text:
Datalog is a really cool technology because it is both declarative, intuitive and high performance. The core algorithms can be accomplished using database technology, which has become very refined over the years.

Datalog is a relative of prolog. Prolog and Datalog programs are roughly described as being made of Horn clauses, which can be considered a subset of first order logic.

Datalog as a theorem prover is discussed even less than prolog’s character as a theorem prover. The Nadathur and Miller book Programming with Higher Order Logic has an excellent discussion of the logical semantics of prolog. This book is a classic. It is so rich with insight, I’m hard pressed to think of another book that is its equal.

This blog post is trying to illuminate the class of formulas that Datalog or Datalog with light preprocessing is a prover for.

# Prolog

Prolog works via what can be described as top-down, backwards, or goal-directed inference, trying to construct a proof tree backwards from a given goal formula.

Prolog has a few pieces.

• Clauses or rules which consist of a head and body.
• A notion of query which you enter in at the repl. It’s easy to forget about this part, but it is kind of crucial for what follows.
• Unification variables, which are denoted by constants that start with a capitalized letter.

Prolog works by maintaining a goal list state which is initialized by the user given query. It matches (unifies) these goals against the heads of rules, and if successful replaces that goal with the body of the rule in the state. It can choose a rule that leads down a blind alley, so it maintains the ability to backtrack these choices and try another rule.

Prolog as described above can be seen as logical. :- is like implication. Unification variables are implicitly forall quantified in the program clauses and implicitly existentially quantified in the goal formula. But there is a slightly richer logical formula representation.

## Horn Clauses

One presentation of Horn clauses is (Section 2.3 of Nadathur and Miller)

G ::= True | A | G ∧ G | G ∨ G | ∃τ x G
D ::= A | G ⊃ D | D ∧ D | ∀x, D

When you give prolog a program clause file P, and state a query G in the repl it can be thought of as setting up a sequent P |- G for which you want prolog to find a proof tree.

An operational interpretation can be given to breaking down the goal formula in this sequents in prolog. Breaking down a compound goal is automatic and does not represent a backtracking step. These rules are invertible, meaning we have not committed to a possibly bad thing by taking them.

• to prove the goal P |- G1 /\ G1, prove P |- G1 and then P |- G2

In inference rule form this looks like ths right introduction rule

P |- G1           P |- G2
----------------------------------- /\ R
P |- G1 /\ G1

• to prove the goal P |- G1 \/ G1, try proving P |- G1 and if that doesn’t work try P |- G2. Ok this is kind of a backtracking step

• P |- exists X, G introduce a fresh unification X.

These unification variables introduced here are really metavariables, standing for an as yet undetermined actual term.

There is something a bit weird about not tracking existential metavariables in the sequent. They allow spooky action at a distance in the proof search, which can be disconcerting. Minikanren is more explicit on this point, purely functionally carrying a substitution dictionary.

Prolog for free will prove a stronger theorem than the one asked for. A metavariable in the answer substitution represents a universal quantification when all you asked for as an existential one. This is very cool.

A subtle point. If the answer is ungrounded, this is proving a universal quantification, but a universal only proves an existential if the sort in question is non empty, so actually it is returning a stronger theorem and a proof obligation for non emptiness. In the unityped setting this is less apparent.

# Harrop Clauses

There is an extension of Horn clauses called hereditary Harrop clauses which also have an operational character. Lambda prolog directly supports these. Enriching the set of formula you support is good.

G ::= True | A | G ∧ G | G ∨ G | ∃τ x G | D ⊃ G | ∀x G
D ::= A | G ⊃ D | D ∧ D | ∀x D

The additions compared to the above Horn clause structure is implication and forall are allowed in the goal formula. We now add a parameter S to the sequent to represent the signature.

• P ; S |- D -> G puts a new program clause into the database (D :: P) ; S |- G.

• P ; S |- forall x, G introduce a new fresh constant P ; (x :: S) |- G

We can see that implication and forall are related. In dependent type theory implication is just a forall where you ignore the argument in the rest of the type A -> B = forall x : A, B. Likewise bounded quantifiers $\forall_S x, P$ can be usefully expanded as $\forall x, S \rightarrow P$ when you are modeling this construct in a first order theorem prover.

The forall rule also gives us a principled way to generate well scoped fresh names. This interplays with lambda prolog’s ability to talk about lambda terms.

What’s really beautiful and powerful seeming is that the program and goal formulas are mutually recursively defined.

# Backchaining

The above rules break down the goal formula, but eventually it becomes atomic. What then? Then comes the backchaining phase where the logic program selects a program clause to resolve against the goal.

This has something to do with focusing

Prolog’s default depth first search strategy hurts it’s character as a theorem prover. There is much to be gained from this convention, as it increases it’s predictable character as a operational programming language.

Datalog is superior on this count. It breadth first searches through forward inference and cannot get caught digging down as dusty alley forever.

# Datalog

Datalog works via bottom up / forward inference.

What Datalog support.

• We can assert facts in the database likeedge(1,2)..
• We can assert rules like path(x,z) :- edge(x,y), path(y,z).. Operationally speaking, the body is a SQL-like query matched (not unified!) against the database, for which the query result is used to fill out the head. Because of how this works, every variable occurring in the head must be bound in the body. If it isn’t, what are you going to fill it out with? Well we can imagine some ideas (to be discussed later), but that is not stock Datalog.

# Bounded Horn Clauses

What is the set of formulas that Datalog can prove?

Datalog does not support Horn clauses. “Outrage!” you scream at me, readying to tear me apart limb from limb.

Datalog does not really support either of Horn or Harrop, but can handle restrictions.

The issue with horn clauses is the universal quantifier in D formula. In Datalog, it isn’t a full universal quantifier. It is more like a bounded quantifier which is a much weaker notion. What is kind of interesting is that the bound relation for the quantifier is mutually recursively defined, and not necessarily some a priori fixed range.

G ::= True | A | G ∧ G | G ∨ G | ∃τ x, G
D ::= A | G ⊃ D | D ∧ D | ∀x y z..:G, D

This is all basically non surprising. D formula can be expanded into a normal form expected by a typical datalog engine. Souffle datalog supports the disjunction operator in rule bodys as ; and multiheaded conjunction rules ,.

Goals correspond to reasonable queries you can make over a database. Existentials in G are variables in the query, and the query can be expanded into disjunctive normal form. You can also turn goals into rules via introducing a proven predicate. Then a goal G can be expanded into clauses producing thi predicate

// exists x, g(x)
proven() :- g(_x).

// G1 /\ G2
proven() :- g1, g2.

// G1 \/ G2
proven() :- g1.
proven() :- g2.
// or identically
proven() :- g1; g2.

Likewise the expansion of rules can be written a a set rewriting system

D1 /\ D2 :- G    --> {D2 :- G, D2 :- G}
D :- (G1 \/ G2)  --> {D :- G1, D :- G2}
D :- (exists x, G1) --> {D :- G1} // alpha renaming for any variable clashes
D :- (G1 /\ G2) --> {D :- G1, G2}

# Harrop-Lite

Ok, so how much of Harrop clauses can we do? Well, I’ve been on a wild goose chase on this one, but I think the following is the simple point in the design space.

We stratify goals into two kinds. One is in essence the same as it appears in horn clauses, which I will now call Q for queries and one contains the extra Harrop formula.

G ::= True | Q | G ∧ G | G ∨ G | ∃x G | D ⊃ G | ∀x G
D ::= A | | Q ⊃ D | D ∧ D | ∀x..:Q, D
Q ::= True | A | Q ∧ Q | Q ∨ Q | ∃x Q

• D -> G can be simply inverted by inserting the clause D into the datalog program.
• ∀x G can be dealt with by creating a fresh constant and instantiating it in the formula.

Perhaps less interesting and more optional are

• G /\ G is dealt with by forking the database into two process and waiting for both to come back true.
• G \/ G is dealt with by forking the database search into two processes and seeing if one comes back proven

Note that despite the richness of G, this set of formula is not mutually recursive. So it is weaker than Harrop clauses because the constant generating universal quantification can only happen in G formula and not Q.

One way of describing what is happening here is we are permitting 1 pass of goal breakdown as generative metaprogramming of a datalog program.

The D rule Q -> D is redundant as it represents a bounded quantifier that does not use it’s bound variables.

Edit: I glid right over ∃x G. It is not clear how to implement this. What if this captured variable is in a D? If it isn’t, it’s basically a ∃x Q. I suspect one way out of this conundrum is to insist on Q quantification here too. ∃x:Q, G. Fork the database every time there is a suggestion coming out of Q. Kind of ludicrous. Maybe here is an opportunity for magic set since this is where prolog would introduce a unification variable?

# Is that it?

No, but this is essentially the set of formula that can be operationally interpreted in normal stock datalog without too much agony. By agony, I mean with the true bulk search occurring in the datalog engine itself and where the proving process proceeds via generative metaprogramming and not deeper matching on the structure of the program clauses.

Going beyond that requires funky features or weird encodings and it appears thus far that the farther you go afield the harder the going is.

## Existentials

This is a really cool one actually. Can you add existentials to the heads of program clauses? It appears so. These are rules that are called tuple generating dependencies in the database community and are one of the subjects of the topic known as the chase. Some other topics to search for “Existential Datalog”, “datalog +/-“

G ::= Q | G ∧ G | G ∨ G | ∃x G | D ⊃ G | ∀x G
D ::= A | D ∧ D | ∀x..:PG, D | ∃x D
Q ::= True | A | Q ∧ Q | Q ∨ Q | ∃x Q

This can be encoded using gensym/autoinc counters to create fresh terms for the existentials, or datatypes to create skolem symbols, for which you probably get better performance and termination properties. Datatypes are a bit problematic because the skolem symbols aren’t persay not equal to other constants, but datatypes are. A slightly sticky point.

## Equality

Equality is arguably part of the formula A. This is a modelling choice of how you want to define your formula. Sometimes, it is useful to consider equality so important as to lift it out of the atoms and put it on the same level as /\ or ->. These are called equality generating dependencies. Equality in the queries Q is unproblematic until you add equality to D. Now the two interplay and you’ve got some challenges.

This is also the subject of the chase and egglog.

G ::= Q | G ∧ G | G ∨ G | ∃x G | D ⊃ G | ∀x G
D ::= A | t1 = t2 | D ∧ D | ∀x..:Q, D
Q ::= True | A | Q ∧ Q | Q ∨ Q | ∃x Q | t1 = t2

## Hypotheses / Contexts

Can Q ⊃ D becomes G ⊃ D? It’d be really nice. That would get us back to something that is closer to full Harrop clauses.

This is one of the extensions that is most tempting and still pretty cloudy how to do it right or even really what it is I even want. I currently have some reason to believe that I don’t want set-like contexts but instead something more like evaluation contexts.

Some terms for this are Contextual datalog, hypothetical datalog, or scoped datalog.

# Negation

Negation is a whole thing and has sent the logic programming into fits since the beginning. There are stories about how to do it right, but it’s nuanced.

Negation as failure

Many datalog’s support stratified negation. Under some situations, negation can be used to model swap and/or and forall/exists. This is a tricky topic.

# Disjunction

Disjunction in the head of rules (In the D formula) also requires more magumbo than a stock datalog allows. This is something like what answer set programming (notes) supports. ASP is something akin to slamming a datalog ad SAT solver together. See also DLV, disjunctive datalog

# Lambda Terms / Higher Order Logic

Lambda prolog of course has built in stuff for this. But, for datalog, I don’t even know, man.

Also how does nominal logic interact with datalog? Good, bad? Dunno. See alpha prolog and alpha kanren for nominal logic in the prolog like setting.

# Are formulas even the thing to be talking about

I actually kind personally de-emphasize this perspective. Instead I like the think of datalog/prolog rules as corresponding to inference rules (the horizontal line) and datalog/prolog predicates correspond to judgements. See these Pfenning notes.

Datalog is an engine for breadth first exploration of forward inferences. Every piece of knowledge that datalog asserts is justified. One can extract a judgemental proof tree for any fact derived by a datalog execution using some lightweight extra tagging. This is in distinction to traditional resolution based theorem provers or SMT solvers which have a much more classical bent. They can make guesses. Datalog has a more constructive feel from this perspective.

From a pragmatic standpoint, every datalog operation has an operational as well as logical character. Only inferring justifiable rules enables the datalog inference process to be more predictable and controllable in my experience than classical solvers. Perhaps as Zach put it “You can play computer”.

An alternative point made by Burak Emir is that Datalog should be thought of in terms of first order logic + least fixed points. I don’t get this one. I do understand that you can’t really express a transitive closure using only (finite?) first order axioms. Writing down path <- edge, path actually allows models that over approximate the closure, which sometimes is unacceptable. I think fol+lfp’s relation to datalog has something to do more with model checking rather than proof search, but I’m not sure. It definitely has something to do with descriptive complexity

# Bits and Bobbles

Alternative suggested names: Bounded Harrop formula, Non-recursive Harrop formula

Constrained horn clause solvers are model finders, not provers like above.

It seems plausible to me that harrop-lite can extended to a hierarchy that is still finite by deeper.

An interplay of a stronger prolog process as a metaprogram or subprogram of the datalog program could be an interesting thing. Souffle’s inline relatins can be seen as a weak terminating prolog metaprogram.

Contexts also have a hierarchy that may be useful. 1-contexts, 2-contexts, and so on and then sut them off. This is remininscent of how context is treated in abstract interpretation

Really a list of these, which represent the current leaves of a partially constructed proof tree.

G ::= Top | A | G ∧ G | G ∨ G | ∃τ x G
D ::= A | G ⊃ D | D ∧ D | ∀x:bounds, D

In a sense, the requirement of needing a G is

The truth values aren’t booleans. The two values are unknown and proven, not true and false. They are like Marshall B’s sierpinski trth values

CoInductive Sierp := Unsure Sierp | Proven.

def goldbach():
i = 4
while True:
i += 2 # even numbers
for j in range(i):
if isprime(j) and isprime(j - i):
return True
yield None

where G is a formula that grounds the variables being introduced. (Hmm. But is it ok tht G includes \/? The DNF expansion must include the variables being introduced. So undert more complicated conditions that A can become a The bounds can also be a primitive enumerator like range or nats. Using A is a bummer because forall x (b(x) \/ c(x)), q(x) is fine but we have to reformulate it as forall x: b(x), q(x) /\ forall x: c(x), q(x)

)

A bottom up technique

Each one of these additions is hard fought.

The chase skolemization egglog

A gensym or autoinc counter can be used to invent new symbols. Significantly better is to use skolem functions. These essentially record the reason by . What is weak or subtly undesirable about this is that creating a skolem function as a datatype is that the existential doesn’t guarantee a new thing. There may be something that already exists to satisfy the need of the existential. It also is somewhat implying that there is a unique element that satisfies the existential. This is why we can’t use egglog union to “delete” the skolem. It is not easy to tell when you can soundly assume the outputs of the skolem function are != from other types.

g and equality generating in the database community. They can be used to specify functional dependencies. a = b :- f(x,a), f(x,b). f(x,a) != :- a != b. // coinductive rule

Support higher order constructs. Curried rules can be supported in a different way via defining new predciates. There’s kind f a cute normal form for datalog programs of binary lcayses. Ths presuppose a query ordering, which maye be indesirable since a benefit of the datalog approach is to have big queries that the database has descided how to break up.

Egglog supports an intrinsic notion of equality. It is a presentational choice to decide whether equaliy is part of the logic or not. It is such an important notion it often is.

slog

# Example Rust Implementation for Egglog

https://github.com/mwillsey/egg-smol/pull/79 The Egraph data structure itself is holding P as rules and Sig as functions There are perhaps some seperated sig and fact data structures.

use crate::*;
impl EGraph {
fn calc_helper(
&mut self,
idents: Vec<IdentSort>,
exprs: Vec<Expr>,
depth: &mut i64,
) -> Result<(), Error> {
self.push();
*depth += 1;
// Insert fresh symbols for locally universally quantified reasoning.
for IdentSort { name, sort } in idents {
let sort = self.sorts.get(&sort).unwrap().clone();
self.declare_const(name, &sort)?;
}
// Insert each expression pair and run until they match.
for ab in exprs.windows(2) {
let a = &ab[0];
let b = &ab[1];
self.push();
*depth += 1;
self.eval_expr(a, None, true)?;
self.eval_expr(b, None, true)?;
let cond = Fact::Eq(vec![a.clone(), b.clone()]);
self.run_command(
Command::Run(RunConfig {
limit: 100000,
until: Some(cond.clone()),
}),
true,
)?;
self.run_command(Command::Check(cond), true)?;
self.pop().unwrap();
*depth -= 1;
}
self.pop().unwrap();
*depth -= 1;
Ok(())
}

// Prove a sequence of equalities universally quantified over idents
pub fn calc(&mut self, idents: Vec<IdentSort>, exprs: Vec<Expr>) -> Result<(), Error> {
if exprs.len() < 2 {
Ok(())
} else {
let mut depth = 0;
let res = self.calc_helper(idents, exprs, &mut depth);
if res.is_err() {
// pop egraph back to original state if error
for _ in 0..depth {
self.pop()?;
}
} else {
assert!(depth == 0);
}
res
}
}

fn run_query_formula(&mut self, goal: Query) -> Result<(), Error> {
match goal {
Query::Atom(fact) => {
if self.check_fact(&fact, true).is_err() {
println!("{}", self.summary());
// I should actually have run check until first if it doesn't already
self.run_command(
Command::Run(RunConfig {
limit: 100000,
until: Some(fact.clone()),
}),
true,
)?;
self.check_fact(&fact, true)
} else {
Ok(())
}
}
Query::And(goals) => {
for goal in goals {
self.run_query_formula(goal)?;
}
Ok(())
}
}
}
fn body_from_query(body: &mut Vec<Fact>, sig: &HashSet<Symbol>, g: Query) {
match g {
Query::Atom(f) => {
// TODO: check all pattern variables in f are in sig.
body.push(f)
}
Query::And(goals) => {
for goal in goals {
EGraph::body_from_query(body, sig, goal);
}
}
}
}
fn assert_prog_helper(
&mut self,
body: &mut Vec<Fact>,
sig: &mut HashSet<Symbol>,
prog: Prog,
) -> Result<(), Error> {
match prog {
Prog::Atom(action) => {
if body.is_empty() {
self.eval_actions(&[action])
} else {
body: body.clone(),
})?; // TODO: allow duplicate rule error?
Ok(())
}
}
Prog::And(progs) => {
for prog in progs {
self.assert_prog_helper(&mut body.clone(), &mut sig.clone(), prog)?;
}
Ok(())
}
Prog::ForAll(idents, prog) => {
for ident in idents {
assert!(!self.functions.contains_key(&ident.name));
// TODO: This should actually be shadowing.
assert!(!sig.contains(&ident.name));
}
self.assert_prog_helper(body, sig, *prog)
}
Prog::Implies(goal, prog) => {
EGraph::body_from_query(body, sig, goal);
self.assert_prog_helper(body, sig, *prog)
}
}
}

pub fn assert_prog(&mut self, prog: Prog) -> Result<(), Error> {
self.assert_prog_helper(&mut vec![], &mut Default::default(), prog)
}
fn reduce_goal(&mut self, goal: Goal) -> Result<(), Error> {
match goal {
Goal::ForAll(idents, goal) => {
for IdentSort { name, sort } in idents {
let sort = self.sorts.get(&sort).unwrap().clone();
self.declare_const(name, &sort)?;
}
self.reduce_goal(*goal)
}
Goal::Query(goal) => self.run_query_formula(goal),
Goal::Implies(prog, goal) => {
self.assert_prog(prog)?;
self.reduce_goal(*goal)
}
}
}
pub fn prove_goal(&mut self, goal: Goal) -> Result<(), Error> {
self.push();
let res = self.reduce_goal(goal);
self.pop().unwrap();
res
}
}
`


text:
# minimalPresentation(ZZ,LieAlgebra) -- compute a minimal presentation

## Description

The optional input given above is not relevant for Lie algebras. A minimal set of generators and relations for the Lie algebra $L$ (without differential) is given. In general the presentation applies to $H_0(L)$. The example $L$ below is the Lie algebra of strictly upper triangular $4\times 4$-matrices given by its multiplication table on the natural basis.

 i1 : L=lieAlgebra({e12,e23,e34,e13,e24,e14},Weights=>{1,1,1,2,2,3})/ {e12 e34,e12 e13,e12 e14, e23 e13,e23 e24,e23 e14, e34 e24,e34 e14,e13 e24, e13 e14,e24 e14, e12 e23 - e13, e12 e24 - e14, e13 e34 - e14, e23 e34 - e24} o1 = L o1 : LieAlgebra i2 : M=minimalPresentation(3,L) o2 = M o2 : LieAlgebra i3 : describe M o3 = generators => {e12, e23, e34} Weights => {{1, 0}, {1, 0}, {1, 0}} Signs => {0, 0, 0} ideal => {(e34 e12), (e34 e34 e23), (e23 e34 e23), (e23 e23 e12), (e12 e23 e12)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i4 : dims(1,4,M) o4 = {3, 2, 1, 0} o4 : List

format: plain


text:
# Mathematical Enigma - X

Number Theory Level 4

Find the sum of all integers $$n$$ for which $$n^{2}+n+41$$ is a perfect square.

×


text:
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If the maximum shear stress Fig. 7.5 Examples of Stress Paths . 7-8 This demonstrates that q is the same regardless of whether total stresses or effective Shear stress refers to a type of shear which makes the shearing surfaces to glidetoward each other Benchmarking for milling machine + example of the required

2/11/2015В В· Normal & Shear Stress Old Exam Question Lesson 3 - Shear Stress Example, Single and Double - Duration: Average Shear Stress and Simple Connections Statics and Mechanics of Materials Stress, вЂў The corresponding average shear stress is, Definition of stress.

Shear Stress and Shear Strain: When Ok. letвЂ™s take an example to understand the scenario. Shear Stress Rigidity is by definition вЂњThe ratio of the shear Shearing forces are unaligned forces pushing one part of a body in one specific direction, and another part of the body in the opposite direction.

Looking for online definition of shear in the Medical Dictionary? shear An example of a shearing force is seen when a patient slumps in shear stress. the Home В» SHEAR CENTRE -WITH EXAMPLES Assuming a uniform distribution of shear stress EXAMPLE вЂ“ 2. TO DETERMINE THE SHEAR CENTRE FOR THE SECTION SHOWN IN FIGURE:

Calculation of Bed Shear Stress. Reach-Averaged Method. Mean Bed Shear Stress - force per unit area exerted by a вЂњblockвЂќ of water on the channel boundary as it moves 29/01/2015В В· http://goo.gl/z20oaz for more FREE video tutorials covering Fluid Mechanics. In this video we are presented with an example of a 0.5m wide, 5m long belt

Pressure is an example of a normal stress, and acts inward, toward the surface, Definition of shear stress - Shear stress is defined as a force per unit area, Determination of shear stress Shear rate and viscosity are directly related to the properties of the fluid. Non-Newtonian fluids are governed by a non-linear

## Defining Yield Stress and Failure Stress (Strength)

Stress University of Sydney. Shear stress definition: the form of stress in a body, part, etc, that tends to produce cutting rather than... Meaning, pronunciation, translations and examples, Home В» SHEAR CENTRE -WITH EXAMPLES Assuming a uniform distribution of shear stress EXAMPLE вЂ“ 2. TO DETERMINE THE SHEAR CENTRE FOR THE SECTION SHOWN IN FIGURE:.

### Definition of Shear Stress Chegg.com

Defining Yield Stress and Failure Stress (Strength). Define Wall shear stress. English dictionary definition of Wall shear stress. n. in terms of the relationship of the wall shear stress to the wall shear, Chapter 1 Tension, Compression, and Shear Example 1-1 for a hollow definition of true stress "t = P / A.

Shear force definition: Shear force is force that makes one surface of a substance move over another parallel... Meaning, pronunciation, translations and examples Define shear stress. shear stress synonyms, shear stress pronunciation, shear stress translation, English dictionary definition of shear stress. n.

normal stress, shear stress, bulk stress, Lecture 10 (Elasticity) Example S2. A glass cube One book defines the shear stress $\tau$ of a (Newtonian) fluid as $$\tau = \eta \frac{\partial v}{\partial r}$$ where $\eta$ is the viscosity. There is not much

Shear Stress Definition - Shear Stress is the shear force per unit area acted on by shear force to sustain a constant rate of fluid movement. It... Statics and Mechanics of Materials Stress, вЂў The corresponding average shear stress is, Definition of stress.

Shear modulus: Shear modulus by definition, x/y. The shear modulus itself may be expressed mathematically as. shear modulus = (shear stress)/(shear strain) = Shear definition is an action or stress resulting from applied forces that causes or tends to cause two contiguous parts of a body Examples of shear in a

Shear stress definition, the external force acting on an object or surface parallel to the slope or plane in which it lies; the stress tending to produce shear. See more. Looking for online definition of shear stress in the Medical Dictionary? shear stress and self-doubt are all examples of conditions shear stress Shear.

Frequently it is necessary to calculate the normal and the shear stress on an Example. The stress state at a point is Plane stress. Definition of plane Looking for online definition of Wall shear stress in the Medical Dictionary? shear stress Shear. (for example, Rayz, 2010). Using

Determination of shear stress Shear rate and viscosity are directly related to the properties of the fluid. Non-Newtonian fluids are governed by a non-linear Define Wall shear stress. English dictionary definition of Wall shear stress. n. in terms of the relationship of the wall shear stress to the wall shear

Define shear stress. shear stress synonyms, shear stress pronunciation, shear stress translation, English dictionary definition of shear stress. n. Looking for online definition of shear in the Medical Dictionary? shear An example of a shearing force is seen when a patient slumps in shear stress. the

Shear stress: Shear stress, by definition, yield to shear stresses no matter how small the force per unit of area is called shear stress. For example, Define shear stress. shear stress synonyms, shear stress pronunciation, shear stress translation, English dictionary definition of shear stress. n.

Shear Stress and Shear Strain: When Ok. letвЂ™s take an example to understand the scenario. Shear Stress Rigidity is by definition вЂњThe ratio of the shear Here the concepts of stress analysis will be stated in a finite element context. That definition of the shear strain. For example, if

Yield Stress Definition uniaxial compression, shear, only a convenient approximation to the rigorous definition of yield stress in (1). In the examples the Define shear stress and strain. For example a strain of 0.000068 could be written as 68 x 10-6 but Shear stress is the force per unit area carrying

Frequently it is necessary to calculate the normal and the shear stress on an Example. The stress state at a point is Plane stress. Definition of plane BEAMS: BENDING STRESS by Dr. Ibrahim A. Assakkaf Definition A beam may be axial loading and shear stress due to

Chapter 1 Tension, Compression, and Shear Example 1-1 for a hollow definition of true stress "t = P / A The mathematical definition of stress is As with the last example В the cube, the stress ellipsoid is none of the infinite plans feel a shear stress,

Normal & Shear Stress Old Exam Question YouTube. One book defines the shear stress $\tau$ of a (Newtonian) fluid as $$\tau = \eta \frac{\partial v}{\partial r}$$ where $\eta$ is the viscosity. There is not much, normal stress, shear stress, bulk stress, Lecture 10 (Elasticity) Example S2. A glass cube.

### Wall shear stress definition of Wall shear stress by

Viscosity and Shear Stress Example Fluid Mechanics YouTube. Shear Stress and Shear Strain: When Ok. letвЂ™s take an example to understand the scenario. Shear Stress Rigidity is by definition вЂњThe ratio of the shear, A form of stress that subjects an object to which force is applied to skew, tending to cause shear strain. For example, shear stress on a block of wood would arise by.

What is Shear stress and Shear strain? ExtruDesign. Shear in Bending. Internal stresses a definition of shear force in a beam. as you might have guessed, shear stress it is not uniform throughout the cross-section., 3 Components of Stress On a real or imaginary plane through a material, there can be normal forces and shear forces. These forces create the stress.

### Shear stress definition and meaning Collins English

Shear force definition and meaning Collins English. Unlike shear force, Shear stress acts in a parallel to the surface. It causes one object to slip over another. Q&A related to Shear Force And Shear Stress. Shear stress definition: the form of stress in a body, part, etc, that tends to produce cutting rather than... Meaning, pronunciation, translations and examples.

• Vascular Wall Shear Stress Basic Principles and Methods
• What is Fluid Mechanics? mne.psu.edu
• Shear stress definition and meaning Collins English

• Mechanical Properties - Stresses & Strains By definition, What is the shear stress П„ along the direction m = 0, 1, Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam. Example. Considering a 2D space in cartesian

Chapter 1 Tension, Compression, and Shear Example 1-1 for a hollow definition of true stress "t = P / A Here the concepts of stress analysis will be stated in a finite element context. That definition of the shear strain. For example, if

The mathematical definition of stress is As with the last example В the cube, the stress ellipsoid is none of the infinite plans feel a shear stress, Shear stress is calculated by dividing the force exerted on an object by that object's How Do I Calculate Shear Stress? What Is the Definition of Internal

Define shear stress and strain. For example a strain of 0.000068 could be written as 68 x 10-6 but Shear stress is the force per unit area carrying Shear Stress and Shear Strain: When Ok. letвЂ™s take an example to understand the scenario. Shear Stress Rigidity is by definition вЂњThe ratio of the shear

This is the definition of stress What is the stress? Worked Example 2: Tensile, Compressive and Shear stress. normal stress, shear stress, bulk stress, Lecture 10 (Elasticity) Example S2. A glass cube

Statics and Mechanics of Materials Stress, вЂў The corresponding average shear stress is, Definition of stress. Pressure is an example of a normal stress, and acts inward, toward the surface, Definition of shear stress - Shear stress is defined as a force per unit area,

Pressure is an example of a normal stress, and acts inward, toward the surface, Definition of shear stress - Shear stress is defined as a force per unit area, Shear force definition: Shear force is force that makes one surface of a substance move over another parallel... Meaning, pronunciation, translations and examples

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text:
## Connecting To WiFi Networks

The Omega comes ready with a command-line tool called wifisetup that makes it easy to connect your Omega to WiFi networks. This article will cover what wifisetup is, as well as how you can use it to connect your Omega to a WiFi network near you.

### Using wifisetup

To begin setting up your connection to the Internet, enter wifisetup in a terminal, and you’ll see the following output:

root@Omega-2757:/# wifisetup
Onion Omega Wifi Setup

Select from the following:
1) Scan for Wifi networks
2) Type network info
q) Exit

Selection:

You can enter 1, and your Omega will scan for available networks:

Selection: 1
Scanning for wifi networks...

Select Wifi network:
1) BYB
2) studio sixteen
3) EG Energy
4) mayaaa
5) Authentic
6) OnionWiFi
7) Orpheus
8) Omega-18C2

Selection:

Enter your selection and you will be prompted for a password if required. Your network authentication type will be automatically detected in the scan:

Selection: 6
Network: OnionWiFi
Authentication type: WPA2PSK
Enter password:

Since the network adapter is restarting, the Omega’s AP will go down and be inaccessible during this period, but it will come back up in roughly 30 seconds.

The Omega’s Network Manager
The Omega’s network manager will attempt to the networks in range that are configured on your Omega. For example, if you have three configured networks (school, home, friend’s house) and you’re currently at your friend’s house, the network manager will immediately attempt to connect to that network.

On the other hand, if you have multiple configured networks in range the Omega will attempt to connect to the network with the highest priority. Priority is defined as most recently configured networks having the lowest priority. This means that adding a network configuration immediately assigns it as the lowest priority.

### Entering Network Info Manually

Alternatively, you can choose to type in the network info yourself by entering 2 as your selection.

Selection: 2
Enter network name:

Enter your network’s name (SSID) and hit enter

Selection: 2
Enter network name: OnionWiFi

Select network authentication type:
1) WPA2
2) WPA
3) WEP
4) None

Selection:

Select the network authentication type

Selection: 2
Enter network name: OnionWiFi

Select network authentication type:
1) WPA2
2) WPA
3) WEP
4) None

Selection: 1
Enter password:

Enter your password, and hit enter. Your Omega’s network adapter will restart, causing the AP to go down for roughly 30 seconds. Once your network adapter is back up, it will attempt to connect to the network.

The Omega’s Network Manager
The Omega’s network manager will attempt to the networks in range that are configured on your Omega. For example, if you have three configured networks (school, home, friend’s house) and you’re currently at your friend’s house, the network manager will immediately attempt to connect to that network.

On the other hand, if you have multiple configured networks in range the Omega will attempt to connect to the network with the highest priority. Priority is defined as most recently configured networks having the lowest priority. This means that adding a network configuration immediately assigns it as the lowest priority.

### Using the wifisetup Command-Line Arguments

You can also use wifisetup in the command line to directly add configurations without having to go through the user input method. To begin lets show the available usage for wifisetup with the following command:

wifisetup -h

Going through the available commands we see that we can add new networks, edit current networks, remove current networks, change the priority of networks, list the networks, and list the info about a specific network.

#### Adding or Editing a Network

To add or edit a network use the following:

wifisetup add -ssid <SSID> -encr <ENCRYPTION TYPE> -password <PASSWORD>
wifisetup edit -ssid <SSID> -encr <ENCRYPTION TYPE> -password <PASSWORD>

For example to add a network named myNetwork enter the following:

wifisetup add -ssid myNetwork -encr psk2 -password mynetworkpassword

And if you wanted to edit the network after adding it, you would enter:

wifisetup edit -ssid myNetwork -encr psk2 -password myNewNetworkPasswordWithCaps

#### Removing a Network

The command for removing a network takes in the ssid as the only parameter:

wifisetup remove -ssid <SSID>

From the above example, if you were to remove myNetwork from my list of configured networks you would enter the following

wifisetup remove -ssid myNetwork

Note if you have multiple networks with the same SSID, this command would remove the highest priority network from your list of configured networks.

#### Changing a Network’s Priority

The network priority determines the order in which the Omega’s network manager will attempt to connect, assuming that the network is within range. A network of higher priority will have a lower number. For example, your highest priority network has priority 1, and your Omega will attempt to connect to that network first if it is available.

You can use wifisetup priority -ssid <SSID> -move <up|down> to change the priority of a specified network.

For example, to move the priority of myNetwork up you would enter:

wifisetup priority -ssid myNetwork -move up

and to move it down you would enter:

wifisetup priority -ssid myNetwork -move down

### Disabling the WiFi Client Interface

To disable the Omega’s WiFi client interface, run the following commands:

uci set wireless.sta.disabled='1'
uci commit wireless
wifi

Note that the Omega’s WiFi Access Point will still be active. In this case, the Omega will be hosting a WiFi network, but will not attempt to connect to other networks.

To re-enable the WiFi client interface, run the same commands but instead set the wireless.sta.disabled option back to 0.

### Disabling the WiFi Radio Completely

The Omega’s entire WiFi radio can be disabled by running these commands:

uci set wireless.radio0.disabled='1'
uci commit wireless
wifi

This will entirely disable the WiFi radio, including both the WiFi AP and client interface. Users may want to do this to take advantage of the power savings that come from disabling the relatively power-hungry WiFi radio.

To re-enable the WiFi radio, run the same commands but instead set the wireless.radio0.disabled option back to 0.


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# How can I generate (suitably random) symplectic matrices?

I would like to write a computer script to generate a lot of symplectic matrices. How can I do this? Is there a parameterization of all symplectic matrices?

-

http://www-stat.stanford.edu/~cgates/PERSI/papers/what-is.pdf

-
Choose a subgroup that is easy to generate, say $Sp(2)$, and pick a random pair of coordinates $i < j$ and a random element in $Sp(2)$ spanning the subspace spanned by those two coordinates. This gives a markov chain analogous to the Kac random walk. It is known that this procedure converges. In fact if it measures the convergence rate in the transportation distance, then the rate is $n^2 \log n$.


text:
# 最完整的检测模型评估指标mAP计算指南(附代码)在这里！

2018.07.15 22:33 9771浏览

image

# 评估目标检测模型

1. 为什么是mAP?

2. 关于Ground Truth

image

image

image

03

# mAP含义及计算

image

1. IoU

IoU是预测框与ground truth的交集和并集的比值。这个量也被称为Jaccard指数，并于20世纪初由Paul Jaccard首次提出。为了得到交集和并集，我们首先将预测框与ground truth放在一起，如图所示。

image

image

image

2. 鉴别正确的检测结果并计算precision和recall

image

image

3. 计算mAP

mAP这个术语有不同的定义。此度量指标通常用于信息检索和目标检测领域。然而这两个领域计算mAP的方式却不相同。这里我们只谈论目标检测中的mAP计算方法。

For a given task and class, the precision/recall curve is computed from a method’s ranked output. Recall is defined as the proportion of all positive examples ranked above a given rank. Precision is the proportion of all examples above that rank which are from the positive class. The AP summarises the shape of the precision/recall curve, and is defined as the mean precision at a set of eleven equally spaced recall levels [0,0.1,…,1]:

image

The precision at each recall level r is interpolated by taking the maximum precision measured for a method for which the corresponding recall exceeds r:The intention in interpolating the precision/recall curve in this way is to reduce the impact of the “wiggles” in the precision/recall curve, caused by small variations in the ranking of examples.

image

Up until 2009 interpolated average precision (Salton and Mcgill 1986) was used to evaluate both classification and detection. However, from 2010 onwards the method of computing AP changed to use all data points rather than TREC-style sampling (which only sampled the monotonically decreasing curve at a fixed set of uniformly-spaced recall values 0, 0.1, 0.2,…, 1). The intention in interpolating the precision–recall curve was to reduce the impact of the ‘wiggles’ in the precision–recall curve, caused by small variations in the ranking of examples. However, the downside of this interpolation was that the evaluation was too crude to discriminate between the methods at low AP.

• mAP通常是在一个数据集上计算得到的。

• 虽然解释模型输出的绝对量化并不容易，但mAP作为一个相对较好的度量指标可以帮助我们。 当我们在流行的公共数据集上计算这个度量时，该度量可以很容易地用来比较目标检测问题的新旧方法。

• 根据训练数据中各个类的分布情况，mAP值可能在某些类（具有良好的训练数据）非常高，而其他类（具有较少/不良数据）却比较低。所以你的mAP可能是中等的，但是你的模型可能对某些类非常好，对某些类非常不好。因此，建议在分析模型结果时查看各个类的AP值。这些值也许暗示你需要添加更多的训练样本。

04

```# 按照置信度降序排序sorted_ind = np.argsort(-confidence)BB = BB[sorted_ind, :]   # 预测框坐标image_ids = [image_ids[x] for x in sorted_ind] # 各个预测框的对应图片id# 便利预测框，并统计TPs和FPsnd = len(image_ids)tp = np.zeros(nd)
fp = np.zeros(nd)for d in range(nd):
R = class_recs[image_ids[d]]
bb = BB[d, :].astype(float)
ovmax = -np.inf
BBGT = R['bbox'].astype(float)  # ground truth

if BBGT.size > 0:        # 计算IoU
# intersection
ixmin = np.maximum(BBGT[:, 0], bb[0])
iymin = np.maximum(BBGT[:, 1], bb[1])
ixmax = np.minimum(BBGT[:, 2], bb[2])
iymax = np.minimum(BBGT[:, 3], bb[3])
iw = np.maximum(ixmax - ixmin + 1., 0.)
ih = np.maximum(iymax - iymin + 1., 0.)
inters = iw * ih        # union
uni = ((bb[2] - bb[0] + 1.) * (bb[3] - bb[1] + 1.) +
(BBGT[:, 2] - BBGT[:, 0] + 1.) *
(BBGT[:, 3] - BBGT[:, 1] + 1.) - inters)

overlaps = inters / uni
ovmax = np.max(overlaps)
jmax = np.argmax(overlaps)    # 取最大的IoU
if ovmax > ovthresh:  # 是否大于阈值
if not R['difficult'][jmax]:  # 非difficult物体
if not R['det'][jmax]:    # 未被检测
tp[d] = 1.
R['det'][jmax] = 1    # 标记已被检测
else:
fp[d] = 1.    else:
fp[d] = 1.# 计算precision recallfp = np.cumsum(fp)tp = np.cumsum(tp)
rec = tp / float(npos)# avoid divide by zero in case the first detection matches a difficult# ground truthprec = tp / np.maximum(tp + fp, np.finfo(np.float64).eps)```

```def voc_ap(rec, prec, use_07_metric=False):
"""Compute VOC AP given precision and recall. If use_07_metric is true, uses
the VOC 07 11-point method (default:False).
"""
if use_07_metric:  # 使用07年方法
# 11 个点
ap = 0.
for t in np.arange(0., 1.1, 0.1):            if np.sum(rec >= t) == 0:
p = 0
else:
p = np.max(prec[rec >= t])  # 插值
ap = ap + p / 11.
else:  # 新方式，计算所有点
# correct AP calculation
# first append sentinel values at the end
mrec = np.concatenate(([0.], rec, [1.]))
mpre = np.concatenate(([0.], prec, [0.]))                # compute the precision 曲线值（也用了插值）
for i in range(mpre.size - 1, 0, -1):
mpre[i - 1] = np.maximum(mpre[i - 1], mpre[i])                # to calculate area under PR curve, look for points
# where X axis (recall) changes value
i = np.where(mrec[1:] != mrec[:-1])[0]                # and sum (\Delta recall) * prec
ap = np.sum((mrec[i + 1] - mrec[i]) * mpre[i + 1])    return ap```

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text:
findMax {DisimForMixed} R Documentation

## Calculate Distance Between given Attribute Values by considering only a pair of attributes.

### Description

Takes in two lists Ai and Aj, representing values of two attributes, two values x and y from Ai. Quantitative attributes are accepted only after descretization. Calculates distance between x and y for Aj with respect to Ai.

### Usage

findMax(Ai, Aj, x, y)

### Arguments

 Ai A list consisting values of a selected attribute Aj A list consisting values of another selected attribute x Value from Ai y Another value from Ai

### Details

findMax is the implementation of find_max() function proposed by Ahmad & Dey (2007). See Ahmad & Dey (2007) for more datails.

### Value

distance between x and y for Aj with respect to Ai.

### References

Ahmad, A., & Dey, L. (2007). A k-mean clustering algorithm for mixed numeric and categorical data. Data & Knowledge Engineering, 63(2), 503-527.

### Examples

Attrib_i <- c("A","B","A","C")
Attrib_j <- c("Q","Q","R","Q")
xVal <- "A"
yVal <- "B"
QualiVars <- data.frame(Qlvar1 = c("A","B","A","C"), Qlvar2 = c("Q","Q","R","Q"))
library(dplyr)
distBetXY <- findMax(Attrib_i,Attrib_j,xVal,yVal)

[Package DisimForMixed version 0.2 Index]


text:
# zbMATH — the first resource for mathematics

## Springer Proceedings in Mathematics & Statistics

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#### Authors / Editors

 10 Pardalos, Panos M. 7 Pinelas, Sandra 6 Dobrev, Vladimir K. 6 Kalyagin, Valery A. 6 Pinto, Alberto Adrego 6 Silvestrov, Sergei D. 5 Beilina, Larisa 5 Elaydi, Saber Nasr 5 Gonçalves, Patrícia C. 5 Terlaky, Tamás 5 Wiberg, Marie 4 Awrejcewicz, Jan 4 Bolt, Daniel M. 4 Mohapatra, Ram Narayan 4 Rančić, Milica 4 Ruzhansky, Michael V. 4 Sidoravicius, Vladas 4 Soares, Ana Jacinta 4 van der Ark, L. Andries 4 Wang, Wen-Chung 3 Agrawal, Purshottam Narain 3 Al-Baali, Mehiddin 3 Baklouti, Ali 3 Bohner, Martin J. 3 Culpepper, Steven Andrew 3 Cushing, Jim M. 3 Dasgupta, Ratan 3 Douglas, Jeffrey A. 3 Fasshauer, Gregory E. 3 Fuhrmann, Jürgen 3 González, Jorge A. 3 Gupta, Vijay 3 Kloeden, Peter Eris 3 Kotsireas, Ilias S. 3 Lanzarone, Ettore 3 Melnik, Roderick V. Nicholas 3 Molenaar, Dylan 3 Nathanson, Melvyn Bernard 3 Prokopyev, Oleg Alexan 3 Schumaker, Larry L. 3 Srivastava, Hari Mohan 3 Toni, Bourama 3 Zilberman, David 2 Abdi, Hervé 2 Acu, Ana Maria 2 Albeverio, Sergio A. 2 Almeida, Joao Paulo Andrade 2 Anastassiou, George Angelos 2 Antoniano-Villalobos, Isadora 2 Argiento, Raffaele 2 Bailey, David Harold 2 Băleanu, Dumitru I. 2 Banasiak, Jacek 2 Bobrowski, Adam 2 Byun, Jisoo 2 Cancès, Clément 2 Chakraborty, Mihir Kumar 2 Chen, Gui-Qiang G. 2 Chhetri, Maya 2 Deo, Naokant 2 Došlá, Zuzana 2 Došlý, Ondřej 2 Duman, Oktay 2 Durante, Daniele 2 Ebrahimi-Fard, Kurusch 2 Esposito Vinzi, Vincenzo 2 Ferrando, Juan Carlos 2 Fontana, Marco 2 Fournier-Prunaret, Danièle 2 Friz, Peter Karl 2 Garvan, Frank G. 2 Geroldinger, Alfred 2 Giri, Debasis 2 Glau, Kathrin 2 Grácio, Clara 2 Graef, John R. 2 Grandinetti, Lucio 2 Heim, Bernhard Ernst 2 Jaiani, George V. 2 Janssen, Rianne 2 Kallel, Sadok 2 Kar, Samarjit 2 Kauffman, Louis Hirsch 2 Kim, Byung Hak 2 Lachowicz, Mirosław 2 Lahiri, Soumendra Nath 2 Lambropoulou, Sofia S. F. 2 Lee, Hyunjin 2 Londoño, Jaime A. 2 Louzada, Francisco 2 Löwe, Matthias 2 Maciak, Matúš 2 Makarov, Roman N. 2 Malyarenko, Anatoliy A. 2 Manna, Santanu 2 Melão, Nuno 2 Melas, Viatcheslav B. 2 Migdalas, Athanasios 2 Miller, John James Henry 2 Millsap, Roger E. ...and 657 more Authors
all top 5

#### Fields

 311 General and overarching topics; collections (00-XX) 58 Statistics (62-XX) 52 Numerical analysis (65-XX) 51 Partial differential equations (35-XX) 41 Operations research, mathematical programming (90-XX) 39 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 38 Probability theory and stochastic processes (60-XX) 34 Biology and other natural sciences (92-XX) 30 Dynamical systems and ergodic theory (37-XX) 24 Number theory (11-XX) 22 Algebraic geometry (14-XX) 21 Quantum theory (81-XX) 20 Combinatorics (05-XX) 20 Ordinary differential equations (34-XX) 20 Computer science (68-XX) 18 Calculus of variations and optimal control; optimization (49-XX) 16 Commutative algebra (13-XX) 15 Operator theory (47-XX) 15 Differential geometry (53-XX) 15 Statistical mechanics, structure of matter (82-XX) 14 Difference and functional equations (39-XX) 14 Fluid mechanics (76-XX) 13 Nonassociative rings and algebras (17-XX) 13 Information and communication theory, circuits (94-XX) 12 Functional analysis (46-XX) 11 Systems theory; control (93-XX) 10 Several complex variables and analytic spaces (32-XX) 10 Mechanics of deformable solids (74-XX) 9 Associative rings and algebras (16-XX) 9 Group theory and generalizations (20-XX) 9 Global analysis, analysis on manifolds (58-XX) 8 History and biography (01-XX) 8 Topological groups, Lie groups (22-XX) 8 Approximations and expansions (41-XX) 8 Mechanics of particles and systems (70-XX) 6 Real functions (26-XX) 6 Harmonic analysis on Euclidean spaces (42-XX) 6 Convex and discrete geometry (52-XX) 5 Measure and integration (28-XX) 5 Special functions (33-XX) 5 Geometry (51-XX) 5 Optics, electromagnetic theory (78-XX) 5 Astronomy and astrophysics (85-XX) 4 Potential theory (31-XX) 3 Mathematical logic and foundations (03-XX) 3 General topology (54-XX) 3 Algebraic topology (55-XX) 3 Manifolds and cell complexes (57-XX) 2 Field theory and polynomials (12-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 $$K$$-theory (19-XX) 2 Functions of a complex variable (30-XX) 2 Abstract harmonic analysis (43-XX) 2 Relativity and gravitational theory (83-XX) 2 Geophysics (86-XX) 2 Mathematics education (97-XX) 1 General algebraic systems (08-XX) 1 Category theory; homological algebra (18-XX) 1 Sequences, series, summability (40-XX) 1 Integral equations (45-XX) 1 Classical thermodynamics, heat transfer (80-XX)

#### Citations contained in zbMATH Open

49 Publications have been cited 96 times in 92 Documents Cited by Year
Multiplicative ideal theory and factorization theory. Commutative and non-commutative perspectives. Selected papers based on the presentations at the meeting ‘Arithmetic and ideal theory of rings and semigroups’, Graz, Austria, September 22–26, 2014. Zbl 1346.13002
Chapman, Scott (ed.); Fontana, Marco (ed.); Geroldinger, Alfred (ed.); Olberding, Bruce (ed.)
2016
Large deviations and asymptotic methods in finance. Zbl 1322.60003
Friz, Peter K. (ed.); Gatheral, Jim (ed.); Gulisashvili, Archil (ed.); Jacquier, Antoine (ed.); Teichmann, Josef (ed.)
2015
Marshall-Olkin distributions – advances in theory and applications. Selected papers based on the presentations at the conference, Bologna, Italy, October 2–3, 2013. Zbl 1322.60002
Cherubini, Umberto (ed.); Durante, Fabrizio (ed.); Mulinacci, Sabrina (ed.)
2015
Set optimization and applications – the state of the art. From set relations to set-valued risk measures. Zbl 1335.49003
Hamel, Andreas H. (ed.); Heyde, Frank (ed.); Löhne, Andreas (ed.); Rudloff, Birgit (ed.); Schrage, Carola (ed.)
2015
Mathematical models of tumor-immune system dynamics. Proceedings of the international workshop, Sydney, Australia, January 7–10, 2013. Zbl 1303.92004
2014
Descriptive topology and functional analysis. In honour of Jerzy Kąkol’s 60th birthday. Proceedings of the 1st meeting in topology and functional analysis, Elche, Spain, September 27–28, 2013. Zbl 1297.46001
Ferrando, Juan Carlos (ed.); López-Pellicer, Manuel (ed.)
2014
Recent advances in delay differential and difference equations. Research papers based on the presentations at the international conference on delay differential and difference equations and applications, Balatonfüred, Hungary, July 15–19, 2013. Zbl 1297.34002
Hartung, Ferenc (ed.); Pituk, Mihály (ed.)
2014
Noncommutative Iwasawa main conjectures over totally real fields. Based on a workshop, Münster, Germany, April 25–30, 2011. Zbl 1253.11002
Coates, John (ed.); Schneider, Peter (ed.); Sujatha, R. (ed.); Venjakob, Otmar (ed.)
2013
Ergodic theory, open dynamics, and coherent structures. Proceedings of the workshop at the Banff International Research Station (BIRS), Banff, Canada, April 9–15, 2012. Zbl 1290.37001
Bahsoun, Wael (ed.); Bose, Christopher (ed.); Froyland, Gary (ed.)
2014
New directions in paraconsistent logic. Collected papers of the 5th world congress on paraconsistency, WCP, Kolkata, India, February 13–17, 2014. Zbl 1338.03003
Beziau, Jean-Yves (ed.); Chakraborty, Mihir (ed.); Dutta, Soma (ed.)
2015
New perspectives in partial least squares and related methods. Based on the presentation at the 7th partial least squares (PLS) conference, Houston, TX, USA, May 19–22, 2012. Zbl 1276.62007
Abdi, Hervé (ed.); Chin, Wynne W. (ed.); Esposito Vinzi, Vincenzo (ed.); Russolillo, Giorgio (ed.); Trinchera, Laura (ed.)
2013
Lie theory and its applications in physics. IX international workshop. Based on the 9th workshop on Lie theory and its applications in physics, Varna, Bulgaria, June 20–26, 2011. Zbl 1266.00027
2013
Descriptive topology and functional analysis. II. In honour of Manuel López-Pellicer mathematical work. Proceedings of the 2nd meeting in topology and functional analysis, Elche, Spain, June 7–8, 2018. Zbl 1419.46002
Ferrando, Juan Carlos (ed.)
2019
Nonlinear maps and their applications. Selected contributions from the NOMA 2013 international workshop, Zaragoza, Spain, September 3–4, 2013. Zbl 1309.37004
López-Ruiz, Ricardo (ed.); Fournier-Prunaret, Danièle (ed.); Nishio, Yoshifumi (ed.); Grácio, Clara (ed.)
2015
Beauville surfaces and groups. Proceedings of the conference, Newcastle, UK, June 7–9, 2012. Zbl 1316.14005
Bauer, Ingrid (ed.); Garion, Shelly (ed.); Vdovina, Alina (ed.)
2015
Monte Carlo and quasi-Monte Carlo methods 2012. Proceedings of the 10th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Sydney, Australia, February 13–17, 2012. Zbl 1282.65007
Dick, Josef (ed.); Kuo, Frances Y. (ed.); Peters, Gareth W. (ed.); Sloan, Ian H. (ed.)
2013
Fractional calculus. Selected papers from the international conference on fractional differentiation and its applications (ICFDA 2018), Amman, Jordan, July 16–18, 2018. Zbl 1444.26001
Agarwal, Praveen (ed.); Baleanu, Dumitru (ed.); Chen, YangQuan (ed.); Momani, Shaher (ed.); Tenreiro Machado, José António (ed.)
2019
Monte Carlo and quasi-Monte Carlo methods, MCQMC 2016. Proceedings of the 12th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Stanford, CA, August 14–19, 2016. Zbl 1400.65006
Owen, Art B. (ed.); Glynn, Peter W. (ed.)
2018
Singularities and foliations. Geometry, topology and applications. BMMS 2/NBMS 3, Salvador, Brazil, 2015. Proceedings of the 3rd singularity theory meeting, ENSINO, July 8–11, 2015 and the Brazil-Mexico 2nd meeting of singularities, July 13–17, 2015. Zbl 1392.58001
Araújo dos Santos, Raimundo Nonato (ed.); Menegon Neto, Aurélio (ed.); Mond, David (ed.); Saia, Marcelo J. (ed.); Snoussi, Jawad (ed.)
2018
Patterns of dynamics. In honour of Bernold Fiedler’s 60th birthday, Berlin, Germany, July 25–29, 2016. Zbl 1388.93001
Gurevich, Pavel (ed.); Hell, Juliette (ed.); Sandstede, Björn (ed.); Scheel, Arnd (ed.)
2017
Mathematical technology of networks. Bielefeld, December 2013. Selected papers based on the presentations at the conference, Bielefeld, Germany, December 4–7, 2013. Zbl 1342.94006
Mugnolo, Delio (ed.)
2015
Proceedings of the international conference on health care systems engineering, Milan, Italy, May 22–24, 2013. Zbl 1308.92007
Matta, Andrea (ed.); Li, Jingshan (ed.); Sahin, Evren (ed.); Lanzarone, Ettore (ed.); Fowler, John (ed.)
2014
Algebraic design theory and Hadamard matrices. ADTHM, Lethbridge, Alberta, Canada, July 8–11, 2014. Selected papers based on the presentations at the workshop and at the workshop on algebraic design theory with Hadamard matrices: applications, current trends and future directions, Banff International Research Station, Alberta, Canada, July 11–13, 2014. Zbl 1329.05003
Colbourn, Charles J. (ed.)
2015
Mathematical analysis and its applications. Proceedings of the international conference on recent trends in mathematical analyis and its applications, ICRTMAA 2014, Roorkee, India, December 21–23, 2014. Zbl 1331.00047
Agrawal, P. N. (ed.); Mohapatra, R. N. (ed.); Singh, Uaday (ed.); Srivastava, H. M. (ed.)
2015
Theory and applications of difference equations and discrete dynamical systems. Proceedings of the 19th international conference on difference equations and applications, ICDEA, Muscat, Oman, May 26–30, 2013. Zbl 1297.39001
AlSharawi, Ziyad (ed.); Cushing, Jim M. (ed.); Elaydi, Saber (ed.)
2014
Innovations in derivatives markets. Fixed income modeling, valuation adjustments, risk management, and regulation. Proceedings of the conference, Munich, Germany, March 30 – April 1, 2015. Zbl 1366.91005
Glau, Kathrin (ed.); Grbac, Zorana (ed.); Scherer, Matthias (ed.); Zagst, Rudi (ed.)
2016
Monte Carlo and quasi-Monte Carlo methods. MCQMC. Proceedings of the 11th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Leuven, Belgium, April 6–11, 2014. Zbl 1347.65003
Cools, Ronald (ed.); Nuyens, Dirk (ed.)
2016
Difference equations, discrete dynamical systems and applications, ICDEA, Barcelona, Spain, July 23–27, 2012. Proceedings of the 18th international conference. Zbl 1357.39001
Alsedà i Soler, Lluís (ed.); Cushing, Jim M. (ed.); Elaydi, Saber (ed.); Pinto, Alberto A. (ed.)
2016
Applied analysis in biological and physical sciences. ICMBAA, Aligarh, India, June 4–6, 2015. Zbl 1361.92003
Cushing, Jim M. (ed.); Saleem, M. (ed.); Srivastava, H. M. (ed.); Khan, Mumtaz Ahmad (ed.); Merajuddin, M. (ed.)
2016
Compositional data analysis. CoDaWork, L’Escala, Spain, June 2015. Selected contributions based on the presentations at the workshop. Zbl 1359.62020
Martín-Fernández, Josep Antoni (ed.); Thió-Henestrosa, Santiago (ed.)
2016
Recent developments in integrable systems and related topics of mathematical physics. Kezenoi-Am, Russia, 2016. Zbl 1412.37003
Buchstaber, Victor M. (ed.); Konstantinou-Rizos, Sotiris (ed.); Mikhailov, Alexander V. (ed.)
2018
Reality and measurement in algebraic quantum theory. NWW 2015, Nagoya, Japan, March 9–13, 2015. Selected papers based on presentations at the Nagoya winter workshop 2015: reality and measurement in algebraic quantum theory. Zbl 1408.81007
Ozawa, Masanao (ed.); Butterfield, Jeremy (ed.); Halvorson, Hans (ed.); Rédei, Miklós (ed.); Kitajima, Yuichiro (ed.); Buscemi, Francesco (ed.)
2018
Applied mathematics in Tunisia. International conference on advances in applied mathematics (ICAAM), Hammamet, Tunisia, December 16–19, 2013. Zbl 1325.00029
Jeribi, Aref (ed.); Hammami, Mohamed Ali (ed.); Masmoudi, Afif (ed.)
2015
Finite volumes for complex applications VIII – hyperbolic, elliptic and parabolic problems. FVCA 8, Lille, France, June 12–16, 2017. Zbl 1371.65001
Cancès, Clément (ed.); Omnes, Pascal (ed.)
2017
From particle systems to partial differential equations. Proceedings of the conference ‘Particle systems and partial differential equations I’, Braga, Portugal, December 5–7, 2012. Zbl 1290.35004
Bernardin, Cedric (ed.); Gonçalves, Patricia (ed.)
2014
Geometry and its applications. Selected papers based on the presentations at the 2nd international workshop on geometry and symbolic computation, Haifa, Israel, May 15–18, 2013. Zbl 1290.53002
Rovenski, Vladimir (ed.); Walczak, Paweł (ed.)
2014
Finite volumes for complex applications VII – elliptic, parabolic and hyperbolic problems. Proceedings of the FVCA 7, Berlin, Germany, June 15–20, 2014. Vol. II. Zbl 1291.65005
Fuhrmann, Jürgen (ed.); Ohlberger, Mario (ed.); Rohde, Christian (ed.)
2014
Modern problems of stochastic analysis and statistics. Selected contributions in honor of Valentin Konakov’s 70th birthday, Moscow, Russia, May 29 – June 2, 2016. Zbl 1387.60008
2017
Functional analysis in interdisciplinary applications, Astana, Kazakhstan, October 2–5, 2017. Zbl 1392.46001
Kalmenov, Tynysbek Sh. (ed.); Nursultanov, Erlan D. (ed.); Ruzhansky, Michael V. (ed.); Sadybekov, Makhmud A. (ed.)
2017
Dynamics of information systems: mathematical foundations. Zbl 1255.93007
Sorokin, Alexey (ed.); Murphey, Robert (ed.); Thai, My T. (ed.); Pardalos, Panos M. (ed.)
2012
Random matrices and iterated random functions. Selected papers based on the presentations at the workshop, Münster, Germany, October 4–7, 2011. Zbl 1272.60003
Alsmeyer, Gerold (ed.); Löwe, Matthias (ed.)
2013
Operator algebra and dynamics. Nordforsk network closing conference, Gjógv, Faroe Islands, May 2012. Zbl 1280.47001
Carlsen, Toke M. (ed.); Eilers, Søren (ed.); Restorff, Gunnar (ed.); Silvestrov, Sergei (ed.)
2013
Modeling and optimization. Theory and applications. Selected contributions from the MOPTA 2012 conference, Bethlehem, PA, USA, July 30 – August 1, 2012. Zbl 1279.90007
Zuluaga, Luis F. (ed.); Terlaky, Tamás (ed.)
2013
Advances in applied mathematics and approximation theory. Contributions from AMAT 2012 – International conference on applied mathematics and approximation theory, Ankara, Turkey, May 17–20, 2012. Zbl 1264.00039
Anastassiou, George A. (ed.); Duman, Oktay (ed.)
2013
Seki, founder of modern mathematics in Japan. A commemoration on his tercentenary. Collected papers of the international conference on the history of mathematics in memory of Seki Takakazu, Tokyo, Japan, August 25–31, 2008. Zbl 1266.01005
Knobloch, Eberhard (ed.); Komatsu, Hikosaburo (ed.); Liu, Dun (ed.)
2013
Computational and analytical mathematics. In Honor of Jonathan Borwein’s 60th birthday. Selected papers based on the presentations at the workshop, also known as JonFest, Simon Fraser University, BC, Canada, May 16–20, 2011. Zbl 1276.00018
Bailey, David H. (ed.); Bauschke, Heinz H. (ed.); Borwein, Peter (ed.); Garvan, Frank (ed.); Théra, Michel (ed.); Vanderwerff, Jon D. (ed.); Wolkowicz, Henry (ed.)
2013
Differential and difference equations with applications. Contributions from the international conference on differential and difference equations and applications in honour of Ravi P. Agarwal, Ponta Delgada, Portugal, July 4–8, 2011. Zbl 1277.00035
Pinelas, Sandra (ed.); Chipot, Michel (ed.); Dosla, Zuzana (ed.)
2013
Recent trends in dynamical systems. Proceedings of the international conference, Munich, Germany, January 11–13, 2012, in honor of Jürgen Scheurle on the occasion of his 60th birthday. Zbl 1278.37005
Johann, Andreas (ed.); Kruse, Hans-Peter (ed.); Rupp, Florian (ed.); Schmitz, Stephan (ed.)
2013
Monte Carlo and quasi-Monte Carlo methods 2010. Selected papers based on the presentations at the 9th international conference on Monte Carlo and quasi Monte Carlo in scientific computing (MCQMC 2010), Warsaw, Poland, August 15–20, 2010. Zbl 1252.65004
Plaskota, Leszek (ed.); Woźniakowski, Henryk (ed.)
2012
Descriptive topology and functional analysis. II. In honour of Manuel López-Pellicer mathematical work. Proceedings of the 2nd meeting in topology and functional analysis, Elche, Spain, June 7–8, 2018. Zbl 1419.46002
Ferrando, Juan Carlos
2019
Fractional calculus. Selected papers from the international conference on fractional differentiation and its applications (ICFDA 2018), Amman, Jordan, July 16–18, 2018. Zbl 1444.26001
Agarwal, Praveen; Baleanu, Dumitru; Chen, YangQuan; Momani, Shaher; Tenreiro Machado, José António
2019
Monte Carlo and quasi-Monte Carlo methods, MCQMC 2016. Proceedings of the 12th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Stanford, CA, August 14–19, 2016. Zbl 1400.65006
Owen, Art B.; Glynn, Peter W.
2018
Singularities and foliations. Geometry, topology and applications. BMMS 2/NBMS 3, Salvador, Brazil, 2015. Proceedings of the 3rd singularity theory meeting, ENSINO, July 8–11, 2015 and the Brazil-Mexico 2nd meeting of singularities, July 13–17, 2015. Zbl 1392.58001
Araújo dos Santos, Raimundo Nonato; Menegon Neto, Aurélio; Mond, David; Saia, Marcelo J.; Snoussi, Jawad
2018
Recent developments in integrable systems and related topics of mathematical physics. Kezenoi-Am, Russia, 2016. Zbl 1412.37003
Buchstaber, Victor M.; Konstantinou-Rizos, Sotiris; Mikhailov, Alexander V.
2018
Reality and measurement in algebraic quantum theory. NWW 2015, Nagoya, Japan, March 9–13, 2015. Selected papers based on presentations at the Nagoya winter workshop 2015: reality and measurement in algebraic quantum theory. Zbl 1408.81007
Ozawa, Masanao; Butterfield, Jeremy; Halvorson, Hans; Rédei, Miklós; Kitajima, Yuichiro; Buscemi, Francesco
2018
Patterns of dynamics. In honour of Bernold Fiedler’s 60th birthday, Berlin, Germany, July 25–29, 2016. Zbl 1388.93001
Gurevich, Pavel; Hell, Juliette; Sandstede, Björn; Scheel, Arnd
2017
Finite volumes for complex applications VIII – hyperbolic, elliptic and parabolic problems. FVCA 8, Lille, France, June 12–16, 2017. Zbl 1371.65001
Cancès, Clément; Omnes, Pascal
2017
Modern problems of stochastic analysis and statistics. Selected contributions in honor of Valentin Konakov’s 70th birthday, Moscow, Russia, May 29 – June 2, 2016. Zbl 1387.60008
2017
Functional analysis in interdisciplinary applications, Astana, Kazakhstan, October 2–5, 2017. Zbl 1392.46001
Kalmenov, Tynysbek Sh.; Nursultanov, Erlan D.; Ruzhansky, Michael V.; Sadybekov, Makhmud A.
2017
Multiplicative ideal theory and factorization theory. Commutative and non-commutative perspectives. Selected papers based on the presentations at the meeting ‘Arithmetic and ideal theory of rings and semigroups’, Graz, Austria, September 22–26, 2014. Zbl 1346.13002
Chapman, Scott; Fontana, Marco; Geroldinger, Alfred; Olberding, Bruce
2016
Innovations in derivatives markets. Fixed income modeling, valuation adjustments, risk management, and regulation. Proceedings of the conference, Munich, Germany, March 30 – April 1, 2015. Zbl 1366.91005
Glau, Kathrin; Grbac, Zorana; Scherer, Matthias; Zagst, Rudi
2016
Monte Carlo and quasi-Monte Carlo methods. MCQMC. Proceedings of the 11th international conference on ‘Monte Carlo and quasi-Monte Carlo methods in scientific computing’, Leuven, Belgium, April 6–11, 2014. Zbl 1347.65003
Cools, Ronald; Nuyens, Dirk
2016
Difference equations, discrete dynamical systems and applications, ICDEA, Barcelona, Spain, July 23–27, 2012. Proceedings of the 18th international conference. Zbl 1357.39001
Alsedà i Soler, Lluís; Cushing, Jim M.; Elaydi, Saber; Pinto, Alberto A.
2016
Applied analysis in biological and physical sciences. ICMBAA, Aligarh, India, June 4–6, 2015. Zbl 1361.92003
Cushing, Jim M.; Saleem, M.; Srivastava, H. M.; Khan, Mumtaz Ahmad; Merajuddin, M.
2016
Compositional data analysis. CoDaWork, L’Escala, Spain, June 2015. Selected contributions based on the presentations at the workshop. Zbl 1359.62020
Martín-Fernández, Josep Antoni; Thió-Henestrosa, Santiago
2016
Large deviations and asymptotic methods in finance. Zbl 1322.60003
Friz, Peter K.; Gatheral, Jim; Gulisashvili, Archil; Jacquier, Antoine; Teichmann, Josef
2015
Marshall-Olkin distributions – advances in theory and applications. Selected papers based on the presentations at the conference, Bologna, Italy, October 2–3, 2013. Zbl 1322.60002
Cherubini, Umberto; Durante, Fabrizio; Mulinacci, Sabrina
2015
Set optimization and applications – the state of the art. From set relations to set-valued risk measures. Zbl 1335.49003
Hamel, Andreas H.; Heyde, Frank; Löhne, Andreas; Rudloff, Birgit; Schrage, Carola
2015
New directions in paraconsistent logic. Collected papers of the 5th world congress on paraconsistency, WCP, Kolkata, India, February 13–17, 2014. Zbl 1338.03003
Beziau, Jean-Yves; Chakraborty, Mihir; Dutta, Soma
2015
Nonlinear maps and their applications. Selected contributions from the NOMA 2013 international workshop, Zaragoza, Spain, September 3–4, 2013. Zbl 1309.37004
López-Ruiz, Ricardo; Fournier-Prunaret, Danièle; Nishio, Yoshifumi; Grácio, Clara
2015
Beauville surfaces and groups. Proceedings of the conference, Newcastle, UK, June 7–9, 2012. Zbl 1316.14005
Bauer, Ingrid; Garion, Shelly; Vdovina, Alina
2015
Mathematical technology of networks. Bielefeld, December 2013. Selected papers based on the presentations at the conference, Bielefeld, Germany, December 4–7, 2013. Zbl 1342.94006
Mugnolo, Delio
2015
Algebraic design theory and Hadamard matrices. ADTHM, Lethbridge, Alberta, Canada, July 8–11, 2014. Selected papers based on the presentations at the workshop and at the workshop on algebraic design theory with Hadamard matrices: applications, current trends and future directions, Banff International Research Station, Alberta, Canada, July 11–13, 2014. Zbl 1329.05003
Colbourn, Charles J.
2015
Mathematical analysis and its applications. Proceedings of the international conference on recent trends in mathematical analyis and its applications, ICRTMAA 2014, Roorkee, India, December 21–23, 2014. Zbl 1331.00047
Agrawal, P. N.; Mohapatra, R. N.; Singh, Uaday; Srivastava, H. M.
2015
Applied mathematics in Tunisia. International conference on advances in applied mathematics (ICAAM), Hammamet, Tunisia, December 16–19, 2013. Zbl 1325.00029
Jeribi, Aref; Hammami, Mohamed Ali; Masmoudi, Afif
2015
Mathematical models of tumor-immune system dynamics. Proceedings of the international workshop, Sydney, Australia, January 7–10, 2013. Zbl 1303.92004
2014
Descriptive topology and functional analysis. In honour of Jerzy Kąkol’s 60th birthday. Proceedings of the 1st meeting in topology and functional analysis, Elche, Spain, September 27–28, 2013. Zbl 1297.46001
Ferrando, Juan Carlos; López-Pellicer, Manuel
2014
Recent advances in delay differential and difference equations. Research papers based on the presentations at the international conference on delay differential and difference equations and applications, Balatonfüred, Hungary, July 15–19, 2013. Zbl 1297.34002
Hartung, Ferenc; Pituk, Mihály
2014
Ergodic theory, open dynamics, and coherent structures. Proceedings of the workshop at the Banff International Research Station (BIRS), Banff, Canada, April 9–15, 2012. Zbl 1290.37001
Bahsoun, Wael; Bose, Christopher; Froyland, Gary
2014
Proceedings of the international conference on health care systems engineering, Milan, Italy, May 22–24, 2013. Zbl 1308.92007
Matta, Andrea; Li, Jingshan; Sahin, Evren; Lanzarone, Ettore; Fowler, John
2014
Theory and applications of difference equations and discrete dynamical systems. Proceedings of the 19th international conference on difference equations and applications, ICDEA, Muscat, Oman, May 26–30, 2013. Zbl 1297.39001
AlSharawi, Ziyad; Cushing, Jim M.; Elaydi, Saber
2014
From particle systems to partial differential equations. Proceedings of the conference ‘Particle systems and partial differential equations I’, Braga, Portugal, December 5–7, 2012. Zbl 1290.35004
Bernardin, Cedric; Gonçalves, Patricia
2014
Geometry and its applications. Selected papers based on the presentations at the 2nd international workshop on geometry and symbolic computation, Haifa, Israel, May 15–18, 2013. Zbl 1290.53002
2014
Finite volumes for complex applications VII – elliptic, parabolic and hyperbolic problems. Proceedings of the FVCA 7, Berlin, Germany, June 15–20, 2014. Vol. II. Zbl 1291.65005
Fuhrmann, Jürgen; Ohlberger, Mario; Rohde, Christian
2014
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text:
Precalculus

Review Exercises

PrecalculusReview Exercises

Review Exercises

Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval $[ 0,2π ). [ 0,2π ).$

1.

$csc 2 t=3 csc 2 t=3$

2.

$cos 2 x= 1 4 cos 2 x= 1 4$

3.

$2sinθ=−1 2sinθ=−1$

4.

$tanxsinx+sin( −x )=0 tanxsinx+sin( −x )=0$

5.

$9sinω−2=4 sin 2 ω 9sinω−2=4 sin 2 ω$

6.

$1−2tan(ω)= tan 2 (ω) 1−2tan(ω)= tan 2 (ω)$

For the following exercises, use basic identities to simplify the expression.

7.

$secxcosx+cosx− 1 secx secxcosx+cosx− 1 secx$

8.

$sin 3 x+ cos 2 xsinx sin 3 x+ cos 2 xsinx$

For the following exercises, determine if the given identities are equivalent.

9.

$sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x$

10.

$tan 3 x csc 2 x cot 2 xcosxsinx=1 tan 3 x csc 2 x cot 2 xcosxsinx=1$

Sum and Difference Identities

For the following exercises, find the exact value.

11.

$tan( 7π 12 ) tan( 7π 12 )$

12.

$cos( 25π 12 ) cos( 25π 12 )$

13.

$sin( 70 ∘ )cos( 25 ∘ )−cos( 70 ∘ )sin( 25 ∘ ) sin( 70 ∘ )cos( 25 ∘ )−cos( 70 ∘ )sin( 25 ∘ )$

14.

$cos( 83 ∘ )cos( 23 ∘ )+sin( 83 ∘ )sin( 23 ∘ ) cos( 83 ∘ )cos( 23 ∘ )+sin( 83 ∘ )sin( 23 ∘ )$

For the following exercises, prove the identity.

15.

$cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x$

16.

$cos(3x)− cos 3 x=−cosx sin 2 x−sinxsin(2x) cos(3x)− cos 3 x=−cosx sin 2 x−sinxsin(2x)$

For the following exercise, simplify the expression.

17.

$tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x ) tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x )$

For the following exercises, find the exact value.

18.

$cos( sin −1 ( 0 )− cos −1 ( 1 2 ) ) cos( sin −1 ( 0 )− cos −1 ( 1 2 ) )$

19.

$tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) ) tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) )$

Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

20.

Find $sin( 2θ ), cos( 2θ ), sin( 2θ ), cos( 2θ ),$ and $tan( 2θ ) tan( 2θ )$ given $cosθ=− 1 3 cosθ=− 1 3$ and $θ θ$ is in the interval $[ π 2 ,π ]. [ π 2 ,π ].$

21.

Find $sin( 2θ ), cos( 2θ ), sin( 2θ ), cos( 2θ ),$ and $tan( 2θ ) tan( 2θ )$ given $secθ=− 5 3 secθ=− 5 3$ and $θ θ$ is in the interval $[ π 2 ,π ]. [ π 2 ,π ].$

22.

$sin( 7π 8 ) sin( 7π 8 )$

23.

$sec( 3π 8 ) sec( 3π 8 )$

For the following exercises, use Figure 1 to find the desired quantities.

Figure 1
24.

25.

For the following exercises, prove the identity.

26.

$2cos( 2x ) sin( 2x ) =cotx−tanx 2cos( 2x ) sin( 2x ) =cotx−tanx$

27.

$cotxcos(2x)=−sin(2x)+cotx cotxcos(2x)=−sin(2x)+cotx$

For the following exercises, rewrite the expression with no powers.

28.

$cos 2 x sin 4 (2x) cos 2 x sin 4 (2x)$

29.

$tan 2 x sin 3 x tan 2 x sin 3 x$

Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

30.

$cos( π 3 )sin( π 4 ) cos( π 3 )sin( π 4 )$

31.

$2sin( 2π 3 )sin( 5π 6 ) 2sin( 2π 3 )sin( 5π 6 )$

32.

$2cos( π 5 )cos( π 3 ) 2cos( π 5 )cos( π 3 )$

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

33.

$sin( π 12 )−sin( 7π 12 ) sin( π 12 )−sin( 7π 12 )$

34.

$cos( 5π 12 )+cos( 7π 12 ) cos( 5π 12 )+cos( 7π 12 )$

For the following exercises, change the functions from a product to a sum or a sum to a product.

35.

$sin(9x)cos(3x) sin(9x)cos(3x)$

36.

$cos(7x)cos(12x) cos(7x)cos(12x)$

37.

$sin(11x)+sin(2x) sin(11x)+sin(2x)$

38.

$cos(6x)+cos(5x) cos(6x)+cos(5x)$

Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval $[ 0,2π ). [ 0,2π ).$

39.

$tanx+1=0 tanx+1=0$

40.

$2sin(2x)+ 2 =0 2sin(2x)+ 2 =0$

For the following exercises, find all exact solutions on the interval $[ 0,2π ). [ 0,2π ).$

41.

$2 sin 2 x−sinx=0 2 sin 2 x−sinx=0$

42.

$cos 2 x−cosx−1=0 cos 2 x−cosx−1=0$

43.

$2 sin 2 x+5sinx+3=0 2 sin 2 x+5sinx+3=0$

44.

$cosx−5sin( 2x )=0 cosx−5sin( 2x )=0$

45.

$1 sec 2 x +2+ sin 2 x+4 cos 2 x=0 1 sec 2 x +2+ sin 2 x+4 cos 2 x=0$

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $[0,2π). [0,2π).$ Round to four decimal places.

46.

$3 cot 2 x+cotx=1 3 cot 2 x+cotx=1$

47.

$csc 2 x−3cscx−4=0 csc 2 x−3cscx−4=0$

For the following exercises, graph each side of the equation to find the zeroes on the interval $[0,2π). [0,2π).$

48.

$20 cos 2 x+21cosx+1=0 20 cos 2 x+21cosx+1=0$

49.

$sec 2 x−2secx=15 sec 2 x−2secx=15$

Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.

50.
 $x x$ $y y$ $0 0$ $1 1$ $1 1$ $6 6$ $2 2$ $11 11$ $3 3$ $−6 −6$ $4 4$ $−1 −1$ $5 5$ $6 6$
51.
 $x x$ $y y$ $0 0$ $−2 −2$ $1 1$ $1 1$ $2 2$ $−2 −2$ $3 3$ $−5 −5$ $4 4$ $−2 −2$ $5 5$ $1 1$
52.
 $x x$ $y y$ $−3 −3$ $3+2 2 3+2 2$ $−2 −2$ $3 3$ $−1 −1$ $2 2 −1 2 2 −1$ $0 0$ $1 1$ $1 1$ $3−2 2 3−2 2$ $2 2$ $−1 −1$ $3 3$ $−1−2 2 −1−2 2$
53.

A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

54.

Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

For the following exercises, construct functions that model the described behavior.

55.

A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to $t, t,$ the month.

56.

Daily temperatures in the desert can be very extreme. If the temperature varies from $90°F 90°F$ to $30°F 30°F$ and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the following exercises, find the amplitude, frequency, and period of the given equations.

57.

$y=3cos(xπ) y=3cos(xπ)$

58.

$y=−2sin(16xπ) y=−2sin(16xπ)$

For the following exercises, model the described behavior and find requested values.

59.

An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to $t, t,$ the number of years from now.

60.

The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to $t, t,$ the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.

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Something half baked. Let $s$ be some integer, multiple of $3$, in the order of $2^{50}$. Let the set be triplets $(x,y,z)\in[0,s)^3$, with $N=s^3\approx2^{150}$ elements. Define f(x,y,z)=\begin{...