text:
### Strutt Help

CNOSSOS Ground Effect    1/1,1/3

Strutt|Environmental Noise|CNOSSOS Ground Attenuation inserts attenuation (dB) into the active row of the worksheet calculated using the CNOSSOS EU noise propagation model.

CNOSSOS is developed for propagation above varying-height ground planes, and uses the concept of equivalent height z_s,z_r - i.e. the source and receiver heights are defined relative to the average height of the ground plane, and are not necessarily the same as the local source/receiver heights h_s,h_r:

As well as the source-receiver geometry, the CNOSSOS model uses the ground factor G over the ground plane (i.e. proportion of porous ground). In cases where the ground plane consists of multiple ground types, the average bar G (weighted by the extent of each ground type) should be used.

Typical values of G are as follows:

• Soft, uncompacted ground (pasture, loose soil); snow etc: G=1.0
• Compacted soft ground (lawns, park areas): G=0.7
• Compacted dense ground (gravel road, compacted soil): G=0.3
• Hard surfaces (asphalt, concrete): G=0.0
The ground factor G_S for the source region (the zone within a horizontal distance <= 30(z_S+z_R) from the source) is also required, to account for propagation where the source and receiver are close, where the reflection properties of the source region dominate.

In cases where the source and receiver are close (within 30(z_s+z_r) horizontal distance), Strutt modifies the user-entered average ground factor as follows:

bar G prime = bar G (d_(SR))/(30(z_S+z_R))+G_s (1-d_(SR)/(30(z_s+z_r)))

d_(SR) is the horizontal source-receiver distance along the average ground plane

The CNOSSOS model predicts the attenuation under neutral (or "homogenous") atmospheric conditions (constant sound speed where sound propagation paths are straight rays), as well as adverse (or "favourable") conditions where a sound speed gradient results in curved ray paths.

The effect of the curved ray paths is to increase the average propagation height and reduce the strength of the ground effect.

Under neutral conditions, the ground attenuation is calculated as:

A_(ground) = -"Max" {(-10 log_10 (4 k^2/d_(SR)^2(z_s^2-sqrt((2 C_f)/k) z_s + C_f/k)*(z_r^2-sqrt((2 C_f)/k)z_r + C_f/k))),(-3(1-bar G_w)) :}

where:
k is the wavenumber
C_f=d_(SR) (1+3 w d_(SR) e^(-sqrt(w d_(SR))))/(1+w d_(SR))
w = 0.0185 (f^2.5 bar G_w^2.6)/(f^1.5 bar G_w^2.6 + 1.3*10^3 f^0.75 bar G_w^1.3 + 1.16*10^6)
bar G_w is either the corrected ground factor bar G prime or the uncorrected ground factor bar G, depending on the calculation type:

(The CNOSSOS ground module is also used by the CNOSSOS diffraction module to calculate the ground reflection from the source to the diffraction edge Delta_(ground(S,O)) and from the diffraction edge to the receiver Delta_(ground(O,R); the ground factor term is taken as either the modified or the unmodified term as outlined in the table below)

Under adverse conditions, the ground attenuation is calculated as:
A_(ground) = -"Max" {(-10 log_10 (4 k^2/d_(SR)^2(hat z_s ^2-sqrt((2 C_f)/k) hat z_s + C_f/k)*(hat z_r^2-sqrt((2 C_f)/k) hat z_r + C_f/k))),(-3(1-bar G_m) \ " for " \ d_(SR)<=30(z_s+z_r)),(-3(1-bar G_m)*(1+2*(1-(30(z_s+z_r))/d_(SR))) " otherwise") :}

where:
hat z_s = z_s+ Delta z_s + Delta z_t is the effective source height under adverse conditions

hat z_r = z_r + Delta z_r + Delta z_t is the effective receiver height under adverse conditions

Delta z_s = a_0(z_s/(z_s+z_r))d_(SR)^2/2 is the change in effective source height due to the curvature of the sound paths

Delta z_r = a_0(z_r/(z_s+z_r))d_(SR)^2/2 is the change in effective receiver height due to the curvature of the sound paths

Delta z_t = 6*10^-3 d_(SR)/(z_s+z_r) is the change in effective source/receiver height due to atmospheric turbulence

a_0 is the inverse of the radius of curvature of the sound paths. A default value of a_0 = 2*10^-4 is provided in CNOSSOS, however this could potentially be user-modified in Strutt if required.

bar G_m is either the corrected ground factor bar G prime or the uncorrected ground factor bar G, depending on the calculation type.

References:

• Stylianos Kephalopoulos, Marco Paviotti, Fabienne Anfosso-Lédée (2012) Common Noise Assessment Methods in Europe (CNOSSOS-EU) EUR 25379 EN. Luxembourg: Publications Office of the European Union, 2012, 180 pp.


text:
# Compact Quantum Groups from Hopf Algebras

For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around, ie given a Hopf algebra $H$ (say a Drinfeld--Jimbo algebra if it makes things easier) can it always be completed to give a compact quantum group? If so, is this completion unique in anyway, or are there many ways to get a cqg from a Hopf algebra?

-
You definitely need to assume some sort of unitartity here. –  Noah Snyder Jun 12 '11 at 6:49

I think this was solved in the paper:

MR1310296 (95m:16029)
Dijkhuizen, Mathijs S.(NL-MATH); Koornwinder, Tom H.(NL-AMST-CS)
CQG algebras: a direct algebraic approach to compact quantum groups. (English summary)
Lett. Math. Phys. 32 (1994), no. 4, 315–330.

They show that a Hopf $*$-algebra $A$ is the maximal Hopf $*$-algebra of a compact quantum group in the Woronowicz sense if (and only if) $A$ is spanned by the matrix coefficients of its finite-dimensional unitary (irreducible) corepresentations. The "only if" part was shown by Woronowicz, of course.

They also show that a Hopf $*$-algebra has this property if and only if it admits a positive definite Haar functional (in the sense Andreas talks about).

You also ask about uniqueness-- this occurs if and only if $A$ is "coamenable" (see various papers by Bedos, Tuset and coauthors). A classical example comes from a discrete group $G$ with Hopf $*$-algebra $\mathbb C[G]$ (under convolution product). Then there is the maximal $C^*$ completion $C^*(G)$ and the completion of $\mathbb C[G]$ acting on $\ell^2(G)$ by the left-regular representation, leading to $C^*_r(G)$. The quotient map $C^*(G) \rightarrow C^*_r(G)$ is injective if and only if $G$ is amenable.

-
Do the Drinfeld--Jimbo algebras fit into this category? –  John McCarthy Jun 12 '11 at 20:03
I know very little about the more algebraic side of things-- but you might understand, better than I do, the contents of Section 6 of: maths.tcd.ie/pub/ims/bull44/lars2.ps Or, hopefully someone else will come along with a more definitive answer... –  Matthew Daws Jun 12 '11 at 20:07
I do not believe that Drinfel'd-Jimbo algebras fit in (though they are Hopf duals to the algebras which do fit if equipped with a correct $\ast$-structure). –  Zoran Skoda Jun 12 '11 at 22:11
They do -- see Chari & Pressley's book Chap. 10-13. They are also amenable as proved by Banica's work. –  Makoto Yamashita Jun 12 '11 at 22:47

First say we're taking about $*$-Hopf algebras, to start with.

Then, in any case, the answer is no. Any compact quantum group admits a biinvariant Haar measure. On a general Hopf algebra not even the existence of a left integral is granted (Hopf algebra with a left integral, if I remember correctly, are called coFrobenius). Even if left and right integral exists it is not granted that they coincide (again I think it is quite easy to find finite-dimensional examples of this: see van Daele The Haar measure on finite quantum groups, Proc. Amer. Math. Soc. 125, 3489-3500 (1997)).

One has certainly to enlarge the theory to locally compact quantum groups; I'm not completely sure whether one can obtain existence in such a case. I strongly doubt about unicity.

ADDED Drinfel'd-Jimbo quantization, by which I mean quantized universal enveloping algebras, are examples of non commutative Hopf * algebras that do not fall in the CQG setting; in fact they have a real group-like element $K$ and and easy computation shows that any left invariant integral $\phi$ should satisfy $\phi(K^2)=\phi(KK^*)=0$, thus cannot be positive and therefore there is no Haar measure. This should come as no surprise. The quantum duality principle allows to interpretate quantized universal enveloping algebras as quantizationof dual Poisson-Lie groups. Dual Poisson Lie groups of (standard Poisson-Lie) compact groups are not compact. This already shows that the setting of compact quantum groups is too narrow for the whole quantum group thing. If I remember correctly quantized universal enveloping algebras are algebraic quantum groups in the sense of Kustermans - van Daele, i.e. they have a Haar measure on the associated multiplier Hopf * algebra (which is non unital), in perfect agreement with the fact that they quantize non compact unimodular Lie groups.

-
The new comment is really useful; thanks for adding it. –  Matthew Daws Jun 13 '11 at 8:30

The key is not only the existence and identification of left and right integral, but also that they define a positive linear functional on the $\ast$-algebra, i.e. $$\phi(a^*a)\geq 0, \quad \forall a \in A.$$

This allows to construct a Hilbert space representation via the GNS-construction, and hence yields the desired analytic picture.

-

A little side remark, concerning the last part of your question: Marc Rosso describes in

Marc Ross, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif, Duke Math. J. Volume 61, Number 1 (1990), 11-40.

another way to get compact quantum groups from Hopf algebras. He defines inner products on the representation spaces of the deformed enveloping algebras of simple Lie groups (Drinfeld and Jimbo's $U_q(g)$'s), such that they form a concrete monoidal $W^*$ category. Applying Woronowicz' Tannaka-Krein duality he can recover compact quantum groups from this: $C(G_q)$, a deformation of the algebra of continuous functions on the simple Lie group $G$. The Hopf algebras $U_q(g)$ do not sit inside $C(G_q)$, but their restricted duals can be identified with the dense Hopf *-algebra contained in $C(G_q)$.

-

format: plain


text:
# a.e. point wise convergence on finite measured set implies convergence in measure

Let $$f,f_k$$ for $$k \in \mathbb{N}$$ be measurable finite a.e. on measurable set $$E$$. Suppose $$f_k \rightarrow f$$ point wise a.e., then $$f_k$$ converges to $$f$$ in measure on $$E$$.

This is a basic result and the proof follows from basic definitions I believe, I want to know why we need finite valued and finite measure, is it for the obvious reasons that infinite measure does not converge and finite valued so that the limit as $$k$$ tends to $$\infty$$ makes sense?

because converging pointwise a.e. means that for $$x \in E$$ with $$E$$ having positive finite measure, one has

lim$$_{k \rightarrow \infty}$$ $$f_k(x) = f(x)$$

so

lim$$_{k \rightarrow \infty}$$ $$f_k(x) - f(x)$$ = $$0$$,

and we wish to show for any $$\epsilon > 0$$ that

$$m(\{x : \vert f_k(x) - f(x) \vert > \epsilon \}) \rightarrow 0$$ as $$k \rightarrow \infty$$

and since $$x$$ is taken out of some set of finite positive measure, we obtain (and I am not sure this works)

$$m(\{x : \vert 0 \vert > \epsilon \}) \rightarrow 0$$ as $$k \rightarrow \infty$$

I get confused here, So sorry for being naive in measure theory, I apologize and thank you in advance! if there is any simple small subtle silly errors I am making, any intuitive assumptions I am incorrectly making please let me know, I am very eager to strengthen my measure theory skills.

• there are a lot of ways to show this, depending on the theoretic background. You can show that $(f_j)\to f$ point-wise a.e. implies that $(f_j)\to f$ almost uniformly when the space have finite measure. And from here that $(f_j)\to f$ almost uniformly implies that $(f_j)\to f$ in measure – Masacroso Dec 4 '18 at 22:40
• is almost uniform like saying uniform almost everywhere? or is it a slightly weaker notion of uniform convergence? – Hossien Sahebjame Dec 4 '18 at 23:04
• a sequence of measurable functions converges almost uniformly in the measure space $(X,\mu)$ if, for any chosen $\epsilon>0$, there is some measurable $A_\epsilon\subset X$ such that $(f_n)\to f$ converges uniformly in $A_\epsilon$ and $\mu(A_\epsilon^\complement)<\epsilon$. However this doesn't mean that $(f_n)\to f$ uniformly a.e.! – Masacroso Dec 4 '18 at 23:11


text: Somewhat guessing, but maybe use dissectortable.get("ip.proto") to get the table of dissectors registered with ip, then use dissectortable::try() to call the appropriate entry from that table with the protocol value you have for the next header.


text:
Note: This is not the latest SHOE specification.

SHOE 0.991

Proposed Specification
Sean Luke
SHOE Project
May 10, 1997

Version 0.99 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.99.html
Version 0.98 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.98.html
Version 0.97 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.97.html
Version 0.96 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.96.html
Version 0.95 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.95.html
Version 0.90 of this document: http://www.cs.umd.edu/projects/plus/SHOE/spec0.9.html

1 Introduction

This specification describes SHOE, an extension to HTML which provides a way to incorporate machine-readable semantic knowledge in HTML or other World-Wide Web documents. This specification describes:

• A hierarchical classification mechanism for HTML documents (and optionally non-HTML documents) or subsections of HTML documents.
• A mechanism for specifying relationships between classified elements and other classified elements or specific kinds of data (numbers, dates, etc.)
• An simple way to specify ontologies containing rules that define valid classifications, relationships, and inferred rules.

The intent of this specification is to make it possible for user-agents, robots, etc., to gather truly meaningful information about web pages and documents, enabling significantly better search mechanisms and knowledge-gathering.

• First, define an ontology describing valid classifications of web objects, and valid relationships between web objects and other web objects or data. This ontology may borrow from other ontologies.
• Annotate HTML pages to describe themselves, other pages, or subsections of themselves, as having attributes as described in one or more ontologies.

We're playing a bit fast-and-loose with the term ontology here. In this specification, ontology'' simply means an ISA hierarchy of classes/categories, plus a set of atomic relations between these categories, and a set of inferential rules in the form of simplified horn clauses. Categories inherit relations defined for parent categories.

User agents following this specification should be aware that assertions made by HTML pages are not facts, but claims. I.e., if element x claims that element y is related with relation r to element z, then the user-agent should not be entering r(y,z) into its database (i.e., "Now I know that y is related to z with the relationship r!!"). Instead, it should be entering something along the lines of r(x,y,z) into its database (i.e., "x is claiming that y is related to z with relationship r."). This is an important distinction: it's perfectly fine for HTML pages out there to be making completely false claims; one shouldn't simply accept them as truth. For similar reasons, HTML pages can only make assertions, not retractions.

1.1 Extensions to HTML

SHOE adds the following tags to the HTML standard:

To make possible ontology declarations, SHOE adds ONTOLOGY, /ONTOLOGY, ONTOLOGY-EXTENDS, ONTDEF, /ONTDEF, ONTIF, and ONTTHEN.

To make possible semantic markup of HTML pages, SHOE adds INSTANCE, /INSTANCE, USE-ONTOLOGY, CATEGORY, RELATION, ATTRIBUTE and /ATTRIBUTE. Additionally, SHOE declares the META HTTP-EQUIV tags "Instance-Key" and "Instance-Delegate".

2 Terms

Terms not described here may be found in the HTML 2.0 specification.

Category
An element under which HTML page instances or subinstances can be classified. Category names are element names, and may be prefixed. Categories may have parent categories. Categories define inheritance: if an instance is classified under a category, keys classified with this category may fill argument positions in relations defined for that category or any of its parent (or ancestor) categories. Multiple inheritance is valid. Categories may not be named STRING, NUMBER, DATE, or TRUTH. Additionally, SHOE reserves for future use all category names that begin with a character other than a number or a letter.
Data
Data which can be placed in an argument of a relationship. Data must be of the following types:

Instances
See "Instance" below.
Strings (STRING)
HTML String Literals, as defined in the HTML 2.0 specification.
Numbers (NUMBER)
Floating-point numerical constants. Knowledge-agents should be able to read common floating-point numbers like 2, 2.0, -1.432e+4, etc. Numbers may be of the form,
0|
([+|-|]
['.'digit*|0'.'digit*|non-zero-digit digit*['.'digit*|]]
[([e|E][+|-]non-zero-digit digit*)])

Dates (DATE)
Date/Timestamps following RFC 1123, as shown in section 3.3.1 of the HTTP/1.0 specification.
Booleans (TRUTH)
HTML String Literals of the form YES or NO, case-insensitive.

Element
A category or relationship name, or one of the following reserved keywords (all caps): STRING, NUMBER, DATE, or TRUTH. Element names are case-sensitive, and may contain only letters, digits, or hyphens.
Instance
A data object which may be classified under zero or more categories, and included as an argument to relationships (along with other forms of data). Some instances, page instances, are associated with World-Wide Web documents. All page instances are automatically of the category PageInstance. Other instances, subinstances, are associated with subsections of HTML page instance documents. Subinstances are automatically of the category Subinstance. Unless they are special subinstances for delegating declaritive power (see section 4.7), subinstances have a parentInstance relationship linking them to their enclosing instance. Lastly, privileged instances are declared within ontologies, and are automatically of the category PrivilegedInstance. Instances form the most common data entities in databases built up from this specification.
Key
A string which uniquely defines a page instance or a subinstance. It is up to you to decide on the keys for your documents. For page instances of SHOE-conformant documents, the proper method is to use a single absolute URL for the document. For example, http://www.cs.umd.edu is a valid key for the document located at that URL.

To create keys for subinstances, add to the page instance's unique key a pound-suffix such as #MyDog. For example, http://www.cs.umd.edu#MyDog is a valid key for a subinstance located at http://www.cs.umd.edu. It's good style for this unique key to correspond with an actual anchor in the document.

The unique key of a non-SHOE-conformant document is defined to be one particular absolute URL of the document, chosen for the document by a SHOE-conformant document which references it.

The SHOE spec reserves the key "me" and any capitalized form of it. "me" (under any capitalized form) may be used as an argument of a claim to refer to the enclosing data instance actually making the claim. The SHOE spec also reserves any keys beginning with "!" for referencing privileged instances. See section 3.7.1 for more information on how to form such keys. Finally, the SHOE spec reserves for future use any other keys whose first character is something other than a number or a letter.

Ontology
As defined in this specification, a description of valid classifications for HTML page instances and subinstances, and valid relationships between instances and elements.
Prefix
A small string attached with a period at the beginning of an instance, category, or relation name. For example,cs is a prefix in cs.junk. Prefixes may also be attached to already-prefixed elements, forming a prefix chain. For example, foo.bar.cs is a prefix chain for foo.bar.cs.junk. A prefix indicates the ontology from which the element (or prefixed element) following it is defined.
Relation (Relationship)
An element which defines a relationship between instances and other instances or data. Relation names are element names, and may be prefixed. A relation is between one or more elements called arguments; if a relation is defined for some set of arguments, this permits SHOE documents to declare this relation among instances of those arguments. Arguments are explicitly ordered, so each has a numbered position (the first is argument 1). Many relations are binary (have exactly two arguments). A binary relation's domain is argument 1 of the relation. A relation's range (the element the relation is to'') is argument 2 of the relation.
Rule
A formal rule in an ontology defining valid classifications (categories) or valid relationships that can be asserted.
Unique Name
A string which uniquely defines an ontology. Unique names are different from keys in that they do not uniquely define instances but rather the ontologies which the instances may use. Different versions of an ontology may have the same unique name so long as they have different version numbers.
Version (Version Number)
A string which describes the version of an ontology. Versions are case-sensitive, and may contain only letters, digits, or hyphens.

3 Declaring Ontologies

Except as specified, all declarations must be made in the BODY section of an HTML document.

3.1 Declaring An Ontology Definition

An HTML document may contain any number of ontology definitions. Each ontology definition should use a unique name. Ontology definitions are accompanied with a version number. If an ontology completely subsumes previous versions of the same ontology (it contains all the rules defined in those versions), it may declare itself to be backward-compatible with those versions. To begin an ontology definition, use:

<ONTOLOGY "ontology-unique-name"
VERSION="version"
[BACKWARD-COMPATIBLE-WITH="version list"]
[DESCRIPTION="text"]>

ontology-unique-name'' (mandatory)
The ontology's unique name.
VERSION (mandatory)
The ontology's version.
BACKWARD-COMPATIBLE-WITH
A whitespace-delimited list of previous versions which this ontology subsumes.
DESCRIPTION
A short, human-readable description of the purpose of the ontology.

To end an ontology definition, use:

</ONTOLOGY>

All rules and extensions in an ontology must appear between the beginning and ending declarations. Ontologies may not be nested or overlap.

3.2 Extending An Existing Ontology

An ontology may be declared to extend one or more existing ontologies. This means that it will use elements in those ontologies in its own rules. To distinguish between those elements and its own elements, an ontology must provide a unique prefix for each ontology it extends. This will be prefixed to elements borrowed from each particular ontology whenever they are referred to. To declare that an ontology is extending another ontology, use:

<ONTOLOGY-EXTENDS "ontology-unique-name"
VERSION="version"
PREFIX="prefix"
[URL="URL"]>

ontology-unique-name'' (mandatory)
The extended ontology's unique name.
VERSION (mandatory)
The extended ontology's version.
PREFIX (mandatory)
The prefix you are assigning the extended ontology. All categories and relations from the extended ontology which are used in your ontology must be prefixed with this prefix. Within an HTML document, a prefix must be different from all prefixes declared with either <USE-ONTOLOGY ...> or <ONTOLOGY-EXTENDS ...> tags.
URL
A URL that points to a document which contains the extended ontology.

3.3 Declaring Classification Rules

Inside an ontology definition, an ontology may declare various new categories which instances can belong to. Categories should descend from one or more parent categories. To declare a new category, or to add new parent categories for a category, use:

<ONTDEF CATEGORY="category-name"
[ISA="parent-category-list"]
[DESCRIPTION="text"]
[SHORT="text"]>

CATEGORY (mandatory)
The newly declared category, or the one being given more parent categories. Newly declared categories should be distinct from all other categories and relationships declared in the ontology.
ISA
A whitespace-delimited list of categories to define as parent categories of this category.
DESCRIPTION
A short, human-readable description of the category's semantics.
SHORT
A phrase which an agent may use to display the category to a user a more understandable fashion than the category's name. In English ontologies, SHORT should be a noun, lower-case unless it is a proper noun. For example, the category "LaunchVehicleNASA" might have SHORT="rocket".

A particular category should not be defined more than once within an ontology's declaration.

3.4 Declaring Relationship Rules

Inside an ontology definition, an ontology may declare various new valid relationships between category instances or between category instances and data. To declare a relationship, use:

<ONTDEF RELATION="relation-name"
ARGS="element-list"
[DESCRIPTION="text"]
[ARGS-DESCRIPTION="term-list"]
[SHORT="text"]>

RELATION (mandatory)
The newly declared relationship name. This should be distinct from all other categories and relationships declared in the ontology.
ARGS (mandatory)
The arguments of the relation. This should be a whitespace-delimited list of declared categories or the following keywords (all caps): STRING, NUMBER, DATE, TRUTH.

DESCRIPTION
A short, human-readable description of the relationship's semantics.
ARGS-DESCRIPTION
A whitespace-delimited list of one-word descriptions of each argument in the relation. This may be useful for an agent to display to the user in a more understandable fashion than "argument 1", "argument 2", and so on. For example, the relation "nameOf" relating a person with a name might have ARGS-DESCRIPTION="person name".
SHORT
A phrase which an agent may use to display the relation to a user a more understandable fashion than the relation's name. In English ontologies, SHORT should be a verb, lower-case, and in the format that makes some sense when appearing after the first argument but before the remaining arguments. For example, the relation "nameOf" might have SHORT="is named".

A particular named relationship should not be defined more than once within an ontology's declaration.

3.5 Renaming Rules

To reduce the number of prefixes, an ontology may rename a category, privileged-instance, or relation (plus its prefix chain) to a simpler name, so long as this name is not used in any other category, privileged-instance, or relation in the ontology. For example, an ontology could rename the category cs.junk.foo.person to simply person, so long as person is not defined elsewhere in the ontology.

Ontologies are not permitted to rename (or rename elements to) the following keywords: STRING, NUMBER, DATE, TRUTH. To rename a category, privileged-instance or relation, use:

<ONTDEF RENAME="element-name"
TO="new-element-name">

RENAME (mandatory)
The element's old name.
TO (mandatory)
The element's new name.L

3.6 Declaring Inferences

An ontology may declare one or more inferences which may be automatically made based on the relationship and category declarations found in marked-up SHOE text.

A SHOE inference clause is similar to a Horn clause in that it consists of a body of one or more subclauses describing claims entities might make, and a head consisting of one or more subclauses describing a claim that may be inferred if all the claims in the body do turn out to be made.

Subclause Rules.Both the head and the body of an inference clause must contain at least one subclause; inference clauses with an empty head or body are incorrect and may be ignored. A subclause in the head may either be a relation declaration or a category declaration. A subclause in the body may be either a relation declaration, a category declaration, or a "special" declaration as described below. Special declarations may not be made in the head. Special declarations permit inferences to include special equals/not-equals/greater-than/less-than relationships between elements in subclauses.

Constants and Variables. The data in a head or body subclause may be either constants or variables. Constant data must be matched exactly as it is. Variables may be matched (bound) to any constant data interned by a SHOE agent, so long as variables of the same name in subclauses within the same inference clause always bind to the same value, and fill argument positions of the same type. Variables are case-insensitive: the variable V matches with the variable v.

Variable Rules . There are two variable rules, intended to eliminate ambiguities and excess computational complexity in the inference clauses:

• The head of the inference clause may not contain variables that do not appear somewhere in the body of the clause.
• If there is more than one variable in the body, then each variable must appear at least once in a relationship-type (<ONTIF RELATION=...>) subclause within the body along with another variable in such a way that one could trace a path from any variable to any other variable through a series of relationships (in other words, the relationship graph for variables in the body is connected). Special declarations (other than "equal") and category declarations do not count as part of this relationship graph.
Inference clauses that do not meet these constraints are incorrect and may be ignored.

A SHOE inference begins with an inference declaration:

<ONTDEF INFERENCE
[DESCRIPTION="text"]>

DESCRIPTION
A short, human-readable description of the inference's semantics.

An inference is ended with

</ONTDEF>

3.6.1 Declaring an Inference Body Subclause

An inference may have zero or more body subclauses, defined with the ONTIF tag. Each body subclause is either a category declaration, or a relationship declaration, but not both. Body subclauses indicate claims that must be actually made by SHOE documents for the inferences in the head subclause to take place. If an inference has no body subclauses, this indicates that the inferences in the head should always be true. However, this method of making assertions should be made sparingly; such claims are better expressed in SHOE documents themselves, not in an ontology.

To define that a body subclause is a category declaration, use:

<ONTIF CATEGORY="prefixed.category"
FOR="variable" | CFOR="data">

prefixed.category'' (mandatory)
A category with full prefix chains showing a path through extended ontologies back to the ontology in which it was defined.
FOR (co-mandatory)
Contains a variable to be bound to an instance which has been declared to belong to this category.
CFOR (co-mandatory)
Contains the key of an actual instance which has been declared to belong to this category.

Either FOR or CFOR must be defined, but not both.

To define that a body subclause is a relationship declaration, use:

<ONTIF RELATION="prefixed.relationship"
[[1="key" | C1="data"]
[2="key" | C2="data"]
[3="key" | C3="data"]
[4="key" | C4="data"]
[5="key" | C5="data"] etc...]>

prefixed.relationship'' (mandatory)
A relationship with full prefix chains showing a path through used and extended ontologies back to the ontology in which it was defined.
C1, C2, C3, C4, C5, C6, C7, ...
Declares the element as a constant in argument position indicated by the tag. For example, C7="George" declares that the constant "George" is argument 7 in the relation. This object must be of the type declared for that particular argument position.
1, 2, 3, 4, 5, 6, 7, ...
Declares the element as a variable in argument position indicated by the tag. For example, C7=VAR1 declares that the variable VAR1 is argument 7 in the relation.

It is incorrect for an ontology to declare relation subclauses with missing arguments; if this happens, the inference clause is incorrect and may be ignored.

To define that a body subclause is a special declaration, use:

<ONTIF SPECIAL="special.declaration"
[[1="key" | C1="data"]
[2="key" | C2="data"]
[3="key" | C3="data"]
[4="key" | C4="data"]
[5="key" | C5="data"] etc...]>

special.declaration" (mandatory)
One of the special declaration key words: equal, notEqual, greaterThan,greaterThanOrEqual,lessThanOrEqual, or lessThan. These are all binary special declarations, and indicate that argument 1 is equal, not equal to, greater than, or less than argument 2. For strings, instance keys, category names, and relation names, these declarations assume case-sensitivity, and greaterThan/lessThan have no meaning. For dates, assume that earlier dates are "less than" later dates. For truths, assume that NO is "less than" YES.
C1, C2, C3, C4, C5, C6, C7, ...
Declares the element as a constant in argument position indicated by the tag. For example, C7="George" declares that the constant "George" is argument 7 in the special declaration predicate. This element must be of the same type as all other argument positions. Since the current key words are all binary, values C3 and beyond currently have no meaning.
1, 2, 3, 4, 5, 6, 7, ...
Declares the element as a variable in argument position indicated by the tag. For example, C7=VAR1 declares that the variable VAR1 is argument 7 in the special declaration predicate. There must be at least one variable as an argument in a special declaration.

It is incorrect for an ontology to declare special declaration subclauses with missing arguments; if this happens, the inference clause is incorrect and may be ignored.

3.6.2 Declaring an Inference Head Subclause

An inference must have at least one head subclause. Each head subclause is either a category declaration, or a relationship declaration. A head subclause indicates a claim that may be inferred if all the claims defined in the body subclauses are met.

To define that a head subclause is a category declaration, use:

<ONTTHEN CATEGORY="prefixed.category"
FOR="variable" | CFOR="data">

The semantics of the tags for this declaration are identical to those declared for the <ONTIF CATEGORY=...> tag format described above.

To define that a head subclause is a relationship declaration, use:

<ONTTHEN RELATION="prefixed.relationship"
[[1="key" | C1="key"]
[2="key" | C2="key"]
[3="key" | C3="key"]
[4="key" | C4="key"]
[5="key" | C5="key"] etc...]>

The semantics of the tags for this declaration are identical to those declared for the <ONTIF RELATION=...> tag format described above.

3.7 Making Declarations from an Ontology

Note: Section 3.7 assumes that the reader is familiar with Section 4 already; you may want to come back to this section later. Ordinarily an ontology cannot {\it declare} relationships or categorizations, only define the rules that govern such declarations. However, an ontology {\it may} state that one or more specific instances are making important standard declarations associated with the ontology. It is important to note that it is the instances themselves that make the declarations, not the ontology (See section 4 to make instances). An ontology makes this statement about an instance using:

<ONTDEF DECLARATOR="declaring-instance">

declaring-instance" (mandatory)
The key for the instance the ontology has associated with itself.

3.7.1 Declaring a Privileged-Instance

<ONTDEF DECLARATOR...> might be used to declare standard constant instances like "Red" which are subclasses of the ontology-defined class Color and used in the colorOf relationship. However, the key for "Red" may then turn out to be be something like "http://www.cs.umd.edu/ontology-associated-instance.html#Red". This is not very nice for common, very standard instances (like colors). To remedy this, SHOE allows ontologies themselves to declare instances, using:

<ONTDEF INSTANCE="privileged-instance-name">

privileged-instance-name" (mandatory)
The name of the instance.

The key for this instance is defined as a "!" followed by the priviledged instance name plus its prefix chain. For example, if an ontology defined "Red" as a privileged instance, and some page instance uses this ontology with the "cs" prefix, then the page instance can reference "Red" with the key "!cs.Red". Note that this means that there may be more than one key referencing the same privileged instance, depending on the particular path of prefixes chosen. All such keys should resolve to the same instance.

If the ontology wants to do anything additional to the instance (such as categorizing it), it must rely on the <ONTDEF DECLARATOR...> mechanism to do this.

4 Marking Up HTML Documents Using Ontologies

Except as specified, all declarations must be made in the BODY section of an HTML document.

4.1 Declaring a Page Instance

SHOE-conformant HTML documents must declare themselves page instances and provide a unique key for themselves. To declare an HTML document to be a page instance, add the following text to the HEAD section of the document:

<META HTTP-EQUIV="Instance-Key"
CONTENT="key">

key'' The page instance's unique key.

4.2 Declaring a Subinstance

A document may declare zero or more subinstances. Subinstances may be nested. To declare the start of a subinstance, use:

<INSTANCE "key">

key'' (mandatory)
The unique key for the instance.

To mark the end of the section of a subinstance, use:

</INSTANCE>

All relationship and category declarations made within a subinstance belong to that subinstance. All relationship and category declarations made outside a subinstance belong to the enclosing instance.

4.3 Declaring Ontology Usage

Before you can classify documents or establish relationships between them, you'll need to define exactly which ontologies these classifications and relations are derived from, and associate with each of these ontologies some prefix unique to that ontology. An HTML document may declare that is using as many ontologies as it likes, as long as each ontology has a unique prefix in the document. To declare that a page instance and all its subinstances use a particular ontology, use:

<USE-ONTOLOGY "ontology-unique-name"
VERSION="version"
PREFIX="prefix"
[URL="URL"]>

ontology-unique-name'' (mandatory)
The ontology's unique name.
VERSION (mandatory)
The ontology's version.
PREFIX (mandatory)
The prefix you are assigning the ontology. All categories and relations from this ontology which are used in this document must be prefixed with this prefix. Within this document, the prefix must be different from all prefixes declared with either <USE-ONTOLOGY ...> or <ONTOLOGY-EXTENDS ...> tags.
URL
A URL that points to a document which contains the used ontology.

4.4 Declaring Categories

Instances may be classified, that is, they may be declared to belong to one or more categories in an ontology, using the CATEGORY tag:

<CATEGORY "prefixed.category"
[FOR="key"]>

prefixed.category'' (mandatory)
A category with full prefix chains showing a path through used and extended ontologies back to the ontology in which it was defined.
FOR
Contains the key of the instance which is being declared to belong the category. If FOR is not declared, then the key is assumed to be that of the enclosing subinstance, or (if there is no enclosing subinstance) the page instance. If FOR is declared, then it provides the key.

4.5 Explicitly Declaring Relationships

Instances may declare relationships with elements. To explicitly declare relationships between elements, use:

<RELATION "prefixed.relationship"
[[FROM="key"] [TO="key"]] |
[[1="key"] [2="key"]
[3="key"] [4="key"]
[5="key"] etc... ] >

prefixed.relationship'' (mandatory)
A relationship with full prefix chains showing a path through used and extended ontologies back to the ontology in which it was defined.
1, 2, 3, 4, 5, 6, 7, ...
Declares the element in argument position indicated by the tag. For example, 7="George" declares that "George" is argument 7 in the relation. This element must be of the type declared for that particular argument position.
FROM
Synonymous with 1. A relation declared with both FROM and any 1, 2, 3, 4 ... tags is invalid.
TO
Synonymous with 2. A relation declared with both TO and any 1, 2, 3, 4 ... tags is invalid.

Explicit declarations take two forms, the "binary" form and the "general" form.

The "binary" form is used only for binary relationships. In this form, the tags FROM and TO are used. A tag (FROM or TO) may be omitted if the element type for that tag is INSTANCE. In this case, the key for the element is assumed to be that of the enclosing subinstance, or (if there is no enclosing subinstance) the page instance. If the tag is declared, and the type of the tag's argument is an instance, then it provides the key.

The "general" form may be used for relationship of any number of arguments (including binary relationships). In this form, the 1, 2, 3, 4 ... tags may be used. If a numbered tag is not declared, and an argument of that number position exists in the relationship, then the information should be considered "unknown". Note that this is different from the assumptions in the "binary" form.

The reason for two different forms is that while the "general" form is necessary to describe all possible relationships, the large majority of relationships are binary relationships between the claimant and some other instance or data. Hence, the "binary" format allows the claimant to declare binary relationships in a more natural format ("to" and "from") and not have to include himself in the claim when it's clear from the context. The "binary" form makes relations feel more like slots in the claimant "object" than ordinary predicate relationships.

4.6 Marking Up Text Relationships

It's possible to wrap existing HTML body text and declare it to be a relationship. This is done by:

<ATTRIBUTE "prefixed.relationship"
attribute text
</ATTRIBUTE>

This is functionally the same as declaring:

<RELATION "prefixed.relationship"
TO="attribute text">

4.7 Delegating Declarative Power to Another Page Instance

A page instance may indicate that it permits another page instance to make declarations on its behalf. The other page instance may then declare a subinstance of the same key name as the permitting page instance's key. Agents should consider all claims made within that subinstance as if they were made on the permitting page instance itself. This might be done to consolidate claims for a web site into a single document, perhaps, or to eliminate a large number of claims from slowing down the download time of a document.

If the other page instance does not in fact declare this special subinstance, then delegating declarative power is simply a pointer to an agent to look elsewhere for relevant SHOE knowledge. To delegate declarative power to another web page, use (in the HEAD section of the HTML document):

<META HTTP-EQUIV="Instance-Delegate"
CONTENT="key">

key'' The key of the page instance to whom we are delegating declaritive power.

Delegated subinstances should be considered to be proxies for the delegating page instance, and any subinstances contained within a delegated subinstance should be considered to be proxies for subinstances within the delegating page instance. An agent can guess that a subinstance is a delegated subinstance instead of an ordinary subinstance by examining its key. If the key is not based on the enclosing page instance URL, it's more than likely a delegated subinstance for some other URL. However, an agent should not take this at face value, but should check the delegating page instance first to make certain that the delegation is valid.

Web Accessibility


text:
## A cylinder has a radius of 3.2 meters its volume is 190 Cubic meters find its height to the nearest 10th of a meter

Question

A cylinder has a radius of 3.2 meters its volume is 190 Cubic meters find its height to the nearest 10th of a meter

in progress 0
2 months 2021-08-01T03:24:14+00:00 1 Answers 5 views 0


text:
# Definition:Density (Physics)

## Definition

Density is a physical quantity.

The density of a body is its mass per unit volume.

For a homogeneous body it is found by finding its total mass and dividing it by its total volume:

$\rho = \dfrac m V$

where:

$m$ is the body's mass
$V$ is the body's volume

However, if the substance of the body varies throughout, then its density may be a function of position within the body.

### Symbol

The usual symbol used to denote density of a body is $\rho$ (Greek letter rho).

Some sources use $d$, but that has so many other uses that it may become confused with other things.

### Dimension

The dimension of density is $\mathsf {M L}^{-3}$: mass per unit volume.

### Units

Thus:

$1 \, \mathrm g \, \mathrm {cm}^{-3} = 1000 \, \mathrm {kg} \, \mathrm m^{-3}$

## Area Density

The area density of a two-dimensional body is its mass per unit area.

## Linear Density

The linear density of a one-dimensional body is its mass per unit length.


text:
Article Contents
Article Contents

# A modification of the forward-backward splitting method for maximal monotone mappings

• In this paper, we propose a modification of the forward-backward splitting method for maximal monotone mappings, where we adopt a new step-size scheme in generating the next iterate. This modification is motivated by the ingenious rule proposed by He and Liao in modified Korpelevich's extra-gradient method [13]. Under suitable conditions, we prove the global convergence of the new algorithm. We apply our method to solve some monotone variational inequalities and report its numerical results. Comparisons with modified Khobotov-Korpelevich's extragradient method [13,14] and Tseng's method [30] show the significance of our work.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

•  [1] D. P. Bertsekas, "Constrained Optimization and Lagarange Multiplier Method," Academic Press, New York, 1982. [2] H. G. Chen, "Forward-Backward Splitting Techniques: Theory and Applications," Ph.D. Thesis, Department of Applied Mathematics, University of Washington, Seattle, WA, 1994. [3] H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM Journal on Control and Optimization, 7 (1997), 421-444.doi: 10.1137/S1052623495290179. [4] J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Mathematical Programming, 83 (1998), 113-123.doi: 10.1007/BF02680553. [5] J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.doi: 10.1007/BF01581204. [6] J. Eckstein and M. C. Ferris, Operator splitting methods for monotone affine variational inequalities, with a parallel application to optimal control, INFORMS Journal on Computing, 10 (1998), 218-235. [7] J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in "Large Scale Optimization: State of the Art", (eds. W. W. Hager, D. W. Hearn and P. M. Pardalos), Kluwer Academic Publishers, Dordrecht, The Netherlands, (1994), 115-134.doi: 10.1007/978-1-4613-3632-7_7. [8] F. Facchinei and J. S. Pang, "Finite-Dimensional Variatiaonal Inequalities and Complementarity Problems," Spring-Verlag, New York, 2003. [9] D. Gabay, Applications of the method of multipliers to variational inequalities, in "Augemented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems" (eds. M. Fortin and R. Glowinski), North Holland, Amsterdam, (1983), 299-331.doi: 10.1016/S0168-2024(08)70034-1. [10] A. A. Goldstein, Convex programming in Hilbert space, Bulletin of the American Mathematical Society, 70 (1964), 709-710.doi: 10.1090/S0002-9904-1964-11178-2. [11] O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419.doi: 10.1137/0329022. [12] B. S. He, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Computational Optimization and Applications, 35 (2006), 19-46.doi: 10.1007/s10589-006-6442-4. [13] B. S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 111-128.doi: 10.1023/A:1013096613105. [14] B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129-143.doi: 10.1023/A:1013048729944. [15] K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, Journal on Control and Optimization, 35 (1997), 1142-1168.doi: 10.1137/S0363012995281742. [16] B. Lemaire, Coupling optimization methods and variational convergence, in "Trends inMathematical Optimization" (eds. K. H. Hoffman, J. B. Hiriart-Urruty and J. Zowe, C. Lemarechal), Birkhauser-Verlage, Basel, (1988), 163-179.doi: 10.1007/978-3-0348-9297-1_12. [17] B. Lemaire, The proximal algorithm, International Series of Numerical Mathematics, 87 (1989), 73-87 [18] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Control and Optimization, 16 (1979), 964-979. [19] Z. Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Annals of Operations Research, 46 (1993), 157-178.doi: 10.1007/BF02096261. [20] B. Martinet, Regularisation d'inéquations variationelles par approximations successives, Rev. Francaise Informat Recherche Opérationnelle (French), 4 (1970), 154-159. [21] G. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal, 29 (1962), 341-346.doi: 10.1215/S0012-7094-62-02933-2. [22] B. Martinet, Determination approchée d'un point fixe d'une application pseudocontractante, Comptes rendus de l'Académie des sciences Paris (French), 274 (1972), 163-165. [23] D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Editura Academiei, Bucharest, 1978. [24] R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability," Springer-Verlag, New York, 1989.doi: 10.1007/BFb0089089. [25] A. Renaud and G. Cohen, Conditioning and regularization of nonsymmetric operators, Journal of Optimization Theory and Applications, 92 (1997), 127-148.doi: 10.1023/A:1022692114480. [26] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970. [27] R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, New York, 1997. [28] P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization, 29 (1991), 119-138.doi: 10.1137/0329006. [29] P. Tseng, On linear convergence of iterative methods for the variational inequality problem, Journal of Computational and Applied Mathematics, 60 (1995), 237-252.doi: 10.1016/0377-0427(94)00094-H. [30] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, 38 (1998), 431-446.doi: 10.1137/S0363012998338806. [31] P. Tseng and M. V. Solodov, Modified projection-type methods for monotone variational inequalities, SIAM Journal on Control and Optimization, 34 (1996), 1814-1830.doi: 10.1137/S0363012994268655. [32] E. Zeidler, "Nonlinear Monotone Operators, Nonlinear Functional Analysis and its Applications," II/B, Springer-Verlag, New York, 1990.doi: 10.1007/978-1-4612-0985-0.


text:
# Show that $V[a,b]$ is a Banach space

Let $V[a,b]$ be the space of all complex-valued functions of bounded variation on $[a,b]$, which we treat naturally as a $\Bbb C$-vector space. In addition, we equip it with the norm: $$\|x\|:=|x(a)|+Vx,$$ in which $Vx$ denotes the total variation of $x(t)$ on $[a,b]$. I have to show that $(V[a,b],\|\cdot\|)$ is a Banach space.

I think I'm almost there but stuck in one critical step, here are my thoughts:

Suppose $\{x_n\}$ is a Cauchy sequence in $V[a,b]$, then it is clear that both $|x_m(a)-x_n(a)|$ and $V(x_m-x_n)$ turn small when $m,n$ turn large. In particular, $\lim_{n\to\infty}x_n(a)$ exists; and given any $\epsilon>0$, we can find:
1). $N_1$ s.t. $\forall m,n>N_1$, $|x_m(a)-x_n(a)|<\epsilon/2$ and $V(x_m-x_n)<\epsilon/2$.
2). $N_2>N_1+1$ s.t. $\forall m,n>N_2$, $|x_m(a)-x_n(a)|<\epsilon/2^2$ and $V(x_m-x_n)<\epsilon/2^2$.
2). $N_3>N_1+1$ s.t. $\forall m,n>N_3$, $|x_m(a)-x_n(a)|<\epsilon/2^3$ and $V(x_m-x_n)<\epsilon/2^3$.
$$\cdots$$ This progress may continue indefinitely and we can pick $n_k\in (N_k,N_{k+1})$ so that $$|x_{n_{k+1}}(a)-x_{n_k}(a)|<\epsilon/2^k,\, V(x_{n_{k+1}}-x_{n_k})<\epsilon/2^k.$$ Define $g_N=x_{n_1}+\sum_{k=1}^N (x_{n_{k+1}}-x_{n_k})$. Note that for all $t\in [a,b]$ we have \begin{align} V(x_{n_{k+1}}-x_{n_k})&\ge |(x_{n_{k+1}}-x_{n_k})(b)-(x_{n_{k+1}}-x_{n_k})(t)|+|(x_{n_{k+1}}-x_{n_k})(t)-(x_{n_{k+1}}-x_{n_k})(a)|\\ &\ge |(x_{n_{k+1}}-x_{n_k})(t)-(x_{n_{k+1}}-x_{n_k})(a)| \\ &\ge |(x_{n_{k+1}}-x_{n_k})(t)|-|(x_{n_{k+1}}-x_{n_k})(b)|. \end{align} Hence we have $$|(x_{n_{k+1}}-x_{n_k})(t)|\le |(x_{n_{k+1}}-x_{n_k})(a)|+V(x_{n_{k+1}}-x_{n_k})<\epsilon/2^{k-1}.$$ Hence by Weierstrass M-test $g_N$ converges absolutely uniformly to some function $g$ on $[a,b]$. The hardest part is: how to show $g_N\to g$ by $\|\cdot\|$?
Naively I guess the following should be right: $$V(g-g_N)=V(\lim_{m\to\infty}g_m-g_N)=\lim_{m\to\infty}V(g_m-g_N).$$ However, even the uniform convergence isn't enough to justify the interchange of limits! So I really don't know any method else to get around it. Any help? Thanks in advance!

• Uniform convergence is more than sufficient. Take any partition, with say $M$ points, and look $g_m$ with $m$ large enough so that $|g_m(x_j)-g(x_j)|<\varepsilon/M$ for each of the points $x_j$ in the chosen partition. – SamM Sep 17 '16 at 14:36
• @SamM This isn't right. One can't vary $m$ while fixing $M$. In fact there is a quick counterexample: $$x_n(t):[0,1]\in t\mapsto \frac1{2^n}\sum_{i=0}^{2^n-1}\mathbb{1}_{[\frac{2i}{2^{n+1}},\frac{2i+1}{2^{n+1}}]}(t)\in \Bbb R,$$ then $x_n$ converges uniformly, yet $V(x_n)\equiv 1$. – Vim Sep 17 '16 at 14:44
• The choice of partition of $[a,b]$ is completely independent of the sequence $g_m$ and its limit. – SamM Sep 17 '16 at 14:47
• @SamM the total variation is defined for each $g_m$. So we may well for each $g_m$ choose a different partition. Anyway my example has furnished a counterexample where $x_m$ uniformly approaching $x$ doesn't mean $(x_m-x)$ has vanishing total variation. – Vim Sep 17 '16 at 14:54
• I believe I expressed myself quite poorly in my above; what I meant is, if you have BV-Cauchy and uniform convergence, then we can conclude that we pass the limit. – SamM Sep 17 '16 at 17:11

Fix a BV-norm Cauchy sequence $(g_n)$, $\varepsilon>0$, and $j,k\in\Bbb N$ such that $||g_j-g_k||<\varepsilon$. Let $R>0$ such that $||g_n||\leq R$ for all $n$.
You have shown that $(g_n)$ converges uniformly to a bounded function $g$ on $[a,b]$. (Here is another proof of this fact which is a bit easier: Let $t\in [a,b]$. We have $$|g_j(t)-g_k(t)|\leq |g_j(t)-g_j(a)|+|g_j(a)-g_k(a)|+|g_k(a)-g_k(t)|\\ \leq V(g_j)+|g_j(a)-g_k(a)|+V(g_k)<3\varepsilon$$ and so $(g_n)$ is uniformly Cauchy.)
Let $a=x_1<x_2<\dots<x_n=b$ be a partition of $[a,b]$. Let $N\in\Bbb N$ large enough so that $|g_k(x_i)-g(x_i)|<\varepsilon/n$ for all $i$ whenever $k\geq N$. (Pointwise convergence is enough for this.) Assuming $k\geq N$, we have $$\sum_{i=1}^{n-1}|g(x_{i+1})-g(x_i)|\leq \sum_{i=1}^{n-1}|g(x_{i+1})-g_k(x_{i+1})|+\sum_{i=1}^{n-1}|g_k(x_{i+1})-g_k(x_i)|+{\sum_{i=1}^{n-1}|g_k(x_{i})-g(x_i)|}\\ \leq 2\varepsilon + \sum_{i=1}^{n-1}|g_k(x_{i+1})-g_k(x_i)|.$$ Now we have $$\sum_{i=1}^{n-1}|g(x_{i+1})-g(x_i)|\leq R+2\varepsilon$$ and we conclude that $g\in V[a,b]$.
Now, by similar calculations assuming both $j\geq N$, we obtain $$\sum_{i=1}^{n-1}|(g_k-g)(x_{i+1})-(g_k-g)(x_i)|\leq \sum_{i=1}^{n-1}|(g_k-g_j)(x_{i+1})-(g_k-g_j)(x_i)|+2\varepsilon\\\leq V(g_j-g_k)+2\varepsilon<3\varepsilon.$$ This is independent of the choice of partition, so $V(g_k-g)\leq 3\varepsilon$.


text:
It is currently 23 Jan 2018, 22:03

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi

 post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Manager
Joined: 21 Feb 2010
Posts: 205
If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

03 Sep 2010, 13:45
6
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

56% (01:43) correct 44% (01:11) wrong based on 126 sessions

### HideShow timer Statistics

If equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A. 20
B. 50
C. 100
D. 200
E. 400

(C) 2008 GMAT Club - m25#19

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-equation- ... 01963.html
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 43380
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

03 Sep 2010, 14:18
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
$$|\frac{x}{2}| + |\frac{y}{2}| = 5$$

You will have 4 case:

$$x<0$$ and $$y<0$$ --> $$-\frac{x}{2}-\frac{y}{2}=5$$ --> $$y=-10-x$$;

$$x<0$$ and $$y\geq{0}$$ --> $$-\frac{x}{2}+\frac{y}{2}=5$$ --> $$y=10+x$$;

$$x\geq{0}$$ and $$y<0$$ --> $$\frac{x}{2}-\frac{y}{2}=5$$ --> $$y=x-10$$;

$$x\geq{0}$$ and $$y\geq{0}$$ --> $$\frac{x}{2}+\frac{y}{2}=5$$ --> $$y=10-x$$;

So we have equations of 4 lines. If you draw these four lines you'll see that the figure which is bounded by them is square which is turned by 90 degrees and has a center at the origin. This square will have a diagonal equal to 20, so the $$Area_{square}=\frac{d^2}{2}=\frac{20*20}{2}=200$$.

Or the $$Side= \sqrt{200}$$ --> $$area=side^2=200$$.

Similar problem: graphs-modulus-help-86549.html?hilit=horizontal#p649401

Hope it's clear.
_________________
Senior Manager
Joined: 09 Mar 2017
Posts: 310
Location: India
Concentration: Marketing, Organizational Behavior
WE: Information Technology (Computer Software)
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

27 Nov 2017, 20:31
drawing each line by finding coordinates would take more than 2 minutes in actual gmat test. Is there any other way for dealing such questions? Or I need to increase my speed for such questions?
_________________

------------------------------
"Trust the timing of your life"
Hit Kudus if this has helped you get closer to your goal, and also to assist others save time. Tq

Manager
Joined: 31 Jul 2017
Posts: 169
Location: Malaysia
WE: Consulting (Energy and Utilities)
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

27 Nov 2017, 20:56
4
KUDOS
tt11234 wrote:
If equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A. 20
B. 50
C. 100
D. 200
E. 400

(C) 2008 GMAT Club - m25#19

|x| + |y| = 10.
Take the first quadrant => x + y = 10
Draw a line with (10,0) and (0,10) .. Area of this quadrant will be = 0.5 * 10 * 10 = 50
As this is mod it will be same in all quadrant = 50 * 4 = 200
Senior Manager
Joined: 23 Jun 2012
Posts: 334
Location: Pakistan
Concentration: Strategy, International Business
GPA: 3.76
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

05 Dec 2017, 09:55
Hi Bunuel and other experts,
Can someone help me draw the diagram because with diagram i am getting 40 not 20.
_________________

Push yourself again and again. Don't give an inch until the final buzzer sounds. -Larry Bird
Success isn't something that just happens - success is learned, success is practiced and then it is shared. -Sparky Anderson
-S

Math Expert
Joined: 02 Sep 2009
Posts: 43380
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi [#permalink]

### Show Tags

05 Dec 2017, 10:08
sananoor wrote:
Hi Bunuel and other experts,
Can someone help me draw the diagram because with diagram i am getting 40 not 20.

Here it is:

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-equation- ... 01963.html
_________________
Re: If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi   [#permalink] 05 Dec 2017, 10:08
Display posts from previous: Sort by

# If equation |x/2| + |y/2| = 5 encloses a certain region on the coordi

 post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.


text:
The profit P, in dollars, for any given month at a certain company is defined by P = I - C, where I represents total income, in dollars, and C represents total costs, in dollars, for the month. For each of the first 4 months of the year, C = I + 32,000; and for each of the next 3 months, I = C + 36,000. If I = C + 10,000 for each of the 5 remaining months of the year, what was the company's total profit for the 12-month year?

$10,000$30,000

$40,000$50,000

\$70,000


text:
# Lp weak convergence

1. Feb 10, 2012

### muzialis

Hello,

I am struggling to understand why L1 is not weakly compact, while Lp, p>1, is.

The example I have seen put forward is the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, the function being defined on (0,1).

It is shown this u_n converging to the Dirac measure, and this shows L1 not being weakly compact (as integrating u_n times the charactersitic function of the interval yields 1 as a result, while the result is zero for a function which is zero at the boundary.

I can not see why this should not happen for Lp, p > 1 too. In the attached short notes there is an example after which (pag. 6) it is stated that this function converges weakly to zero in Lp.

I can not understand this.

Many thanks

Regards

#### Attached Files:

• ###### Weakconvergence L1.pdf
File size:
165.6 KB
Views:
151
2. Feb 10, 2012

### kai_sikorski

Sorry, not sure my response makes sense, edited it out. Will think about it some more.

Last edited: Feb 10, 2012
3. Feb 10, 2012

### kai_sikorski

Actually maybe what I said did make sense. So to be weakly compact any bounded sequence has to have a convergent sub-sequence from what I remember right? The sequence you defined is not bounded in Lp for p>1.

4. Feb 10, 2012

### muzialis

Thanks for replying, I am not sure I understand your reply. The sequence i mention is bounded in the L1 norm, and the example I mentioned, u_n = x ^(1/2) if x belongs to (0,1/n), 0 otherwise, is bounded in L2. But L2, on the contrary to L1, is weakly compact.

5. Feb 10, 2012

### kai_sikorski

"u_n (x) = n if x belongs to (0, 1/n), 0 otherwise"

So my point is that this is bounded in L1, and therefore it can be used as a counterexample to show L1 is not weakly compact.

However since it is not bounded in Lp for p>1, it can not be used as a counter example in those cases because the definition of weak compactness only puts restrictions on the behavior of bounded sequences.

6. Feb 10, 2012

### muzialis

Sorry, I was very unclear. I showed the example in L1, giving for granted that when considering the L2 case its analogue, mentioned in the attachment, would have considered.
The function u_n = x^(1/2) if x belongs to (0,1/n), 0 otherwise.
This is bounded in L2 and must converges to an element of L2, given that L2 is weakly compact.

But thank you anyhow, I think I discovered my silly mistake!

Bye


text:
# Existence of a subsequence in compact metric space

Let $$\{x_n\}$$ be a sequence in a compact metric space X such that:

1) For all $$n \in \mathbb{N}$$ the set $$\{ x_n : n \ge N\}$$ is closed.
2) The sequence $$\{x_n\}$$ is Cauchy.

Prove that there exists $$x \in X$$ and a subsequence $$\{x_{n_k}\}$$ such that $$x_{n_k} =x$$ for all $$k \in \mathbb{N}$$.

Since we are in a compact metric space, the Cauchy sequence is convergent to some $$x \in X$$ and as $$\{x_n\}$$ is closed that would mean that $$x= x_M$$ for some $$m\in \mathbb{N}$$. But I'm not sure how to extract a subsequence which is eventually constant.

As you note, we can take $$x$$ to be the limit of the sequence $$x_n$$, which we know exists since compact spaces are complete.
In order to prove that the desired subsequence exist, it suffices to show that infinitely many elements of the sequence satisfy $$x_n = x$$. Suppose (for the purpose of contradiction) that only finitely many elements of the sequence satisfy $$x_n = x$$. Then there exists an $$N$$ such that the set $$\{x_n : n \geq N\}$$ does not contain the element $$x$$. The sequence $$\{x_n\}_{n \geq N}$$ converges to $$x$$, so $$x$$ is a limit point of the set $$\{x_n : n \geq N\}$$. The fact that $$x \notin \{x_n : n \geq N\}$$ contradicts the premise that $$\{x_n : n \geq N\}$$ is closed.