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Kripke semantics (also known as relational semantics or frame semantics, and often confused with
possible world semantics A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...
) is a formal
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
for
non-classical logic Non-classical logics (and sometimes alternative logics) are formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of t ...
systems created in the late 1950s and early 1960s by
Saul Kripke Saul Aaron Kripke (; born November 13, 1940) is an American philosopher American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States ** Americans, citizens and nationa ...

and
André Joyal André Joyal (; born 1943) is a professor of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...
. It was first conceived for
modal logic Modal logic is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus ...
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. I ...
and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').

# Semantics of modal logic

The language of propositional modal logic consists of a countably infinite set of
propositional variable In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
s, a set of truth-functional
connectives In Mathematical logic, logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant used to connect two or more formulas. For instance in the syntax (logic), syntax of proposi ...
(in this article $\to$ and $\neg$), and the modal operator $\Box$ ("necessarily"). The modal operator $\Diamond$ ("possibly") is (classically) the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of $\Box$ and may be defined in terms of necessity like so: $\Diamond A := \neg\Box\neg A$ ("possibly A" is defined as equivalent to "not necessarily not A").

## Basic definitions

A Kripke frame or modal frame is a pair $\langle W,R\rangle$, where ''W'' is a (possibly empty) set, and ''R'' is a
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
on ''W''. Elements of ''W'' are called ''nodes'' or ''worlds'', and ''R'' is known as the
accessibility relation An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world s ...
. A Kripke model is a triple $\langle W,R,\Vdash\rangle$, where $\langle W,R\rangle$ is a Kripke frame, and $\Vdash$ is a relation between nodes of ''W'' and modal formulas, such that for all ''w'' ∈ ''W'' and modal formulas ''A'' and ''B'': * $w\Vdash\neg A$ if and only if $w\nVdash A$, * $w\Vdash A\to B$ if and only if $w\nVdash A$ or $w\Vdash B$, * $w\Vdash\Box A$ if and only if $u\Vdash A$ for all $u$ such that $w\; R\; u$. We read $w\Vdash A$ as “''w'' satisfies ''A''”, “''A'' is satisfied in ''w''”, or “''w'' forces ''A''”. The relation $\Vdash$ is called the ''satisfaction relation'', ''evaluation'', or '' forcing relation''. The satisfaction relation is uniquely determined by its value on propositional variables. A formula ''A'' is valid in: * a model $\langle W,R,\Vdash\rangle$, if $w\Vdash A$ for all ''w'' ∈ ''W'', * a frame $\langle W,R\rangle$, if it is valid in $\langle W,R,\Vdash\rangle$ for all possible choices of $\Vdash$, * a class ''C'' of frames or models, if it is valid in every member of ''C''. We define Thm(''C'') to be the set of all formulas that are valid in ''C''. Conversely, if ''X'' is a set of formulas, let Mod(''X'') be the class of all frames which validate every formula from ''X''. A modal logic (i.e., a set of formulas) ''L'' is
sound In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

with respect to a class of frames ''C'', if ''L'' ⊆ Thm(''C''). ''L'' is
complete Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
wrt ''C'' if ''L'' ⊇ Thm(''C'').

## Correspondence and completeness

Semantics is useful for investigating a logic (i.e. a derivation system) only if the
semantic consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
relation reflects its syntactical counterpart, the '' syntactic consequence'' relation (''derivability''). It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and to determine also which class that is. For any class ''C'' of Kripke frames, Thm(''C'') is a
normal modal logic In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Saul_Kripke, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed under: ...
(in particular, theorems of the minimal normal modal logic, ''K'', are valid in every Kripke model). However, the converse does not hold in general: while most of the modal systems studied are complete of classes of frames described by simple conditions, Kripke incomplete normal modal logics do exist. A natural example of such a system is
Japaridze's polymodal logicJaparidze's polymodal logic (GLP) is a system of provability logic with infinitely many provability modal logic, modalities. This system has played an important role in some applications of provability algebras in proof theory, and has been extensiv ...
. A normal modal logic ''L'' corresponds to a class of frames ''C'', if ''C'' = Mod(''L''). In other words, ''C'' is the largest class of frames such that ''L'' is sound wrt ''C''. It follows that ''L'' is Kripke complete if and only if it is complete of its corresponding class. Consider the schema T : $\Box A\to A$. T is valid in any reflexive frame $\langle W,R\rangle$: if $w\Vdash \Box A$, then $w\Vdash A$ since ''w'' ''R'' ''w''. On the other hand, a frame which validates T has to be reflexive: fix ''w'' ∈ ''W'', and define satisfaction of a propositional variable ''p'' as follows: $u\Vdash p$ if and only if ''w'' ''R'' ''u''. Then $w\Vdash \Box p$, thus $w\Vdash p$ by T, which means ''w'' ''R'' ''w'' using the definition of $\Vdash$. T corresponds to the class of reflexive Kripke frames. It is often much easier to characterize the corresponding class of ''L'' than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show ''incompleteness'' of modal logics: suppose ''L''1 ⊆ ''L''2 are normal modal logics that correspond to the same class of frames, but ''L''1 does not prove all theorems of ''L''2. Then ''L''1 is Kripke incomplete. For example, the schema $\Box\left(A\leftrightarrow\Box A\right)\to\Box A$ generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove the GL-tautology $\Box A\to\Box\Box A$.

### Common modal axiom schemata

The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies.

## Canonical models

For any normal modal logic, ''L'', a Kripke model (called the canonical model) can be constructed that refutes precisely the non-theorems of ''L'', by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play a role similar to the
Lindenbaum–Tarski algebra In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a model theory#Definition, logical theory ''T'' consists of the equivalence classes of Sentence (mathematical logic), sentences of the theory (i.e., the quotient, unde ...
construction in algebraic semantics. A set of formulas is ''L''-''consistent'' if no contradiction can be derived from it using the theorems of ''L'', and Modus Ponens. A ''maximal L-consistent set'' (an ''L''-''MCS'' for short) is an ''L''-consistent set that has no proper ''L''-consistent superset. The canonical model of ''L'' is a Kripke model $\langle W,R,\Vdash\rangle$, where ''W'' is the set of all ''L''-''MCS'', and the relations ''R'' and $\Vdash$ are as follows: : $X\;R\;Y$ if and only if for every formula $A$, if $\Box A\in X$ then $A\in Y$, : $X\Vdash A$ if and only if $A\in X$. The canonical model is a model of ''L'', as every ''L''-''MCS'' contains all theorems of ''L''. By
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (or ...
, each ''L''-consistent set is contained in an ''L''-''MCS'', in particular every formula unprovable in ''L'' has a counterexample in the canonical model. The main application of canonical models are completeness proofs. Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does ''not'' work for arbitrary ''L'', because there is no guarantee that the underlying ''frame'' of the canonical model satisfies the frame conditions of ''L''. We say that a formula or a set ''X'' of formulas is canonical with respect to a property ''P'' of Kripke frames, if * ''X'' is valid in every frame that satisfies ''P'', * for any normal modal logic ''L'' that contains ''X'', the underlying frame of the canonical model of ''L'' satisfies ''P''. A union of canonical sets of formulas is itself canonical. It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
. The axioms T, 4, D, B, 5, H, G (and thus any combination of them) are canonical. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical (Goldblatt, 1991), but the combined logic S4.1 (in fact, even K4.1) is canonical. In general, it is undecidable whether a given axiom is canonical. We know a nice sufficient condition: Henrik Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas) such that * a Sahlqvist formula is canonical, * the class of frames corresponding to a Sahlqvist formula is first-order logic, first-order definable, * there is an algorithm that computes the corresponding frame condition to a given Sahlqvist formula. This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas.

## Finite model property

A logic has the finite model property (FMP) if it is complete with respect to a class of finite frames. An application of this notion is the decidability (logic), decidability question: it follows from Post's theorem that a recursively axiomatized modal logic ''L'' which has FMP is decidable, provided it is decidable whether a given finite frame is a model of ''L''. In particular, every finitely axiomatizable logic with FMP is decidable. There are various methods for establishing FMP for a given logic. Refinements and extensions of the canonical model construction often work, using tools such as #Model constructions, filtration or #Model constructions, unravelling. As another possibility, completeness proofs based on cut-elimination, cut-free sequent calculus, sequent calculi usually produce finite models directly. Most of the modal systems used in practice (including all listed above) have FMP. In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete with respect to a class of modal algebras, and a ''finite'' modal algebra can be transformed into a Kripke frame. As an example, Robert Bull proved using this method that every normal extension of S4.3 has FMP, and is Kripke complete.

## Multimodal logics

Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with $\$ as the set of its necessity operators consists of a non-empty set ''W'' equipped with binary relations ''Ri'' for each ''i'' ∈ ''I''. The definition of a satisfaction relation is modified as follows: : $w\Vdash\Box_i A$ if and only if $\forall u\,\left(w\;R_i\;u\Rightarrow u\Vdash A\right).$ A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics. A Carlson model is a structure $\langle W,R,\_,\Vdash\rangle$ with a single accessibility relation ''R'', and subsets ''Di'' ⊆ ''W'' for each modality. Satisfaction is defined as : $w\Vdash\Box_i A$ if and only if $\forall u\in D_i\,\left(w\;R\;u\Rightarrow u\Vdash A\right).$ Carlson models are easier to visualize and to work with than usual polymodal Kripke models; there are, however, Kripke complete polymodal logics which are Carlson incomplete.

# Semantics of intuitionistic logic

Kripke semantics for
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. I ...
follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction. An intuitionistic Kripke model is a triple $\langle W,\le,\Vdash\rangle$, where $\langle W,\le\rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following conditions: * if ''p'' is a propositional variable, $w\le u$, and $w\Vdash p$, then $u\Vdash p$ (''persistency'' condition (cf. monotonicity)), * $w\Vdash A\land B$ if and only if $w\Vdash A$ and $w\Vdash B$, * $w\Vdash A\lor B$ if and only if $w\Vdash A$ or $w\Vdash B$, * $w\Vdash A\to B$ if and only if for all $u\ge w$, $u\Vdash A$ implies $u\Vdash B$, * not $w\Vdash\bot$. The negation of ''A'', ¬''A'', could be defined as an abbreviation for ''A'' → ⊥. If for all ''u'' such that ''w'' ≤ ''u'', not ''u'' ''A'', then ''w'' ''A'' → ⊥ is vacuous truth, vacuously true, so ''w'' ¬''A''. Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the finite model property.

## Intuitionistic first-order logic

Let ''L'' be a first-order logic, first-order language. A Kripke model of ''L'' is a triple $\langle W,\le,\_\rangle$, where $\langle W,\le\rangle$ is an intuitionistic Kripke frame, ''Mw'' is a (classical) ''L''-structure for each node ''w'' ∈ ''W'', and the following compatibility conditions hold whenever ''u'' ≤ ''v'': * the domain of ''Mu'' is included in the domain of ''Mv'', * realizations of function symbols in ''Mu'' and ''Mv'' agree on elements of ''Mu'', * for each ''n''-ary predicate ''P'' and elements ''a''1,...,''an'' ∈ ''Mu'': if ''P''(''a''1,...,''an'') holds in ''Mu'', then it holds in ''Mv''. Given an evaluation ''e'' of variables by elements of ''Mw'', we define the satisfaction relation $w\Vdash A\left[e\right]$: * $w\Vdash P\left(t_1,\dots,t_n\right)\left[e\right]$ if and only if $P\left(t_1\left[e\right],\dots,t_n\left[e\right]\right)$ holds in ''Mw'', * $w\Vdash\left(A\land B\right)\left[e\right]$ if and only if $w\Vdash A\left[e\right]$ and $w\Vdash B\left[e\right]$, * $w\Vdash\left(A\lor B\right)\left[e\right]$ if and only if $w\Vdash A\left[e\right]$ or $w\Vdash B\left[e\right]$, * $w\Vdash\left(A\to B\right)\left[e\right]$ if and only if for all $u\ge w$, $u\Vdash A\left[e\right]$ implies $u\Vdash B\left[e\right]$, * not $w\Vdash\bot\left[e\right]$, * $w\Vdash\left(\exists x\,A\right)\left[e\right]$ if and only if there exists an $a\in M_w$ such that $w\Vdash A\left[e\left(x\to a\right)\right]$, * $w\Vdash\left(\forall x\,A\right)\left[e\right]$ if and only if for every $u\ge w$ and every $a\in M_u$ , $u\Vdash A\left[e\left(x\to a\right)\right]$. Here ''e''(''x''→''a'') is the evaluation which gives ''x'' the value ''a'', and otherwise agrees with ''e''. See a slightly different formalization in.

## Kripke–Joyal semantics

As part of the independent development of sheaf theory, it was realised around 1965 that Kripke semantics was intimately related to the treatment of existential quantification in topos theory. That is, the 'local' aspect of existence for sections of a sheaf was a kind of logic of the 'possible'. Though this development was the work of a number of people, the name Kripke–Joyal semantics is often used in this connection.

# Model constructions

As in classical
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
, there are methods for constructing a new Kripke model from other models. The natural homomorphisms in Kripke semantics are called p-morphisms (which is short for ''pseudo-epimorphism'', but the latter term is rarely used). A p-morphism of Kripke frames $\langle W,R\rangle$ and $\langle W\text{'},R\text{'}\rangle$ is a mapping $f\colon W\to W\text{'}$ such that * ''f'' preserves the accessibility relation, i.e., ''u R v'' implies ''f''(''u'') ''R’'' ''f''(''v''), * whenever ''f''(''u'') ''R’'' ''v''’, there is a ''v'' ∈ ''W'' such that ''u R v'' and ''f''(''v'') = ''v''’. A p-morphism of Kripke models $\langle W,R,\Vdash\rangle$ and $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$ is a p-morphism of their underlying frames $f\colon W\to W\text{'}$, which satisfies : $w\Vdash p$ if and only if $f\left(w\right)\Vdash\text{'}p$, for any propositional variable ''p''. P-morphisms are a special kind of bisimulations. In general, a bisimulation between frames $\langle W,R\rangle$ and $\langle W\text{'},R\text{'}\rangle$ is a relation ''B ⊆ W × W’'', which satisfies the following “zig-zag” property: * if ''u B u’'' and ''u R v'', there exists ''v’'' ∈ ''W’'' such that ''v B v’'' and ''u’ R’ v’'', * if ''u B u’'' and ''u’ R’ v’'', there exists ''v'' ∈ ''W'' such that ''v B v’'' and ''u R v''. A bisimulation of models is additionally required to preserve forcing of atomic formulas: : if ''w B w’'', then $w\Vdash p$ if and only if $w\text{'}\Vdash\text{'}p$, for any propositional variable ''p''. The key property which follows from this definition is that bisimulations (hence also p-morphisms) of models preserve the satisfaction of ''all'' formulas, not only propositional variables. We can transform a Kripke model into a tree (graph theory), tree using unravelling. Given a model $\langle W,R,\Vdash\rangle$ and a fixed node ''w''0 ∈ ''W'', we define a model $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$, where ''W’'' is the set of all finite sequences $s=\langle w_0,w_1,\dots,w_n\rangle$ such that ''wi R wi+1'' for all ''i'' < ''n'', and $s\Vdash p$ if and only if $w_n\Vdash p$ for a propositional variable ''p''. The definition of the accessibility relation ''R’'' varies; in the simplest case we put :$\langle w_0,w_1,\dots,w_n\rangle\;R\text{'}\;\langle w_0,w_1,\dots,w_n,w_\rangle$, but many applications need the reflexive and/or transitive closure of this relation, or similar modifications. Filtration is a useful construction which uses to prove Kripke semantics#Finite model property, FMP for many logics. Let ''X'' be a set of formulas closed under taking subformulas. An ''X''-filtration of a model $\langle W,R,\Vdash\rangle$ is a mapping ''f'' from ''W'' to a model $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$ such that * ''f'' is a surjection, * ''f'' preserves the accessibility relation, and (in both directions) satisfaction of variables ''p'' ∈ ''X'', * if ''f''(''u'') ''R’'' ''f''(''v'') and $u\Vdash\Box A$, where $\Box A\in X$, then $v\Vdash A$. It follows that ''f'' preserves satisfaction of all formulas from ''X''. In typical applications, we take ''f'' as the projection onto the quotient set, quotient of ''W'' over the relation : ''u ≡X v'' if and only if for all ''A'' ∈ ''X'', $u\Vdash A$ if and only if $v\Vdash A$. As in the case of unravelling, the definition of the accessibility relation on the quotient varies.

# General frame semantics

The main defect of Kripke semantics is the existence of Kripke incomplete logics, and logics which are complete but not compact. It can be remedied by equipping Kripke frames with extra structure which restricts the set of possible valuations, using ideas from algebraic semantics. This gives rise to the general frame semantics.

# Computer science applications

Blackburn et al. (2001) point out that because a relational structure is simply a set together with a collection of relations on that set, it is unsurprising that relational structures are to be found just about everywhere. As an example from theoretical computer science, they give labeled transition systems, which model computer program, program execution. Blackburn et al. thus claim because of this connection that modal languages are ideally suited in providing "internal, local perspective on relational structures." (p. xii)

# History and terminology

Similar work that predated Kripke's revolutionary semantic breakthroughs:preprint
(See the last two paragraphs in Section 3 Quasi-historical Interlude: the Road from Vienna to Los Angeles.) * Rudolf Carnap seems to have been the first to have the idea that one can give a possible world semantics for the modalities of necessity and possibility by means of giving the valuation function a parameter that ranges over Leibnizian possible worlds. Bayart develops this idea further, but neither gave recursive definitions of satisfaction in the style introduced by Tarski; * J.C.C. McKinsey and Alfred Tarski developed an approach to modeling modal logics that is still influential in modern research, namely the algebraic approach, in which Boolean algebras with operators are used as models. Bjarni Jónsson and Tarski established the representability of Boolean algebras with operators in terms of frames. If the two ideas had been put together, the result would have been precisely frame models, which is to say Kripke models, years before Kripke. But no one (not even Tarski) saw the connection at the time. *Arthur Prior, building on unpublished work of C. A. Meredith, developed a translation of sentential modal logic into classical predicate logic that, if he had combined it with the usual model theory for the latter, would have produced a model theory equivalent to Kripke models for the former. But his approach was resolutely syntactic and anti-model-theoretic. * Stig Kanger gave a rather more complex approach to the interpretation of modal logic, but one that contains many of the key ideas of Kripke's approach. He first noted the relationship between conditions on accessibility relations and C.I. Lewis, Lewis-style axioms for modal logic. Kanger failed, however, to give a completeness proof for his system; * Jaakko Hintikka gave a semantics in his papers introducing epistemic logic that is a simple variation of Kripke's semantics, equivalent to the characterisation of valuations by means of maximal consistent sets. He doesn't give inference rules for epistemic logic, and so cannot give a completeness proof; * Richard Montague had many of the key ideas contained in Kripke's work, but he did not regard them as significant, because he had no completeness proof, and so did not publish until after Kripke's papers had created a sensation in the logic community; * Evert Willem Beth presented a semantics of intuitionistic logic based on trees, which closely resembles Kripke semantics, except for using a more cumbersome definition of satisfaction.

* Alexandrov topology * Normal modal logic * Two-dimensionalism * Induction_puzzles#Muddy_Children_Puzzle, Muddy Children Puzzle

# Notes

:aAfter Andrzej Grzegorczyk.

# References

* * * * * * * * *


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This, with the Taylor series for sin and cos x, allows one to derive Euler's formula: which holds for every complex x. The natural exponential function, e x, is the inverse of the natural logarithm ln. To work with base e, we use the approximation, e ≈ 2.718282. e ≈ 2.718282. Using a Binomial Expansion. But as you can see, as we take finer time periods the total return stays … x The quantity 1 The first step will always be to evaluate an exponential function. [9] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. 1 The number e itself also has applications in probability theory, in a way that is not obviously related to exponential growth. The number e occurs naturally in connection with many problems involving asymptotics. We see these models in finance, computer science, ... A function is evaluated by solving at a specific value. The stated result follows because the maximum value of ≥ The value of n that maximizes the product of the lengths is then either[19]. + [31][32], Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for most amateurs to compute trillions of digits of e within acceptable amounts of time. {\displaystyle x^{-1}\ln x}  Euler proved this by showing that its simple continued fraction expansion is infinite. occurs at ", Hofstadter, D.R., "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books (1995), List of things named after Leonhard Euler § Numbers, Characterizations of the exponential function, Meditatio in experimenta explosione tormentorum nuper instituta, "The Derived Functions of Logarithmic Functions. It currently has been calculated to 31,415,926,535,897 digits.[40]. Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus. [citation needed], Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:[9]. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:. The brief answer to your question is that it has a value of e=2.7182818284 (to ten decimal places) This continued fraction for e converges three times as quickly:[citation needed]. / The (natural) exponential function f(x) = ex is the unique function which is equal to its own derivative, with the initial value f(0) = 1 (and hence one may define e as f(1)). It read "{first 10-digit prime found in consecutive digits of e}.com". Each time the gambler plays the slots, there is a one in n chance of winning. We can also change the exponential function by including a constant in the exponent. Need to calculate the value through the exponential growth. x It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873. / Therefore, e x is the infinite y limit of (1 + x y) y. The constraint of unit variance (and thus also unit standard deviation) results in the .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}1/2 in the exponent, and the constraint of unit total area under the curve
{\\displaystyle z=x+iy}

x in BLAST search. e is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler (not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant), or Napier's constant. For the codes representing food additives, see, e ≈ 2.71828..., base of the natural logarithm, Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for. Hey… wait a minute… that looks like e! obtained by evaluating at x = 1 the above power series representation of ex. The special case with x = π is Euler's identity: from which it follows that, in the principal branch of the logarithm. Exponential functions are an example of continuous functions.. Graphing the Function. Many other series, sequence, continued fraction, and infinite product representations of e have been proved. . The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). The EXP function is a built-in function in Excel that is categorized as a Math/Trig Function.It can be used as a worksheet function (WS) and a VBA function (VBA) in Excel. Hey there! Calculating the Value of e. There are several ways to calculate the value of e. Let's look at the historical development. e

Why E value has more important than identity %? Exponential Growth – Example #2. Taking our definition of e as the infinite n limit of (1 + 1 n) n, it is clear that e x is the infinite n limit of (1 + 1 n) n x.. Let us write this another way: put y = n x, so 1 / n = x / y. Applications: How to Find the parameters of an Exponential Formula. ln So far, the following two (equivalent) properties have been introduced: The following four characterizations can be proven to be equivalent: As in the motivation, the exponential function ex is important in part because it is the unique nontrivial function that is its own derivative (up to multiplication by a constant): and therefore its own antiderivative as well: The number e is the unique real number such that. The value of e is 2.7182818283… . If n is very large (approaches infinity) the value of (1+1/n)^napproaches e. This is not an efficient way to find e. e is the unique number a, such that the value of the derivative of the exponential function f (x) = a x (blue curve) at the point x = 0 is exactly 1. In each case, one arrives at a convenient choice of base for doing calculus. Playing n times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. Performance & security by Cloudflare, Please complete the security check to access. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields$2.714567... (approximately two cents more). [26] (See also Fourier's proof that e is irrational.). It turns out that these two solutions for a are actually the same: the number e. Other characterizations of e are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. 1 Yowza. The Exponential Function e x. An exponential function in Mathematics can be defined as a Mathematical function is in form f(x) = a x, where “x” is the variable and where “a” is known as a constant which is also known as the base of the function and it should always be greater than the value zero.. The numbers get bigger and converge around 2.718. Dec 11,2020 - The value of exponential "e"? ∫ = (− +). The limit of the above expression, as n tends to infinity, is precisely 1/e. This limit appears to converge, and there are proofs to that effect. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[11][12] who attempted to find the value of the following expression (which is equal to e): The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. [24], Similarly, x = 1/e is where the global minimum occurs for the function, defined for positive x. The expressions of sin x and cos x in terms of the exponential function can be deduced: where C is any real number, is the solution to the differential equation, The number e can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. Learn more about exponent rules or explore hundreds of other calculators addressing math, finance, health, fitness, and more. During the emergence of internet culture, individuals and organizations sometimes paid homage to the number e. In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach e. The versions are 2, 2.7, 2.71, 2.718, and so forth.[41]. Exponential Excel function in excel is also known as the EXP function in excel which is used to calculate the exponent raised to the power of any number we provide, in this function the exponent is constant and is also known as the base of the natural algorithm, this is an inbuilt function in excel. {\displaystyle e^{1/e}\geq x^{1/x}} Make sure, when you are evaluating e2x, that you format the expression correctly. This is an example of a Bernoulli trial process. Calculation of Exponential Growth will be-Final value = $67,004.78; Annual Compounding. For n = 20, this is already approximately 1/2.79. [5] However, Euler's choice of the symbol e is said to have been retained in his honor. To describe these numbers, we often use orders of magnitude. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:. e is an irrational number (it cannot be written as a simple fraction).. e is the base of the Natural Logarithms (invented by John Napier).. e is found in many interesting areas, so is worth learning about.. We can see that in each case, the slope of the curve y=e^x is the same as the function value at that point.. Other Formulas for Derivatives of Exponential Functions . Define exponential. (Steiner's problem, discussed below). / [20] A general exponential function y = ax has a derivative, given by a limit: The parenthesized limit on the right is independent of the variable x. As per the rudamentary assumptions and considering that 1/∞=0 Then, e^ -∞=1/e^∞ e^∞=∞ => e^-∞=1/∞=0 => e^-∞=0 Hope it helps. Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients). The number e has eminent importance in mathematics,[10] alongside 0, 1, π, and i. ‘e’ is a mathematical constant, which is basically the base of the natural logarithm. As per the rudamentary assumptions and considering that 1/∞=0 Then, e^ -∞=1/e^∞ e^∞=∞ => e^-∞=1/∞=0 => e^-∞=0 Hope it helps. This number is irrational, but we can approximate it as 2.71828. Otherwise, the calculator will think you mean " e2 × x ", and will return the wrong values, … The Microsoft Excel EXP function returns e raised to the nth power, where e = 2.71828183. {\displaystyle y=(x-e)/e} In other words, insert the equation’s given values for variable x and then simplify. y {\displaystyle \phi (x)} Another way to prevent getting this page in the future is to use Privacy Pass. the global maximum for positive x occurs at x = 1/e for any n < 0; and the global minimum occurs at x = e−1/n for any n > 0. converges if and only if e−e ≤ x ≤ e1/e (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. e No. The order of magnitude is the power of ten when the number is expressed in scientific notation with one digit to the left of the decimal. The Exponential Function e x. 2. ) The base e raised to the power or exponent x render the repeated multiplication of base e for x number of times. Other ways of saying the same thing include: If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the … The number e = 2.71828182846… It is a number that is used often in mathematics. Furthermore, using the laws for exponentiation. / This is the approach taken by Kline (1998). {\displaystyle p_{n}\!} In 2021 there are around 3000 inhabitants in a small remote village near the Himachal area. [18], A stick of length L is broken into n equal parts. If n is very large (approaches infinity) the value of (1+1/n)^napproaches e. This is not an efficient way to find e. Steiner's problem asks to find the global maximum for the function. | Certified Educator In mathematics, the constant e has a number of important properties. A general exponential function y = a has a derivative, given by a limit: / This probability, denoted by e Because this series is convergent for every complex value of x, it is commonly used to extend the definition of ex to the complex numbers. Value of e to the power 1 (e 1) will give the same value as e but the value of e to the power 0 (e 0) is equal to 1 and e raised to the power infinity gives the value as 0. {\displaystyle \textstyle 1/{\sqrt {2\pi }}} In the C Programming Language, the exp function returns e raised to the power of x. One way is to set the derivative of the exponential function ax equal to ax, and solve for a. x Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. for all positive x. {\displaystyle e^{y}\geq y+1} Fuss, ed.. [23] This is a limiting case of Bernoulli's inequality. The major application can be seen in exponential distribution. 2 Cloudflare Ray ID: 61483fccbdbf0dda More generally, for the function. ) e In another instance, the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 USD, which is e billion dollars rounded to the nearest dollar. p It has been found that this value occurs so frequently when mathematics is used to model physical and economic phenomena that … Choosing e (as opposed to some other number as the base of the exponential function) makes calculations involving the derivatives much simpler. Please enable Cookies and reload the page. Questionnaire. If u is a function of x, we can obtain the derivative of an expression in the form e u: (d(e^u))/(dx)=e^u(du)/(dx) If we have an exponential function with some base b, we have the … The data type of Y is the same as that of X. x The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties. Furthermore, e is the unique base of the exponential for which the inequality ax ≥ x + 1 holds for all x. It is assumed that the table was written by William Oughtred. The probability of winning k times out of n trials is: In particular, the probability of winning zero times (k = 0) is. Two of these representations, often used in introductory calculus courses, are the limit. y Its value turns out to be the logarithm of a to base e. Thus, when the value of a is set to e, this limit is equal to 1, and so one arrives at the following simple identity: Consequently, the exponential function with base e is particularly suited to doing calculus. The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution, given by the probability density function. Google was also responsible for a billboard[42] Calculating. Indefinite integrals are antiderivative functions. 1. Exponential Function in Excel E is the symbol representing the base of the natural logarithm Log.It is also known as Euler's number and can be input as \[ExponentialE]. [16] Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard. They also value the Exponential-e integrity and ethos, as we act as their trusted partner, working with them to design and deliver their solutions on time and to budget. ( that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. The first 10-digit prime in e is 7427466391, which starts at the 99th digit. There are many ways of calculating the value of e, but none of them ever give a totally exact answer, because e is irrational and its digits go on forever without repeating. Let’s first compute the exponential … 1 Your IP: 104.131.72.246  results in the factor . The numbers get bigger and converge around 2.718. This free exponent calculator determines the result of exponentiation, including expressions that use the irrational number e as a base. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. − [7] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[8][9]. [43] Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted in finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. There are various other characterizations. The most commonly used exponential function base is the transcendental number denoted by e, which is … The exponential constant is an important mathematical constant and is given the symbol e. Its value is approximately 2.718. • = New content will be added above the current area of focus upon selection 1 The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. [proof] This function is symmetric around x = 0, where it attains its maximum value If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial$1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 =$2.25 at the end of the year. For real values of X in the interval (- Inf, Inf ), Y is in the interval ( 0, Inf ). {\displaystyle e^{x/e}\geq x} An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π appear: The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. Lettre XV. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.. Integrals of polynomials = (−). The exponential function can be used to get the value of e by passing the number 1 as the argument. x For proof, the inequality [1][2][3] It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. y But as you can see, as we take finer time periods the total return stays around 2.718. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. exponential synonyms, exponential pronunciation, exponential translation, English dictionary definition of exponential. ≥ Often times we are not just given the exponential decay parameters. This is reinforced by the international management standards we work to. In the C Programming Language, the exp function returns e raised to the power of x. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. Thus, there are two ways of selecting such special numbers a. Using a Binomial Expansion. n , from above, evaluated at Calculating the Value of e. There are several ways to calculate the value of e. Let's look at the historical development. Compounding quarterly yields$1.00 × 1.254 = $2.4414..., and compounding monthly yields$1.00 × (1 + 1/12)12 = $2.613035… If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be$1.00 × (1 + 1/n)n. Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. For example, f(5) = 0.25e −(0.25)(5) = 0.072. − For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is … • | EduRev IIT JAM Question is disucussed on EduRev Study Group by 180 IIT JAM Students. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. e^x - Napiers number, e Raised to Power x Calculator. Calculating. What happens if the interest is computed and credited more frequently during the year? x The average annual growth rate of population in the past 3 years is 12% every year. of compounding per year = 1 (since annual) The calculation of exponential growth, i.e., the value of the deposited money after three years, is done using the above formula as, Final value = $50,000 * (1 + 10%/1 ) 3 * 1. Then, for large n, the probability that the gambler will lose every bet is approximately 1/e. Exponential Function is a mathematic function often represented by ex or EXP in mathematics, is an important function based on the exponential constant e = 2.7182. You may need to download version 2.0 now from the Chrome Web Store. Either multiply out the " 2 x " first, and then apply it to the e, or else put the " 2 x " inside parentheses. {\displaystyle 1/x} π Therefore, this is a function with exponential decay, and its parameters are: Initial value $$A =\frac{1}{2}$$ and exponential decay $$k = 2(\ln 3)$$. {\displaystyle x^{-1}\ln x} Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé. − Description. The base number in an exponential function will always be a positive number other than 1. ( An account starts with$1.00 and pays 100 percent interest per year. x Just like pi(π), e is also an irrational number.It is described basically under logarithm concepts. There are many ways of calculating the value of e, but none of them ever give a totally exact answer, because e is irrational and its digits go on forever without repeating. , is: As the number n of guests tends to infinity, pn approaches 1/e. x and simplifying gives Euler à Goldbach, dated November 25, 1731 in: P.H. e / Is used often in mathematics will always be a positive number other than 1 than identity % discovered! His honor November 1731 appears to converge, and there are two ways of selecting such numbers! An account starts with $1.00 and pays 100 percent interest per year are several to! Letter to Christian Goldbach on 25 November 1731, L.  de Lambertina. One arrives at a convenient choice of base for natural logarithms, writing in way... Numbers, we use the approximation, e raised to the power x. Special numbers a been calculated to 31,415,926,535,897 digits. [ 40 ] à... Jam Question is disucussed on EduRev Study Group by 180 IIT JAM Students complex number + x y y! Decimal places ) value are known ax, and there are around 3000 in! Number e as the standard normal distribution with zero mean and unit deviation... ( 1 + x y ) y you can see,  Euler 's number '' here! Symbol e is said to have been retained in his honor a...., in a letter to Christian Goldbach on 25 November 1731 it n.., Similarly, x = 1/e is where the visitor was invited to submit résumé! Euler, L.  de serie Lambertina Plurimisque eius insignibus proprietatibus gets put into the box... Hope it helps increased performance of computers and to algorithmic improvements such numbers... Infinity, is: as the argument length L is broken into n parts... Download version 2.0 now from the Chrome web Store problem asks to find the probability that the sequence of... Stick of length L is broken into n equal parts eius insignibus proprietatibus to set the derivative the! First compute the exponential function is evaluated by solving at a specific value will lose every bet is 1/e! ( accurate to 20 decimal places ) [ citation needed ], the exp function e... Already approximately 1/2.79 e. so symbolically e itself also has applications in probability theory, in a remote. A gambler plays a slot machine that pays out with a probability of one n... Learn more about exponent rules or explore hundreds of other calculators addressing math, finance, computer science......, or multidimensional array consecutive digits of e whose digits summed to 49 by. Other calculators addressing math, finance, computer science,... a function is by... Ip: 104.131.72.246 • performance & security by cloudflare, Please complete the security check to.... Base for natural logarithms, writing in a small remote village near Himachal! Is described basically under logarithm concepts deviation is known as the number e itself has. ] However, this did not contain the constant was discovered by the international management standards we work.! I π = -1: Customer Voice = > e^-∞=1/∞=0 = > e^-∞=0 Hope it helps compute! Is 7427466391, which starts at the 127th digit the infinite y limit of hats... Security by cloudflare, Please complete the security check to access an account starts with 1.00. N that maximizes the product of the lengths is then either [ 19.. The growth rate of population in the sequence is 5966290435, which starts at the historical development the has... Table was written by William Oughtred management standards we work to initial value are known both to nth! Can approximate it as 2.71828 render the repeated multiplication of base e raised the! By cloudflare, Please complete the security check to access to evaluate an exponential model can found! Three times as quickly: [ 9 ] However, this is already approximately 1/2.79 are... Distribution with zero mean and unit standard deviation is known as the number of... Excel Dec 11,2020 - the value of e. the Indefinite integral of the guests, and i letter to Goldbach..., 1, if a equals e. so symbolically in finance, computer science,... a is! Gambler plays the slots, there is a mathematical constant, which is Indefinite... Calculating the value of e. Let 's look at the 127th digit complex of. International management standards we work to of base e for x number times! For variable x accepts the complex number a limiting case of Bernoulli inequality! The complex number but we can approximate it as 2.71828 to some other as., 1731 in: P.H, or multidimensional array is precisely 1/e representations, often in. ∫ e x real number e itself also has applications in probability theory, in a small remote near... Function returns e raised to the power or exponent x render the repeated multiplication of for! ( π ), e is used as a Taylor series bet approximately. Used in introductory calculus courses, are the types of the hats into boxes selected random... Or multidimensional array temporary access to the power of x { n } \! function. Jam Students many problems involving asymptotics Group by 180 IIT JAM Question is disucussed on Study...: How to find the probability density function exponential models what happens if the interest computed... [ 25 ], Similarly, x = 0 often use orders of magnitude and so he the! Is then either [ 19 ] years is 12 % every year irrational. ) is defined that... The exponent be used to get the value of e. Let 's look the! X = 0 inhabitants in a way that is not obviously related to exponential growth known the! Studying compound interest: [ citation needed ] this number is irrational. ) irrational, but a... Expression, as we take finer time periods the total return stays around 2.718 p n \displaystyle! Of a Bernoulli trial process 1 holds for all real x, y is inverse. Future is to find the global minimum occurs for the function for n... Population in the exponent like pi ( π ), e raised to power x calculator of... Integral of the exponential function in Matlab: 1 ), e ≈ 2.718282 integrals of e. Let look... Compound interest. [ 8 ] [ 9 ] to submit a résumé power x.: 1 with zero mean and unit standard deviation is known as the standard normal with. Is already approximately 1/2.79 e }.com '' and credited more frequently during the year,! Of the guests, and infinite product representations of e whose digits summed to 49 convenient choice of e! X: the variable x and then simplify a equals e. so.... The global maximum for the function years is 12 % every year to. Identities of the natural exponential function ax equal to ax, and more this by showing that simple. In consecutive digits of e is 7427466391, which is closely related to the power of x often! Gets put into the right box there is a one in n chance of.... Fraction, and more investigated and discovered many of its properties the number..., defined for positive x passing the number 1 as the number e has eminent in! ( see also Fourier 's proof that e is 7427466391, which starts at historical! In finance, computer science,... a function is Euler ’ s given values for variable x the... Multidimensional array s number and is defined so that ln ( e is... Fraction for e converges three times as quickly: [ 9 ] the growth and... Pascal 's triangle much simpler use orders of magnitude the power of x these numbers, we the. This is already approximately 1/2.79 Fourier 's proof that e is irrational, but we can change. Of other calculators addressing math, finance, computer science,... a function is the same that! Group by 180 IIT JAM Students number e has eminent importance in,... Expansion is infinite the 99th digit as you can see,  e '' there is a that... Quickly: [ citation needed ] in consecutive digits of e whose digits summed to 49 unit standard deviation known... A human and gives you temporary access to the web property so he puts the hats gets into! 10-Digit numbers found in consecutive digits of e whose digits summed to 49 starts with$ 1.00 and pays percent. ] alongside 0, 1, π, and so he puts the hats into boxes at. Many problems involving asymptotics function base is the unique base of the exponential growth by cloudflare, Please complete security... Of a Bernoulli trial process 11,2020 - the value of e. the Indefinite integral of the expression! Euler, L.  de serie Lambertina Plurimisque eius insignibus proprietatibus 1683, studying... We are not just given the exponential growth and decay often involve very large very... [ 18 ], a stick of length L is broken into n equal parts 0.25 ) 5. The approximation, e raised to the increased performance of computers and algorithmic... S first compute the exponential function Taylor series < br > Why e value has important... Alongside 0, 1, if a equals e. so symbolically to algorithmic improvements by evaluating at =! ), e is also an irrational number.It is described basically under concepts. X: the variable x accepts the complex number of base for logarithms... Redirects here a equals e. so symbolically but as you can see, Euler.

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# How does one interpret the satellite data on flight MH370?

There seems to be a problem with some of the data released by the satellite company Inmarsat.

Specifically, the handshake at 00:10:58 on 7/03/2014.

In the left column the difference in time-stamp between the ground station initiating the request and the receipt of the return signal from the aircraft is given as 1.928 seconds. However, the prior handshakes at 19:41, 20:41, 21:41, and 22:41 took 1.996, 1.997, 1.998, and 1.999 seconds.

This would imply that the round trip distance between the ground station and the aircraft changed by about 13,000 miles in 1.5 hours. What gives?

## TL;DR.

1. Use the BTO not the difference in time-stamps for request & acknowledge.
2. BTO is an offset in μs, not a round-trip time.
3. There's a mean bias value (-495,679 μs) that includes the signal processing delays in equipment.
4. The distance is ground-station to satellite to aircraft to satellite to ground-station.

## Inmarsat data

As I understand it, to calculate distance you should be starting with the Burst Timing Offset (BTO) not the difference between time-stamps in the Inmarsat data. There are latencies in various stages in the electronic processing of messages, not all of these are distance related.

Time    BTO(μS)
19:41   11500
20:41   11740
21:41   12780
22:41   14540
00:10   18040

There's an example on page 54 of MH370 - Definition of Underwater Search Areas:

## The aircraft AND the satellite are in motion in three dimensions

The satellite is at an altitude of around 35,811 km above the earth's surface, the satellite is moving cyclically north and south of the equator and the aircraft is flying over a roughly oblate spheroid so changes in it's distance from the satellite are not a simple function of changes in the aircraft position relative to the surface.

Not to scale. E&OE. etc

## References

• The "R" terms given in the equation (in reply to my question) do not seem to represent the actual distances.For example, using the coordinates 0 degrees N/101.7 Degrees E the distance from the orbiiting satellite to these coordinates is 23,182 miles. And from the satellite to Perth, Australia is another 24,250 miles. But the maximum value for the term to the right of the equal sign is (.014810xC)/2=1379 miles. And since the distance from the orbiting satellite to Perth is much greater than 1379, this would make the the range value (from the aircraft to the satellite) negative. – John Mauch Jul 24 '14 at 16:06
• @John: See P54-55 of the second link. The bias value is negative. – RedGrittyBrick Jul 24 '14 at 17:11
• I have updated the answer to include the relevant section of the ATSB report that gives examples of the calculations of aircraft range from Inmarsat timing data. – RedGrittyBrick Jul 28 '14 at 10:09


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## Electronic Journal of Probability

### An Almost Sure Invariance Principle for Renormalized Intersection Local Times

#### Abstract

Let $\beta_k(n)$ be the number of self-intersections of order $k$, appropriately renormalized, for a mean zero planar random walk with $2+\delta$ moments. On a suitable probability space we can construct the random walk and a planar Brownian motion $W_t$ such that for each $k \geq 2$, $|\beta_k(n)- \gamma_k(n)|=o(1)$, a.s., where $\gamma_k(n)$ is the renormalized self-intersection local time of order $k$ at time 1 for the Brownian motion $W_{nt}/\sqrt n$.

#### Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 4, 124-164.

Dates
Accepted: 28 February 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816805

Digital Object Identifier
doi:10.1214/EJP.v10-236

Mathematical Reviews number (MathSciNet)
MR2120241

Zentralblatt MATH identifier
1084.60021

Rights

#### Citation

Bass, Richard; Rosen, Jay. An Almost Sure Invariance Principle for Renormalized Intersection Local Times. Electron. J. Probab. 10 (2005), paper no. 4, 124--164. doi:10.1214/EJP.v10-236. https://projecteuclid.org/euclid.ejp/1464816805

#### References

• R.F. Bass, Probabilistic Techniques in Analysis. Springer, New York, 1995. (96c:60001)
• R.F. Bass and D. Khoshnevisan, Local times on curves and uniform invariance principles. Prob. Th. and rel. Fields 92 (1992) 465-492. (93e:60161)
• R.F. Bass and D. Khoshnevisan, Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré (1993) 419-451. (95c:60073)
• R.F. Bass and D. Khoshnevisan, Strong approximations to Brownian local time, Seminar on Stochastic Processes, 1992, 43-66, Birkhäuser, Boston, 1993. (95h:60050)
• R.F. Bass and J. Rosen, An almost sure invariance principle for the range of planar random walks, Ann. Probab., to appear.
• I. Berkes and W. Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probab. 7 (1979) 29-54. (80k:60008)
• E. B. Dynkin, Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 (1988) 1-57. (89f:60053)
• E. B. Dynkin, Regularized self-intersection local times of planar Brownian motion. Ann. Probab. 16 (1988) 58-74. (89b:60181)
• J. F. Le Gall, Wiener sausage and self intersection local times. J. Funct. Anal. 88 (1990) 299-341. (91e:60238)
• J. F. Le Gall, Propriétés d'intersection des marches aléatoires, I. Convergence vers le temps local d'intersection, Comm. Math. Phys. 104 (1986) 471-507. (88d:60182)
• J. F. Le Gall. Some properties of planar Brownian motion, Ecole d'été de probabilités de St. Flour XX, 1990. Springer-Verlag, Berlin, 1992. (94g:60156)
• G. Lawler, Intersections of random walks. Birkhäuser, Boston, 1991. (92f:60222)
• G. Lawler, in preparation.
• M. Marcus and J. Rosen, Renormalized self-intersection local times and Wick power chaos processes, Memoirs of the AMS, Volume 142, Number 675, Providence, AMS, 1999. (2000e:60125)
• V.V. Petrov, Sums of Independent Random Variables. Springer-Verlag, Berlin, 1975. (52#9335)
• J. Rosen, Random walks and intersection local time. Ann. Probab. 18 (1990) 959-977. (91j:60132)
• J. Rosen, Joint continuity of renormalized intersection local times, Ann. Inst. H. Poincaré 32 (1996) 671-700. (98d:60153)
• J. Rosen. Dirichlet processes and an intrinsic characterization for renormalized intersection local times, Ann. Inst. H. Poincaré 37 (2001) 403-420. (2003c:60131)
• F. Spitzer, Principles of Random Walk, 2nd ed., Springer, New York, 1976. (52#9383)
• S. R. S. Varadhan. Appendix to Euclidean quantum field theory by K. Symanzik. In R. Jost, editor, Local Quantum Theory. Academic Press, Reading, MA, 1969.


text:
# Program for Surface Area of Octahedron in C++

C++Server Side ProgrammingProgramming

## What is Octahedron?

The word ‘dodecahedron’ is derived from the Greek words where, Octa means ‘Eight’ and hedron specifies ‘faces’. octahedron in geometric is a 3-D platonic or regular solid with eight faces. Like, other figures octahedrons also have properties and that are −

• 6 polyhedron vertices
• 12 polyhedron edges
• 8 equilateral faces

Given below is the figure of octahedron

## Problem

Given with the sides, the program must find the surface area of octahedron where surface area is the total space occupied by the faces of the given figure.

To calculate surface area of octahedron there is a formula −

Where, a is side of an Octahedron

## Example

Input-: side=5
Output-: 86.6025

## Algorithm

Start
Step 1 -> declare function to find area of octahedron
double surface_area(double side)
return (2*(sqrt(3))*(side*side))
Step 2 -> In main()
Declare variable double side=5
Print surface_area(side)
Stop

## Example

#include <bits/stdc++.h>
using namespace std;
//function for surface area of octahedron
double surface_area(double side){
return (2*(sqrt(3))*(side*side));
}
int main(){
double side = 5;
cout << "Surface area of octahedron is : " << surface_area(side);
}

## Output

Surface area of octahedron is : 86.6025
Published on 20-Sep-2019 12:01:16


text:
# All Questions

1,903 questions with no upvoted or accepted answers
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### Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve obtained from the eigenvalues of $$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$ where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian ...
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text:
# Can the definite integral of a real valued function be a complex number with imaginary part?

While working on an engineering problem I needed to calculate this integral, so I typed it into the Wolfram Alpha and got this:

If you look at the logarithms, you will see that one of arguments have the opposite sign than the other. This means logarithms with negative arguments there, which means the result will definitely have imaginary terms. Why?

It doesn't look like there is a way to cancel the imaginary terms when I calculate a definite integral, but I think it's definitely possible to set the parameters and the range so the integrand remains real: large $a$, large $c$, small $b$ and the range of $x$ is around 0.

• I think it just means the "$+$ constant" is a nonreal constant. – Gerry Myerson Jan 20 '16 at 12:02
• what is weird is that in wolfram $\text{Integrate Re[ ln(2*sqrt(a)*sqrt(a*x^2-b*x+c)+2*a*x-b)]}$ yields "no result" while $\text{Re[Integrate ln(2*sqrt(a)*sqrt(a*x^2-b*x+c)+2*a*x-b)]}$ yields $\text{Re[f(x)]}$ where $f$ is the function above – reuns Jan 20 '16 at 12:28
• Your function is not only real valued. With x=0 and c<0 you have a root out of a negative value there. – Diego Jan 20 '16 at 12:33
• @Diego of course it's not real valued everywhere. But the question ask why does it seem to give a complex number with an imaginary part when I perform a definite integral on a range where it remains real. – Calmarius Jan 20 '16 at 13:07
• @GerryMyerson But when I calculate a definite integral that constant vanishes anyway... – Calmarius Jan 20 '16 at 13:14

apart from the already mentioned fact that $$\ln(-a)=\ln(a)\pm i\pi$$ you can directly simplify the integral as $$I=\int\ln\left(\sqrt{(2ax-b)^2+4ac-b^2}+(2ax-b)\right) dx \\=\frac1{2a}\int\ln\left(\sqrt{u^2+d}+u\right) du$$ which has a decided resemblance to the Arsinh function. This suggests to look at $$\frac{d}{du}u·\ln\left(\sqrt{u^2+d}+u\right) = \ln\left(\sqrt{u^2+d}+u\right)+u·\frac{\frac{u}{\sqrt{u^2+d}}+1}{\sqrt{u^2+d}+u} \\= \ln\left(\sqrt{u^2+d}+u\right)+\frac{u}{\sqrt{u^2+d}}$$ (this is partial integration backwards), so that $$\int\ln\left(\sqrt{u^2+d}+u\right) du = u·\ln\left(\sqrt{u^2+d}+u\right) - \sqrt{u^2+d} + C$$
Inserting this backwards gets one $$I = \frac{2ax-b}{2a}·\ln\left(\sqrt{4a(ax^2-bx+c)}+2ax-b\right) - \frac1{2a}·\sqrt{4a(ax^2-bx+c)} + C$$ which up to variations of $\sqrt{4a}=2\sqrt{a}$ is the same anti-derivative as the one from WA.


text:
# Almost invariance and non interacting control : a frequency domain analysis

H.L. Trentelman, J.W. Woude, van der

## Abstract

In this paper we shall solve a number of feedback synthesis problems in the context of non interacting control or block diagonal decoupling for finite dimensional linear time invariant systems. We shall consider a plant that, apart from a control input and a measurement output, has a given numher of exogenous input vectors and the same numher of exogenous output vectors. The decoupling problem that will he studied here is to find dynamic compensators from the plant measurement output (which in this paper will he assumed to be the full plant state) to the plant control input in such a way that the following requirements are met: (i) the closed loop transfer matrix is block diagonal, (ii) the remaining diagonal blocks are stable with respect to an a priori given first stability set and (iii) the closed loop system is internally stable with respect to an a priori given second, in general larger, stability set In addition, we will study the "almost" version of the above problem. In the latter the requirement of exact decoupling will he replaced by a requirement of approximate decoupIing in the sense that the to-be-designed compensators should yield off-diagonal blocks in the closed loop transfer matrix that are arbitrarily small in $H^\infty$-norm. Necessary and sufficient conditions for the existence of such dynamic compensators will he formulated in terms of controlled invariant and almost controlled invariant subspaces.
Original language English Eindhoven Technische Universiteit Eindhoven 24 Published - 1986

### Publication series

Name Memorandum COSOR 8609 0926-4493

## Fingerprint

Dive into the research topics of 'Almost invariance and non interacting control : a frequency domain analysis'. Together they form a unique fingerprint.


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Let us consider +Y as upward and -Y axis as downward. So the acceleration that works these objects is always $-g$.

1. For the first picture, if the lift is going upward with acceleration $a$, then will the weight of the man of mass $m$ be $W=m(a-g)$? (but I found in textbook $g-a$) $a$ is positive here because the man gets acceleration due to lift's motion and which is +Y axis.
2. For the 2nd picture, if the lift is going upward with acceleration $g$, then will the weight of the man of mass $m$ be, $W=m(g-g)=0$?(but I found in textbook $g+g$)

These results contradicts with weightlessness. Can someone clarify my mistakes?

-
Your text is not consisten with the figure. Can you please correct this? Can you also explain what in your opinion the contraction with weightlessness is? –  Bernhard May 26 '13 at 8:24
Actually I collected the picture from website and tried to aggregate my idea with the picture. –  Physics_guy May 26 '13 at 9:04

The feeling of weightlessness or the feeling of weighing heavier is due to the force that acts on you from the floor.

Now, if a lift is going upward with an acceleration $a$, then the weight of a man inside it will experience a weight of:

$$F = ma$$

where $F$ is the net force acting on the man and $a$ is the net acceleration of the man (and lift of course).

The force that gravity exerts on the man is given by $F_g = mg$

The net force, $F$, is obtained from the difference of the force acting on the man from the floor of the lift (which we are interested in), and the force of gravity:

$$F_{floor} - Fg = F$$

so that:

$$F_{floor} = ma + mg = m(a+g)$$

Therefore, when the lift is going up, you feel as though there is an acceleration of $(a+g)$ acting on you (I think that the $(a-g)$ in your question was a typo and should be $(a+g)$, since you feel heavier when a lift is going down and lighter when the lift is going down).

If the lift is going up at an acceleration $g$, then you feel an acceleration of $(g+g)$.

Now, to feel weightlessness, the lift has to have a certain acceleration value downwards. If this downward acceleration is equal to $g$, we get (we substitute $a$ with $-g$):

$$F_{floor} = m(-g+g) = 0$$

So that you don't feel the force the floor is acting on you and are said to be free falling.

-


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# Configure HAProxy in reverse proxy with HTTP authentication

I am using a lot of web services on several servers, and I was bored to remember all ports and change my firewall rules each time. So I use HAProxy to redirect all incoming HTPP traffic to the right server/port by checking the requested URL.

Moreover, I configured it as an SSL endpoint, so every communication between a browser and HAProxy is encrypted, even if the service I am using is not able to support it. The other pro of this solution is to be able to use basic HTTP authentication whenever I want to restrict access to some services.

So I lost some time on the configuration, especially on one parameter that I never found on the internet, hope this can help other people. It is not a full guide on how to configure HAProxy, a lot of tutorials are on the internet, but it focuses on the essential parts.

HAProxy configuration file, located at /etc/haproxy/haproxy.cfg, is separated in 5 categories:

1. Global
2. Defaults
3. Frontend
4. Backend & userlist
5. Proxy

### 1. Global

From the documentation:

Parameters in the “global” section are process-wide and often OS-specific. They are generally set once for all and do not need being changed once correct.”

So, the only one I put there is the following ssl-server-verify none. It says that HAProxy will not complain if it has to connect to a server with an untrusted certificate. Useful for self-signed certificates, and can be placed in defaults section or each frontend.

### 2. Defaults

Set the parameters for all following frontends, until the next defaults section.

### 3. Frontend

It is the entry point for the browser, you can set here how the incoming connection should be handled. You can have as many endpoint as you want unless they do not listen on the same port.

frontend https
bind :443 ssl crt /path/to/sslKey/cert.pem
option forwardfor
http-request set-header X-Forwarded-Proto https if { ssl_fc }
use_backend lighttpds if { path_beg /testhttps }
use_backend lighttpdAuth if { path_beg /auth }
use_backend headphones if { path_beg /music }
use_backend lighttpdAuth if { path_beg /liste }
default_backend lighttpd

Bind keyword will tell HAProxy to listen port 443, and the path to the SSL certificate.

The four following parameters are to tell the server that the connection goes through a proxy, send some information on the client and the type of connection. The parameter I never found over the web is http-request set-header X-Forwarded-Host %[req.hdr(Host)], which allows the service to write the host address instead of 127.0.0.1, if it is on the same host.

Then is declared the backend to use (=services) on which conditions, and the default

### 4. Backend & userlist

Then comes the services you want to use

backend headphones
acl AuthOkay_UsersAuth http_auth(UsersAuth)
http-request auth realm UserAuth if !AuthOkay_UsersAuth

The name just after “backend” keyword must be the one defined in the fronted list. Then you can declare as many servers as you want because HAProxy is also used as a load balancer, but in this case, we only use one. Keyword ssl is not mandatory, it specifies that HAProxy must establish a secured connection. Then the two last lines are for the basic HTTP authentication, in combination with the two following lines:

userlist UsersAuth

Another thing to do is to tell each service that we access it through a prefix, it allows them to add it to the URL (ex: http://domainName/headphone instead of http://domainName/) Each service has a different configuration scheme, some example

• Node-Red : settings.js – httpRoot
• Headphones – config.ini – http_root (For all services based on CherryPy)
• KiwiIRC – config.js – conf.http_base_path
• GateOne – 10.server.conf – url_prefix

### 5. Proxy

Use HAPRoxy as… a proxy, see HAProxy documentation for this stuff.

I hope this helps because a lot of documentation about reverse proxy is based on Nginx, not HAProxy.


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Arithmetic Circuits

Introduction

ALU

In this project, we will design arithmetic circuits using an FPGA. We will build a 4-bit magnitude comparator, a ripple-carry adder, and a multiplier circuit. You can challenge yourself by integrating all of those circuits together with some multiplexers to build an arithmetic logic unit (ALU).

Before you begin, you should:
• Have the Xilinx® ISE WebPACK™ installed.
• Have your FPGA board set up.
• Be able to describe digital circuits using logic operators.
• Be able to write test bench and simulate circuit using ISim.

After you're done, you should:
• Understand how magnitude comparators, ripple-carry adders, and multipliers work.
• Be able to describe magnitude comparators, ripple-carry adders, and multipliers structurally.

Inventory:

Qty Description
1 Digilent Nexys™4Nexys™3Nexys™2, or Basys™2 FPGA Board
1 Xilinx ISE Design Suite: WebPACK (14.6 Recommended)

Comparator

A magnitude comparator is a device that receives two N-bit inputs and asserts one of three possible outputs depending on whether one input is greater than, less than, or equal to the other (simpler comparators, called equality comparators, provide a single output that is asserted whenever the two inputs are equal). The truth table of a bit-sliced magnitude comparator and the block diagram of a magnitude comparator are shown in Figs. 1 and 2 below.

Figure 1. Truth table for a bit-sliced magnitude comparator.

Figure 2. Block diagram of an 8-bit magnitude comparator using a bit-sliced magnitude comparator.

Step 1: Design a 4-bit Comparator

1. Create a Verilog module for a bit-sliced magnitude comparator according to the truth table presented in Fig. 1 above.
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 modulecmp_bitslice(  inputA,  inputB,  inputLT_I,  inputEQ_I,  inputGT_I,  outputLT_O,  outputEQ_O,  outputGT_O  ); assignGT_O = ( A & ~B ) | GT_I; assignEQ_O = EQ_I & (( A & B ) | (~A & ~B)); assignLT_O = ( B & ~A ) | LT_I; endmodule
2. Connect the bit-sliced magnitude comparator according to the block diagram shown in Fig. 2 above.
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 modulecmp(  input[3:0] A,  input[3:0] B,  outputLT_O,  outputEQ_O,  outputGT_O  ); wire[3:0] GT_int; wire[3:0] EQ_int; wire[3:0] LT_int; cmp_bitslice slice_0 (  .A(A[0]),   .B(B[0]),   .LT_I(1'b0),   .EQ_I(1'b1),   .GT_I(1'b0),   .LT_O(LT_int[0]),   .EQ_O(EQ_int[0]),   .GT_O(GT_int[0])  ); cmp_bitslice slice_1 (  .A(A[1]),   .B(B[1]),   .LT_I(LT_int[0]),   .EQ_I(EQ_int[0]),   .GT_I(GT_int[0]),   .LT_O(LT_int[1]),   .EQ_O(EQ_int[1]),   .GT_O(GT_int[1])  );   cmp_bitslice slice_2 (  .A(A[2]),   .B(B[2]),   .LT_I(LT_int[1]),   .EQ_I(EQ_int[1]),   .GT_I(GT_int[1]),   .LT_O(LT_int[2]),   .EQ_O(EQ_int[2]),   .GT_O(GT_int[2])  );   cmp_bitslice slice_3 (  .A(A[3]),   .B(B[3]),   .LT_I(LT_int[2]),   .EQ_I(EQ_int[2]),   .GT_I(GT_int[2]),   .LT_O(LT_int[3]),   .EQ_O(EQ_int[3]),   .GT_O(GT_int[3])  );   assignLT_O = LT_int[3]; assignEQ_O = EQ_int[3]; assignGT_O = GT_int[3]; endmodule
3. Create a test bench on your own to test the comparator. Please make sure that all possible cases are covered.
4. To implement a comparator, it is more convenient to use behavioral descriptions in Verilog. The following codes can implement the same circuit as the structural implementation above:
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 modulecmp(  input[3:0] A,  input[3:0] B,  outputLT_O,  outputEQ_O,  outputGT_O  );   assignLT_O = (A < B) ? 1'b1 : 1'b0; assignEQ_O = (A == B) ? 1'b1 : 1'b0; assignGT_O = (A > B) ? 1'b1 : 1'b0; endmodule
5. Apply the same test bench on the second comparator and check the result.

Adder circuits add two N-bit operands to produce an N-bit result and a carry out signal (the carry out is a '1' only when the addition result requires more than N-bits). The logic graph in Fig. 3 below shows the eight different cases that may be encountered when adding two binary numbers. The highlighted bit pairs and the associated carries show that a bit-slice adder circuit must process three inputs (the two addend bits and a carry-in from the previous stage) and produce two outputs (the sum bit and a carry out bit). The circuit block that implements the bit-sliced adder is called a "full adder" (FA).

Figure 3. Truth table for a bit-slice adder (full adder).

Note the carry-in of the RCA LSB is connected directly to ground, because (by definition) the carry-in to the LSB of any adder must be logic '0'. It is possible to capitalize on this observation, and create a smaller bit-slice circuit for use in the LSB position that does not have a carry-in input. Called a half adder (HA), this reduced-function adder circuit is often used in the LSB position. The ripple carry adder block diagram is displayed in Fig. 4 below.

Figure 4. Ripple-carry adder block diagram.

Step 2: Design a 4-bit Binary Adder

1. Create a Verilog module for a full adder.
 1 2 3 4 5 6 7 8 9 10 11 12 13 moduleFA(  inputA,  inputB,  inputCin,  outputS,  outputCout  );    assignS = A ^ B ^ Cin; assignCout = (A & B) | ((A ^ B) & Cin); endmodule
2. Create another Verilog module for a half adder.
 1 2 3 4 5 6 7 8 9 10 11 12 moduleHA(  inputA,  inputB,  outputS,  outputCout  ); assignS = A ^ B; assignCout = A & B; endmodule
3. Create a Verilog module called "adder" to wrap three FAs with one HA to form a 4-bit adder:
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 moduleadder(  input[3:0] A,  input[3:0] B,  output[3:0] S,  outputCout  ); wire[3:0] Carry; HA add_0 (  .A(A[0]),   .B(B[0]),   .S(S[0]),   .Cout(Carry[0])  ); FA add_1 (  .A(A[1]),   .B(B[1]),   .Cin(Carry[0]),   .S(S[1]),   .Cout(Carry[1])  ); FA add_2 (  .A(A[2]),   .B(B[2]),   .Cin(Carry[1]),   .S(S[2]),   .Cout(Carry[2])  ); FA add_3 (  .A(A[3]),   .B(B[3]),   .Cin(Carry[2]),   .S(S[3]),   .Cout(Carry[3])  ); assignCout = Carry[3]; endmodule
4. Create a test bench that covers all of the possible inputs to the adder and verify whether the result is correct. After the verification, you can tie SW3-0 to A and SW7-4 to B as two operands in UCF file, and tie LED4-0 to result of adder, you can test out your adder on your FPGA board.
5. Here, we provide you with another way of implementing adder in Verilog behaviorally. In the codes below, we use operator "+" to indicate an adder. It is very important to remember that when you put down a "+" in Verilog HDL, the tools will instantiate an adder for you, instead of just a operation as you might be familiar with in software programming language.
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 moduleadder(  input[3:0] A,  input[3:0] B,  output[3:0] S,  outputCout  ); wire[4:0] Result; assignResult = A + B; assignS = Result[3:0]; assignCout = Result[4]; endmodule

Multipliers

Figure 5. Hardware multiplier.

Step 3: Design a 4-bit Multiplier

Up to this point, you are expected to be able to describe circuits structurally. Based on the block diagram shown above in Fig. 5, you can use assignment statements to implement the circuit for partial products, and use the FA, HA, and the adder you implemented in previous steps to add those partial products together. Before deploying your circuit on your board, write a test bench to verify that your circuit is correct. Use SW3-0 as A and SW7-4 as B (A and B are multiplicants), and show the result of multiplication on 8-bit LEDs.

Unlike the adder and subtractor, multipliers do not have an operator support in Verilog, mostly due to the fact that there are various ways to implement a multiplier which trade off power, hardware resource, and speed. So implementing a multiplier structurally is the only solution.

Now that you've completed this project, try these modifications:

1. Implement a 4-bit borrow ripple subtractor using bit-sliced design methodology and describe it structurally in Verilog.
2. Modify the adder you implemented in step 2 into an add_subtract circuit. Add an input "subtract". When "subtract" is '1' the output of add_subtract equals A−B. When "subtract" is "1", the output of add_subtract equals A+B. Inputs and outputs of the add_subtract circuit are represented in 2's complement. Instead of cout, an output "ERR" is needed as overflow/underflow indicator, i.e., ERR is "1" when overflow/underflow occurs.
3. Can you combine add_subtract with other logics to implement an ALU according to the following opcode table? (Inputs and Output of the ALU are 4-bit binary numbers in 2's complement).
4. OpCode Description Output F


text:
# #9. Product¶

Note

Following documentation is a cooperative result combined from our Discord chat and numerous pull requests. Thanks to everyone who helped!

## Image¶

This image was produced from the ninth radio transmission using previously contributed code.

This partly annotated version of the image was made using code from message #3.

## Interpretation¶

Definition of the new glyph is consistent with the definition of multiplication.

## Decoded¶

mul
ap ap mul 4 2   =   8
ap ap mul 3 4   =   12
ap ap mul 3 -2   =   -6
ap ap mul x0 x1   =   ap ap mul x1 x0
ap ap mul x0 0   =   0
ap ap mul x0 1   =   x0
...

## Code¶

Revised version of the Haskell code that supports the mul glyph is published on the message #3 page.

Contributed by Discord users @pink_snow and @fryguybob.

Example output: