> (~WearKilt -> ~Scottish)).
apply ContraPos with (A:=Scottish) (B:=WearKilt).
pose (H rule6) as H1.
pose (H1 NoKiltHypothesis) as H2.
pose (rule1 H2) as H3.
destruct rule2 as [H4 | H5].
pose (NoKiltHypothesis H4) as CONTRA.
contradiction CONTRA.
pose (H5 H3) as CONTRA.
contradiction CONTRA.
Qed.
End clubExplicit.
assert ((Scottish -> WearKilt) -> (~WearKilt -> ~Scottish)).
apply ContraPos with (A:=Scottish) (B:=WearKilt).
pose (H rule6) as H1.
pose (H1 NoKiltHypothesis) as H2.
pose (rule1 H2) as H3.
destruct rule2 as [H4 | H5].
pose (NoKiltHypothesis H4) as CONTRA.
contradiction CONTRA.
pose (H5 H3) as CONTRA.
contradiction CONTRA.
Qed.
End clubExplicit.
Hypotheses can be replaced by conditions
Lemma ScottishClub:
forall Scottish RedSocks WearKilt Married GoOutSunday: Prop,
((~ Scottish -> RedSocks) /\
(WearKilt \/ ~ RedSocks) /\
(Married -> ~ GoOutSunday) /\
(GoOutSunday <-> Scottish) /\
(Scottish /\ Married) /\
(Scottish -> WearKilt)) -> False.
Proof.
intros Scottish RedSocks WearKilt Married GoOutSunday.
intros H.
destruct H as [rule1 [ rule2 [rule3 [rule4 [rule5 rule6]]]]].
Admitted.
After the "destruct H" we get the same
initial state: Prove "False", if there are 6 hypotheses or
rules r1, r2, r3, r4, r5, r6.
The statement to prove looks like this:
1 subgoal Scottish, RedSocks, WearKilt, Married, GoOutSunday : Prop rule1 : ~ Scottish -> RedSocks rule2 : WearKilt \/ ~ RedSocks rule3 : Married -> ~ GoOutSunday rule4 : GoOutSunday <-> Scottish rule5 : Scottish /\ Married rule6 : Scottish -> WearKilt ____________________________________(1/1) False


class SleeperRb::Resources::Draft
Represents a draft for a given user.
Attributes
created[R]
creators[R]
draft_id[R]
draft_order[R]
last_message_id[R]
last_message_time[R]
last_picked[R]
league[R]
league_id[R]
season[R]
season_type[R]
sport[R]
start_time[R]
status[R]
type[R]
Public Instance Methods
auction?()
click to toggle source
# File lib/sleeper_rb/resources/draft.rb, line 101 def auction? type == "auction" end
draft_picks()
click to toggle source
All picks in the Draft
.
@return [SleeperRb::Resources::Draft::Pick
]
# File lib/sleeper_rb/resources/draft.rb, line 97 cached_association :draft_picks do retrieve_picks! end
metadata()
click to toggle source
Metadata
for the draft.
@return [SleeperRb::Resources::Draft::Metadata
]
# File lib/sleeper_rb/resources/draft.rb, line 73 cached_attr :type, :status, :start_time, :sport, :season_type, :season, :league_id, :last_picked, :league, :last_message_time, :last_message_id, :draft_order, :draft_id, :creators, :created, :slot_to_roster_id, settings: ->(hash) { Settings.new(hash) }, metadata: ->(hash) { Metadata.new(hash) }
settings()
click to toggle source
Settings
for the draft.
@return [SleeperRb::Resources::Draft::Settings
]
# File lib/sleeper_rb/resources/draft.rb, line 62
snake?()
click to toggle source
# File lib/sleeper_rb/resources/draft.rb, line 105 def snake? type == "snake" end
traded_picks()
click to toggle source
All picks in the Draft
that were traded.
@return [SleeperRb::Resources::TradedPickArray
]
# File lib/sleeper_rb/resources/draft.rb, line 88 cached_association :traded_picks do retrieve_traded_picks! end


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Class Block (v2.2.2)
Logical element on the page.
Generated from protobuf message google.cloud.vision.v1.Block
Methods
__construct
Constructor.
| Name | Description |
| data |
array
Optional. Data for populating the Message object. |
| ↳ property |
Google\Cloud\Vision\V1\TextAnnotation\TextProperty
Additional information detected for the block. |
| ↳ bounding_box |
Google\Cloud\Vision\V1\BoundingPoly
The bounding box for the block. The vertices are in the order of top-left, top-right, bottom-right, bottom-left. When a rotation of the bounding box is detected the rotation is represented as around the top-left corner as defined when the text is read in the 'natural' orientation. For example: * when the text is horizontal it might look like: 0----1 | | 3----2 * when it's rotated 180 degrees around the top-left corner it becomes: 2----3 | | 1----0 and the vertex order will still be (0, 1, 2, 3). |
| ↳ paragraphs |
array<Google\Cloud\Vision\V1\Paragraph>
List of paragraphs in this block (if this blocks is of type text). |
| ↳ block_type |
int
Detected block type (text, image etc) for this block. |
| ↳ confidence |
float
Confidence of the OCR results on the block. Range [0, 1]. |
getProperty
Additional information detected for the block.
Generated from protobuf field .google.cloud.vision.v1.TextAnnotation.TextProperty property = 1;
| Type | Description |
| Google\Cloud\Vision\V1\TextAnnotation\TextProperty|null |
hasProperty
clearProperty
setProperty
Additional information detected for the block.
Generated from protobuf field .google.cloud.vision.v1.TextAnnotation.TextProperty property = 1;
| Name | Description |
| var |
Google\Cloud\Vision\V1\TextAnnotation\TextProperty
|
| Type | Description |
| $this |
getBoundingBox
The bounding box for the block.
The vertices are in the order of top-left, top-right, bottom-right, bottom-left. When a rotation of the bounding box is detected the rotation is represented as around the top-left corner as defined when the text is read in the 'natural' orientation. For example:
- when the text is horizontal it might look like: 0----1 | | 3----2
- when it's rotated 180 degrees around the top-left corner it becomes: 2----3 | | 1----0 and the vertex order will still be (0, 1, 2, 3).
Generated from protobuf field .google.cloud.vision.v1.BoundingPoly bounding_box = 2;
| Type | Description |
| Google\Cloud\Vision\V1\BoundingPoly|null |
hasBoundingBox
clearBoundingBox
setBoundingBox
The bounding box for the block.
The vertices are in the order of top-left, top-right, bottom-right, bottom-left. When a rotation of the bounding box is detected the rotation is represented as around the top-left corner as defined when the text is read in the 'natural' orientation. For example:
- when the text is horizontal it might look like: 0----1 | | 3----2
- when it's rotated 180 degrees around the top-left corner it becomes: 2----3 | | 1----0 and the vertex order will still be (0, 1, 2, 3).
Generated from protobuf field .google.cloud.vision.v1.BoundingPoly bounding_box = 2;
| Name | Description |
| var |
Google\Cloud\Vision\V1\BoundingPoly
|
| Type | Description |
| $this |
getParagraphs
List of paragraphs in this block (if this blocks is of type text).
Generated from protobuf field repeated .google.cloud.vision.v1.Paragraph paragraphs = 3;
| Type | Description |
| Google\Protobuf\Internal\RepeatedField |
setParagraphs
List of paragraphs in this block (if this blocks is of type text).
Generated from protobuf field repeated .google.cloud.vision.v1.Paragraph paragraphs = 3;
| Name | Description |
| var |
array<Google\Cloud\Vision\V1\Paragraph>
|
| Type | Description |
| $this |
getBlockType
Detected block type (text, image etc) for this block.
Generated from protobuf field .google.cloud.vision.v1.Block.BlockType block_type = 4;
| Type | Description |
| int |
setBlockType
Detected block type (text, image etc) for this block.
Generated from protobuf field .google.cloud.vision.v1.Block.BlockType block_type = 4;
| Name | Description |
| var |
int
|
| Type | Description |
| $this |
getConfidence
Confidence of the OCR results on the block. Range [0, 1].
Generated from protobuf field float confidence = 5;
| Type | Description |
| float |
setConfidence
Confidence of the OCR results on the block. Range [0, 1].
Generated from protobuf field float confidence = 5;
| Name | Description |
| var |
float
|
| Type | Description |
| $this |


Two terrorists killed after encounter breaks out in Kashmir's Bandipora
While one terrorist was killed on Sunday during a gunfight between security forces in Lawdara village, about 55 km from Srinagar, another was killed this morning
by Press Trust of IndiaA terrorist was killed in an encounter with security forces in Bandipora district of north Kashmir on Monday morning, police said.
While one terrorist was killed on Sunday during a gunfight between security forces in Lawdara village, about 55 km from Srinagar, another was killed this morning, they said.
The encounter broke out when security forces launched a search operation following specific information about the presence of some terrorists.
More details are awaited.


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Class OrganizationInboxProjection
Organization team inbox
Inherited Members
Namespace: mailslurp.Model
Assembly: mailslurp.dll
Syntax
[DataContract(Name = "OrganizationInboxProjection")]
public class OrganizationInboxProjection : IEquatable<OrganizationInboxProjection>, IValidatableObject
Constructors
View SourceOrganizationInboxProjection()
Initializes a new instance of the OrganizationInboxProjection class.
Declaration
[JsonConstructor]
protected OrganizationInboxProjection()
OrganizationInboxProjection(Guid, Guid?, DateTime, string, string, bool, List<string>, bool, InboxTypeEnum?, bool, bool)
Initializes a new instance of the OrganizationInboxProjection class.
Declaration
public OrganizationInboxProjection(Guid id = default, Guid? domainId = null, DateTime createdAt = default, string name = null, string emailAddress = null, bool favourite = false, List<string> tags = null, bool teamAccess = false, OrganizationInboxProjection.InboxTypeEnum? inboxType = null, bool readOnly = false, bool virtualInbox = false)
Parameters
| Type | Name | Description |
|---|---|---|
| Guid | id | ID of the inbox. The ID is a UUID-V4 format string. Use the inboxId for calls to Inbox and Email Controller endpoints. See the emailAddress property for the email address or the inbox. To get emails in an inbox use the WaitFor and Inbox Controller methods |
| Guid? | domainId | ID of custom domain used by the inbox if any. |
| DateTime | createdAt | When the inbox was created. Time stamps are in ISO DateTime Format |
| string | name | Name of the inbox and used as the sender name when sending emails .Displayed in the dashboard for easier search. |
| string | emailAddress | The inbox's email address. Inbox projections and previews may not include the email address. To view the email address fetch the inbox entity directly. Send an email to this address and the inbox will receive and store it for you. Note the email address in MailSlurp match characters exactly and are case sensitive so |
| bool | favourite | Is the inbox a favorite inbox. Make an inbox a favorite is typically done in the dashboard for quick access or filtering (required). |
| List<string> | tags | Tags that inbox has been tagged with. Tags can be added to inboxes to group different inboxes within an account. You can also search for inboxes by tag in the dashboard UI.. |
| bool | teamAccess | Does inbox permit team access for organization team members. If so team users can use inbox and emails associated with it. See the team access guide at https://www.mailslurp.com/guides/team-email-account-sharing/ (required). |
| OrganizationInboxProjection.InboxTypeEnum? | inboxType | Type of inbox. HTTP inboxes are faster and better for most cases. SMTP inboxes are more suited for public facing inbound messages (but cannot send).. |
| bool | readOnly | Is the inbox readOnly for the caller. Read only means can not be deleted or modified. This flag is present when using team accounts and shared inboxes. (required). |
| bool | virtualInbox | Virtual inbox can receive email but will not send emails to real recipients. Will save sent email record but never send an actual email. Perfect for testing mail server actions. (required). |
Properties
View SourceCreatedAt
When the inbox was created. Time stamps are in ISO DateTime Format yyyy-MM-dd'T'HH:mm:ss.SSSXXX
e.g. 2000-10-31T01:30:00.000-05:00
.
Declaration
[DataMember(Name = "createdAt", IsRequired = true, EmitDefaultValue = true)]
public DateTime CreatedAt { get; set; }
Property Value
| Type | Description |
|---|---|
| DateTime | When the inbox was created. Time stamps are in ISO DateTime Format |
DomainId
ID of custom domain used by the inbox if any
Declaration
[DataMember(Name = "domainId", EmitDefaultValue = true)]
public Guid? DomainId { get; set; }
Property Value
| Type | Description |
|---|---|
| Guid? | ID of custom domain used by the inbox if any |
EmailAddress
The inbox's email address. Inbox projections and previews may not include the email address. To view the email address fetch the inbox entity directly. Send an email to this address and the inbox will receive and store it for you. Note the email address in MailSlurp match characters exactly and are case sensitive so +123
additions are considered different addresses. To retrieve the email use the Inbox and Email Controller endpoints with the inbox ID.
Declaration
[DataMember(Name = "emailAddress", EmitDefaultValue = true)]
public string EmailAddress { get; set; }
Property Value
| Type | Description |
|---|---|
| string | The inbox's email address. Inbox projections and previews may not include the email address. To view the email address fetch the inbox entity directly. Send an email to this address and the inbox will receive and store it for you. Note the email address in MailSlurp match characters exactly and are case sensitive so |
Favourite
Is the inbox a favorite inbox. Make an inbox a favorite is typically done in the dashboard for quick access or filtering
Declaration
[DataMember(Name = "favourite", IsRequired = true, EmitDefaultValue = true)]
public bool Favourite { get; set; }
Property Value
| Type | Description |
|---|---|
| bool | Is the inbox a favorite inbox. Make an inbox a favorite is typically done in the dashboard for quick access or filtering |
Id
ID of the inbox. The ID is a UUID-V4 format string. Use the inboxId for calls to Inbox and Email Controller endpoints. See the emailAddress property for the email address or the inbox. To get emails in an inbox use the WaitFor and Inbox Controller methods waitForLatestEmail
and getEmails
methods respectively. Inboxes can be used with aliases to forward emails automatically.
Declaration
[DataMember(Name = "id", IsRequired = true, EmitDefaultValue = true)]
public Guid Id { get; set; }
Property Value
| Type | Description |
|---|---|
| Guid | ID of the inbox. The ID is a UUID-V4 format string. Use the inboxId for calls to Inbox and Email Controller endpoints. See the emailAddress property for the email address or the inbox. To get emails in an inbox use the WaitFor and Inbox Controller methods |
InboxType
Type of inbox. HTTP inboxes are faster and better for most cases. SMTP inboxes are more suited for public facing inbound messages (but cannot send).
Declaration
[DataMember(Name = "inboxType", EmitDefaultValue = true)]
public OrganizationInboxProjection.InboxTypeEnum? InboxType { get; set; }
Property Value
| Type | Description |
|---|---|
| OrganizationInboxProjection.InboxTypeEnum? | Type of inbox. HTTP inboxes are faster and better for most cases. SMTP inboxes are more suited for public facing inbound messages (but cannot send). |
Name
Name of the inbox and used as the sender name when sending emails .Displayed in the dashboard for easier search
Declaration
[DataMember(Name = "name", EmitDefaultValue = true)]
public string Name { get; set; }
Property Value
| Type | Description |
|---|---|
| string | Name of the inbox and used as the sender name when sending emails .Displayed in the dashboard for easier search |
ReadOnly
Is the inbox readOnly for the caller. Read only means can not be deleted or modified. This flag is present when using team accounts and shared inboxes.
Declaration
[DataMember(Name = "readOnly", IsRequired = true, EmitDefaultValue = true)]
public bool ReadOnly { get; set; }
Property Value
| Type | Description |
|---|---|
| bool | Is the inbox readOnly for the caller. Read only means can not be deleted or modified. This flag is present when using team accounts and shared inboxes. |
Tags
Tags that inbox has been tagged with. Tags can be added to inboxes to group different inboxes within an account. You can also search for inboxes by tag in the dashboard UI.
Declaration
[DataMember(Name = "tags", EmitDefaultValue = true)]
public List<string> Tags { get; set; }
Property Value
| Type | Description |
|---|---|
| List<string> | Tags that inbox has been tagged with. Tags can be added to inboxes to group different inboxes within an account. You can also search for inboxes by tag in the dashboard UI. |
TeamAccess
Does inbox permit team access for organization team members. If so team users can use inbox and emails associated with it. See the team access guide at https://www.mailslurp.com/guides/team-email-account-sharing/
Declaration
[DataMember(Name = "teamAccess", IsRequired = true, EmitDefaultValue = true)]
public bool TeamAccess { get; set; }
Property Value
| Type | Description |
|---|---|
| bool | Does inbox permit team access for organization team members. If so team users can use inbox and emails associated with it. See the team access guide at https://www.mailslurp.com/guides/team-email-account-sharing/ |
VirtualInbox
Virtual inbox can receive email but will not send emails to real recipients. Will save sent email record but never send an actual email. Perfect for testing mail server actions.
Declaration
[DataMember(Name = "virtualInbox", IsRequired = true, EmitDefaultValue = true)]
public bool VirtualInbox { get; set; }
Property Value
| Type | Description |
|---|---|
| bool | Virtual inbox can receive email but will not send emails to real recipients. Will save sent email record but never send an actual email. Perfect for testing mail server actions. |
Methods
View SourceEquals(OrganizationInboxProjection)
Returns true if OrganizationInboxProjection instances are equal
Declaration
public bool Equals(OrganizationInboxProjection input)
Parameters
| Type | Name | Description |
|---|---|---|
| OrganizationInboxProjection | input | Instance of OrganizationInboxProjection to be compared |
Returns
| Type | Description |
|---|---|
| bool | Boolean |
Equals(object)
Returns true if objects are equal
Declaration
public override bool Equals(object input)
Parameters
| Type | Name | Description |
|---|---|---|
| object | input | Object to be compared |
Returns
| Type | Description |
|---|---|
| bool | Boolean |
Overrides
View SourceGetHashCode()
Gets the hash code
Declaration
public override int GetHashCode()
Returns
| Type | Description |
|---|---|
| int | Hash code |
Overrides
View SourceToJson()
Returns the JSON string presentation of the object
Declaration
public virtual string ToJson()
Returns
| Type | Description |
|---|---|
| string | JSON string presentation of the object |
ToString()
Returns the string presentation of the object
Declaration
public override string ToString()
Returns
| Type | Description |
|---|---|
| string | String presentation of the object |
Overrides
View SourceValidate(ValidationContext)
To validate all properties of the instance
Declaration
public IEnumerable<ValidationResult> Validate(ValidationContext validationContext)
Parameters
| Type | Name | Description |
|---|---|---|
| ValidationContext | validationContext | Validation context |
Returns
| Type | Description |
|---|---|
| IEnumerable<ValidationResult> | Validation Result |


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Function for plotting a customized version of the Skew-T diagram. Please note that drawing Skew-T may require increasing size or modifying aspect ratio of plotting window in order to provide readable results.
skewt_plot( ptop = 100, isoterms_col = "#d8be9b", temp_stripes = FALSE, mixing_ratio_col = "#8470FF90", dry_adiabats_col = "#d6878750", moist_adiabats_col = "#00FF0095", deg45 = FALSE, isotherm0 = TRUE, close_par = TRUE, ... )
| ptop | Pressure top level to be used for plotting diagram. Valid options: 200, 150, 100 (default) and 50 hPa |
|---|---|
| isoterms_col | color to be used for drawing dry isoterms |
| temp_stripes | logical, whether to draw color stripes for isotherms |
| mixing_ratio_col | color to be used for drawing mixing ratio isolines and labels. If set to NA or empty string isolines are not drawn |
| dry_adiabats_col | color to be used for drawing dry adiabats. If set to NA or not provided drawing lines skipped |
| moist_adiabats_col | color to be used for drawing moist adiabats. If set to NA or not provided drawing lines skipped |
| deg45 | whether to preserve 45 degrees for diagonal isolines on Skew-T diagram regardless ploting window aspect ratio. [logical, default: FALSE] |
| isotherm0 | whether to deliminate 0 degree Celsius isother [logical, default: TRUE] |
| close_par | if plot will be modified in next steps storing par settings is needed. This logical argument is turned on by default. If you want to modify Skew-T plot in next step set it to FALSE |
| ... | additional (mostly graphical) parameters to be passed |
Draws a Skew-T log-p diagram
skewt_plot(ptop = 100)skewt_plot(ptop = 150, temp_stripes = TRUE) # add color stripes for temperatureskewt_plot(ptop = 100, close_par = FALSE)#>#> #>#>#> #>output = sounding_export(pressure, altitude, temp, dpt, wd, ws) skewt_lines(output$dpt, output$pressure,type='l',col='forestgreen',lwd = 2.5)


Let's Go To Create, Design, Code, Build For Everyone !
✅ Let's the taks.go : it's a todo management which base on vue, gin framework and sqlite3 database.
📁 A bijou file management tool and nothing framework be used on it so run it only by one command !
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hey guy if you want to join us please concat me via email or QQ Group


Sets the editing mode of the sketchpad.
The sketchpad has to be initialized as an editor.
There are three modes:
true - drawing mode
false - viewer mode
erase - erase mode
Parameters
mode string or boolean
{% capture js %}
// When the sketchpad is in drawing mode.
editor.editing(true);
// When the sketchpad is in viewer mode.
editor.editing(false);
// When the sketchpad is in erase mode.
editor.editing("erase");
{% endcapture %}


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Lets create some waves. Try the following applet and create waves of all kind on this string. Make sure to try the pulse button with no damping. You can try to find how to make waves move faster!
The Speed Equation
- S: So I see that, if I increase the tension in the string, the pulse or wave moves faster (try it out above); but I don't understand why that is...
- M: Good. The key idea is that the speed of a wave is determined by the medium in which it propagates. If you think of a string as a long line of atoms attached by springs, then the higher the tension, the more stiff the springs are. If you displace one atom at one end of the rope, it will pull on its neighbor which will then pull on the next one, etc... The stiffer the springs are (the higher the tension), the faster the disturbance will travel.
- S: Hmmm I see.
- M: Here is another simple (but super important) equation.
Starting from
- wavelength \\lambda\ = (distance)/(oscillation): The wavelength is the distance traveled per oscillation or cycle.
- Frequency f = (Oscillation)/(time): The frequency is the number of oscillation you do per unit of time. (usually in Hz so per second).
one gets:
(Wavelength)(frequency) = (distance)/(time) = speed v
or speed of wave \v = \lambda f\. There is a subtle point here that the speed v is determined by the medium. The speed is usually constant. The wavelength and frequency are the things that adjust themselves such that the product always equals v.
Example:


Deep Linear Neural Networks or Deep Matrix Factorization
- NEW Bregman Proximal Framework for Deep Linear Neural Networks joint work with Felix Westerkamp, Emanuel Laude, Daniel Cremers, Peter Ochs is online at arXiv.
Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization
- NEW Beyond Alternating Updates for Matrix Factorization with Inertial Bregman Proximal Gradient Algorithms joint work with Peter Ochs is accepted at NeurIPS 2019.
- Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization joint work with Peter Ochs, Thomas Pock and Shoham Sabach.
Deep Learning Theory
- On the Loss Landscape of a class of Deep neural networks with no bad local valleys joint work with Quynh Nguyen and Matthias Hein is accepted at International Conference on Learning Representations (ICLR) 2019.
- Neural Networks Should Be Wide Enough to Learn Disconnected Decision Regions joint work with Quynh Nguyen and Matthias Hein is accepted at International Conference on Machine Learning (ICML) 2018.
Stochastic/Online Optimization for Deep Learning
- Variants of RMSProp and Adagrad with Logarithmic Regret Bounds joint work with
Matthias Hein is accepted at International Conference on Machine Learning (ICML) 2017.
Also see long version containing all proofs here


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