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Reinforcement Learning (English)
Hello and welcome to our course; Reinforcement Learning. Reinforcement Learning is a very exciting and important field of Machine Learning and AI. Some call it the crown jewel of AI. In this course,
we will cover all the aspects related to Reinforcement Learning or RL. We will start by defining the RL problem, and compare it to the Supervised Learning problem, and discover the areas of
applications where RL can excel. This includes the problem formulation, starting from the very basics to the advanced usage of Deep Learning, leading to the era of Deep Reinforcement Learning. In our
journey, we will cover, as usual, both the theoretical and practical aspects, where we will learn how to implement the RL algorithms and apply them to the famous problems using libraries like OpenAI
Gym, Keras-RL, TensorFlow Agents or TF-Agents and Stable Baselines. The course is divided into 6 main sections: 1- We start with an introduction to the RL problem definition, mainly comparing it to
the Supervised learning problem, and discovering the application domains and the main constituents of an RL problem. We describe here the famous OpenAI Gym environments, which will be our playground
when it comes to practical implementation of the algorithms that we learn about. 2- In the second part we discuss the main formulation of an RL problem as a Markov Decision Process or MDP, with
simple solution to the most basic problems using Dynamic Programming. 3- After being armed with an understanding of MDP, we move on to explore the solution space of the MDP problem, and what the
different solutions beyond DP, which includes model-based and model-free solutions. We will focus in this part on model-free solutions, and defer model-based solutions to the last part. In this part,
we describe the Monte-Carlo and Temporal-Difference sampling based methods, including the famous and important Q-learning algorithm, and SARSA. We will describe the practical usage and implementation
of Q-learning and SARSA on control tabular maze problems from OpenAI Gym environments. 4- To move beyond simple tabular problems, we will need to learn about function approximation in RL, which leads
to the mainstream RL methods today using Deep Learning, or Deep Reinforcement Learning (DRL). We will describe here the breakthrough algorithm of DeepMind that solved the Atari games and AlphaGO,
which is Deep Q-Networks or DQN. We also discuss how we can solve Atari games problems using DQN in practice using Keras-RL and TF-Agents. 5- In the fifth part, we move to Advanced DRL algorithms,
mainly under a family called Policy based methods. We discuss here Policy Gradients, DDPG, Actor-Critic, A2C, A3C, TRPO and PPO methods. We also discuss the important Stable Baseline library to
implement all those algorithms on different environments in OpenAI Gym, like Atari and others. 6- Finally, we explore the model-based family of RL methods, and importantly, differentiating
model-based RL from planning, and exploring the whole spectrum of RL methods. Hopefully, you enjoy this course, and find it useful. | {"url":"https://coursatai.azurewebsites.net/course/Reinforcement%20Learning%20(English)","timestamp":"2024-11-03T03:30:26Z","content_type":"text/html","content_length":"16038","record_id":"<urn:uuid:4bd8b907-ba28-410e-b584-fb2a13beb8aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00420.warc.gz"} |
How does R calculate histogram break points?
Thursday December 25, 2014
Break points make (or break) your histogram. R's default algorithm for calculating histogram break points is a little interesting. Tracing it includes an unexpected dip into R's C implementation.
# set seed so "random" numbers are reproducible
# generate 100 random normal (mean 0, variance 1) numbers
x <- rnorm(100)
# calculate histogram data and plot it as a side effect
h <- hist(x, col="cornflowerblue")
The hist function calculates and returns a histogram representation from data. That calculation includes, by default, choosing the break points for the histogram. In the example shown, there are ten
bars (or bins, or cells) with eleven break points (every 0.5 from -2.5 to 2.5). With break points in hand, hist counts the values in each bin. The histogram representation is then shown on screen by
(By default, bin counts include values less than or equal to the bin's right break point and strictly greater than the bin's left break point, except for the leftmost bin, which includes its left
break point.)
The choice of break points can make a big difference in how the histogram looks. Badly chosen break points can obscure or misrepresent the character of the data. R's default behavior is not
particularly good with the simple data set of the integers 1 to 5 (as pointed out by Wickham).
hist(1:5, col="cornflowerblue")
A manual choice like the following would better show the evenly distributed numbers.
hist(1:5, breaks=0.5:5.5, col="cornflowerblue")
It might be even better, arguably, to use more bins to show that not all values are covered.
hist(1:5, breaks=seq(0.55, 5.55, 0.1), col="cornflowerblue")
In any event, break points matter. When exploring data it's probably best to experiment with multiple choices of break points. But in practice, the defaults provided by R get seen a lot.
So how does R choose break points?
By default, inside of hist a two-stage process will decide the break points used to calculate a histogram:
1. The function nclass.Sturges receives the data and returns a recommended number of bars for the histogram. The documentation says that Sturges' formula is "implicitly basing bin sizes on the range
of the data" but it's just based on the number of values, as ceiling(log2(length(x)) + 1). This is really fairly dull.
2. Then the data and the recommended number of bars gets passed to pretty (usually pretty.default), which tries to "Compute a sequence of about n+1 equally spaced ‘round’ values which cover the
range of the values in x. The values are chosen so that they are 1, 2 or 5 times a power of 10." This ends up calling into some parts of R implemented in C, which I'll describe a little below.
Note: In what follows I'll link to a mirror of the R sources because GitHub has a nice, familiar interface. I'll point to the most recent version of files without specifying line numbers. You'll want
to search within the files to what I'm talking about. To see exactly what I saw go to commit 34c4d5dd.
The source for nclass.Sturges is trivial R, but the pretty source turns out to get into C. I hadn't looked into any of R's C implementation before; here's how it seems to fit together:
The source for pretty.default is straight R until:
z <- .Internal(pretty( # ... cut
This .Internal thing is a call to something written in C. The file names.c can be useful for figuring out where things go next. We find this line:
{"pretty", do_pretty, 0, 11, 7, {PP_FUNCALL, PREC_FN, 0}},
So it goes to a C function called do_pretty. That can be found in util.c. This is a lot of very Lisp-looking C, and mostly for handling the arguments that get passed in. For example:
int n = asInteger(CAR(args)); args = CDR(args);
That's kind of neat, but the actual work is done somewhere else again. The body of do_pretty calls a function R_pretty like this:
R_pretty(&l, &u, &n, min_n, shrink, REAL(hi), eps, 1);
The call is interesting because it doesn't even use a return value; R_pretty modifies its first three arguments in place. Gross.
The function R_pretty is in its own file, pretty.c, and finally the break points are made to be "nice even numbers" and there's a result.
I was surprised by where the code complexity of this process is. | {"url":"https://planspace.org/20141225-how_does_r_calculate_histogram_break_points/","timestamp":"2024-11-05T03:12:06Z","content_type":"text/html","content_length":"9388","record_id":"<urn:uuid:bf85de9b-c655-4afd-8f6b-aa05f81f1435>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00624.warc.gz"} |
Intuitionistic Logic
First published Wed Sep 1, 1999; substantive revision Fri Jan 2, 2015
Intuitionistic logic encompasses the principles of logical reasoning which were used by L. E. J. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Because these principles
also underly Russian recursive analysis and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics.
Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing constructive reasoning about
uncountable structures (e.g. monotone bar induction on the tree of potentially infinite sequences of natural numbers); and from platonism by viewing mathematical objects as mental constructs with no
independent ideal existence. Hilbert's formalist program, to justify classical mathematics by reducing it to a formal system whose consistency should be established by finitistic (hence constructive)
means, was the most powerful contemporary rival to Brouwer's developing intuitionism. In his 1912 essay Intuitionism and Formalism Brouwer correctly predicted that any attempt to prove the
consistency of complete induction on the natural numbers would lead to a vicious circle.
Brouwer rejected formalism per se but admitted the potential usefulness of formulating general logical principles expressing intuitionistically correct constructions, such as modus ponens. Formal
systems for intuitionistic propositional and predicate logic and arithmetic were developed by Heyting [1930], Gentzen [1935] and Kleene [1952]. Gödel [1933] proved the equiconsistency of
intuitionistic and classical theories. Kripke [1965] provided a semantics with respect to which intuitionistic logic is correct and complete.
Intuitionistic logic can be succinctly described as classical logic without the Aristotelian law of excluded middle (LEM): (A ∨ ¬A) or the classical law of double negation elimination (¬ ¬A → A), but
with the law of contradiction (A → B) → ((A → ¬B) → ¬A) and ex falso quodlibet: (¬A → (A → B)). Brouwer [1908] observed that LEM was abstracted from finite situations, then extended without
justification to statements about infinite collections. For example, let x, y range over the natural numbers 0, 1, 2, … and B(x) abbreviate the property expressed by the following claim in which the
variable x is free: there is a y greater than x such that both y and y+2 are prime numbers, i.e.,
∃y(y>x & Prime(y) & Prime(y+2))
Then we have no general method for deciding whether B(x) is true or false for arbitrary x, so ∀x(B(x) ∨ ¬B(x)) cannot be asserted in the present state of our knowledge. And if A abbreviates the
statement ∀xB(x), then (A ∨ ¬A) cannot be asserted because neither A nor (¬A) has yet been proved.
One may object that these examples depend on the fact that the Twin Primes Conjecture has not yet been settled. A number of Brouwer's original “counterexamples” depended on problems (such as Fermat's
Last Theorem) which have since been solved. But to Brouwer the general LEM was equivalent to the a priori assumption that every mathematical problem has a solution — an assumption he rejected,
anticipating Gödel's incompleteness theorem by a quarter of a century.
The rejection of LEM has far-reaching consequences. On the one hand,
• Intuitionistically, Reductio ad absurdum only proves negative statements, since ¬ ¬A → A does not hold in general. (If it did, LEM would follow by modus ponens from the intuitionistically
provable ¬ ¬ (A ∨ ¬A).)
• Not every propositional formula has an intuitionistically equivalent disjunctive or conjunctive normal form.
• Not every predicate formula has an intuitionistically equivalent prenex form.
• While ∀x ¬ ¬ (A(x) ∨ ¬A(x)) is a theorem of intuitionistic predicate logic, ¬ ¬ ∀x(A(x) ∨ ¬A(x)) is not.
• Pure intuitionistic logic is axiomatically incomplete. Infinitely many intermediate axiomatic extensions of intuitionistic propositional and predicate logic are contained in classical logic.
On the other hand,
• Every intuitionistic proof of a closed statement of the form A ∨ B can be effectively transformed into an intuitionistic proof of A or an intuitionistic proof of B, and similarly for closed
existential statements.
• Classical logic is finitistically interpretable in the negative fragment of intuitionistic logic.
• Arithmetical formulas have relatively simple intuitionistic normal forms.
• Intuitionistic arithmetic can consistently be extended by axioms (such as Church's Thesis) which contradict classical arithmetic, enabling the formal study of recursive mathematics.
• Brouwer's controversial intuitionistic analysis, which conflicts with LEM, can be formalized and shown consistent relative to a classically and intuitionistically correct subtheory.
Formalized intuitionistic logic is naturally motivated by the informal Brouwer-Heyting-Kolmogorov explanation of intuitionistic truth, outlined in Section 3.1 of the entry on intuitionism in the
philosophy of mathematics and discussed extensively in the entry on the development of intuitionistic logic. The constructive independence of the logical operations &, ∨, →, ¬, ∀, ∃ contrasts with
the classical situation, where e.g., (A ∨ B) is equivalent to ¬ (¬A & ¬B), and ∃xA(x) is equivalent to ¬ ∀x ¬A(x). From the B-H-K viewpoint, a sentence of the form (A ∨ B) asserts that either a proof
of A, or a proof of B, has been constructed; while ¬ (¬A & ¬B) asserts that an algorithm has been constructed which would effectively convert any pair of constructions proving ¬A and ¬B respectively,
into a proof of a known contradiction.
Following is a Hilbert-style formalism H, from Kleene [1952], for intuitionistic first-order predicate logic IQC. The language L has predicate letters P, Q(.),… of all arities and individual
variables a, b, c,… (with or without subscripts 1,2,…), as well as symbols &, ∨, →, ¬, ∀, ∃ for the logical connectives and quantifiers, and parentheses (, ). The prime formulas of L are expressions
such as P, Q(a), R(a, b, a) where P, Q(.), R(…) are 0-ary, 1-ary and 3-ary predicate letters respectively; that is, the result of filling each blank in a predicate letter by an individual variable
symbol is a prime formula. The (well-formed) formulas of L are defined inductively as follows.
• Each prime formula is a formula.
• If A and B are formulas, so are (A & B), (A ∨ B), (A → B) and ¬A.
• If A is a formula and x is a variable, then ∀xA and ∃xA are formulas.
In general, we use A, B, C as metavariables for well-formed formulas and x, y, z as metavariables for individual variables. Anticipating applications (for example to intuitionistic arithmetic) we use
s, t as metavariables for terms; in the case of pure predicate logic, terms are simply individual variables. An occurrence of a variable x in a formula A is bound if it is within the scope of a
quantifier ∀x or ∃x, otherwise free. Intuitionistically as classically, “(A ↔ B)” abbreviates “(A → B) & (B → A),” and parentheses are omitted when this causes no confusion.
There are three rules of inference:
• Modus Ponens: From A and (A → B), conclude B.
• ∀-Introduction: From (C → A(x)), where x is a variable which does not occur free in C, conclude (C → ∀xA(x)).
• ∃-Elimination: From (A(x) → C), where x is a variable which does not occur free in C, conclude (∃xA(x) → C).
The axioms are all formulas of the following forms, where in the last two schemas the subformula A(t) is the result of substituting an occurrence of the term t for every free occurrence of x in A(x),
and no variable free in t becomes bound in A(t) as a result of the substitution.
• A → (B → A).
• (A → B) → ((A → (B → C)) → (A → C)).
• A → (B → A & B).
• A & B → A.
• A & B → B.
• A → A ∨ B.
• B → A ∨ B.
• (A → C) → ((B → C) → (A ∨ B → C)).
• (A → B) → ((A → ¬B) → ¬A).
• ¬A → (A → B).
• ∀xA(x) → A(t).
• A(t) → ∃xA(x).
A proof is any finite sequence of formulas, each of which is an axiom or an immediate consequence, by a rule of inference, of (one or two) preceding formulas of the sequence. Any proof is said to
prove its last formula, which is called a theorem or provable formula of first-order intuitionistic predicate logic. A derivation of a formula E from a collection F of assumptions is any sequence of
formulas, each of which belongs to F or is an axiom or an immediate consequence, by a rule of inference, of preceding formulas of the sequence, such that E is the last formula of the sequence. If
such a derivation exists, we say E is derivable from F.
Intuitionistic propositional logic IPC is the subtheory which results when the language is restricted to formulas built from proposition letters P, Q, R,… using the propositional connectives &, ∨, →
and ¬, and only the propositional postulates are used. Thus the last two rules of inference and the last two axiom schemas are absent from the propositional theory.
If, in the given list of axiom schemas for intuitionistic propositional or first-order predicate logic, the law expressing ex falso sequitur quodlibet:
¬A → (A → B).
is replaced by the classical law of double negation elimination:
¬ ¬A → A.
(or,equivalently, if the intuitionistic law of negation introduction
(A → B) → ((A → ¬B) → ¬A)
is replaced by LEM), a formal system for classical propositional logic CPC or classical predicate logic CQC results. Since the law of contradiction is a classical theorem, intuitionistic logic is
contained in classical logic. In a sense, classical logic is also contained in intuitionistic logic; see Section 4.1 below.
It is important to note that while LEM and the law of double negation are equivalent as schemas, the implication (¬ ¬A → A) → (A ∨ ¬A) is not a theorem schema of IPC. For theories T based on
intuitionistic logic, if E is an arbitrary formula of L(T) then by definition
• E is decidable in T if and only if T proves (E ∨ ¬E).
• E is stable in T if and only if T proves (¬ ¬E → E).
• E is testable in T if and only if T proves (¬E ∨ ¬ ¬E).
Decidability implies stability, but not conversely. The conjunction of stability and testability is equivalent to decidability. By Brouwer's first published logical theorem ¬ ¬ ¬A → ¬A, every formula
of the form ¬A is stable; but in IPC and IQC prime formulas and their negations are undecidable, as shown in Section 5.1 below.
The Hilbert-style system H is useful for metamathematical investigations of intuitionistic logic, but its forced linearization of deductions and its preference for axioms over rules make it an
awkward instrument for establishing derivability. A natural deduction system I for intuitionistic predicate logic results from the deductive system D, presented in Section 3 of the entry on classical
logic in this Encyclopedia, by omitting the symbol and rules for identity, and replacing the classical rule (DNE) of double negation elimination by the intuitionistic negation elimination rule
(INE) If F entails A and F entails ¬A, then F entails B.
While identity can of course be added to intuitionistic logic, for applications (e.g., to arithmetic) the equality symbol is generally treated as a distinguished predicate constant satisfying
nonlogical axioms (e.g., the primitive recursive definitions of addition and multiplication) in addition to reflexivity, symmetry and transitivity. Identity is decidable, intuitionistically as well
as classically, but intuitionistic extensional equality is not always decidable; see the discussion of Brouwer's continuity axioms in Section 3 of the entry on intuitionism in the philosophy of
mathematics. The keys to proving that H is equivalent to I are modus ponens and its converse, the
Deduction Theorem: If B is derivable from A and possibly other formulas F, with all variables free in A held constant in the derivation (that is, without using the second or third rule of
inference on any variable x occurring free in A, unless the assumption A does not occur in the derivation before the inference in question), then (A → B) is derivable from F.
This fundamental result, roughly expressing the rule (→I) of I, can be proved for H by induction on the definition of a derivation. The other rules of I hold for H essentially by modus ponens, which
corresponds to (→E) in I. To illustrate the usefulness of the Deduction Theorem, consider the (apparently trivial) theorem schema (A → A) of IPC. A correct proof in H takes five lines:
1. A → (A → A)
2. (A → (A → A)) → ((A → ((A → A) → A)) → (A → A))
3. (A → ((A → A) → A)) → (A → A)
4. A → ((A → A) → A)
5. A → A
where 1, 2 and 4 are axioms and 3, 5 come from earlier lines by modus ponens. However, A is derivable from A (as assumption) in one obvious step, so the Deduction Theorem allows us to conclude that a
proof of (A → A) exists. (In fact, the formal proof of (A → A) just presented is part of the constructive proof of the Deduction Theorem!)
It is important to note that, in the definition of a derivation from assumptions in H, the assumption formulas are treated as if all their free variables were universally quantified, so that ∀x A(x)
is derivable from the hypothesis A(x). However, the variable x will be varied (not held constant) in that derivation, by use of the rule of ∀-introduction; and so the Deduction Theorem cannot be used
to conclude (falsely) that A(x) → ∀x A(x) (and hence, by ∃-elimination, ∃x A(x) → ∀x A(x)) are provable in H. As an example of a correct use of the Deduction Theorem for predicate logic, consider the
implication ∃x A(x) → ¬∀x¬A(x). To show this is provable in IQC, we first derive ¬∀x¬A(x) from A(x) with all free variables held constant:
1. ∀x¬A(x) → ¬A(x)
2. A(x) → (∀x¬A(x) → A(x))
3. A(x) (assumption)
4. ∀x¬A(x) → A(x)
5. (∀x¬A(x) → A(x)) → ((∀x¬A(x) → ¬A(x)) → ¬∀x¬A(x))
6. (∀x¬A(x) → ¬A(x)) → ¬∀x¬A(x)
7. ¬∀x¬A(x)
Here 1, 2 and 5 are axioms; 4 comes from 2 and 3 by modus ponens; and 6 and 7 come from earlier lines by modus ponens; so no variables have been varied. The Deduction Theorem tells us there is a
proof P in IQC of (A(x) → ¬∀x¬A(x)), and one application of ∃-elimination converts P into a proof of ∃x A(x) → ¬∀x¬A(x). The converse is not provable in IQC, as shown in Section 5.1 below.
Intuitionistic (Heyting) arithmetic HA and classical (Peano) arithmetic PA share the same first-order language and the same non-logical axioms; only the logic is different. In addition to the logical
connectives, quantifiers and parentheses and the individual variables a, b, c, … (with metavariables x, y, z as usual), the language L(HA) of arithmetic has a binary predicate symbol =, individual
constant 0, unary function constant S, and finitely or countably infinitely many additional constants for primitive recursive functions including addition and multiplication; the precise choice is a
matter of taste and convenience. Terms are built from variables and 0 using the function constants; in particular, each natural number n is expressed in the language by the numeral n obtained by
applying S n times to 0 (e.g., S(S(0)) is the numeral for 2). Prime formulas are of the form (s = t) where s, t are terms, and compound formulas are obtained from these as usual.
The logical axioms and rules of HA are those of first-order intuitionistic predicate logic IQC. The nonlogical axioms include the reflexive, symmetric and transitive properties of =, primitive
recursive defining equations for each function constant, the axioms characterizing 0 as the least natural number and S as a one-to-one function:
• ∀x¬(S(x) = 0),
• ∀x∀y(S(x) = S(y) → x = y),
the extensional equality axiom for S:
• ∀x∀y (x = y → S(x) = S(y)),
and the (universal closure of the) schema of mathematical induction, for arbitrary formulas A(x):
• A(0) & ∀x(A(x) → A(S(x))) → ∀x A(x).
Extensional equality axioms for all the other function constants are derivable by mathematical induction from the equality axiom for S and the primitive recursive function axioms.
The natural order relation x < y can be defined in HA by ∃z(S(z) + x = y), or by a quantifier-free formula if the symbol and defining axioms for cutoff subtraction are present in the formalism. HA
proves the comparative law
∀x ∀y (x < y ∨ x = y ∨ y < x)
and an intuitionistic form of the least number principle, for arbitrary formulas A(x):
∀x[∀y (y < x → A(y) ∨ ¬A(y)) → ∃y (y < x & A(y) & ∀z(z < y → ¬A(z))) ∨ ∀y(y < x → ¬A(y))].
The hypothesis is needed because not all arithmetical formulas are decidable in HA. However, ∀x∀y(x = y ∨ ¬(x = y)) can be proved directly by mathematical induction, and so
□ Prime formulas (and hence all quantifier-free formulas) are decidable and stable in HA.
If A(x) is decidable in HA, then by induction on x so are ∀y (y < x → A(y)) and ∃y (y < x & A(y)). Hence
□ Formulas in which all quantifiers are bounded are decidable and stable in HA.
The collection Δ[0] of arithmetical formulas in which all quantifiers are bounded is the lowest level of a classical arithmetical hierarchy based on the pattern of alternations of quantifiers in a
prenex formula. In HA not every formula has a prenex form, but Burr [2004] discovered a simple intuitionistic arithmetical hierarchy corresponding level by level to the classical. For the purposes of
the next two definitions only, ∀x denotes a block of finitely many universal number quantifiers, and similarly ∃x denotes a block of finitely many existential number quantifiers. With these
conventions, Burr's classes Φ[n] and Ψ[n] are defined by
• Φ[0] = Ψ[0] = Δ[0],
• Φ[1] is the class of all formulas of the form ∀x A(x) where A(x) is in Ψ[0]. For n ≥ 2, Φ[n] is the class of all formulas of the form ∀x [A(x) →∃y B(x,y)] where A(x) is in Φ[n-1] and B(x,y) is in
• Ψ[1] is the class of all formulas of the form ∃x A(x) where A(x) is in Φ[0]. For n ≥ 2, Ψ[n] is the class of all formulas of the form A → B where A is in Φ[n] and B is in Φ[n-1].
The corresponding classical prenex classes are defined more simply:
• Π[0] = Σ[0] = Δ[0],
• Π[n+1] is the class of all formulas of the form ∀x A(x) where A(x) is in Σ[n],
• Σ[n+1] is the class of all formulas of the form ∃x A(x) where A(x) is in Π[n].
Peano arithmetic PA comes from Heyting arithmetic HA by adding LEM or (¬ ¬A → A) to the list of logical axioms, i.e., by using classical instead of intuitionistic logic. The following results hold
even in the fragments of HA and PA with the induction schema restricted to Δ[0] formulas.
Burr's Theorem:
□ Every arithmetical formula is provably equivalent in HA to a formula in one of the classes Φ[n].
□ Every formula in Φ[n] is provably equivalent in PA to a formula in Π[n], and conversely.
□ Every formula in Ψ[n] is provably equivalent in PA to a formula in Σ[n], and conversely.
HA and PA are proof-theoretically equivalent, as will be shown in Section 4. Each is capable of (numeralwise) expressing its own proof predicate. By Gödel's famous Incompleteness Theorem, if HA is
consistent then neither HA nor PA can prove its own consistency.
A fundamental fact about intuitionistic logic is that it has the same consistency strength as classical logic. For propositional logic this was first proved by Glivenko [1929].
Glivenko's Theorem: An arbitrary propositional formula A is classically provable, if and only if ¬¬A is intuitionistically provable.
Glivenko's Theorem does not extend to predicate logic, although an arbitrary predicate formula A is classically provable if and only if ¬¬A is provable in intuitionistic predicate logic plus the
“double negation shift” schema
(DNS) ∀x¬¬B(x) → ¬¬∀x B(x).
The more sophisticated negative translation of classical into intuitionistic theories, due independently to Gödel and Gentzen, associates with each formula A of the language L another formula g(A)
(with no ∨ or ∃), such that
(I) Classical predicate logic proves A ↔ g(A).
(II) Intuitionistic predicate logic proves g(A) ↔ ¬ ¬ g(A).
(III) If classical predicate logic proves A, then intuitionistic predicate logic proves g(A).
The proofs are straightforward from the following inductive definition of g(A) (using Gentzen's direct translation of implication, rather than Gödel's in terms of ¬ and &):
• g(P) is ¬ ¬ P, if P is prime.
• g(A & B) is (g(A) & g(B)).
• g(A ∨ B) is ¬ (¬g(A) & ¬g(B)).
• g(A → B) is (g(A) → g(B)).
• g(¬A) is ¬ g(A).
• g(∀xA(x)) is ∀x g(A(x)).
• g(∃xA(x)) is ¬∀x¬g(A(x)).
For each formula A, g(A) is provable intuitionistically if and only if A is provable classically. In particular, if (B & ¬B) were classically provable for some formula B, then (g(B) & ¬g(B)) (which
is g(B & ¬B)) would in turn be provable intuitionistically. Hence
(IV) Classical and intuitionistic predicate logic are equiconsistent.
The negative translation of classical into intuitionistic number theory is even simpler, since prime formulas of intuitionistic arithmetic are stable. Thus g(s=t) can be taken to be (s=t), and the
other clauses are unchanged. The negative translation of any instance of mathematical induction is another instance of mathematical induction, and the other nonlogical axioms of arithmetic are their
own negative translations, so
(I), (II), (III) and (IV) hold also for number theory.
Gödel [1933e] interpreted these results as showing that intuitionistic logic and arithmetic are richer than classical logic and arithmetic, because the intuitionistic theory distinguishes formulas
which are classically equivalent, and has the same consistency strength as the classical theory.
Direct attempts to extend the negative interpretation to analysis fail because the negative translation of the countable axiom of choice is not a theorem of intuitionistic analysis. However, it is
consistent with intuitionistic analysis, including Brouwer's controversial continuity principle, by the functional version of Kleene's recursive realizability (Section 5.2 below).
Gödel [1932] observed that intuitionistic propositional logic has the disjunction property:
(DP) If (A ∨ B) is a theorem, then A is a theorem or B is a theorem.
Gentzen [1935] established the disjunction property for closed formulas of intuitionistic predicate logic. From this it follows that if intuitionistic logic is consistent, then (P ∨ ¬P) is not a
theorem if P is prime. Kleene [1945, 1952] proved that intuitionistic first-order number theory also has the related (cf. Friedman [1975]) existence property:
(ED) If ∃xA(x) is a closed theorem, then for some closed term t, A(t) is a theorem.
The disjunction and existence properties are special cases of a general phenomenon peculiar to nonclassical theories. The admissible rules of a theory are the rules under which the theory is closed.
For example, Harrop [1960] observed that the rule
If (¬A → (B ∨ C)) is a theorem, so is (¬A → B) ∨ (¬A → C)
is admissible for intuitionistic propositional logic IPC because if A, B and C are any formulas such that (¬A → (B ∨ C)) is provable in IPC, then also (¬A → B) ∨ (¬A → C) is provable in IPC. Harrop's
rule is not derivable in IPC because (¬A → (B ∨ C)) → (¬A → B) ∨ (¬A → C) is not intuitionistically provable. Another important example of an admissible nonderivable rule of IPC is Mints' rule:
If ((A → B) → A ∨ C) is a theorem, so is ((A → B) → A) ∨ ((A → B) → C).
The two-valued truth table interpretation of classical propositional logic CPC gives rise to a simple proof that every admissible rule of CPC is derivable: otherwise, some assignment to A, B, etc.
would make the hypothesis true and the conclusion false, and by substituting e.g. (P → P) for the letters assigned “true” and (P & ¬ P) for those assigned “false” one would have a provable hypothesis
and unprovable conclusion. The fact that the intuitionistic situation is more interesting leads to many natural questions, some of which have recently been answered.
By generalizing Mints' Rule, Visser and de Jongh identified a recursively enumerable sequence of successively stronger admissible rules (“Visser's rules”) which, they conjectured, formed a basis for
the admissible rules of IPC in the sense that every admissible rule is derivable from the disjunction property and one of the rules of the sequence. Building on work of Ghilardi [1999], Iemhoff
[2001] succeeded in proving their conjecture. Rybakov [1997] proved that the collection of all admissible rules of IPC is decidable but has no finite basis. Visser [2002] showed that his rules are
also the admissible propositional rules of HA, and of HA extended by Markov's Principle MP (defined in Section 5.2 below). More recently, Jerabek [2008] found a different basis for the admissible
rules of IPC with the property that no rule in the basis derives another.
Much less is known about the admissible rules of intuitionistic predicate logic. Pure IQC, without individual or predicate constants, has the following remarkable admissible rule for A(x) with no
variables free but x:
If ∃x A(x) is a theorem, so is ∀x A(x).
Not every admissible predicate rule of IQC is admissible for all formal systems based on IQC; for example, HA evidently violates the rule just stated. Visser proved in [1999] that the property of
being an admissible predicate rule of HA is Π[2] complete, and in [2002] that HA + MP has the same predicate admissible rules as HA. Plisko [1992] proved that the predicate logic of HA + MP (the
set of sentences in the language of IQC all of whose uniform substitution instances in the language of arithmetic are provable in HA + MP) is Π[2] complete; Visser [2006] extended this result to some
constructively interesting consistent extensions of HA which are not contained in PA.
While they have not been completely classified, the admissible rules of intuitionistic predicate logic are known to include Markov's Rule for decidable predicates:
If ∀x(A(x) ∨ ¬A(x)) & ¬∀x¬A(x) is a theorem, so is ∃x A(x)
and the following Independence-of-Premise Rule (where y is assumed not to occur free in A(x)):
If ∀x(A(x) ∨ ¬A(x)) & (∀x A(x) → ∃y B(y)) is a theorem, so is ∃y (∀x A(x) → B(y)).
Both rules are also admissible for HA. The corresponding implications (MP and IP respectively), which are not provable intuitionistically, are verified by Gödel's “Dialectica” interpretation of HA
(cf. Section 6.3 below). So is the implication (CT) corresponding to one of the most interesting admissible rules of Heyting arithmetic, let us call it the Church-Kleene Rule:
If ∀x ∃y A(x, y) is a closed theorem of HA then there is a number n such that, provably in HA, the partial recursive function with Gödel number n is total and maps each x to a y satisfying A(x, y
) (and moreover A(x,y) is provable, where x is the numeral for the natural number x and y is the numeral for y).
Combining Markov's Rule with the negative translation gives the result that classical and intuitionistic arithmetic prove the same formulas of the form ∀x ∃y A(x, y) where A(x, y) is quantifier-free.
In general, if A(x, y) is provably decidable in HA and if ∀x ∃y A(x, y) is a closed theorem of classical arithmetic PA, the conclusion of the Church-Kleene Rule holds even in intuitionistic
arithmetic. For if HA proves ∀x ∀y (A(x,y) ∨ ¬A(x,y)) then by the Church-Kleene Rule the characteristic function of A(x,y) has a Gödel number m, provably in HA; so HA proves ∀x ∃y A(x,y) ↔ ∀x ∃y ∃z B
(m,x,y,z) where B is quantifier-free, and the adjacent existential quantifiers can be contracted in HA. It follows that HA and PA have the same provably recursive functions.
Here is a proof that the rule “If ∀x (A ∨ B(x)) is a theorem, so is A ∨ ∀x B(x)” (where x is not free in A) is not admissible for HA, if HA is consistent. Gödel numbering provides a quantifier-free
formula G(x) which (numeralwise) expresses the predicate “x is the code of a proof in HA of (0 = 1).” By intuitionistic logic with the decidability of quantifier-free arithmetical formulas, HA proves
∀x(∃yG(y) ∨ ¬G(x)). However, if HA proved ∃yG(y) ∨ ∀x¬G(x) then by the disjunction property, HA must prove either ∃yG(y) or ∀x¬G(x). The first case is impossible, by the existence property with the
consistency assumption and the fact that HA proves all true quantifier-free sentences. But the second case is also impossible, by Gödel's second incompleteness theorem, since ∀x¬G(x) expresses the
consistency of HA.
Intuitionistic systems have inspired a variety of interpretations, including Beth's tableaus, Rasiowa and Sikorski's topological models, formulas-as-types, Kleene's recursive realizabilities, the
Kleene and Aczel slashes, and models based on sheafs and topoi. Kripke's [1965] possible-world semantics, with respect to which intuitionistic predicate logic is complete and consistent, most
resembles classical model theory.
A Kripke structure K for L consists of a partially ordered set K of nodes and a domain function D assigning to each node k in K an inhabited set D(k), such that if k ≤ k′, then D(k) ⊆ D(k′). In
addition K has a forcing relation determined as follows.
For each node k let L(k) be the language extending L by new constants for all the elements of D(k). To each node k and each 0-ary predicate letter (each proposition letter) P, either assign f(P, k) =
true or leave f(P, k) undefined, consistent with the requirement that if k ≤ k′ and f(P, k) = true then f(P, k′) = true also. Say that
k forces P if and only if f(P, k) = true.
To each node k and each (n+1)-ary predicate letter Q(…), assign a (possibly empty) set T(Q, k) of (n+1)-tuples of elements of D(k) in such a way that if k ≤ k′ then T(Q, k) ⊆ T(Q, k′). Say that
k forces Q(d[0],…,d[n]) if and only if (d[o],…,d[n]) ∈ T(Q, k).
Now define forcing for compound sentences of L(k) inductively as follows:
• k forces (A & B) if k forces A and k forces B.
• k forces (A ∨ B) if k forces A or k forces B.
• k forces (A → B) if, for every k′ ≥ k, if k′ forces A then k′ forces B.
• k forces ¬A if for no k′ ≥ k does k′ force A.
• k forces ∀xA(x) if for every k′ ≥ k and every d ∈ D(k′), k′ forces A(d).
• k forces ∃xA(x) if for some d ∈ D(k), k forces A(d).
Any such forcing relation is consistent and monotone:
• for no sentence A and no k does k force both A and ¬A.
• if k ≤ k′ and k forces A then k′ forces A.
Kripke's Soundness and Completeness Theorems establish that a sentence of L is provable in intuitionistic predicate logic if and only if it is forced by every node of every Kripke structure. Thus to
show that (¬∀x¬P(x) → ∃xP(x)) is intuitionistically unprovable, it is enough to consider a Kripke structure with K = {k, k′}, k < k′, D(k) = D(k′) = {0}, T(P, k) empty but T(P, k′) = {0}. And to show
the converse is intuitionistically provable (without actually exhibiting a proof), one only needs the consistency and monotonicity properties of arbitrary Kripke models, with the definition of
Kripke models for languages with equality may interpret = at each node by an arbitrary equivalence relation, subject to monotonicity. For applications to intuitionistic arithmetic, normal models
(those in which equality is interpreted by identity at each node) suffice because equality of natural numbers is decidable.
Propositional Kripke semantics is particulary simple, since an arbitrary propositional formula is intuitionistically provable if and only if it is forced by the root of every Kripke model whose frame
(the set K of nodes together with their partial ordering) is a finite tree with a least element (the root). For example, the Kripke model with K = {k, k′, k ′′}, k < k′ and k < k′′, and with P true
only at k′, shows that both P ∨ ¬P and ¬P ∨ ¬¬P are unprovable in IPC.
Each terminal node or leaf of a Kripke model is a classical model, because a leaf forces every formula or its negation. Only those proposition letters which occur in a formula E, and only those nodes
k′ such that k≤k′, are relevant to deciding whether or not k forces E. Such considerations allow us to associate effectively with each formula E of L(IPC) a finite class of finite Kripke structures
which will include a countermodel to E if one exists. Since the class of all theorems of IPC is recursively enumerable, we conclude that
IPC is effectively decidable. There is a recursive procedure which determines, for each propositional formula E, whether or not E is a theorem of IPC, concluding with either a proof of E or a
Kripke countermodel.
The decidability of IPC was first obtained by Gentzen in 1933 as an immediate corollary of his Hauptsatz. The undecidability of IQC follows from the undecidability of CQC by the negative
Familiar non-intuitionistic logical schemata correspond to structural properties of Kripke models, for example
• DNS holds in every Kripke model with finite frame.
• (A → B) ∨ (B → A) holds in every Kripke model with linearly ordered frame. Conversely, every propositional formula which is not derivable in IPC + (A → B) ∨ (B → A) has a Kripke countermodel with
linearly ordered frame.
• If x is not free in A then (∀x(A∨B(x)) → (A ∨ ∀x B(x))) holds in every Kripke model K with constant domain (so that D(k) = D(k′) for all k, k′ in K). The same is true for MP.
Kripke models are a powerful tool for establishing properties of intuitionistic formal systems; cf. Troelstra and van Dalen [1988], Smorynski [1973], de Jongh and Smorynski [1976], Ghilardi [1999]
and Iemhoff [2001], [2005]. Following Gödel, Kreisel [1962] argued that Kripke-completeness of intuitionistic logic entailed Markov's Principle. By modifying the definition of a Kripke model to allow
“exploding nodes” which force every sentence, Veldman [1976] found an intuitionistic completeness proof avoiding (the informal use of) MP.
One way to implement the B-H-K explanation of intuitionistic truth for arithmetic is to associate with each sentence E of HA some collection of numerical codes for algorithms which could establish
the constructive truth of E. Following Kleene [1945], a number e realizes a sentence E of the language of arithmetic by induction on the logical form of E:
• e realizes (r = t), if (r = t) is true.
• e realizes (A & B), if e codes a pair (f,g) such that f realizes A and g realizes B.
• e realizes A∨B, if e codes a pair (f,g) such that if f = 0 then g realizes A, and if f > 0 then g realizes B.
• e realizes A→B, if, whenever f realizes A, then the eth partial recursive function is defined at f and its value realizes B.
• e realizes ¬A, if no f realizes A.
• e realizes ∀x A(x), if, for every n, the eth partial recursive function is defined at n and its value realizes A(n).
• e realizes ∃x A(x), if e codes a pair (n,g) and g realizes A(n).
An arbitrary formula is realizable if some number realizes its universal closure. Observe that not both A and ¬A are realizable, for any formula A. The fundamental result is
Nelson's Theorem [1947]. If A is derivable in HA from realizable formulas F, then A is realizable.
Some nonintuitionistic principles can be shown to be realizable. For example, Markov's Principle (for decidable formulas) can be expressed by the schema
(MP) ∀x(A(x) ∨ ¬A(x)) & ¬∀x¬A(x) → ∃x A(x).
Although unprovable in HA, MP is realizable by an argument which uses Markov's Principle informally. But realizability is a fundamentally nonclassical interpretation. In HA it is possible to express
an axiom of recursive choice CT (for “Church's Thesis”), which contradicts LEM but is (constructively) realizable. Hence by Nelson's Theorem, HA + MP + CT is consistent.
Kleene used a variant of number-realizability to prove HA satisfies the Church-Kleene Rule; the same argument works for HA with MP and/or CT. In Kleene and Vesley [1965] and Kleene [1969], functions
replace numbers as realizing objects, establishing the consistency of formalized intuitionistic analysis and its closure under a second-order version of the Church-Kleene Rule.
De Jongh [1970] combined realizability with Kripke modeling to show that intuitionistic predicate logic is arithmetically complete for HA. If, to each n-place predicate letter P(…), a formula f(P) of
L(HA) with n free variables is assigned, and if the formula f(A) of L(HA) comes from the formula A of L by replacing each prime formula P(x[1],…, x[n]) by f(P)(x[1],…, x[n]), then f(A) is called an
arithmetical substitution instance of A. A uniform assignment of simple existential formulas to predicate letters suffices to prove
De Jongh's Theorem. If a sentence A of the language L is not provable in IQC, then some arithmetical substitution instance of A is not provable in HA.
For example, if P(x, y) expresses “x codes a proof in HA of the formula with code y,” then ∀y (∃x P(x, y) ∨ ¬∃x P(x, y)) is unrealizable, hence unprovable in HA, and so is its double negation. (The
proof of de Jongh's Theorem for IPC does not need realizability; cf. Smorynski [1973]. As an example, Rosser's form of Gödel's Incompleteness Theorem provides a sentence C of L(HA) such that PA
proves neither C nor ¬C, so by the disjunction property HA cannot prove (C ∨ ¬C).)
Without claiming that number-realizability coincides with intuitionistic arithmetical truth, Nelson observed that for each formula A of L(HA) the predicate “y realizes A” can be expressed in HA by
another formula (abbreviated “y re A”), and the schema A ↔ ∃y (y re A) is consistent with HA. Troelstra [1973] showed that HA + (A ↔ ∃y (y re A)) is equivalent to HA + ECT, where ECT is a
strengthened form of CT. In HA + MP + ECT, which Troelstra considers to be a formalization of Russian recursive mathematics (cf. section 3.2 of the entry on constructive mathematics), every formula
of the form (y re A) has an equivalent “classical” prenex form A′(y) consisting of a quantifier-free subformula preceded by alternating “classical” quantifiers of the forms ¬¬∃x and ∀z¬¬, and so ∃y A
′(y) is a kind of prenex form of A.
At present there are several other entries in this encyclopedia treating intuitionistic logic in various contexts, but a general treatment of intermediate logics appears to be lacking so a brief one
is included here. A subintuitionistic propositional logic can be obtained from IPC by restricting the language, or weakening the logic, or both. An extreme example of the first is RN, intuitionistic
logic with a single propositional variable P, which is named after its discoverers Rieger and Nishimura [1960]. RN is characterized by the Rieger-Nishimura lattice of infinitely many nonequivalent
formulas F[n] such that every formula whose only propositional variable is P is equivalent by intuitionistic logic to some F[n]. Nishimura's version is
• F[∞] = P → P.
• F[0] = P & ¬ P.
• F[1] = P.
• F[2] = ¬ P.
• F[2n+3] = F[2n+1] ∨ F[2n+2].
• F[2n+4] = F[2n+3] → F[2n+1].
In RN neither F[2n + 1] nor F[2n + 2] implies the other; but F[2n] implies F[2n+1], and F[2n+1] implies each of F[2n+3] and F[2n+4].
Fragments of IPC missing one or more logical connective restrict the language and incidentally the logic, since the intuitionistic connectives &, ∨, →, ¬ are logically independent over IPC. Rose
[1953] proved that the implicationless fragment (without →) is complete with respect to realizability, in the sense that if every arithmetical substitution instance of a propositional formula E
without → is (number)-realizable then E is a theorem of IPC. This result contrasts with
Rose's Theorem [1953]. IPC is incomplete with respect to realizability. Let F be the propositional formula
((¬ ¬ D → D) → (¬ ¬ D ∨ ¬ D)) → (¬ ¬ D ∨ ¬ D)
where D is (¬ P ∨ ¬ Q) and P, Q are prime. Every arithmetical substitution instance of F is realizable (using classical logic), but F is not provable in IPC.
It follows that IPC is arithmetically incomplete for HA + ECT (cf. Section 5.2).
An intermediate propositional logic is any consistent collection of propositional formulas containing all the axioms of IPC and closed under modus ponens and substitution of arbitrary formulas for
proposition letters. Each intermediate propositional logic is contained in CPC. Some particular intermediate propositional logics, characterized by adding one or more classically correct but
intuitionistically unprovable axiom schemas to IPC, have been studied extensively.
One of the simplest intermediate propositional logics is the Gödel-Dummett logic LC, obtained by adding to IPC the schema (A → B) ∨ (B → A) which is valid on all and only those Kripke frames in which
the partial order of the nodes is linear. Gödel [1932] used an infinite sequence of successively stronger intermediate logics to show that IPC has no finite truth-table interpretation. For each
positive integer n, let G[n] be LC plus the schema (A[1] → A[2]) ∨ ... ∨ (A[1] & ... & A[n] → A[n+1]). Then G[n] is valid on all and only those linearly ordered Kripke frames with no more than n
The Kreisel-Putnam logic KP, obtained by adding to IPC the schema (¬ A → B ∨ C) → ((¬ A → B) ∨ (¬ A → C)), has the disjunction property but does not satisfy all the Visser rules. The intermediate
logic obtained by adding the schema ((¬ ¬ D → D) → (D ∨ ¬ D)) → (¬ ¬ D ∨ ¬ D), corresponding to Rose's counterexample, to IPC also has the disjunction property. Iemhoff [2005] proved that IPC is the
only intermediate propositional logic with the disjunction property which is closed under all the Visser rules. Iemhoff and Metcalfe [2009] developed a formal calculus for generalized admissibility
for IPC and some intermediate logics.
An intermediate propositional logic L is said to have the finite frame property if there is a class of finite frames on which the Kripke-valid formulas are exactly the theorems of L. Many
intermediate logics, including LC (the class of finite linear frames) and KP, have this property. De Jongh, Verbrugge and Visser [2009] proved that every intermediate logic L with the finite frame
property is the propositional logic of HA(L), that is, the class of all formulas in the language of IPC all of whose arithmetical substitution instances are provable in the logical extension of HA by
Some intermediate predicate logics, extending IQC and closed under substitution, are IQC + DNS (Section 4.1), IQC + MP (cf. Section 5.2), IQC + MP + IP (cf. Section 6.2), and the
intuitionistic logic of constant domains CD obtained by adding to IQC the schema ∀x(A∨B(x)) → (A ∨ ∀x B(x)) for all formulas A, B(x) with x not occurring free in A. Mints, Olkhovikov and Urquhart
[2012, Other Internet Resources] showed that CD does not have the interpolation property, refuting earlier published proofs by other authors.
Brouwer's influence on Gödel was significant, although Gödel never became an intuitionist. Gödel's [1933f] translation of intuitionistic propositional logic into the modal logic S4 is described in
Section 2.5 of the entry on Gödel and in Troelstra's introductory note to the translation of [1933f] in Volume I of Gödel's Collected Works. See also Mints [2012]. Kripke models for modal logic
predated those for intuitionistic logic.
An alternative to realizability semantics for intuitionistic arithmetic is Gödel's [1958] “Dialectica” interpretation, which associates with each formula B of L(HA) a quantifier-free formula B[D] in
the language of intuitionistic arithmetic of all finite types. The “Dialectica” interpretation of B, call it B^D, is ∃Y∀x B[D](Y, x). If B is a closed theorem of HA, then B[D](F, x) is provable for
some term F in Gödel's theory T of “primitive recursive” functionals of higher type. The translation from B to B^D requires the axiom of choice (at all finite types), MP and IP, so is not strictly
constructive; however, the number-theoretic functions expressible by terms F of T are precisely the provably recursive functions of HA (and of PA). The interpretation was extended to analysis by
Spector [1962]; cf. Howard [1973]. Clear expositions, and additional references, are to be found in Troelstra's introduction to the English translation in Gödel [1990] of the original Dialectica
article, and in Avigad and Feferman [1998].
While HA is a proper part of classical arithmetic, the intuitionistic attitude toward mathematical objects results in a theory of real numbers (cf. sections 3.4–3.7 of the entry on intuitionism in
the philosophy of mathematics) diverging from the classical. Kleene's function-realizability interpretation, developed to prove the consistency of his formalization FIM of the intuitionistic theory
of sequences (“intuitionistic analysis”), changes the interpretation of arithmetical formulas; for example, ¬ ¬∀x (A(x) ∨ ¬A(x)) is function-realizable for every arithmetical formula A(x). In the
language of analysis, Markov's Principle and the negative translation of the countable axiom of choice are among the many non-intuitionistic principles which are function-realizable (by classical
arguments) and hence consistent with FIM; cf. Kleene [1965], Vesley [1972] and Moschovakis [2003].
Concrete and abstract realizability semantics for a wide variety of formal systems have been developed and studied by logicians and computer scientists; cf. Troelstra [1998] and van Oosten [2002] and
[2008]. Variations of the basic notions are especially useful for establishing relative consistency and relative independence of the nonlogical axioms in theories based on intuitionistic logic; some
examples are Moschovakis [1971], Lifschitz [1979], and the realizability notions for constructive and intuitionistic set theories developed by Rathjen [2006, 2012] and Chen [2012]. Early abstract
realizability notions include the slashes of Kleene [1962, 1963] and Aczel [1968], and Läuchli [1970]. Kohlenbach, Avigad and others have developed realizability interpretations for parts of
classical mathematics.
Artemov's justification logic is an alternative interpretation of the B-H-K explanation of the intuitionistic connectives and quantifiers, with (idealized) proofs playing the part of realizing
objects. See also Artemov and Iemhoff [2007].
Another line of research in intuitionistic logic concerns Brouwer's very controversial “creating subject counterexamples” to principles of classical analysis (such as Markov's Principle) which could
not be refuted on the basis of the theory FIM of Kleene and Vesley [1965]. By weakening Kleene's form of Brouwer's principle of continuous choice, and adding an axiom he called Kripke's Schema (KP),
Myhill managed to formalize the creating subject arguments. KP is inconsistent with FIM, but Vesley [1970] found an alternative principle (Vesley's Schema (VS)) which can consistently be added to FIM
and implies all the counterexamples for which Brouwer required a creating subject. Krol and Scowcroft gave topological consistency proofs for intuitionistic analysis with Kripke's Schema and weak
The entry on L. E. J. Brouwer discusses Brouwer's philosophy and mathematics, with a chronology of his life and a selected list of publications including translations and secondary sources. The best
way to learn more is to read some of the original papers. English translations of Brouwer's doctoral dissertation and other papers which originally appeared in Dutch, along with a number of articles
in German, can be found in L. E. J. Brouwer: Collected Works [1975], edited by Heyting. Benacerraf and Putnam's essential source book contains Brouwer [1912] (in English translation), Brouwer [1949]
and Dummett [1975]. Mancosu's [1998] provides English translations of many fundamental articles by Brouwer, Heyting, Glivenko and Kolmogorov, with illuminating introductory material by W. van Stigt
whose [1990] is another valuable resource.
The third edition [1971] of Heyting's classic [1956] is an attractive introduction to intuitionistic philosophy, logic and mathematical practice. As part of the formidable project of editing and
publishing Brouwer's Nachlass, van Dalen [1981] provides a comprehensive view of Brouwer's own intuitionistic philosophy. The English translation, in van Heijenoort [1969], of Brouwer's [1927] (with
a fine introduction by Parsons) is still an indispensable reference for Brouwer's theory of the continuum. Veldman [1990] and [2005] are authentic modern examples of traditional intuitionistic
mathematical practice. Troelstra [1991] places intuitionistic logic in its historical context as the common foundation of constructive mathematics in the twentieth century. Bezhanishvili and de Jongh
[2005, Other Internet Resources] includes very recent developments in intuitionistic logic.
Kleene and Vesley's [1965] gives a careful axiomatic treatment of intuitionistic analysis, a proof of its consistency relative to a classically correct subtheory, and an extended application to
Brouwer's theory of real number generators. Kleene's [1969] formalizes the theory of partial recursive functionals, enabling precise formalizations of the function-realizability interpretation used
in [1965] and of a related q-realizability interpretation which gives the Church-Kleene Rule for intuitionistic analysis.
Troelstra's [1973], Beeson's [1985] and Troelstra and van Dalen's [1988] stand out as the most comprehensive studies of intuitionistic and semi-intutionistic formal theories, using both constructive
and classical methods, with useful bibliographies. Troelstra's [1998] presents formulas-as-types and (Kleene and Aczel) slash interpretations for propositional and predicate logic, as well as
abstract and concrete realizabilities for a multitude of applications. Martin-Löf's constructive theory of types [1984] (cf. Section 3.4 of the entry on constructive mathematics) provides another
general framework within which intuitionistic reasoning continues to develop.
• Aczel, P., 1968, “Saturated intuitionistic theories,” in H.A. Schmidt, K. Schütte, and H.-J. Thiele (eds.), Contributions to Mathematical Logic, Amsterdam: North-Holland: 1–11.
• Artemov, S. and Iemhoff, R., 2007, “The basic intuitionistic logic of proofs,” Journal of Symbol Logic, 72: 439–451.
• Avigad, J. and Feferman, S., 1998, “Gödel's functional (”Dialectica“) interpretation,” Chapter V of Buss (ed.) 1998: 337–405.
• Bar-Hillel, Y. (ed.), 1965, Logic, Methodology and Philosophy of Science, Amsterdam: North Holland.
• Beeson, M. J., 1985, Foundations of Constructive Mathematics, Berlin: Springer.
• Benecerraf, P. and Hilary Putnam (eds.), 1983, Philosophy of Mathematics: Selected Readings, 2nd Edition, Cambridge: Cambridge University Press.
• Brouwer, L. E. J., 1907, “On the Foundations of Mathematics,” Thesis, Amsterdam; English translation in Heyting (ed.) 1975: 11–101.
• Brouwer, L. E. J., 1908, “The Unreliability of the Logical Principles,” English translation in Heyting (ed.) 1975: 107–111.
• Brouwer, L. E. J., 1912, “Intuitionism and Formalism,” English translation by A. Dresden, Bulletin of the American Mathematical Society, 20 (1913): 81–96, reprinted in Benacerraf and Putnam
(eds.) 1983: 77–89; also reprinted in Heyting (ed.) 1975: 123–138.
• Brouwer, L. E. J., 1923, 1954, “On the significance of the principle of excluded middle in mathematics, especially in function theory,” “Addenda and corrigenda,” and “Further addenda and
corrigenda,” English translation in van Heijenoort (ed.) 1967: 334–345.
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Peters, 51–59.
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Gödel 1986: 222–225.
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• Gödel, K., 1933f, “Eine Interpretation des intuitionistischen Aussagenkalküls,” reproduced and translated with an introductory note by A. S. Troelstra in Gödel 1986: 296–304.
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• Iemhoff, R., 2005, “Intermediate logics and Visser's rules,” Notre Dame Journal of Formal Logic, 46: 65–81.
• Iemhoff, R. and Metcalfe, G., 2009, “Proof theory for admissible rules,” Annals of Pure and Applied Logic, 159: 171–186.
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• de Jongh, D. H. J., and Smorynski, C., 1976, “Kripke models and the intuitionistic theory of species,” Annals of Mathematical Logic, 9: 157–186.
• de Jongh, D., Verbrugge, R. and Visser, A., 2011, “Intermediate logics and the de Jongh property,” Archive for Mathematical Logic, 50: 197–213.
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• Kleene, S. C., 1963, “An addendum,” Journal of Symbolic Logic, 28: 154–156.
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• Läuchli, H., 1970, “An abstract notion of realizability for which intuitionistic predicate calculus is complete,” in A. Kino et. al. (eds.) 1965: 227–234.
• Lifschitz, V., 1979, “CT[0] is stronger than CT[0]!,” Proceedings of the American Mathematical Society, 73: 101–106.
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Springer-Verlag: 487-491.
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• Moschovakis, J. R., 2003, “Classical and constructive hierarchies in extended intuitionistic analysis,” Journal of Symbolic Logic, 68: 1015–1043.
• Moschovakis, J. R., 2009, “The logic of Brouwer and Heyting,” in Logic from Russell to Church (Handbook of the History of Logic, Volume 5), J. Woods and D. Gabbay (eds.), Amsterdam: Elsevier:
• Nishimura, I., 1960, “On formulas of one variable in intuitionistic propositional calculus,” Journal of Symbolic Logic, 25: 327–331.
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How to cite this entry.
Preview the PDF version of this entry at the Friends of the SEP Society.
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Enhanced bibliography for this entry at PhilPapers, with links to its database.
Over the years, many readers have offered corrections and improvements. This revision owes special thanks to Edward Horton for observing that replacing ex falso quodlibet by the LEM in the axioms for
IPC does not yield all of CPC, and for providing the correct substitutions. I am also indebted to Chen Huan, Willemien and Tim Kirschner for corrections, to Nikos Vaporis for his MPLA master's thesis
on the admissible rules of intermediate logics, to Rosalie Iemhoff for recent developments in intuitionistic proof theory, and no doubt to others whose names I have forgotten. | {"url":"https://plato.stanford.edu/archivES/FALL2017/Entries/logic-intuitionistic/","timestamp":"2024-11-12T06:52:48Z","content_type":"text/html","content_length":"107354","record_id":"<urn:uuid:4ec7d50c-d92c-4da2-81ed-be3f9ab9529b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00174.warc.gz"} |
When is the algorithm concept pertinent – and when not? Thoughts about algorithms and paradigmatic examples, and about algorithmic and non-algorithmic mathematical cultures
Until some decades ago, it was customary to discuss much pre-Modern
mathematics as “algebra”, without agreement between workers about what was
to be understood by that word. Then this view came under heavy fire, often with
no more precision.
Now it has instead become customary to classify pre-Modern practical
arithmetic as “algorithmic mathematics”. In so far as any computation in several
steps can be claimed to follow an underlying algorithm (just as it can be
explained from an “underlying theorem”, for instance from proportion theory),
this is certainly justified. Traditionally, however, historians as well as the sources
would speak of a rule.
The paper first goes through some of the formative appeals to the algebraic
interpretation – Eisenlohr, Zeuthen, Neugebauer – as well as some of the better
argued attacks on it (Rodet, Mahoney).
Next it asks for the reasons to introduce the algorithmic interpretation, and
discusses the adequacy or inadequacy of some uses.
Finally, it investigates in which sense various pre-modern mathematical
cultures can be characterized globally as “algorithmic”, concluding that this
characterization fits ancient Chinese and Sanskrit mathematics but neither early
second-millennium Mediterranean practical arithmetic (including Fibonacci and
the Italian abbacus tradition), nor the Old Babylonian corpus.
Konference International Conference on the History of Ancient Mathematics and Astronomy
Lokation Northwest University, Xi'an, China
Land/Område Kina
By Xi'an
Periode 23/08/2015 → 29/08/2015
Andet In memory of Professor Li Jimin (1938-1993) | {"url":"https://forskning.ruc.dk/da/publications/when-is-the-algorithm-concept-pertinent-and-when-not-thoughts-abo","timestamp":"2024-11-08T21:50:28Z","content_type":"text/html","content_length":"47144","record_id":"<urn:uuid:be5955e7-1aeb-43da-a063-9dc00ed7a52b>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00050.warc.gz"} |
CS 3200: Introduction to Scientific Computing Assignment 5 solved
1. Solve the coffee cup problem analytically for 5 minutes based upon Newton’s law of cooling:
π π π π π π
π π π π = β π π (π π π π β π π π π )
π π π π = 19β , π π π π = 84β , π π = 0.025/π π π π π π π π π π π π
By making sure the supplied program runs using Matlab. Write the function Tsexact which is
the analytical solution for use in the supplied matlab code. Use the solution to the in class
activity to help you write this function.
2. Using the supplied program to solve the coffee cup problem using the Forward Euler
Method calculate and compare your results to the analytical answer given by the provided
β ’ Plot the results for all algorithms using several different values for the step size β (β =
30π π , 15π π , 10π π , 5π π , 1π π , 0.5π π , 0.25π π )
β ’ Using these results estimate the order of the error after the first step and at the end of
the integration. Describe how the error changes with changes in β . Use a graph
3. Implement the ODE23 method described in the book Moler in the test program you have
been given . For the standard equation dy/dt = f(t,y) (note y here is Tc above) the method
is given by
β ’ S1 = f(tn,yn)
β ’ S2 = f(tn + h/2, yn+h/2 S1)
β ’ S3 = f(tn + 3h/4, yn+3h/4 S2)
β ’ tn+1 = tn + h
β ’ yn+1 = yn + h/9 ( 2S1 + 3S2 + 4S3)
β ’ S4 = f(tn+1, yn+1)
β ’ Errorn+1 = h/72 ( -5S1 + 6S2 + 8S3 -9 S4)
4. Plot the results for this algorithms using several different values for the step size β (β =
30π π , 15π π , 10π π , 5π π , 1π π , 0.5π π , 0.25π π )
5. Using these results estimate the order of the error after the first step and at the end of the
integration. Compare the actual error on the first step with the predicted error on the first
6. Change the value of r in the problem being solved to r = 0.6. Does the error estimator blow
up in the same way as the solution when the solution becomes unstable?
What to turn in
For these assignments, we expect both SOURCE CODES and a written REPORT be uploaded
as a zip or tarball file to Canvas.
β ’ Source code for all programs that you write, thoroughly documented.
o Include a README file describing how to compile and run your code.
β ’ Your report should be in PDF format and should stand on its own.
o It should describe the methods used.
o It should explain your results and contain figures.
o It should also answer any questions asked above.
o It should cite any sources used for information, including source code.
o It should list all of your collaborators.
This homework is due on April 14th by 11:59 pm. If you don’t understand these directions, please
send questions to teach-cs3200@list.eng.utah.edu or come see one of the TAs or the instructor
during office hours well in advance of the due date. | {"url":"https://codeshive.com/questions-and-answers/cs-3200-introduction-to-scientific-computing-assignment-5-solved/","timestamp":"2024-11-08T18:05:50Z","content_type":"text/html","content_length":"103521","record_id":"<urn:uuid:7f271a48-0ec2-4fd2-a47f-18b2e2f1eff7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00385.warc.gz"} |
How do you calculate how tall you will grow?
How do you calculate how tall you will grow?
There are also some very simple, but less accurate, methods available. One of them is adding 2.5 inches (7.6 cm) to the average of the parent’s height for a boy and subtracting 2.5 inches (7.6 cm)
for a girl. The second calculator above is based on this method.
What is the most accurate height calculator?
The Khamis-Roche child height predictor calculates your child’s future height by using parents’ height, child’s current height, child’s current weight, and child’s gender. It’s the most accurate
method of predicting a child’s height without determining the skeletal age.
How can you tell if your still growing?
Look for signs of growth.
1. Short pant legs are an easy way to tell that you must be growing. If the jeans you used to have to roll up now make you look like you’re ready for a flood, it may be time to take a height
measurement (as well as buy some new jeans).
2. Foot growth is another likely sign of height growth.
How do I know if my child will be tall?
What’s the best way to predict a child’s adult height?
• Add the mother’s height to the father’s height in either inches or centimeters.
• Add 5 inches (13 centimeters) for boys or subtract 5 inches (13 centimeters) for girls.
• Divide by 2.
How do I use the height calculator?
Enter their height in feet and inches or in cm or meters. If doing this for your child or a child patient, enter their age and height. Finally, press “Predict Adult Height” to get the estimated adult
height in centimeters and feet and inches from the height predictor. The calculator works great when used to predict the height of a teenager as well.
How does the child growth calculator work?
The calculator is a straightforward tool where you enter your child’s birthday, gender, height, weight (in kg or lbs) and head circumference. Once all the data is completely entered, the calculator
interprets these values using different growth charts by the WHO. All the charts are interpreted similarly.
Can a growth chart predict the adult height of a child?
Measurements such as height, weight, and head circumference of a child can be compared to the expected values based on data from these growth charts of children of the same age and sex. In general,
children maintain a fairly constant growth curve, which is why these charts can be used to predict the adult height of a child to a certain extent.
What is the height percentile calculator?
Height Percentile Calculator. Use this height percentile calculator to calculate how tall or short you are relative to the general population (select ‘The World’) or to people of a specific gender,
age, or country. Child height percentiles are only available for U.S. citizens (based on CDC data). Newborns, toddlers, and infants data is based | {"url":"https://www.tonyajoy.com/2022/11/01/how-do-you-calculate-how-tall-you-will-grow/","timestamp":"2024-11-03T00:05:24Z","content_type":"text/html","content_length":"48284","record_id":"<urn:uuid:372e2b9d-dec5-4c9b-a278-0fe3f5d97852>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00232.warc.gz"} |
# -*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
from keras.layers import Dense, Activation, SimpleRNN
from keras.models import Sequential
import codecs
INPUT_FILE = "C:\\Users\\admin\\log\\11-0.txt"
# extract the input as a stream of characters
print("Extracting text from input...")
with codecs.open(INPUT_FILE, "r", encoding="utf-8") as f:
lines = [line.strip().lower() for line in f
if len(line) != 0]
text = " ".join(lines)
# creating lookup tables
# Here chars is the number of features in our character "vocabulary"
chars = set(text)
nb_chars = len(chars)
char2index = dict((c, i) for i, c in enumerate(chars))
index2char = dict((i, c) for i, c in enumerate(chars))
# create inputs and labels from the text. We do this by stepping
# through the text ${step} character at a time, and extracting a
# sequence of size ${seqlen} and the next output char. For example,
# assuming an input text "The sky was falling", we would get the
# following sequence of input_chars and label_chars (first 5 only)
# The sky wa -> s
# he sky was ->
# e sky was -> f
# sky was f -> a
# sky was fa -> l
print("Creating input and label text...")
SEQLEN = 10
STEP = 1
input_chars = []
label_chars = []
for i in range(0, len(text) - SEQLEN, STEP):
input_chars.append(text[i:i + SEQLEN])
label_chars.append(text[i + SEQLEN])
# vectorize the input and label chars
# Each row of the input is represented by seqlen characters, each
# represented as a 1-hot encoding of size len(char). There are
# len(input_chars) such rows, so shape(X) is (len(input_chars),
# seqlen, nb_chars).
# Each row of output is a single character, also represented as a
# dense encoding of size len(char). Hence shape(y) is (len(input_chars),
# nb_chars).
print("Vectorizing input and label text...")
X = np.zeros((len(input_chars), SEQLEN, nb_chars), dtype=np.bool)
y = np.zeros((len(input_chars), nb_chars), dtype=np.bool)
for i, input_char in enumerate(input_chars):
for j, ch in enumerate(input_char):
X[i, j, char2index[ch]] = 1
y[i, char2index[label_chars[i]]] = 1
# Build the model. We use a single RNN with a fully connected layer
# to compute the most likely predicted output char
HIDDEN_SIZE = 128
BATCH_SIZE = 128
NUM_ITERATIONS = 25
NUM_EPOCHS_PER_ITERATION = 1
NUM_PREDS_PER_EPOCH = 100
model = Sequential()
model.add(SimpleRNN(HIDDEN_SIZE, return_sequences=False,
input_shape=(SEQLEN, nb_chars),
model.compile(loss="categorical_crossentropy", optimizer="rmsprop")
# We train the model in batches and test output generated at each step
for iteration in range(NUM_ITERATIONS):
print("=" * 50)
print("Iteration #: {}".format(iteration))
model.fit(X, y, batch_size=BATCH_SIZE, epochs=NUM_EPOCHS_PER_ITERATION)
# testing model
# randomly choose a row from input_chars, then use it to
# generate text from model for next 100 chars
test_idx = np.random.randint(len(input_chars))
test_chars = input_chars[test_idx]
print("Generating from seed: {}".format(test_chars))
print(test_chars, end="")
for i in range(NUM_PREDS_PER_EPOCH):
Xtest = np.zeros((1, SEQLEN, nb_chars))
for j, ch in enumerate(test_chars):
Xtest[0, j, char2index[ch]] = 1
pred = model.predict(Xtest, verbose=0)[0]
ypred = index2char[np.argmax(pred)]
print(ypred, end="")
# move forward with test_chars + ypred
test_chars = test_chars[1:] + ypred
Extracting text from input...
Creating input and label text...
Vectorizing input and label text...
Iteration #: 0
Epoch 1/1
162739/162739 [==============================] - 9s 55us/step - loss: 2.3730
Generating from seed: as gone, a
as gone, and the the the the the the the the the the the the the the the the the the the the the the the the t
Iteration #: 1
Epoch 1/1
162739/162739 [==============================] - 9s 56us/step - loss: 2.0644
Generating from seed: ing, the q
ing, the queen the har she for she for she for she for she for she for she for she for she for she for she for
Iteration #: 2
Epoch 1/1
162739/162739 [==============================] - 10s 60us/step - loss: 1.9566
Generating from seed: so i shoul
so i should the was an the was an the was an the was an the was an the was an the was an the was an the was an
Iteration #: 3
Epoch 1/1
162739/162739 [==============================] - 9s 58us/step - loss: 1.8721
Generating from seed: of her fav
of her fave and the say har she said the doong to the pooked and the say har she said the doong to the pooked
Iteration #: 4
Epoch 1/1
162739/162739 [==============================] - 9s 58us/step - loss: 1.8049
Generating from seed: ole pack o
ole pack out of the said the was the was the was the was the was the was the was the was the was the was the w
Iteration #: 5
Epoch 1/1
162739/162739 [==============================] - 10s 62us/step - loss: 1.7491
Generating from seed: d a vague
d a vague to see the said the doon and the said the doon and the said the doon and the said the doon and the s
Iteration #: 6
Epoch 1/1
162739/162739 [==============================] - 10s 61us/step - loss: 1.7034
Generating from seed: can;--but
can;--but it was the dore of the the say and when the dormouse was the dore of the the say and when the dormou
Iteration #: 7
Epoch 1/1
162739/162739 [==============================] - 10s 61us/step - loss: 1.6628
Generating from seed: cense. 1.
cense. 1.e. ‘which was it was the project gutenberg-tm electronic works in a little began the was the projec
Iteration #: 8
Epoch 1/1
162739/162739 [==============================] - 10s 62us/step - loss: 1.6296
Generating from seed: e a secure
e a secures and she can the grope a growing the gryphon the growes and the gryphon the growes and the gryphon
Iteration #: 9
Epoch 1/1
162739/162739 [==============================] - 10s 63us/step - loss: 1.6003
Generating from seed: thing i a
thing i and and all the was and she said the dormouse she had for the ward alice a look at the rabbit harder
Iteration #: 10
Epoch 1/1
162739/162739 [==============================] - 10s 62us/step - loss: 1.5755
Generating from seed: call it s
call it she went on an the court in a monether she can the hatter was she can the hatter was she can the hatt
Iteration #: 11
Epoch 1/1
162739/162739 [==============================] - 10s 64us/step - loss: 1.5533
Generating from seed: at’s all t
at’s all the same a down and the dittle beanther a look to the choored at the mouse she was she cat all the fi
Iteration #: 12
Epoch 1/1
162739/162739 [==============================] - 11s 65us/step - loss: 1.5351
Generating from seed: aw alice.
aw alice. ‘i could be no mary dore or a little beat had not and be not lear the was of the pare of the same as
Iteration #: 13
Epoch 1/1
162739/162739 [==============================] - 10s 62us/step - loss: 1.5164
Generating from seed: ou our cat
ou our cat a little she said to herself and the project gutenberg-tm electronic works in a little she said to
Iteration #: 14
Epoch 1/1
162739/162739 [==============================] - 10s 61us/step - loss: 1.5011
Generating from seed: her alarme
her alarmed in the dormouse the office the dormouse the office the dormouse the office the dormouse the office
Iteration #: 15
Epoch 1/1
162739/162739 [==============================] - 10s 62us/step - loss: 1.4859
Generating from seed: ‘shan’t,’
‘shan’t,’ said the caterpillar to the court in the court in the court in the court in the court in the court i
Iteration #: 16
Epoch 1/1
162739/162739 [==============================] - 11s 65us/step - loss: 1.4737
Generating from seed: e last con
e last confuring the project gutenberg-tm electronic works in a mare or a little bean to the queen of the mous
Iteration #: 17
Epoch 1/1
162739/162739 [==============================] - 11s 67us/step - loss: 1.4626
Generating from seed: st at firs
st at first the was a long as it was got the caterpillar with the caterpillar with the caterpillar with the ca
Iteration #: 18
Epoch 1/1
162739/162739 [==============================] - 10s 64us/step - loss: 1.4519
Generating from seed: anxiously
anxiously at the mock turtle sure to herself the king of the mock turtle sure to herself the king of the mock
Iteration #: 19
Epoch 1/1
162739/162739 [==============================] - 11s 67us/step - loss: 1.4414
Generating from seed: ith no oth
ith no other look at the was the was the was the was the was the was the was the was the was the was the was t
Iteration #: 20
Epoch 1/1
162739/162739 [==============================] - 11s 68us/step - loss: 1.4332
Generating from seed: the white
the white rabbit hard alice the thing it so election in an a moraly and alice was a little be the way was a l
Iteration #: 21
Epoch 1/1
162739/162739 [==============================] - 11s 65us/step - loss: 1.4251
Generating from seed: t on plann
t on planned and alice a little she said to herself ‘what is a grow in a come to got be a looking and alice a
Iteration #: 22
Epoch 1/1
162739/162739 [==============================] - 10s 59us/step - loss: 1.4172
Generating from seed: the hatte
the hatter was a little book the read the read the read the read the read the read the read the read the read
Iteration #: 23
Epoch 1/1
162739/162739 [==============================] - 9s 57us/step - loss: 1.4103
Generating from seed: are went ‘
are went ‘and they went on, ‘i peniest it, and they went on, ‘i peniest it, and they went on, ‘i peniest it, a
Iteration #: 24
Epoch 1/1
162739/162739 [==============================] - 9s 57us/step - loss: 1.4035
Generating from seed: the hatte
the hatter was the look of the state her full she had not a large and making of the mock turtle she had not a | {"url":"https://www.erestage.com/develop/tkp7/","timestamp":"2024-11-11T10:33:01Z","content_type":"text/html","content_length":"157548","record_id":"<urn:uuid:2cb1bde5-9031-4acb-805e-bdc1e9bd8802>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00292.warc.gz"} |
change all the negative value into positive value
Last seen 3.4 years ago
I have a protein matrix, and want to use the justvsn function to normalize it.
After normalization, I found all the data are negative, the downstream function only works on positive value.
How can I change the parameter of vsn , them the output can be positive value?
Thank you in advance for great help!
Entering edit mode
Dear Wolfgang,
Thank you so much for your so detail explanation!
My data is protein SILAC ratio light/heavy data. Just as yo said, the value is too small.
I I tried to normalize the data, and then use Deseq2 to find the diffferentially expressed proteins. I used VSN, Rlr(performGlobalRLRNormalization), LoessF (performCyclicLoessNormalization), median
(globalIntensityNormalization). Of course, vsn2 has the best perform.
Would you please give me some ideas to deal with this kind of data?
Thank you in advance for our great help and really appreciated!
Entering edit mode
DESeq2 is strictly for count data, which you can recognize by the fact that they are non-negative integer numbers. It is not intended for (quasi)-continuous data. For general differential abundance
testing, please use limma.
A workflow of vsn2 followed by limma seems reasonable for your use case, although of course I cannot comment on the suitability (i.e., quality) of your specific data. The package arrayQualityMetrics
or at least some of the plots it suggests (e.g. https://www.huber.embl.de/arrayQualityMetrics/Report_for_nCCl4/) may be useful.
Entering edit mode
Hello Wolfgang,
Thank you so much for your great help!
Thank you1 | {"url":"https://support.bioconductor.org/p/p133410/#p133522","timestamp":"2024-11-04T18:40:11Z","content_type":"text/html","content_length":"23570","record_id":"<urn:uuid:6aba8fb8-c12b-4678-9ec0-d4aa4087010f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00544.warc.gz"} |
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Modified Eccentric Connectivity Index and Polynomial of Tetragonal Carbon Nanocones
Advances in Analysis
Modified Eccentric Connectivity Index and Polynomial of Tetragonal Carbon Nanocones CNC[4][n]
• Linli Zhu^*
School of Computer Engineering, Jiangsu University of Technology, Changzhou 213001, China
• Wei Gao
School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
Chemical compounds and drugs are often modeled as graphs where each vertex represents an atom of molecule and covalent bounds between atoms are represented by edges between the corresponding
vertices. This graph derived from a chemical compounds is called its molecular graph. The modified eccentric connectivity index defined over this molecular graph has been shown to be strongly
correlated to oxidizing properties of the compounds. In this article, by virtue of molecular structural analysis, the modified eccentric connectivity index and modified eccentric connectivity
polynomial of tetragonal carbon nanocones CNC[4][n] are reported. The theoretical results achieved in this article illustrate the promising prospects of the application to the chemical and pharmacy
Theoretical chemistry, modified eccentric connectivity index, tetragonal carbon nanocone
[1] M. R. Farahani, W. Gao, “On Multiplicative and Redefined Version of Zagreb Indices of V-Phenylenic Nanotubes and Nanotorus”, British Journal of Mathematics & Computer Science, vol. 13, no. 5, pp.
1–8, 2016.
[2] M. R. Farahani, W. Gao, M. R. Rajesh Kanna, “On the Omega Polynomial of A Family of Hydrocarbon Moleculs ‘Polycyclic Aromatic Hydrocarbons PAHk’”, Asian Academic Research Journal of
Multidisciplinary, vol. 2, no. 7, pp. 263-268, 2015.
[3] M. R. Farahani, W. Gao, M. R. Rajesh Kanna, “The Connective Eccentric Index for An Infinite Family of Dendrimers”, Indian Journal of Fundamental and Applied Life Sciences, vol. 5, no. S4, pp.
766-771, 2015.
[4] M. R. Farahani, M. R. Rajesh Kanna, W. Gao, “The Edge-Szeged Index of the Polycyclic Aromatic Hydrocarbons PAHk”, Asian Academic Research Journal of Multidisciplinary, vol. 5, no. 6, pp. 136-142,
[5] M. R. Farahani, W. Gao, “The multiply version of Zagreb indices of a family of molecular graph”, Journal of Chemical and Pharmaceutical Research, vol. 7, no. 10, pp. 535-539, 2015.
[6] M. R. Farahani, W. Gao, “The Schultz Index and Schultz Polynomial of the Jahangir Graphs J5,m,”, Applied Mathematics, no. 6, pp. 2319-2325, 2015.
[7] M. R. Farahani, M. R. Rajesh Kanna, W. Gao, “The Schultz, modified Schultz indices and their polynomials of the Jahangir graphs Jn,m for integer numbers n=3, m ≥ 3”, Asian Journal of Applied
Sciences, vol. 3, no. 6, pp. 823-827, 2014.
[8] W. Gao, M. R. Farahani, “Degree-based indices computation for special chemical molecular structures using edge dividing method”, Applied Mathematics and Nonlinear Sciences, vol. 1, no. 1, pp.
94–117, 2015.
[9] W. Gao, M. R. Farahani, “The Theta polynomial Θ(G,x) and the Theta index Θ(G) of molecular graph Polycyclic Aromatic Hydrocarbons PAHk”, Journal of Advances in Chemistry, vol. 12, no. 1, pp.
3934- 3939, 2015.
[10] W. Gao, Y. Gao, “A Note on Connectivity and λ-Modified Wiener Index”, Iranian Journal of Mathematical Chemistry, vol. 6, no. 2, pp. 137-143, 2015.
[11] J. A. Bondy, U. S. R. Mutry. “Graph Theory”, Springer, Berlin, pp. 1-40, 2008.
[12] A. R. Ashrafi, M. Ghorbani, “A study of fullerenes by MEC polynomials”, Electronic Materials Letters, vol. 6, no. 2, pp. 87-90, 2010.
[13] M. Alaeiyan, J. Asadpour, R. Mojarad, “A numerical method for MEC polynomial and MEC index of one-pentagonal carbon nanocones”, Fullerenes, Nanotubes and Carbon Nanostructures, vol. 21, no. 10,
pp. 825–835, 2013.
[14] A. R. Ashrafi, M. Ghorbani, M. A. Hossein-Zadeh, “The eccentric connectivity polynomial of some graph operations”, Serdica Journal of computing, vol. 5, no. 2, pp. 101–116, 2011.
[15] N. Trinajstic, “Chemical Graph Theory”. CRC Press, Boca Raton, 1992.
[16] V. Kumar, A. K. Modan, “Apllication of graph theory: Models for prediction of carbonic anhydrase inhibitory activity of sulfonamides”, J Math Chem., no. 42, pp. 925-940, 1991. | {"url":"http://isaacpub.org/3/1167/2/1/01/2017/AAN.html","timestamp":"2024-11-09T07:07:33Z","content_type":"text/html","content_length":"6919","record_id":"<urn:uuid:61a87e60-7392-4a2f-8aad-c091bdc1bb51>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00805.warc.gz"} |
Volume 51, pp. 1-14, 2019.
Bernstein fractal approximation and fractal full Müntz theorems
Vijender Nallapu
Fractal interpolation functions defined by means of suitable Iterated Function Systems provide a new framework for the approximation of continuous functions defined on a compact real interval.
Convergence is one of the desirable properties of a good approximant. The goal of the present paper is to develop fractal approximants, namely Bernstein $\alpha$-fractal functions, which converge to
the given continuous function even if the magnitude of the scaling factors does not approach zero. We use Bernstein $\alpha$-fractal functions to construct the sequence of Bernstein Müntz fractal
polynomials that converges to either $f\in \mathcal{C}(I)$ or $f\in L^p(I), 1 \le p < \infty.$ This gives a fractal analogue of the full Müntz theorems in the aforementioned function spaces. For a
given sequence $\{f_n(x)\}^{\infty}_{n=1}$ of continuous functions that converges uniformly to a function $f\in \mathcal{C}(I),$ we develop a double sequence $\big\{\{f_{n,l}^{\alpha}(x)\}^\infty_{l=
1}\big\}^\infty_{n=1}$ of Bernstein $\alpha$-fractal functions that converges uniformly to $f$. By establishing suitable conditions on the scaling factors, we solve a constrained approximation
problem of Bernstein $\alpha$-fractal Müntz polynomials. We also study the convergence of Bernstein fractal Chebyshev series.
Full Text (PDF) [503 KB], BibTeX
Key words
Bernstein polynomials, fractal approximation, convergence, full Müntz theorems, Chebyshev series, box dimension.
AMS subject classifications
41A30, 28A80, 41A17, 41A50.
Links to the cited ETNA articles
[16] Vol. 20 (2005), pp. 64-74 M. A. Navascues: Fractal trigonometric approximation
[29] Vol. 41 (2014), pp. 420-442 Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand: $\alpha$-fractal rational splines for constrained interpolation
< Back | {"url":"https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol51/abstract.php?vol=51&pages=1-14","timestamp":"2024-11-07T10:43:10Z","content_type":"application/xhtml+xml","content_length":"9426","record_id":"<urn:uuid:877f89ea-3ca7-44f5-a9cb-81fe16ba6c82>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00027.warc.gz"} |
Davide DAL MARTELLO - Papers
D. Dal Martello and M. Mazzocco
Generalised double affine Hecke algebras, their representations, and higher Teichmüller theory
Advances in Mathematics (2024) 10.1016/j.aim.2024.109763
Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In
this paper, we first construct a functor that sends representations of the \tilde{D}_4-type GDAHA to representations of the \tilde{E}_6-type one for specialised parameters. Then, under no
restrictions on the parameters, we construct embeddings of both GDAHAs of type \tilde{D}_4 and \tilde{E}_6 into matrix algebras over quantum cluster \mathcal{X}-varieties, thus linking to the theory
of higher Teichmüller spaces. For \tilde{E}_6, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations. | {"url":"https://www.davidedalmartello.com/math/papers","timestamp":"2024-11-09T22:09:21Z","content_type":"text/html","content_length":"94100","record_id":"<urn:uuid:07ab9b50-01cf-4aea-93cc-880f4a2d5a3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00105.warc.gz"} |
Multiplication of a Fraction by a Fraction |Multiplying Mixed Fraction
Multiplication of a Fraction by a Fraction
This topic deals with multiplication of a fraction by another fraction. For eg : \(\frac{3}{7}\) by \(\frac{5}{11}\). Here both the numbers are in numerator and denominator form.
Here are few examples to show multiplication of fraction by another fraction
There are few steps or rules that are to be kept in mind while carrying out multiplication of fraction with another fraction.
Step 1:
First change both the fractions into improper fraction if they are mixed fraction
Step 2:
Now we will have to multiply numerator with numerator and denominator with denominator. Or, it can be said product of numerators and product of denominators
Step 3:
Then will have to reduce the fraction that is the numerator and denominator into lowest term
Step 4:
Then the final answer is expressed in whole number or mixed fraction (if it is not in proper fraction) or in proper fraction. The final answer is usually not left in improper fraction we have to
either change it into mixed fraction or whole number (if possible)
Here are few examples to show multiplication of a fraction by another fraction:
Multiplication of proper fraction by another proper fraction
Multiplication of mixed fraction by a proper fraction
Multiplication of mixed fraction by another mixed fraction
1: Multiplying proper fraction by proper fraction.
\(\frac{5}{8}\) × \(\frac{3}{7}\)
= \(\frac{5 × 3}{8 × 7}\); [multiplying numerator by numerator and denominator by denominator]
= \(\frac{15}{56}\); [This is in proper fraction and hence cannot be changed into mixed fraction]
2. Multiplying mixed fraction with proper fraction
7\(\frac{1}{4}\) × \(\frac{6}{7}\)
= \(\frac{4 × 7 + 1}{4}\) × \(\frac{6}{7}\); [Changing the first fraction into improper factor as it is in mixed fraction]
= \(\frac{29}{4}\) × \(\frac{6}{7}\)
= \(\frac{29 × 6}{4 × 7}\); [multiplying numerator by numerator and denominator by denominator]
= \(\frac{174}{28}\)
= \(\frac{174 ÷ 2}{28 ÷ 2}\); [Changing into lowest terms]
= \(\frac{87}{14}\)
Changing it into mixed fraction as it is in improper fraction
= 6\(\frac{3}{14}\)
3. Multiplying mixed fraction with another mixed fraction
4\(\frac{1}{3}\) × 5\(\frac{1}{4}\)
= \(\frac{4 × 3 + 1}{3}\) × \(\frac{4 × 5 + 1}{4}\); [Changing into improper factor as it is in mixed fraction]
= \(\frac{13}{3}\) × \(\frac{21}{4}\)
= \(\frac{21 × 13}{4 × 3}\); [multiplying numerator by numerator and denominator by denominator]
= \(\frac{273}{12}\)
= 22\(\frac{9}{12}\)
Here to write into mixed fraction first we will have to consider the quotient and that has to be written in whole number form, then remainder divided by the divisor.
= 22\(\frac{3}{4}\); [Here \(\frac{9}{12}\) changed into lowest term \(\frac{3}{4}\)]
4. Multiplying a proper fraction with mixed fraction
\(\frac{1}{8}\) × 7\(\frac{14}{15}\)
= \(\frac{1}{8}\) × \(\frac{7 × 15 + 14}{15}\); [Changing the second fraction into improper factor as it is in mixed fraction]
= \(\frac{1}{8}\) × \(\frac{119}{15}\)
= \(\frac{1 × 119}{15 × 8}\); [Multiplying numerator by numerator and denominator by denominator]
= \(\frac{119}{120}\)
The above problems are examples of multiplication of a fraction by another fraction. These fractions can be in any form whether mixed fraction or proper fraction.
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Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness
Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, and David P. Woodruff. Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness. In 9th Innovations in
Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.8","timestamp":"2024-11-07T12:19:19Z","content_type":"text/html","content_length":"158265","record_id":"<urn:uuid:7faf2c8e-b80a-4ea9-b63e-60138f709a3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00877.warc.gz"} |
2023 COURSES
Representation of a Number as a Sum of Powers
Instructor: Rena Chu
In 2025, we will reach the first “square” year of this millennium and the only “square” year of this century. Indeed 2025=45^2, preceded by 1936 and followed by 2116. More common are years which are
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two squares? If not, what about a sum of three squares, or four? How can we be sure—must we check every possibility? Together we will explore these mysteries of sums of squares and their role in
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Instructor: Aygul Galimova
We’ll explore the field of topology, the study of shapes up to stretching and squeezing, comes into play here. Topology studies the properties of a geometric object that are preserved under
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course will cover how to play tic-tac-toe on a donut, glue spaces to get new ones, orientation, knots, the rent-sharing problem, and a few applications. No prerequisites except curiosity. | {"url":"https://sites.duke.edu/swim/program/courses/","timestamp":"2024-11-10T14:19:55Z","content_type":"text/html","content_length":"80248","record_id":"<urn:uuid:4e4bf994-69f8-479e-8d9d-fc8de3383d88>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00247.warc.gz"} |
Methods for robustly measuring the minimum spanning tree and other field level statistics from galaxy surveys (2024)
Methods for robustly measuring the minimum spanning tree and other field level statistics from galaxy surveys–Methods for robustly measuring the minimum spanning tree and other field level statistics
from galaxy surveys
Krishna Naidoo^1,2and Ofer Lahav^2
^1Institute of Cosmology and GravitationE-mail: krishna.naidoo@port.ac.uk University of Portsmouth Burnaby Road Portsmouth PO1 3FX UK
^2Department of Physics & Astronomy University College London Gower Street London WC1E 6BT UK
(Accepted XXX. Received YYY; in original form ZZZ; 2024)
Field level statistics, such as the minimum spanning tree (MST), have been shown to be a promising tool for parameter inference in cosmology. However, applications to real galaxy surveys are
challenging, due to the presence of small scale systematic effectsand non-trivial survey selection functions. Since many field level statistics are ‘hard-wired’, the common practice is to forward
model survey systematic effects to synthetic galaxy catalogues. However, this can be computationally demanding and produces results that are a product of cosmology and systematic effects, making it
difficult to directly compare results from different experiments. We introduce a method for inverting survey systematic effects through a Monte Carlo subsampling technique where galaxies are assigned
probabilities based on their galaxy weight and survey selection functions. Small scale systematic effects are mitigated through the addition of a point-process smoothing technique called jittering.
The inversion technique removes the requirement for a computational and labour intensive forward modelling pipeline for parameter inference. We demonstrate that jittering can mask small scale
theoretical uncertainties and survey systematic effects like fibre collisions and we show that Monte Carlo subsampling can remove the effects of survey selection functions. We outline how to measure
field level statistics from future surveys.
Data Methods – Numerical methods – large-scale structure of Universe – cosmology: observations
1 Introduction
The next generation of cosmological galaxy surveys; such as the Dark Energy Spectroscopic Instrument^1^11https://www.desi.lbl.gov/ (DESI; DESI Collaboration etal. 2016), Euclid^2^22https://
www.euclid-ec.org/(Laureijs etal., 2011), the Rubin Observatory’s Legacy Survey of Space and Time^3^33https://www.lsst.org/ (LSST; Ivezić etal. 2019) and the Wide field Spectroscopic Instrument^4^44
https://www.wstelescope.com/ (WST; Mainieri etal. 2024); will provide a deeper and more resolved view of the universe than has even been available in the past. To maximise the information extracted
from this data requires exploring statistics beyond the two-point correlation function (2PCF), including the 2PCF in different densities or cosmic web environments (Bonnaire etal., 2022, 2023;
Paillas etal., 2023; Massara etal., 2023), and higher-order statistics; such as the minimum spanning tree (MST; Naidoo etal., 2020, 2022), Minkowski functionals (Liu etal., 2022, 2023), critical
points (Moon etal., 2023), topological persistence homology (Pranav etal., 2017; Jalali Kanafi etal., 2023), graph representations (Makinen etal., 2022), voids (Kreisch etal., 2022), wavelet
scattering transforms (Valogiannis & Dvorkin, 2022), and methods directly using field level inference (Fluri etal., 2018; Lemos etal., 2023; Jeffrey etal., 2024). However, many of these techniques
cannot be modelled analytically, and instead require predictions from simulation suites exploring a large cosmological parameter space. To complicate the matter, since these are often performed on
galaxy catalogues, additional modelling are required to explore the halo-to-galaxy relation, often carried out by marginalising over halo occupation distribution (HOD) parameters.
One such example of a field level statistic is the MST, a highly optimised graph built from a set of points and first introduced to astronomy by Barrow etal. (1985). In graph theory, a tree is
defined as a loop-free structure, while ‘spanning’ refers to a graph connecting all points in a single structure. The MST is the spanning tree with the shortest possible total length. This
optimisation leads to a graph, the MST, that very effectively traces filaments in the cosmic web (see Libeskind etal., 2018) and, more recently, has been shown to be highly sensitive to neutrino mass
(Naidoo etal., 2022). Unlike the $N$-point correlation function, the MST is sensitive to density, an effect that cannot be removed by the inclusion of randoms to account for survey selection effects.
Additionally, because the MST operates directly on galaxies, we cannot exclude small scales by pixelising the galaxy distribution onto a grid. Furthermore, there is no mechanism to directly
incorporate galaxy weights, which are typically used to correct for observational systematic effects, into the MST graph construction. Exploiting ‘hard-wired’ statistics like the MST for parameter
inference presents a unique challenge for observational cosmology. While the MST is the statistic of interest in this paper, hard-wired artificial intelligence and machine learning algorithms, such
as graph neural networks, will benefit from resolving these current limitations.
The standard approach for using hard-wired statistics like the MST for parameter inference would be to forward model survey systematic effects and selection functions to synthetic galaxy catalogues.
However, forward modelling survey systematic effects to the levels of accuracy needed for the MST has yet to be performed and would be computationally and labour intensive,. The benefits of forward
modelling is that it enables inference in scenarios that appear intractable, see Lemos etal. (2023) and Jeffrey etal. (2024) for powerful demonstrations of field level simulation based inference
techniques used in cosmology. However, in forward modelling, we lose the ability to clearly delineate a cosmological measurement from survey systematic effects – we can only compare and contrast
measurements at the parameter level. Being able to look and directly interpret measurements will become particularly important when measurements are in ‘tension’ with other probes or if they suggest
the discovery of new physics.
In this paper, we outline how to make robust measurements of the MST and other hard-wired field level statistics from real galaxy surveys. We illustrate how to marginalise over survey selection
functions, such as the redshift selection function and variabilities in completeness, and how to mitigate small scale systematic effects. The techniques introduced in this paper remove the necessity
to forward model survey geometry, selection functions and systematic effects to synthetic galaxy catalogues. Our techniques involve inverting survey systematic effects on real data and comparing to
synthetic galaxy catalogues without survey systematic effects. Since the methods are imposed at the catalogue level they can be applied generally to any technique. In section2 we describe the mock
galaxy catalogues used, the summary statistics measured and the methods for inverting survey selection functions and mitigating small scale systematic effects. In section3 we validate the techniques,
in4 we discuss the methods and outline how to use them for measuring the MST from future galaxy redshift surveys and in5 we summarise the results of the paper.
2 Methods
In this section we summarise the simulations, summary statistics and techniques used throughout this paper. We then introduce the Monte Carlo subsampling technique used to indirectly apply galaxy
weights and mitigating survey systematic effects at the catalogue level.
2.1 Simulations
2.1.1 Millennium XXL galaxy lightcone
We use the Millennium XXL galaxy lightcone catalogue of Smith etal. (2022). The lightcones are produced from the Millennium XXL simulation, computed in a box of $3\,h^{-1}{\rm Gpc}$ in a flat $\
Lambda$CDM cosmology (Planck Collaboration etal., 2020) with matter density $\Omega_{\rm m}=0.25$, dark energy density $\Omega_{\rm\Lambda}=0.75$, amplitude of fluctuations at $8\,h^{-1}{\rm Mpc}$$\
sigma_{8}=0.9$, the Hubble constant $H_{0}=73$ and primordial spectral tilt $n_{\rm s}=1$. For this analysis we limit the catalogue to halos of mass greater than $\geq 10^{13}h^{-1}M_{\odot}$ and
only consider galaxies either in a single quadrant (i.e. $0^{\circ}\leq{\rm RA}\leq 90^{\circ}$ and $0^{\circ}\leq{\rm Dec.}\leq 90^{\circ}$) or inside the Baryonic Oscillation Spectroscopic Survey
(BOSS) LOWZ north footprint.
2.1.2 Lévy flight random walk distribution
We use a set of Lévy flight random walk simulations. Unlike cosmological simulations the points produced have no higher-order information, since the only input is the size of the steps used in the
random walk. We use two sets of distributions, a standard technique which we will refer to as Lévy flight (LF Mandelbrot, 1982) and the Adjusted Lévy flight (ALF) which we developed in Naidoo etal. (
2020). These distributions are approximately equal on large scales but have significantly different small scale distributions. Both the LF and ALF random walk distributions are implemented in the
MiSTree python package (Naidoo, 2019). The parameters used for the LF are the minimum step-size $t_{0}=0.2$ and tilt $\alpha=1$, and for the ALF are the step-size parameters $t_{0}=0.325$ and $t_{s}=
0.015$, and the step-size shape parameters $\alpha=1.5$, $\beta=0.45$ and $\gamma=1.3$. Both distributions are produced in a periodic box of length $75\,h^{-1}{\rm Mpc}$.
2.2 Summary statistics
We describe the summary statistics used in this paper and the jackknife resampling technique used for error estimation.
2.2.1 Two-point correlation function
The 2PCF (Peebles, 1980), is a tried-and-tested method for computing clustering properties of an input dataset. It is widely used in cosmology and generally at the forefront of any observation in
large scale structure. Analytical predictions for observations can be made from a given power spectra and galaxy bias prescription. Galaxy survey geometries and systematic effects are routinely
mitigated through the use of randoms to account for the anisotropic and inhomogeneous distribution of galaxies from a galaxy survey, while small scale systematic effects can be masked by limited the
analysis to separations beyond some input scale (say $r_{\min}$). The 2PCF can be computed from the Landy & Szalay (1993) estimator
$\xi(r)=\left(\frac{n_{R}}{n_{D}}\right)^{2}\frac{DD(r)}{RR(r)}-2\left(\frac{n_%{R}}{n_{D}}\right)\frac{DR(r)}{RR(r)}+1,$ (1)
where $r$ is the distance between points, $DD(r)$ the number of galaxy-galaxy pairs at a distance $r$, $DR(r)$ the number of galaxy-random pairs, $RR(r)$ the number of random-random pairs, $n_{D}$
the mean number density of galaxies and $n_{R}$ the mean number density of randoms.
To compute the 2PCF multipoles we measure the 2PCF binned according to the distance parallel to the line-of-sight of the observer $s_{\parallel}$ and the distance perpendicular to the line-of-sight
$\xi(s_{\parallel},s_{\perp})=\left(\frac{n_{R}}{n_{D}}\right)^{2}\frac{DD(s_{%\parallel},s_{\perp})}{RR(s_{\parallel},s_{\perp})}-2\left(\frac{n_{R}}{n_{D}}%\right)\frac{DR(s_{\parallel},s_{\ (2)
The multipoles are then computed from
$\xi_{\ell}(s)=\frac{2\ell+1}{2}\int^{\pi}_{0}\sqrt{1-\mu^{2}}\,\xi(s_{%\parallel},s_{\perp})\,P_{\ell}(\mu){\rm d}\theta,$ (3)
where $s=\sqrt{s_{\parallel}^{2}+s_{\perp}^{2}}$, $\mu=\cos\theta$ and $P_{\ell}$ is the Legendre polynomial. The monopole is computed by setting $\ell=0$, the quadrupole with $\ell=2$ and
hexadecapole with $\ell=4$.
2.2.2 Minimum spanning tree
The MST is computed on a 3-dimensional distribution of points using the MiSTree python package (Naidoo, 2019). From the constructed MST graph we measure:
• •
Degree ($d$): the number of connected edges to each node/point.
• •
Edge length ($l$): the length of each edge.
• •
Branches: edges connected in chains with intermediate nodes of $d=2$. From branches we measure:
□ –
Branch length ($b$): the total length of member edges.
□ –
Branch shape ($s$): the straight line distance between the branch ends divided by the total branch length.
We are interested in the probability distribution function (PDF) of $d$, $l$, $b$ and $s$ that are obtained by binning the MST statistics into histograms. While this is not an exhaustive set of
statistics to measure from the MST, we have found the PDFs to be highly constraining (Naidoo etal., 2022) for cosmology, in particular the PDFs of edge length $l$ and branch length $b$.
2.2.3 Error estimation with jackknife resampling
We use the jackknife resampling technique to compute errors. We first divide our original dataset into $N_{\rm JK}$ jackknife regions. For datasets inside a periodic box, the box is divided into $N_
{\rm JK}^{1/3}$ segments along each axis, creating $N_{\rm JK}$ smaller cubic boxes within the full box. For datasets from a lightcone we segment the footprint into $N_{\rm JK}$ regions using the
binary-partition method in Naidoo et al.(in prep). We then compute the statistics of interest with points in the $i^{\rm th}$ jackknife segment removed. This gives us $N_{\rm JK}$ estimates of the
statistic, which we will refer to as $\boldsymbol{y}$, where we take the mean
$\bar{\boldsymbol{y}}=\frac{1}{N_{\rm JK}}\sum_{i=1}^{N_{\rm JK}}\boldsymbol{y}%_{i},$ (4)
to be the statistic for the full sample. This can be validated, to test for biases from removing a jackknife segment, by comparing to the statistic measured on the full sample. The jackknife estimate
of variance for the statistics is given by
$\boldsymbol{\Delta y}^{2}_{\rm JK}=\frac{N_{\rm JK}-1}{N_{\rm JK}}\sum_{i=1}^{%N_{\rm JK}}(\boldsymbol{y}_{i}-\bar{\boldsymbol{y}})^{2}.$ (5)
Note, in practice we can compute the variance by assuming the samples are independent and then correcting the variance obtained by multiplying by the Jackknife pre-factor $(N_{\rm JK}-1)^{2}/N_{\rm
2.3 Survey systematic effects and theoretical limitations
Cosmological datasets, whether they come from real galaxy surveys, or theoretical data from simulations will always come with uncertainties. It is important that these uncertainties are accounted
for, particularly within an inference pipeline. Accounting for these uncertainties is particularly challenging for hard-wired algorithms considered in this paper. In this section, we explore how
galaxy surveys and theoretical data from $N$-body simulations can introduce uncertainties in both our model and observational measurements.
2.3.1 Small scale theoretical unknowns
Simulations will be integral to making predictions for higher-order cosmic web statistics, but simulations have uncertainties of their own that need to be accounted for and mitigated in any
cosmological analysis. The most problematic are uncertainties on small scales, that are dictated by the force and mass resolution of the simulation – an effect that can be larger for approximate $N$
-body solvers such as COLA (Tassev etal., 2013). This can result in simulations where only massive halos are resolved reliably, making predictions difficult for observations of galaxies in small mass
halos. This is conceptually a rather simple limitation to solve – simply discard low mass galaxies or use higher resolution simulations. A more pressing issue is the uncertainty in the physics of
small scales, $\lesssim 10h^{-1}{\rm Mpc}$. This is the domain where we expect baryonic physics and feedback mechanisms to be important and where there remains uncertainties on the galaxy formation
prescriptions used for hydrodynamic simulations. How these scales are incorporated in any cosmological analysis needs to be treated with great care. Uncertainty on small scales are further enhances
by the fact that simulations with hydrodynamic galaxy formation physics are expensive to run, and not always available, especially for the large $N$-body simulation suites used for cosmological
inference. Typically in cosmology, galaxies are ‘painted’ onto dark matter only simulations using a halo-to-galaxy prescription; this is often constructed using HODs or semi-analytic halo abundance
matching (SHAM) or both. Marginalising over the uncertainties in these semi-analytic models will remove some of the uncertainties but eventually these models will break down. Therefore, we can only
ever expect simulations to provide reliable galaxy catalogues up to some theoretical lower bound, a lower bound which cannot (at the moment) be masked from a hard-wired algorithm.
2.3.2 Small scale observational systematic effects
In observations, small scales provide a different challenge. In spectroscopic surveys, galaxies that are close together on the sky may not be assigned individual fibres for their respective spectras
to be measured. This problem, known as fibre collisions, leads to galaxies that cannot be assigned correct redshifts. To correct for these missing galaxies, the nearest galaxy is often re-weighted,
to account for the fact that there are more galaxies nearby. Nevertheless, this means on angular scales smaller than the fibre collision length, the distribution of galaxies cannot be trusted. From
photometric surveys, a different problem occurs on small scales, and that is a problem caused by the blending and crowding of galaixes in close proximity. In effect, galaxies become difficult to
resolve, making photometry and shape measurements difficult and unreliable. This can lead to inaccuracies in photometric redshift estimates and can introduce systematic errors in shape measurements
relevant for weak lensing studies. In both cases we cannot trust the information on these small scales and need to be able to remove them.
2.3.3 Radial selection function
Depending on the characteristics of the survey, galaxies will be observed with a characteristic radial redshift selection function. This is substantially different from what we have from simulations
where the distribution of galaxies is homogeneous. Not only is the distribution inhomogeneous, but so too are the galaxy properties. For example we will often find only small galaxies being observed
at lower redshifts and only massive galaxies at higher redshifts. If a hard-wired algorithm is applied directly to this distribution of galaxies, it may be overwhelmed by the inhomogeneous redshift
selection function and the biased galaxy redshift evolution. This reduces the relevant cosmological information in the hard-wired algorithm, with a significant portion of the algorithm just picking
up survey systematic. This makes comparisons between different experiments extremely challenging, as it becomes difficult to delineate the cosmological result from the survey’s systematic effects.
2.3.4 Angular selection function
Galaxy surveys will not be observed isotropically, but will instead be observed along the galaxy survey’s footprint. Surveys typically have non-trivial boundaries which may induce significant biases
in the results from hard-wired algorithms. In addition, galaxies observed across the sky will be subject to the seeing conditions during observations and galactic foregrounds, such as zodiacal light
and stellar density from our own galaxy that will induce anisotropies in the distribution of galaxies observed. This effect will need to be replicated in a forward modelled approach.
2.4 Monte Carlo subsampling
In $2$-point (or even $N$-point) statistics galaxy survey systematic effects are mitigated by assigning weights to each galaxy. The weights capture observational biases, such as the angular and
radial selection functions, and on small scales can account for missing galaxies, for e.g. from fibre collisions or blending of galaxies in crowded fields. Including weights into $2$-point clustering
measurements is trivial but for the MST and hard-wired algorithms it is not clear how weights should be included. The only approach to apply these promising techniques to real data for parameter
inference is to carry out a full forward modelling approach. This will mean the creation of realistic galaxy mock catalogues for a given survey and accurate forward modelling of the survey’s
systematic effects and selection functions. This is both computationally expensive and labour intensive, and because the measurements are a product of both cosmological and systematic information, it
becomes difficult to interpret results and thereby the parameter constraints they leads to. For these reasons, we look for a different approach, one where survey systematic effects are mitigated at
the catalogue level – removing the need for the direct inclusion of weights in the calculation of the MST or other hard-wired algorithms.
2.4.1 Systematic weighted probabilities
To incorporate systematic effects at the catalogue level we subsample a galaxy catalogue, treating the galaxy weights $w_{\rm g}$ (which are constructed to include various systematic properties of a
galaxy survey) as probabilities $\mathds{P}_{\rm g}=w_{\rm g}$. To subsample the galaxies we draw a random uniform number $u\in[0,1]$ and only keep galaxies when $\mathds{P}_{\rm g}\leq u$. We then
measure the statistics of interest, which we will refer to as $\boldsymbol{d}$ (a vector of length $n_{\rm d}$), measured for the subsampled galaxy distribution, which we will refer to as $\tilde{\
boldsymbol{d}}$. We repeat the process over many iterations, taking the mean of the measured statistic ($\boldsymbol{d}_{n}$) evaluated over the ensemble of $n$ iterations,
$\boldsymbol{d}_{n}=\frac{1}{n}\sum_{i=1}^{n}\tilde{\boldsymbol{d}}_{i},\quad{%\rm where}\quad\lim_{n\to\infty}\boldsymbol{d}_{n}\equiv\boldsymbol{d},$ (6)
which is only true if $\tilde{\boldsymbol{d}}$ is unbiased. In this way we are able to incorporate the galaxy weights indirectly, since when $n\to\infty$ a galaxy will be sampled $\approx n\mathds{P}
_{\rm g}$, averaged over $n$ iterations gives the galaxy the correct weight $\approx\mathds{P}_{\rm g}=w_{\rm g}$. This process is akin to the superposition of different quantum states in quantum
mechanics, each iteration gives us a discrete set of observations following an underlying probability distribution function. A distribution that becomes clearer once the average is taken over many
discrete observations.
Treating $w_{\rm g}$ as probabilities $\mathds{P}_{\rm g}$ only make sense if $w_{\rm g}\in[0,1]$, if $w_{\rm g}>1$ then this galaxy will always be assigned the incorrect weight, biasing the
analysis. To avoid this potential issue, we replicate galaxies with a weight greater than one by $n_{g}$ times, where $n_{g}=\lceil w_{\rm g}\rceil$, and $\lceil x\rceil$ is the ceiling function
(which rounds $x$ to the nearest integer greater than $x$). The new weights for the replicated galaxies are now $\mathds{P}_{\rm g}=w_{\rm g}/n_{\rm g}$. This results in subsampled galaxies that have
the correct weights. Galaxy weights greater than one, often occur when nearby galaxies cannot be observed, so it is important in practice that the replication of galaxies with weights greater than
one happens in combination with the addition of a jitter (described below). This also ensures galaxy replicates do not lie on top of each other.
2.4.2 Point-process smoothing with jittering
To remove systematic effects and theoretical uncertainties at small scales, we add noise to the three dimensional position of the galaxy $\boldsymbol{p}$, a process we call jitter
$\boldsymbol{p}_{\rm J}=\boldsymbol{p}+\mathcal{N}(0,\,\sigma_{\rm J}\cdot%\boldsymbol{I}_{3}).$ (7)
$\mathcal{N}$ is the normal distribution with mean zero and standard deviation given by the dot product of the three dimensional identity matrix $\boldsymbol{I}_{3}$ and jitter dispersion $\sigma_{\
rm J}$. This process is the point-process equivalent to smoothing a field. Like smoothing, we expect this procedure to remove systematic effects at small scales and remove any sensitivity to small
scales. This is crucial for the MST where small scales cannot be post-processed. Therefore, we must be able to hide galaxy survey systematic effects and theoretical uncertainties from simulations to
be able to use the MST for parameter inference from galaxy surveys.
2.4.3 Removing selection functions with inverse density weights
Galaxy surveys have radial and angular selection functions, the former being characterised by the redshift galaxy density distribution $\rho(z)$ and the latter characterised by the surveys footprint/
mask $M({\rm RA},{\rm Dec})$ and completeness $C({\rm RA},{\rm Dec})$. Both properties create variations in the distribution of galaxies, variations which are properties of the survey and are not
cosmological in origin. In $2$-point or $N$-point statistics, the effect of the surveys selection function can be removed with the inclusion of randoms which allow one to measure the extra clustering
with respect to a uniform sample. If we perform the MST, or other statistics, directly on galaxies we will get results which are a combination of both the cosmology and the survey selection function.
This makes the analysis of such results difficult and interpretation challenging. To remove this issue we subsample galaxies to produce a uniform distribution of galaxies both radially and across the
sky. This process involves applying what we call inverse density weights, which assigns a galaxy a probability that is a function of its radial and angular densities,
$\mathds{P}_{\rm g}=w_{\rm g}\,\left(\frac{\rho_{\rm target}}{\rho(z_{\rm g})}%\right)\,\left(\frac{C_{\rm target}}{C({\rm RA}_{\rm g},{\rm Dec}_{\rm g})}%\right).$ (8)
Where the subscript $\rm g$ simply refers to the respective values for a single galaxy, $\rho_{\rm target}$ and $C_{\rm target}$ are target radial densities and target completeness values for the
subsampled uniform distribution of galaxies. The galaxies being considered for subsampling should be choosen such that $\rho(z_{\rm g})\geq\rho_{\rm target}$ and $C({\rm RA}_{\rm g},{\rm Dec}_{\rm
g})\geq C_{\rm target}$. The effective density of the subsampled distribution is $\rho_{\rm eff}=\rho_{\rm target}C_{\rm target}$. This means theoretical predictions from simulations can aim to
measure the MST or other hard-wired algorithm for a well defined galaxy distribution at an effective density $\rho_{\rm eff}$, a much simpler task than attempting to forward model the surveys
complicated radial and angular selection functions.
2.4.4 Convergence criteria
To estimate the statistics $\boldsymbol{d}_{n}$ requires applying the iterative probabilistic subsampling scheme over $n$ iterations. If the application of weights is unbiased then we expect for $n\
to\infty$, $\boldsymbol{d}_{n}\equiv\boldsymbol{d}$. However, for this procedure to be tractable $n$ should be small but large enough for $\boldsymbol{d}_{n}\approx\boldsymbol{d}$. So, how do we
determine the value of $n$? To understand this it is useful to first approach this assuming we have access to $\boldsymbol{d}$. One way to define $n$ is to look at the fractional difference
$\epsilon_{f}=\frac{1}{n_{\rm d}}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d}^{(j%)}_{n}-\boldsymbol{d}^{(j)}}{\boldsymbol{d}^{(j)}},$ (9)
where the subscript $(j)$ refers to single elements across the data vector. We take $n$ to be when this fractional difference reaches some predefined threshold $\epsilon_{\rm thresh}\leq\epsilon_{f}$
. In reality we do not have access to $\boldsymbol{d}$, but we do have access to $\boldsymbol{d}_{n+1}$. We can then try to redefine the convergence by the iterative fractional difference
$\epsilon_{I}=\frac{1}{n_{\rm d}}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d}^{(j%)}_{n}-\boldsymbol{d}^{(j)}_{n+1}}{\boldsymbol{d}^{(j)}_{n+1}}.$ (10)
For $\epsilon_{f}$ the variance is fully described by the variance in $\boldsymbol{d}_{n}$ which in turn is described by the variance in $\tilde{\boldsymbol{d}}$,
${\rm Var}\left(\boldsymbol{d}_{n}\right)=\frac{1}{n}{\rm Var}\left(\tilde{%\boldsymbol{d}}\right).$ (11)
Therefore the variance of $\epsilon_{f}$ is given by
${\rm Var}(\epsilon_{f})=\frac{1}{n\,n_{\rm d}^{2}}\sum_{j=1}^{n_{\rm d}}\frac{%{\rm Var}\Bigl{(}\tilde{\boldsymbol{d}}^{(j)}\Bigr{)}}{\Bigl{(}\boldsymbol{d}^%{(j)}\Bigr{)}^{2}}.$ (12)
The variance of $\epsilon_{I}$ is a little more complicated to compute since $\boldsymbol{d}_{n}$ and $\boldsymbol{d}_{n+1}$ are strongly correlated since the same $n$ samples are used to compute
both of them. It is useful to remove this correlation and redefine $\epsilon_{I}$ as
$\epsilon_{I}=\frac{1}{n_{\rm d}(n+1)}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d%}_{n}-\tilde{\boldsymbol{d}}_{n+1}}{\boldsymbol{d}_{n+1}}.$ (13)
Assuming $\boldsymbol{d}_{n+1}=\boldsymbol{d}+\boldsymbol{\Delta d}_{n+1}$, we can Taylor expand the denominator to write it in terms of $\boldsymbol{d}^{(j)}$,
$\begin{split}\epsilon_{I}&=\frac{1}{n_{\rm d}\,(n+1)}\sum_{j=1}^{n_{\rm d}}%\frac{\boldsymbol{d}_{n}^{(j)}-\tilde{\boldsymbol{d}}_{n+1}^{(j)}}{\boldsymbol%{d}^{(j)}+\boldsymbol{\Delta d}_
{n+1}^{(j)}},\\&\approx\frac{1}{n_{\rm d}\,(n+1)}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d}_{%n}^{(j)}-\tilde{\boldsymbol{d}}_{n+1}^{(j)}}{\boldsymbol{d}^{(j)}}\left(1+%\mathcal{O}\left(\frac (14)
{\boldsymbol{\Delta d}_{n+1}^{(j)}}{\boldsymbol{d}^{(j)}%}\right)\right).\end{split}$
Finally, since $\boldsymbol{\Delta d}_{n+1}\ll\boldsymbol{d}$ we can make the approximation
$\epsilon_{I}\approx\frac{1}{n_{\rm d}(n+1)}\sum_{j=1}^{n_{\rm d}}\frac{%\boldsymbol{d}_{n}^{(j)}-\tilde{\boldsymbol{d}}_{n+1}^{(j)}}{\boldsymbol{d}^{(%j)}},$ (15)
and therefore the variance is given by
${\rm Var}(\epsilon_{I})\approx\frac{1}{n\,n^{2}_{\rm d}(n+1)}\sum_{j=1}^{n_{%\rm d}}\frac{{{\rm Var}\left(\tilde{\boldsymbol{d}}^{(j)}\right)}}{\left(%\boldsymbol{d}^{(j)}\right)^{2}}$ (16)
and therefore from eq.12 we get
${\rm Var}(\epsilon_{I})\approx\frac{{\rm Var}(\epsilon_{f})}{n+1}.$ (17)
Since the expectation of $\epsilon_{f}$ and $\epsilon_{I}$ is completely described by their variance we can relate the two by their standard deviation and thus
$\epsilon_{f}\approx\sqrt{n+1}\,\epsilon_{I}.$ (18)
Which means we can define a threshold for $\epsilon_{f}$ without knowing $\boldsymbol{d}$.
The choice of threshold here appears to be quite arbitrary, a more robust way to define this would be to define the significant difference
$\varepsilon_{f}=\frac{1}{n_{\rm d}}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d}_%{n}^{(j)}-\boldsymbol{d}^{(j)}}{\boldsymbol{\Delta d}^{(j)}},$ (19)
where $\boldsymbol{\Delta d}$ is the error associated with $\boldsymbol{d}$, which can be measured through jackknife resampling. Similarly, we will not know $\boldsymbol{\Delta d}$ a priori and will
need to instead rely on an iterative estimator
$\varepsilon_{I}=\frac{1}{n_{\rm d}}\sum_{j=1}^{n_{\rm d}}\frac{\boldsymbol{d}^%{(j)}_{n}-\boldsymbol{d}^{(j)}_{n+1}}{\boldsymbol{\Delta d}_{n+1}^{(j)}}.$ (20)
Applying the same logical steps presented before and taking the Taylor expansion of $1/\boldsymbol{\Delta d}_{n+1}\approx 1/\boldsymbol{\Delta d}$ we get the relation
$\varepsilon_{f}\approx\sqrt{n+1}\,\varepsilon_{I}.$ (21)
By calculating $\epsilon_{f}$ and $\varepsilon_{f}$ through the approximations from the iterative $\epsilon_{I}$ and $\varepsilon_{I}$ we can set thresholds on the absolute $\epsilon_{f}$ and $\
varepsilon_{f}$ to estimate how many iterations $n$ are required to get the statistics to the desired level of precision. This can be carried out without knowing the true weighted $\boldsymbol{d}$ or
its true error $\boldsymbol{\Delta d}$.
3 Results
In this section we will explore the performance of the Monte Carlo subsampling and jittering procedure in removing survey systematic effects and masking theoretical uncertainties.
3.1 Jittering performance
3.1.1 Masking small scale theoretical uncertainties
We compare the 2PCF and MST for two random walk distributions LF (Mandelbrot, 1982) and ALF (Naidoo etal., 2020). The distributions have identical higher-order information since they are both random
walks. On scales $\gtrsim 0.3\,h^{-1}{\rm Mpc}$ they, by design, have approximately equal 2PCF – see the left subplot of Fig.1. Despite the agreement of the 2PCF at scales $r\gtrsim 0.3\,h^{-1}{\rm
Mpc}$ the MST shows significant differences in the distribution of $d$, $l$, $b$ and $s$ (dark blue). This agreement between the LF and ALF distributions for the 2PCF and MST is indicated by the
reduced $\chi^{2}$, where errors are computed via jackknife resampling and added in quadrature. The $\chi^{2}_{r}$ are extremely significant for the MST, especially for the edge lengths $l$ where $\
chi^{2}_{r}=207$. The $\chi^{2}_{r}$ is also large for the 2PCF, at $\chi^{2}_{r}=17.1$, but this is due to the inclusion of small scales, if we limit the results to larger scales than $r\gtrsim 0.3
\,h^{-1}{\rm Mpc}$ then it reduces to $\chi^{2}_{r}=0.35$. Unlike the 2PCF, we cannot simply ignore edges of the MST above some threshold scale as these scales percolate into other areas of the
statistics, such as the branch shape and degree, where it is unclear how we could remove these scales retroactively. For this reason we introduce the jittering scheme, which adds a random ‘jitter’ or
displacement to the location of each point in the distribution. The jitter is defined with a jitter dispersion of $\sigma_{\rm J}=1\,h^{-1}{\rm Mpc}$. By adding a random jitter to the positions of
the points, we are effectively applying a point-process equivalent of smoothing a field. The 2PCF with the jitter shows complete consistency between the LF and ALF distributions, with $\chi^{2}_{r}=
0.32$ (light blue line for LF and points for ALF). Similarly we show the distributions of $d$ now has a $\chi^{2}_{r}=0.772$, $l$ a $\chi^{2}_{r}=1.42$, $b$ a $\chi^{2}_{r}=1.5$, and $s$ a $\chi^{2}_
{r}=1.3$. The $\chi^{2}_{r}$ are now all consistent with the same distribution, showing that adding a jitter to the location of points can masks differences on small scales.
3.1.2 Masking small scale survey systematic effects
All galaxy surveys will suffer from small scale systematic effects; for photometric surveys these are typically introduced due to the blending of galaxies in crowded fields, while for spectroscopic
surveys fibre collisions limit the number of spectra that can be obtained for galaxies with close neighbours on the sky. In either case, we can no longer trust the distribution of galaxies on scales
below these angular scales. This is slightly different to the difference in the LF and ALF distributions discussed previously, as in this scenario the systematic occurs on the projected distribution
of galaxies on the sky. This means galaxies that are close by on the sky but actually quite distant can still suffer from these systematic effects.
To understand the consequences of systematic effects on our measurements we take galaxies from the Millennium XXL quadrant described in section2.1.1 and iteratively remove galaxies with an angular
separation of $<30\,{\rm arcsec}$. For each galaxy with more than one close pair (i.e. a galaxy with angular separation $<30\,{\rm arcsec}$) we first match galaxies with their close pairs and
randomly select a galaxy to keep. To ensure the missing galaxies are accounted, we add the weights of the removed galaxies to the weight of galaxy being kept. Each galaxy now has either a weight of
zero (if it has been removed), a weight of one (if it has no close pairs or no pairs were lost), or an integer weight greater than one (to account for the lost pairs). This gives us a catalogue of
galaxies with an approximate fibre collision-like systematic, which we will refer to as the no close pair (No CP) catalogue.
We measure the 2PCF and MST on the Millennium XXL quadrant with all galaxies and with No CP shown in dark blue in Fig.2. For the No CP catalogues, galaxies with integer weights greater than one are
repeated by the weighted number in the catalogue. This ensures they are correctly weighted in the 2PCF, however for the MST, this step has practically no effect on the results, since this introduces
edges of length zero in the MST, which has no physical meaning and is no different to the results we would obtain if galaxies were not replicated. Nevertheless we can see that removing the close
pairs in the 2PCF reduces the amplitude of small scale clustering, although this remains consistent with a $\chi^{2}_{r}=0.76$. For the MST, the distribution of $l$ and $b$ are significantly
different showing $\chi^{2}_{r}=88.2$ for $l$ and $\chi^{2}_{r}=17.8$ for $b$. These distributions are significantly different and highlights the effect that fibre collisions can have on the MST and
other hard-wired algorithms.
We now remeasure the 2PCF and MST with a jitter, defined with a jitter dispersion $\sigma_{\rm J}=3\,h^{-1}{\rm Mpc}$, shown in light blue in Fig.2. This removes any discrepancy in the MST statistics
between the original and No CP catalogues, most significantly reducing $\chi^{2}_{r}=88.2$ for $l$ to $\chi^{2}_{r}=0.41$. The results show that adding a jitter to the position of points is able to
remove the systematic effects caused by fibre collisions. Crucially, it removes a problem caused by galaxies with weights $>1$. In the MST this inevitably leads to edges equal to zero. With a jitter,
this problem is removed entirely, since the repetition of galaxies with a weight $>1$ will never lie exactly on top of each other. This ensures the missing galaxies are correctly accounted for in the
MST and indeed any hard-wired statistic.
In practice the jitter dispersion scale will need to be calibrated to be large enough to ensure sensitivity to small scale systematic effects like fibre collisions are removed. In this current setup
the No CP catalogue suffers from percolation effects since close pairs percolate due to the algorithm used to construct them, this is one way the approximation departs strongly from the real effect
of fibre collisions. In practice we would estimate the jitter dispersion scale by comparing mock datasets with and without fibre collisions as a function of $\sigma_{\rm J}$ and finding the smallest
value that ensures consistency.
3.2 Monte Carlo subsampling performance
We test the performance of the Monte Carlo subsampling scheme, discussed in section2.4, to assign galaxy weights and mitigate survey selection functions at the catalogue level. This is used to
determine the weighted hard-wired statistics without needing to understand how to pass weighted points directly to the hard-wired algorithm itself.
3.2.1 Inverse redshift selection weights
Galaxy surveys observe galaxies inhomogeneously and anisotropically, effects induced by the galaxy survey’s footprint and redshift selection function. The redshift selection function forces
inhomogeneities in the dataset which needs to be accounted for in any cosmological measurement, as we do not want to confuse properties of the survey with cosmology. For the 2PCF and indeed most $N$
-point correlation functions this is readily handled by including a catalogue of randoms which captures the survey’s footprint and selection function. This aptly removes any sensitivity in the 2PCF
to the survey properties. This is not the case for the MST and any hard-wired algorithms, meaning results will be the product of cosmology and the survey’s footprint and selection functions.
To address this we assign galaxies additional weights that re-weight galaxies based on the inverse of the redshift density
$w_{\rm g}=\frac{\rho_{\rm target}}{\rho(z_{\rm g})},$ (22)
where $\rho(z_{\rm g})$ is the redshift selection function density for a galaxy at redshift $z_{\rm g}$ and $\rho_{\rm target}$ the target density for subsampling galaxies (set to $\rho_{\rm target}=
0.001$ in this analysis).
We will first concentrate on the 2PCF and the performance of the weight assignment via Monte Carlo subsampling. The 2PCF is calculated with a random catalogue with 10 times the density of the galaxy
catalogue. The 2PCF is first computed with unit weights, i.e. treating each point equally, indicated in Fig.3 in dark blue. We then compute the 2PCF with inverse density weights applied directly in
red and indirectly, using the Monte Carlo subsampling procedure computed over a 100 iterations, in light blue. Note, for the randoms we directly assign them inverse density weights in the 2PCF to
account for the imposed homogeneity of the distribution being sampled via Monte Carlo subsampling. The 2PCF in Fig.3 shows: (1) weighting galaxies and randoms by the inverse of the density has no
effect on the 2PCF, since randoms are designed to remove any systematic effects from the survey’s footprint and selection function; and (2) the Monte Carlo subsampling procedure is able to impose
weights on the galaxy distribution at the catalogue level and retrieve the directly weighted 2PCF with a $\chi^{2}_{r}=0.00064$.
The MST distributions with (light blue) and without (dark blue) the inverse density weighting scheme is shown in Fig.3 on the right. The figure demonstrates how biased the MST can be if the survey’s
selection function is not accounted for in the measurements. By imposing the inverse density weights we homogenise the distribution at the catalogue level before making the MST measurement. This
ensures the statistic is no longer dependent on the survey’s redshift selection function, which is demonstrated in Fig.4. We plot the mean of the MST statistics as a function of the redshift
selection function in ten redshift slices between $0.1\leq z\leq 0.2$ and plot this against the Millennium XXL density as a function of redshift, $\rho(z)$. The top panels shows the relation with
unit weights, while the bottom panels show the relation when using Monte Carlo subsampling to apply inverse density weights. We fit a linear relation to the points and show that without weighting the
MST the statistics is strongly correlated to the redshift density of the distribution, this is not surprising since we expect the MST to be extremely sensitive to density. Using the inverse density
weights, this sensitivity is completely removed and the MST statistics are no longer correlated with the redshift selection function.
3.2.2 Convergence and signal-to-noise
The Monte Carlo subsampling procedure allows us to place galaxy weights, for mitigating survey systematic effects, at the catalogue level. The iterative procedure allows us to indirectly apply the
weights to the statistics of interest. In Sec.2.4.4 we introduce several convergence criteria that can be used to determine the number of Monte Carlo iterations $n$ required to make the measurement
to the desired accuracy.
The mean fractional difference $\epsilon_{f}$ gives us a measure of the error of the Monte Carlo statistics in relation to the true weighted statistics. In most cases we will not have access to $\
epsilon_{f}$, but instead only have access to $\epsilon_{I}$ which is computed from the sequential iterations of the statistic. This is extremely useful in cases where errors are difficult to obtain
and the convergence criteria can be achieved by requiring the measurement be made to within $1\%$ error. In the cases where we do have access to the error in the measurement we can use the mean
significant difference $\varepsilon_{f}$.
We test the accuracy of the iterative estimators ($\epsilon_{I}$ and $\varepsilon_{I}$) to compute $\epsilon_{f}$ and $\varepsilon_{f}$ respectively, using the 2PCF where galaxy weights can be
assigned directly. In Fig.5 we compare the convergence estimators for the 2PCF multipoles ($\ell=0$ the monopole, $\ell=2$ the quadrupole and $\ell=4$ the hexadecapole). The figure shows the direct
estimator, using the true weighted multipoles, in relation to the iterative estimators, using the Monte Carlo procedure. The iterative estimator follows the distribution for the direct estimator of
the mean fractional difference (top panel) and the mean significant difference (bottom panel). This means we can rely on the iterative estimator to determine the necessary iterations required for
estimating the weighted MST and other hard-wired statistics when we use Monte Carlo subsampling.
In Fig.6 we compute the mean fractional difference and mean significant difference for the MST distributions as a function of Monte Carlo iterations. These are computed using the iterative estimator
since weights cannot be applied directly to the MST. Here we illustrate how the MST statistics converge, showing that in general the degree is the quickest to converge followed by the edge length,
branch length and then branch shape. Some aspect of this is dependent on the bin sizes used for the MST distribution measurements. In the case where iterations are prohibitively expensive, the user
could increase the bin sizes to reduce the iterations required for convergence. The values of $\epsilon_{f}$ and $\varepsilon_{f}$ allow us to know how accurate our estimates of the weighted MST are,
for example we know from $\epsilon_{f}$ (top panel) that we can achieve a $0.1\%$ measurement of the MST statistics after $\gtrsim 50$ iterations and an error of $5\%$ as a fraction of the standard
We measure the mean SNR ratio of the MST
${\rm SNR}=\frac{1}{n_{\rm d}}\sum_{i=1}^{n_{\rm d}}\frac{m_{x}}{\Delta m_{x}},$ (23)
where $x$ is used to denote $d$, $l$, $b$ or $s$. The ${\rm SNR}$ is measured for the MST distributions measured on all the galaxies (${\rm SNR}^{\rm full}$) and then the iterative MST distributions
measured with the Monte Carlo procedure (${\rm SNR}^{\rm iter}$). In Fig.7 we compare the SNR of the full MST in relation to the iterative estimator using inverse density weights. The plot shows that
in all cases the SNR ratio of the measurement is improved by marginalising over the redshift selection function with inverse density weights. A concern in introducing inverse density weights and
homogenising the distribution before applying the MST, is that this process loses information and reduces the information content of the data. This concern only turns out to be true for a few
iterations, after more than a few iterations the SNR shows the opposite effect. Marginalising over the redshift selection function we improve the SNR and are therefore extracting more information
from the data. The improved SNR arises from the fact that our measurement is more well defined, it is a measurement of cosmology at a specific density of galaxies, while prior to this the MST
measurement was a measure of both cosmology and the survey’s redshift selection function – the latter of which is a systematic that we are not interested in measuring.
3.2.3 Survey footprint bias
The footprint for a galaxy survey is dictated by the location of the observations (if on the ground), galactic and zodiacal foregrounds, bright stars and observing conditions. This will create a
footprint which is unique to the survey, non-trivial and sometimes difficult to replicate. For statistics that require $N$-body simulations, we need to decide whether we want to emulate the MST or
hard-wired algorithm with the survey footprint already superimposed on the distribution of galaxies or to emulate it in a topology more readily accessible for simulations, such as a cubic box or for
a lightcone on a quadrant on the sky. The benefit of the latter, is the measurement is independent of the survey and can be ported to multiple different surveys. In all cases to use the emulated
statistic and perform inference with galaxy survey measurement requires learning the footprint bias – i.e. the bias that the footprint imposes on the measured statistic. We will assume the bias is
small and can be learnt from one point in parameter space – in practice this may not be true, in which case it may be beneficial to emulate the statistics with the survey footprint already imposed on
the simulation data.
In Fig.8 we make measurements of the MST on the Millennium XXL galaxy catalogue in a quadrant on the sky and with the BOSS LOWZ North survey imposed (simply referred to as the ‘survey’). The
statistics are measured with inverse density weights to measure the MST with a galaxy density of $0.001\,h^{3}\,{\rm Mpc}^{-3}$ and shown with and without a jitter with dispersion $\sigma_{\rm J}=5
\,h^{-1}{\rm Mpc}$. The measurements show that for the MST statistics, the footprint bias from a quadrant to the LOWZ North survey mask is small, and is smaller after applying a jitter. Since the
footprint bias is small, the functional form of this bias can be learned by applying the MST to a galaxy simulation in a cubic box or quadrant used for the rest of the simulations and then applying
it to the same simulation with the survey footprint imposed. This bias can then be subtracted from the observational measurements or added to the theoretical predictions from simulations prior to
parameter inference.
3.2.4 Inverse angular selection weights
Galaxy surveys will generally aim to observe the sky up to a certain depth across the survey’s footprint. However, seeing conditions, instrumentation and foregrounds can limit the degree to which
this can be achieved. This will create variations in the number of galaxies observed across the sky, which is often characterised by a measure of completeness. If unaccounted, the variation in
galaxies observed across the sky will be a systematic that can alter our results and could lead to biased or incorrect parameter inference. It is therefore important that this effect is carefully
included and mitigated.
To test the effect of completeness we subsample galaxies from the LOWZ North footprint with a completeness fraction that varies linearly in RA, from 1 to 0.7. This is quite an extreme set up designed
to illustrate how such a systematic could bias measurements, but not one we expect to see in real data. We measure the MST distribution for the survey with varying completeness and compare it to a
measurements made on the quadrant with a completeness fraction of 0.85 (shown in blue). The measurements are made with inverse density weights used for the radial selection function and a jitter with
dispersion $\sigma_{\rm J}=5\,h^{-1}{\rm Mpc}$. When we apply the MST to the galaxies with varying completeness we see a more significant offset in relation to a quadrant with an effective
completeness of $0.85$ (i.e. the mean of the variable completeness of the survey). We then apply inverse completeness weights, making full use of Eq.8, setting $C_{\rm target}=0.7$. This homogenises
the distribution of galaxies across the sky, prior to the computation of the MST. This means the completeness systematic does not enter the MST statistics. For comparison we subsample the quadrant
with a completeness fraction of 0.7 (shown in orange). The relations show that correcting for discrepancies in completeness reduces the $\chi^{2}_{r}$. The main discrepancies that still remain are
characteristic of the footprint bias shown in Fig.8, which have not been removed in this analysis.
In Fig.10 we measure the mean of the MST statistics as a function of completeness. Here we use the average completeness in jackknife tiles on the simulated survey. We find that the mean of the
distributions for $l$ and $b$ show a clear relation with respect to completeness, where the significance of the linear gradient is $|m/\sigma_{m}|>2$. After applying the completeness correction, the
linear relation with completeness is greatly reduced, with $|m/\sigma_{m}|<2$.
4 Discussion
The MST and other hard-wired field level statistics offer an alternative probe of large scale structure, capturing higher-order statistical information present in the field and cosmic web which are
not generally attainable from measuring the 2PCF. However, we are yet to capitalise on the additional information in these statistics because making these measurements precisely on real data is
fraught with observational and theoretical systematic effects and uncertainties.
The conventional approach for dealing with hard-wired algorithms such as the MST is to make forward modelled predictions from $N$-body simulations. The simulations would vary both cosmological
parameters and galaxy population parameters (such as HODs) and forward model survey systematic effects. This requires synthetic galaxy catalogues with:
1. 1.
Galaxy weights – used to forward model the galaxy weight bias caused by ignoring or, effectively, using unit weight (i.e.setting all galaxy weights to one) in the MST or hard-wired algorithm.
2. 2.
Small scale systematic effects – such as fibre collisions and systematic effects from crowding need to be injected since the MST and some hard-wired algorithms may not be able to impose small
scale cuts.
3. 3.
Redshift selection function – that may be challenging to impose if the target selection is complex.
4. 4.
Footprint and angular selection functions – that may not be trivial if they are dictated by observational systematic effects.
5. 5.
Galaxy bias evolution – galaxies observed at higher redshift will often be more luminous and more massive, this means our tracer for large scale structure changes as a function of redshift. The
evolution of galaxy bias may need to be forward modelled.
While it is common to reproduce all of these systematic effects in mocks for computing covariances (see Kitaura etal., 2016; Ereza etal., 2024), reproducing these on synthetic data for parameter
inference is much more challenging due to the complexity of the task and the number of simulations utilised. Although all these requirements may be satisfied, we cannot be certain we can trust the
small scale physics from $N$-body simulations. Similarly, even in cases where these are produced from hydrodynamic simulations, we cannot be certain that the galaxy formation and evolution
prescriptions are completely accurate. This issues means our predictions from the MST and hard-wired algorithms may be fatally flawed. Furthermore, forward modelling will need to be carried out
independently for every galaxy survey, due to the specific nature of survey systematic effects. This means the analysis can only be compared to other surveys at the parameter level, since
measurements made are a measure of cosmology, galaxy population and systematic effects – the latter of which will be unique to the survey and will make comparisons between surveys practically
impossible. This will become important if the constraints on cosmological parameters challenges the standard model, requiring deeper investigation and interpretation.
The methods in this paper try to resolve these issues by showing how to incorporate galaxy weights, mitigate small scale uncertainties and galaxy survey systematic effects to produce measurements
from surveys that are solely dependent on cosmology and the galaxy tracers being used. This simplifies the requirements from $N$-body simulations which will no longer be required to reproduce small
scale systematic effects like fibre collisions, and galaxy survey redshift and angular selection functions. This means emulators of the MST statistics or hard-wired algorithm can be built more
generally, substantially reducing the computational requirement for parameter inference. Furthermore, it will allow the measurements to be more readily interpreted, since the measurement is no longer
survey dependent due to survey specific selection functions and systematic effects. The only survey specific issue that will remain is the footprint bias, which is the bias in making a measurement in
the survey’s footprint in comparison to making it in an $N$-body box or lightcone quadrant (or in some cases across the full-sky). The footprint bias can be computed by superimposing synthetic mock
galaxy data into the survey’s footprint and can then be removed from the survey measurement or injected into the predictions depending on the size and nature of the bias.
To implement this method we rely on two concepts, jittering and Monte Carlo subsampling. A jitter is the process of adding noise to the positions of galaxies in data, in our analysis the noise added
follows a Gaussian distribution with a user specified dispersion scale $\sigma_{\rm J}$. Jittering allows us to masks small scale uncertainties from theoretical uncertainties in simulations and small
scale systematic effects from observations, such as fibre collisions. The jittering process is the point-process equivalent to smoothing a field. The method is shown to be able to remove small scale
theoretical differences, as well as fibre collision-like systematic effects in a mock galaxy catalogue. Monte Carlo subsampling is the process by which we can apply weights to galaxy catalogues
indirectly when the algorithm or statistics being measure, like the MST, are unable to include weighted points. Monte Carlo subsampling treats galaxy weights as probabilities, drawing galaxies based
on their probabilities and performing the MST or hard-wired algorithm on the subsampled galaxies iteratively. The mean of the statistics computed from the subsampled galaxies is taken to be the
weighted statistics. We demonstrate the subsampling procedure applies weights correctly by applying the technique to the 2PCF where weights can be applied directly. We also show that without this
indirect application of weights, the measurements of the MST are extremely biased with a reduced $\chi^{2}\approx 400$ in some cases.
We extend the Monte Carlo subsampling to remove survey dependent selection functions, such as the redshift selection function and completeness. To achieve this, we adjust the weights of galaxies by
the inverse of the redshift selection function and completeness. This allows us to draw subsampled galaxies uniformly across the survey at some specified target density. This method removes any
dependence between the survey’s selection functions (redshift and completeness) and the MST or hard-wired statistics measured.
This approach to measuring the MST, and indeed any hard-wired algorithm, has a number of advantages over the traditional forward model approach.
• •
Jittering and Monte Carlo subsampling occur at the catalogue level, meaning they can be relied upon for any statistics, allowing for galaxy weights to be applied correctly and survey systematic
effects to be largely removed.
• •
The method targets a measurement at a specified target galaxy density. This is a better defined measurement then simply making the measurement on all galaxies in the survey and leads to
measurements with higher SNR ratios.
• •
Measurements are not survey specific and can readily be compared to at the data vector level rather than purely at the parameter level. This will greatly improve interpretability and reliability,
especially in cases where the inference from the MST or hard-wired algorithm challenges the standard model.
• •
It removes the heavy cost of forward modelling and enables emulators to be built for a more general use rather than being custom built for a single galaxy survey.
In this analysis we have limited our analysis to galaxies with stellar masses greater than $10^{13}\,h^{-1}{\rm Mpc}$. In practice how galaxies are limited in a cosmological analysis could play a
significant role in the complexity of the galaxy population modelling. A volume limited sample, i.e. a complete sample of galaxies above a certain luminosity or mass, will make modelling
significantly easier. This removes the need to model Malmquist bias and incomplete galaxy populations at the low mass end.
In Fig.11 we compare the schematic approach of a traditional forward modelled parameter inference pipeline in comparison to the inversion model being proposing in this paper. The key difference in
the approaches is that the inversion model removes the impact of survey systematic effects on the MST or hard-wired algorithm on real data, while the forward model approach adds survey systematic
effects to synthetic data.
We outline below the steps required to make measurements of the MST or hard-wired statistic using the inversion model and consequently how to infer parameter constraints from simulations.
1. 1.
Measurements from galaxy survey:
1. (a)
Define the properties of the galaxy population to compute the MST or hard-wired statistic. This could mean creating a volume limited sample of galaxies in mass or luminosity. Note, the more
complex the properties the more complex the galaxy population modelled will need to be.
2. (b)
Measure the redshift selection function on the galaxy population.
3. (c)
Assign galaxies inverse density weights to remove the redshift selection function and variations in completeness across the sky.
4. (d)
Using mocks with and without small scale systematic effects (like fibre collisions) determine the jitter dispersion $\sigma_{\rm J}$ required to mask small scale systematic effects. This is
determined by finding the smallest value of $\sigma_{\rm J}$ which produces consistent measurements with and without small scale systematic effects.
5. (e)
Determine the number of Monte Carlo subsampling iterations required for the data vector to converge.
6. (f)
Test measurements are not correlated with systematic effects, to test systematic corrections have been correctly applied.
2. 2.
Measurements from $N$-body simulations:
1. (a)
Model galaxy population and perform similar cuts to real data.
2. (b)
Assign galaxies subsampling weights to make measurements at the correct effective galaxy density.
3. (c)
Determine the number of Monte Carlo subsampling iterations required for the data vector to converge.
4. (d)
At a fiducial cosmology and galaxy population model, compute the footprint bias of the survey by making the measurement in the original simulation box or lightcone quadrant and then imposing
the survey’s footprint on the sample.
3. 3.
1. (a)
Using a single synthetic catalogue, test that survey systematic effects are correctly removed through the inversion pipeline.
2. (b)
Test predictions from N-body simulations with the same cosmological parameters and galaxy population modelling match measurements from galaxy survey mocks.
3. (c)
Test the parameter inference pipeline is unbiased by constraining cosmological parameters and galaxy population models on mock observations.
4. 4.
Apply parameter inference pipeline to real data.
It is important to note, in all cases we assume the $N$-body measurements are made in redshift space, either using a lightcone or by adding redshift space distortions along a chosen line-of-sight
axis inside an $N$-body periodic box. However, for the MST the measurements are made in comoving distance, where redshifts are converted into comoving distances assuming a fiducial cosmology. For
simulated lightcones a like-for-like analysis can be performed because this conversion can be carried out consistently however for periodic boxes this conversion becomes problematic. Since stretching
the box to carry out the conversion to the fiducial comoving distance will alter the footprint bias. For this reason it may be simpler to project the periodic box across a region on the sky, so that
the positions can be converted into redshift and reprojected into the fiducial comoving distance. To ensure the footprint bias is the same we would limit the analysis to the redshift ranges of the
5 Summary
In this paper, we outline how to make precision measurements of the MST from galaxy surveys. The technique relies on two methods: jittering and Monte Carlo subsampling.
Jittering is a point-process smoothing technique, where the positions of points are given a random ‘jitter’ or noise, that follows a Gaussian with a standard deviation given by the jitter dispersion
$\sigma_{\rm J}$. In Fig.1 we show how jittering can remove the small scale differences of two random walk distributions – illustrating that jittering can mask theoretical small scale uncertainties.
In Fig.2 we show how jittering removes the effects of fibre collisions – illustrating that jittering can mask small scale systematic effects from galaxy surveys.
Monte carlo subsampling is a technique for indirectly incorporating galaxy weights to the MST. The MST is a hard-wired algorithm, meaning its internals and outputs cannot be altered or post-processed
in a consistent way. This presents a unique challenge for galaxy surveys, as there are no methods for directly including galaxy weights to the MST. In Monte Carlo subsampling, galaxy weights are
treated as probabilities for sampling. The MST is then performed on subsampled realisations of galaxies. Taking the mean of the MST distributions over many realisations gives us the MST of the
weighted galaxies. In Fig.3 we show this technique can reproduce the weighted 2PCF and show that failing to include galaxy weights can deeply bias the MST. Furthermore, we show how to alter the
weights of galaxies to correct for variations in the survey’s redshift selection function and completeness across the sky. The fundamental change in this approach is that we target a measurement of
the MST for galaxies at a specified target density. This ensures the MST has no correlations with the redshift selection function (see Fig.4) and completeness (see Fig.10). In Sec.2.4.4 we derive
measurements of convergence which can be used to determine the number of iterations required for Monte Carlo subsampling. We illustrate the convergence as a function of iterations for 2PCF using a
direct and indirect approximation for the convergence in Fig.5, and for the MST using the approximation in Fig.6. In Fig.7 we show that Monte Carlo subsampling improves the SNR of the MST
In Fig.8 we illustrate how the technique allows us to compare predictions from measurements made with quite different footprints on the sky. The only thing that needs to be corrected is the survey’s
footprint bias, which is the bias in making measurements in the survey’s footprint in comparison to making it in a simulation (periodic box or lightcone quadrant). This is a bias that can be learnt
and corrected.
Lastly, in Fig.11 we illustrate how a parameter inference pipeline will need to be altered to include this inversion method and in section4 we outline the steps that need to be taken to make a
measurement and infer cosmological parameter from the MST. The technique introduced in this paper resolves a number of challenges faced by many hard-wired field level statistics, not only the MST,
and because the techniques operate at the catalogue level, they can be generally used for any statistics. By mitigating survey systematic effects we can make measurements that can now be readily
compared between surveys and can rely on emulators and simulations that are not custom built for a single survey. This will improve interpretability and will make it easier for us to reap the rewards
and sensitivities of field level statistics from the next generation of galaxy surveys, such as DESI, Euclid, LSST and WST.
We thank Benjamin Joachimi, Nicolas Tessore, Tessa Baker, Shaun Cole and Grazianno Rossi for stimulating discussions and guidance in the development of this paper. KN acknowledges support from the
Royal Society grant number URF\R\231006. OL acknowledges the STFC Consolidated Grant ST/R000476/1 and visits to All Souls College and to the Physics Department, Oxford University.
Data Availability
All data used in this analysis are produced from publicly available datasets and software packages. The Millennium XXL galaxy lightcone (Smith etal., 2022) can be downloaded from the Millennium
database^5^55http://icc.dur.ac.uk/data/ and random walk Lévy-Flight distributions can be reproduced using the publicly available python package MiSTree (Naidoo, 2019).
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Fernandez-Cobos, R.,Finelli, F., Forastieri, F., Frailis, M., Fraisse, A.A.,Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K.,Génova-Santos, R.T., Gerbino, M., Ghosh, T.,
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• Tassev etal. (2013)Tassev, S., Zaldarriaga, M., & Eisenstein, D.J., 2013.Solving large scale structure in ten easy steps with COLA, J. Cosmology Astropart. Phys, 2013(6), 036.
• Valogiannis & Dvorkin (2022)Valogiannis, G. & Dvorkin, C., 2022.Going beyond the galaxy power spectrum: An analysis of BOSS datawith wavelet scattering transforms, Phys.Rev.D, 106(10), 103509. | {"url":"https://hennesseycap.com/article/methods-for-robustly-measuring-the-minimum-spanning-tree-and-other-field-level-statistics-from-galaxy-surveys","timestamp":"2024-11-06T17:52:56Z","content_type":"text/html","content_length":"485292","record_id":"<urn:uuid:f6a63804-8a7e-4d06-acc5-1ccf91cbc55f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00578.warc.gz"} |
How will you factor difference of two squares step by step?
To factor a difference of squares, the following steps are undertaken: Check if the terms have the greatest common factor (GCF) and factor it out. Remember to include the GCF in your final answer.
Determine the numbers that will produce the same results and apply the formula: a2– b2 = (a + b) (a – b) or (a – b) (a + b)
How do you identify if a quadratic is the difference of two squares?
The Difference of Two Squares theorem tells us that if our quadratic equation may be written as a difference between two squares, then it may be factored into two binomials, one a sum of the square
roots and the other a difference of the square roots. This is sometimes shown by the expression A² – B² = (A + B) (A – B).
What does two squares mean in a text message?
It means that they’re using an emoji/smiley face icon that your device doesn’t support. Square symbols in a text message usually mean the sender has made a mistake.
Is 4000 a perfect square?
Is the number 4000 a Perfect Square? The prime factorization of 4000 = 25 × 53. Here, the prime factor 2 is not in the pair. Therefore, 4000 is not a perfect square.
What are the steps to find the difference of squares?
Every difference of squares problem can be factored as follows: a2 – b2 = (a + b)(a – b) or (a – b)(a + b). So, all you need to do to factor these types of problems is to determine what numbers
squares will produce the desired results. Step 3: Determine if the remaining factors can be factored any further.
Which of the following is an example of difference of two squares?
When an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5).
What is the perfect square rule?
When you FOIL a binomial times itself, the product is called a perfect square. For example, (a + b)2 gives you the perfect-square trinomial a2 + 2ab + b2.
What does the symbol two squares mean?
The protection of appliances marked with this symbol is ensured by double insulation and does not require a safety connection to electrical earth (ground).
What is the meaning of 2 squares?
In math, the squared symbol (2) is an arithmetic operator that signifies multiplying a number by itself. Multiplying a number by itself is called “squaring” the number.
What are all the real square roots of 100?
List of Perfect Squares
97 9,409 9.849
98 9,604 9.899
99 9,801 9.950
100 10,000 10.000 | {"url":"https://www.raiseupwa.com/popular-guidelines/how-will-you-factor-difference-of-two-squares-step-by-step/","timestamp":"2024-11-03T04:28:33Z","content_type":"text/html","content_length":"105317","record_id":"<urn:uuid:8a7194c4-e495-4125-b7b4-56740913cdba>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00264.warc.gz"} |
Cite as
V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya. On Identity Testing and Noncommutative Rank Computation over the Free Skew Field. In 14th Innovations in Theoretical
Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
Copy BibTex To Clipboard
author = {Arvind, V. and Chatterjee, Abhranil and Ghosal, Utsab and Mukhopadhyay, Partha and Ramya, C.},
title = {{On Identity Testing and Noncommutative Rank Computation over the Free Skew Field}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {6:1--6:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-263-1},
ISSN = {1868-8969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.6},
URN = {urn:nbn:de:0030-drops-175093},
doi = {10.4230/LIPIcs.ITCS.2023.6},
annote = {Keywords: Algebraic Complexity, Identity Testing, Non-commutative rank} | {"url":"https://drops.dagstuhl.de/search/documents?author=Ghosal,%20Utsab","timestamp":"2024-11-07T04:32:32Z","content_type":"text/html","content_length":"66744","record_id":"<urn:uuid:2f240bd8-7ac8-401f-b0af-ec1380dcabc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00586.warc.gz"} |
The following statements are given about plant growth hormones:... | Filo
The following statements are given about plant growth hormones:
I. Cytokinins suppress the synthesis of chlorophyll.
II. Auxins control apical dominance.
III. Gibberellins promote shoot elongation.
IV. Abscisic acid enabling seeds to withstand desiccation.
Which of the above statements are correct?
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Cytokinin delays the senescence of leaves and prevents chlorophyll degradation. It can be shown by rapid bioassay technique. Cytokinin treated leaf tips retard the process of chlorophyll degradation
as compared to untreated leaf discs.
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The following statements are given about plant growth hormones:
I. Cytokinins suppress the synthesis of chlorophyll.
II. Auxins control apical dominance.
Question Text III. Gibberellins promote shoot elongation.
IV. Abscisic acid enabling seeds to withstand desiccation.
Which of the above statements are correct?
Updated On Jul 22, 2022
Topic Plant Growth and Development
Subject Biology
Class Class 11
Answer Type Text solution:1 Video solution: 1
Upvotes 121
Avg. Video Duration 7 min | {"url":"https://askfilo.com/biology-question-answers/the-following-statements-are-given-about-plant-growth","timestamp":"2024-11-07T10:14:40Z","content_type":"text/html","content_length":"158478","record_id":"<urn:uuid:6368f2cc-9d7b-41fa-8eec-9eb9f3b60b14>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00339.warc.gz"} |
Now That You Have a Machine Learning Model, It’s Time to Evaluate Your Security Classifier
This is the third installment in a three-part series about machine learning. Be sure to read part one and part two for more context on how to choose the right artificial intelligence solution for
your security problems.
As we move into this third part, we hope we have helped our readers better identify an artificial intelligence (AI) solution and select the right algorithm to address their organization’s security
needs. Now, it’s time to evaluate the effectiveness of the machine learning (ML) model being used. But with so many metrics and systems available to measure security success, where does one begin?
Classification or Regression? Which Can Get Better Insights From Your Data?
By this time, you may have selected an algorithm of choice to use with your machine learning solution. It could fall into one of two categories in general, classification or regression. Here is a
reminder of the main difference: From a security standpoint, these two types of algorithms tend to solve different problems. For example, a classifier might be used as an anomaly detector, which is
often the basis of the new generation of intrusion detection and prevention systems. Meanwhile, a regression algorithm might be better at things such as detecting denial-of-service attacks (DoS)
because these problems tend to involve numbers rather than nominal labels.
At first look, the difference between Classification and Regression might seem complicated, but it really isn’t. It just comes down to what type of value our target variable, also called our
dependent variable, contains. In that sense, the main difference between the two is that the output variable in Regression is numerical while he output for Classification is categorical/discrete.
For our purposes in this blog, we’ll focus on metrics that are used to evaluate algorithms applied to supervised ML. For reference, supervised machine learning is the form of learning where we have
complete labels and a ground truth. For example, we know that the data can be divided into class1 and class2, and each of our training, validation, and testing samples is labeled as belonging to
class1 or class2.
Classification Algorithms – or Classifiers
To have ML work with data, we can select a security classifier, which is an algorithm with a non-numeric class value. We want this algorithm to look at data and classify it into predefined data
“classes.” These are usually two or more categorical, dependent variables.
For example, we might try to classify something as an attack or not an attack. We would create two labels, one for each of those classes. A classifier then takes the training set and tries to learn a
“decision boundary” between the two classes. There could be more than two classes, and in some cases only one class. For example, the Modified National Institute of Standards and Technology (MNIST)
database demo tries to classify an image as one of the ten possible digits from hand-written samples. This demo is often used to show the abilities of deep learning, as the deep net can output
probabilities for each digit rather than one single decision. Typically, the digit with the highest probability is chosen as the answer.
A Regression Algorithm – or Regressor
A Regression algorithm, or regressor, is used when the target variable is a number. Think of a function in math: there are numbers that go into the function and there is a number that comes out of
it. The task in Regression is to find what this function is. Consider the following example:
Y = 3x+9
We will now find ‘Y’ for various values of ‘X’. Therefore:
X = 1 -> y = 12
X = 2 -> y = 15
X = 3 -> y = 18
The regressor’s job is to figure out what the function is by relying on the values of X and Y. If we give the algorithm enough X and Y values, it will hopefully find the function 3x+9.
We might want to do this in cases where we need to calculate the probability of an event being malicious. Here, we do not want a classification, as the results are not fine-grained enough. Instead,
we want a confidence or probability score. So, for example, the algorithm might provide the answer that “there is a 47 percent probability that this sample is malicious.”
In the next section, we will be looking at the various metrics for each, Classification, and Regression, which can help us determine the efficacy of our security posture by using our chosen ML model.
Metrics for Classification
Before we dive into common classification metrics, let’s define some key terms:
• Ground truth is a set of known labels or descriptions of which class or target variable represents the correct solution. In a binary classification problem, for instance, each example in the
ground truth is labeled with the correct classification. This mirrors the training set, where we have known labels for each example.
• Predicted labels represent the classifications that the algorithm believes are correct. That is, the output of the algorithm.
Now let’s take a closer look at some of the most useful metrics against which we can choose to measure the success of our machine learning deployment.
True Positive Rate
This is the ratio of correctly predicted positive examples to the total number of examples in the ground truth. If there are 100 examples in the ground truth and the model correctly predicts 65 of
them as positive, then the true positive rate (TPR) is 65 percent, sometimes written as 0.65.
False Positive Rate
The false positive rate (FPR) is the number of incorrectly predicted examples that are labeled as positive by the algorithm but are actually negative in the ground truth. If we have 100 examples and
15 of them are incorrectly predicted as positive, then the false positive rate would be 15 percent, sometimes written as 0.15.
True Negative Rate
The true negative rate (TNR) is the number of correctly predicted negative examples divided by the number of examples in the ground truth. Let us say that in the scenario of 100 examples that another
15 of these examples were correctly predicted as negative. Therefore, the true negative rate (TNR) is 15 percent, also written as 0.15. Notice here that there were 15 false positives and 15 true
negatives. This makes for a total of 30 negative examples.
False Negative Rate
The false negative rate (FNR) is the ratio of examples predicted incorrectly as belonging to the negative class over the number of examples in the ground truth. Continuing with the aforementioned
case, let’s say that out of 100 examples in the ground truth, the algorithm correctly predicted 65 as positive. We also know that 15 were predicted as false positives and 15 were predicted as true
negatives. This leaves us with 5 examples unaccounted for, so our false negative rate is 5 percent, or 0.05. The false negative rate is the complement to the true positive rate, so the sum of the two
metrics should be 70 percent (0.7), as 70 examples actually belong to the positive class.
Accuracy measures the proportion of correct predictions, both positive and negative, to the total number of examples in the ground truth. This metric can often be misleading if, for instance, there
is a large proportion of positive examples in the ground truth compared to the number of negative examples. Similarly, if the model predicts only the positive class correctly, accuracy will not give
you a sense of how well the model does with negative predictions versus negative examples in the ground truth even though the accuracy could be quite high because the positive examples were
Accuracy = (TP+TN)/(TP+TN+FP+FN)
Before we explore the precision metric, it’s important to define a few more terms:
• TP is the raw number of true positives (in the above example, the TP is 65).
• FP is the raw number of false positives (15 in the above example).
• TN is the raw number of true negatives (15 in the above example).
• FN is the raw number of false negatives (5 in the above example).
Precision, sometimes known as the positive predictive value, is the proportion of true positives predicted by the algorithm over the sum of all examples predicted as positive. That is, precision=TP/
In our example, there were 65 positives in the ground truth that the algorithm correctly labeled as positive. However, it also labeled 15 examples as positive when they were actually negative.
These false positives go into the denominator of the precision calculation. So, we get 65/(65+15), which yields a precision of 0.81.
What does this mean? In brief, high precision means that the algorithm returned far more true positives than false positives. In other words, it is a qualitative measure. The higher the precision,
the better job the algorithm did of predicting true positives while rejecting false positives.
Recall, also known as sensitivity, is the ratio of true positives to true positives plus false negatives: TP/(TP+FN).
In our example, there were 65 true positives and 5 false negatives, giving us a recall of 65/(65+5) = 0.93. Recall is a quantitative measure; in a classification task, it is a measure of how well the
algorithm “memorized” the training data.
Note that there is often a trade-off between precision and recall. In other words, it’s possible to optimize one metric at the expense of the other. In a security context, we may often want to
optimize recall over precision because there are circumstances where we must predict all the possible positives with a high degree of certainty.
For example, in the world of automotive security, where kinetic harm may occur, it is often heard that false positives are annoying, but false negatives can get you killed. That is a dramatic
example, but it can apply to other situations as well. In intrusion prevention, for instance, a false positive on a ransomware sample is a minor nuisance, while a false negative could cause
catastrophic data loss.
However, there are cases that call for optimizing precision. If you are constructing a virus encyclopedia, for example, higher precision might be preferred when analyzing one sample since the missing
information will presumably be acquired from another sample.
An F-measure (or F1 score) is defined as the harmonic mean of precision and recall. There is a generic F-measure, which includes a variable beta that causes the harmonic mean of precision and recall
to be weighted.
Typically, the evaluation of an algorithm is done using the F1 score, meaning that beta is 1 and therefore the harmonic mean of precision and recall is unweighted. The term F-measure is used as a
synonym for F1 score unless beta is specified.
The F1 score is a value between 0 and 1 where the ideal score is 1, and is calculated as 2 * Precision * Recall/(Precision+Recall), or the harmonic mean. This metric typically lies between precision
and recall. If both are 1, then the F-measure equals 1 as well. The F1 score has no intuitive meaning per se; it is simply a way to represent both precision and recall in one metric.
Matthews Correlation Coefficient
The Matthews Correlation Coefficient (MCC), sometimes written as Phi, is a representation of all four values — TP, FP, TN and FN. Unlike precision and recall, the MCC takes true negatives into
account, which means it handles imbalanced classes better than other metrics. It is defined as:
If the value is 1, then the classifier and ground truth are in perfect agreement. If the value is 0, then the result of the classifier is no better than random chance. If the result is -1, the
classifier and the ground truth are in perfect disagreement. If this coefficient seems low (below 0.5), then you should consider using a different algorithm or fine-tuning your current one.
Youden’s Index
Also known as Youden’s J statistic, Youden’s index is the binary case of the general form of the statistic known as ‘informedness’, which applies to multiclass problems. It is calculated as
(sensitivity + specificity–1) and can be seen as the probability of an informed decision verses a random guess. In other words, it takes all four predictors into account.
Remember from our examples that recall=TP/(FP+FN) and specificity, or TNR, is also the complement of the FPR. Therefore, the Youden index incorporates all measures of predictors. If the value of
Youden’s index is 0, then the probability of the decision actually being informed is no better than random chance. If it is 1, then both false positives and false negatives are 0.
Area Under the Receiver Operator Characteristic Curve
This metric, usually abbreviated as AUC or ROC, measures the area under the curve plotted with true positives on the Y-axis and false positives on the X-axis. This metric can be useful because it
provides a single number that lets you compare models of different types. An AUC value of 0.5 means the result of the test is essentially a coin flip. You want the AUC to be as close to 1 as possible
because this enables researchers to make comparisons across experiments.
Area Under the Precision Recall Curve
Area under the precision recall curve (AUPRC) is a measurement that, like MCC, accounts for imbalanced class distributions. If there are far more negative examples than positive examples, you might
want to use AUPRC as your metric and visual plot. The curve is precision plotted against recall. The closer to 1, the better. Note that since this metric/plot works best when there are more negative
predictions than positive predictions, you might have to invert your labels for testing.
Average Log Loss
Average log loss represents the penalty of wrong prediction. It is the difference between the probability distributions of the actual and predicted models.
In deep learning, this is sometimes known as the cross-entropy loss, which is used when the result of a classifier such as a deep learning model is a probability rather than a binary label.
Cross-entropy loss is therefore the divergence of the predicted probability from the actual probability in the ground truth. This is useful in multiclass problems but is also applicable to the
simplified case of binary classification.
By using these metrics to evaluate your ML model, and tailoring them to your specific needs, you could fine-tune the output from the data and essentially get more certain results, thus detecting more
issues/threats, and optimizing controls as needed.
Metrics for Regression
For regression, the goal is to determine the amount of errors produced by the ML algorithm. The model is considered good if the error value between the predicted and observed value is small.
Let’s take a closer look at some of the metrics used for evaluating regression models.
Mean Absolute Error
Mean absolute error (MAE) is the closeness of the predicted result to the actual result. You can think of this as the average of the differences between the predicted value and the ground truth
value. As we proceed along each test example when evaluating against the ground truth, we subtract the actual value reported in the ground truth from the predicted value from the regression algorithm
and take the absolute value. We can then calculate the arithmetic mean of these values.
While the interpretation of this metric is well-defined, because it is an arithmetic mean, it could be affected by very large, or very small differences. Note that this value is scale-dependent,
meaning that the error is on the same scale as the data. Because of this, you cannot compare two MAE values across datasets.
Root Mean Squared Error
Root mean squared error (RMSE) attempts to represent all error across moments in time in one value. This is often the metric that optimization algorithms seek to minimize in regression problems. When
an optimization algorithm is tuning so-called hyperparameters, it seeks to make RMSE as small as possible.
Consider, however, that like MAE, RMSE is both sensitive to large and small outliers and is scale-dependent. Therefore, you have to be careful and examine your residuals to look for outliers — values
that are significantly above or below the rest of the residuals. Also, like MAE, it is improper to compare RMSE across datasets unless the scaling translations have been accounted for, because data
scaling, whether by normalization or standardization, is dependent upon the data values.
For example, in Standardization, the scale from -1 to 1 is determined by subtracting the mean from each value and dividing the value by the standard deviation. This gives the normal distribution. If,
on the other hand, the data is normalized, the scaling is done by taking the current value and subtracting the minimum value, then dividing this by the quantity (maximum value – minimum value). These
are completely different scales, and as a result, one cannot compare the RMSE between these two data sets.
Relative Absolute Error
Relative absolute error (RAE) is the mean difference divided by the arithmetic mean of the values in the ground truth. Note that this value can be compared across scales because it has been
Relative Squared Error
Relative squared error (RSE) is the total squared error of the predicted values divided by the total squared error of the observed values. This also normalizes the error measurement so that it can be
compared across datasets.
Machine Learning Can Revolutionize Your Organization’s Security
Machine learning is integral to the enhancement of cybersecurity today and it will only become more critical as the security community embraces cognitive platforms.
In this three-part series, we covered various algorithms and their security context, from cutting-edge technologies such as generative adversarial networks to more traditional algorithms that are
still very powerful.
We also explored how to select the appropriate security classifier or regressor for your task, and, finally, how to evaluate the effectiveness of a classifier to help our readers better gauge the
impact of optimization. With a better idea about these basics, you’re ready to examine and implement your own algorithms and to move toward revolutionizing your security program with machine
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Instance on Points Node
Instance on Points Node
The Instance on Points node adds a reference to a geometry to each of the points present in the input geometry. Instances are a fast way to add the same geometry to a scene many times without
duplicating the underlying data. The node works on any geometry type with a Point domain, including meshes, point clouds, and curve control points.
Any attributes on the points from the Geometry input will be available on the instance domain of the generated instances.
The Make Instances Real operator can be used to create objects from instances generated with this node.
To instance object types that do not contain geometry, like a light object, the Object Info Node can be used. Other objects like Metaball objects are not supported for instancing.
Standard geometry input. The position of the points of this geometry affect the transforms of each instance output.
If the input geometry contains instances, the node will create more instances on the points inside the instances, creating nested instancing. In this case, each new instance will have the
transform created by the node from the Rotation and Scale inputs, but it will also be transformed based on the parent instances.
Whether to instance on each point. True values mean the an instance will be generated on the point, false values mean the point will be skipped.
The geometry to instance on each selected point. This can contain real geometry, or multiple instances, which can be useful when combined with the Pick Instance option.
Pick Instances
If enabled, instead of adding the entire geometry from the Instance input on every point, choose an instance from the instance list of the geometry based on the Instance Index input. This option
is intended to be used with the Collection Info Node.
Instance Index
The selection of index for every selected point, only used when Pick Instances is true. By default the point ID is used, or the index if that doesn’t exist. Negative values or values that are too
large are wrapped around to the other end of the instance list.
The Euler rotation for every instance. This can use the rotation output of nodes like Distribute Points on Faces and Curve to Points. An Euler rotation can also be created from a direction vector
like the normal with the Align Euler to Vector Node.
The size of each generated instance.
This node has no properties.
Standard geometry output. If the id attribute exists on the input geometry, it will be copied to the result instances. | {"url":"https://docs.blender.org/manual/en/3.3/modeling/geometry_nodes/instances/instance_on_points.html","timestamp":"2024-11-09T23:19:33Z","content_type":"text/html","content_length":"23579","record_id":"<urn:uuid:b4300d35-ca13-47d1-9a4d-06eeb22e69b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00220.warc.gz"} |
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Me[a Posted: Saturday 16th of Feb 14:57
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Algebra filter
This page really needs improving. Please see the page comments for suggestions of what to include, then remove this template when you're done.
Note: Although the Algebra filter is included in Moodle, it is recommended that you use the MathJax filter for writing Mathematical equations and expressions.
Moodle has an "algebra filter" which can display mathematical expressions as if they were typeset. The filter is included in the standard Moodle packages but the administrator must activate it before
you can use it.
NOTE: the use of the algebra filter REQUIRES that the Moodle Tex filter also be working as the algebra filter simply parses math expressions in one syntax and then converts the expressions to LaTex
expressions for the Tex filter to render and display. The filter uses a borrowed perl script
The filter (see http://cvs.moodle.org/moodle/filter/algebra/ ) uses algebra2tex.pl, a perl script (and its associated perl module, AlgParser.pm) to parse text expressions delimited by double@ signs
as tokens (though the tokens employed can be tweaked manually.)
NOTE perl script REQUIRES Perl. This is no problem with most Linux servers, but may be the cause for malfunctions in the Win world like broken connections. Exceptions / failures are NOT catched and
reported via this Moodle filter. If there is no Perl available, activate Moodle Tex (as mentioned above) and use the JAVA Math Formula Editor. It is available after activating Tex. Minor corrections
in the resulting Tex code dont need deeper digging in Tex fundamentals.
While many have argued that the algebra filter is easy to use, there is apparently no Moodle reference on its "grammar" or "syntax". Research in to Webworks produced these two links for using
webworks text expressions (the first one is arguably replaced by the second):
See http://moodle.org/mod/forum/discuss.php?d=126522&parent=554632.
There is a discussion of Algebra filter syntax at http://moodle.org/mod/forum/discuss.php?d=5402 offered by Zbigniew Fiedorowicz.
It appears that as of Moodle.org moving to Moodle 2 (late 2010) the Algebra filter in the forums is no longer working. It also appears that the Algebra filter does not work in Moodle Docs (which is
not Moodle code, but MediaWiki, which I don't believe has a ready filter based on algebra2tex.pl.
The file algebradebug.php is provided in the filter folder for the purpose of debugging. Problems may arise comes from the first tags in algebradebug.php
define('NO_MOODLE_COOKIES', true); // Because it interferes with caching
You can delete the line or set it to false. | {"url":"https://docs.moodle.org/37/en/Algebra_filter","timestamp":"2024-11-08T06:07:17Z","content_type":"text/html","content_length":"30370","record_id":"<urn:uuid:1fbb0f98-4536-4daf-a854-1e05cd0b60aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00236.warc.gz"} |
Comparing obstructions to local-global principles for rational points over semiglobal fields
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Dhruv Ranganathan.
Let K be a complete discretely valued field, let F be the function field of a curve over K, and let Z be a variety over F. When the existence of rational points on Z over a set of local field
extensions of F implies the existence of rational points on Z over F, we say a local-global principle holds for Z.
In this talk, we will compare local-global principles, and obstructions to such principles, for two choices of local field extensions of F. On the one hand we consider completions F_v at valuations
of F, and on the other hand we consider fields F_P which are the fraction fields of completed local rings at points on the special fibre of a regular model of F.
We show that if a local-global principle with respect to valuations holds, then so does a local-global principle with respect to points, for all models of F. Conversely, we prove that there exists a
suitable model of F such that if a local-global principle with respect to points holds for this model, then so does a local-global principle with respect to valuations.
This is joint work with David Harbater, Julia Hartmann, and Florian Pop
This talk is part of the Algebraic Geometry Seminar series.
This talk is included in these lists:
Note that ex-directory lists are not shown. | {"url":"https://talks.cam.ac.uk/talk/index/129859","timestamp":"2024-11-07T16:26:55Z","content_type":"application/xhtml+xml","content_length":"14155","record_id":"<urn:uuid:d89e5320-845c-4f3b-b4ce-560696f15048>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00377.warc.gz"} |
Re: st: tricks to speed up -xtmelogit-
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
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Re: st: tricks to speed up -xtmelogit-
From Stas Kolenikov <[email protected]>
To [email protected]
Subject Re: st: tricks to speed up -xtmelogit-
Date Tue, 21 Dec 2010 14:45:43 -0600
On Tue, Dec 21, 2010 at 2:28 PM, Sergiy Radyakin <[email protected]> wrote:
> Given the rareness of your outcome taking a simple subsample may yield just
> a few positives in the subsample. May I suggest also to consider taking all
> positives and a random subsample of negatives, estimate the candidate and
> then run the full sample on that?
In this case, the estimate of the intercept will be biased (and so
will probably be the estimates of the variance of the random effects),
while the slopes will be OK. You can leave it alone (it will converge
in one or two iterations), or adjust it towards the true proportion of
1's in the sample. Suppose you took 1% sample of 0's and 100% sample
of 1's, so you end up with roughly 0.5%:1% = 1:2 ratio of 0's and 1's.
Then without the regressors, your intercept will be something like
log( odds ratio of 1:2 ) = -0.7, while in reality it should've been
logit( odds ratio of 0.5% to 99.5% ) = -5.3. Thus shifting the
intercept down by 4.6 will take care of most of the bias, at least in
terms of specifying the starting values.
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2010-12/msg00829.html","timestamp":"2024-11-13T15:34:53Z","content_type":"text/html","content_length":"10674","record_id":"<urn:uuid:6d1fc8a9-f60b-4590-b636-87ea9d0cd465>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00310.warc.gz"} |
Notable Points of the Triangle: How to Locate?
You notable triangle points are points that mark the intersection of certain elements of a triangle (polygon that has three sides and three angles). To find the geometric position of each of the four
notable points, it is necessary to know the concepts of median, bisector, perpendicular bisector and height of a triangle.
Read too: What is the condition for the existence of a triangle?
Summary on the notable points of the triangle
• Barycenter, incenter, circumcenter and orthocenter are the notable points of a triangle.
• Barycenter is the point where the medians of the triangle meet.
• The barycenter divides each median in such a way that the largest segment of the median is twice the smallest segment.
• Incenter is the intersection point of the angle bisectors of the triangle.
• The center of the circle inscribed in the triangle is the incenter.
• Circumcenter is the point where the bisectors of the triangle meet.
• The center of the circle circumscribing the triangle is the circumcenter.
• Orthocenter is the intersection point of the heights of the triangle.
Video lesson on the notable points of the triangle
What are the notable points of the triangle?
The four notable points of the triangle are barycenter, incenter, circumcenter and orthocenter. These points are related, respectively, to the median, bisector, perpendicular bisector and height of
the triangle. Let's see what these geometric elements are and what is the relationship of each one with the notable points of the triangle.
→ Barycenter
The barycenter is the notable point of the triangle that is related to the median. The median of a triangle is the segment with one endpoint at one vertex and the other endpoint at the midpoint of
the opposite side. In the triangle ABC below, H is the midpoint of BC and the segment AH is the median relative to vertex A.
In the same way, we can find the medians relative to vertices B and C. In the image below, I is the midpoint of AB and J is the midpoint of AC. Thus, BJ and CI are the other medians of the triangle.
Note that K is the meeting point of the three medians. This point where the medians meet is called the barycenter of triangle ABC..
• Property: the barycenter divides each median of a triangle in a 1:2 ratio.
Consider, for example, the median AH from the previous example. Note that the KH segment is smaller than the AK segment. According to the property, we have
Do not stop now... There's more after the publicity ;)
→ Incenter
The incenter is the notable point of the triangle that is related to the bisector. The bisector of a triangle is the ray with end point at one of the vertices that divide the corresponding interior
angle into congruent angles. In the triangle ABC below, we have the bisector relative to vertex A.
In the same way, we can obtain the bisectors relative to the vertices B and C:
Note that P is the point of intersection of the three bisectors. This point of intersection of the bisectors is called the incenter of the triangle ABC..
• Property: the incenter is equidistant from the three sides of the triangle. So this point is the center of the circumference inscribed in the triangle.
See too: What is the inner bisector theorem?
→ Circumcenter
The circumcenter is the notable point of the triangle that is related to the bisector. The bisector of a triangle is the line perpendicular to the midpoint of one of the sides of the triangle. Ahead,
we have the perpendicular bisector of the segment BC of the triangle ABC.
Constructing the bisectors of the segments AB and AC, we obtain the following figure:
Note that L is the point of intersection of the three bisectors. This intersection pointbisectors is called the circumcenter of triangle ABC.
• Property: the circumcenter is equidistant from the three vertices of the triangle. Thus, this point is the center of the circle circumscribed to the triangle.
→ Orthocenter
The orthocenter is the notable point of the triangle that is related to height. The height of a triangle is the segment whose endpoint is at one of the vertices that form a 90° angle with the
opposite side (or its extension). Below, we have the height relative to vertex A.
Drawing the heights relative to vertices B and C, we produce the following image:
Note that D is the point of intersection of the three heights. This point of intersection of heights is called the orthocenter of triangle ABC..
Important: the triangle ABC used in this text is a scalene triangle (triangle whose three sides have different lengths). The figure below indicates the notable points of the triangle we studied. Note
that, in this case, the points occupy different positions.
In an equilateral triangle (triangle whose three sides are congruent), the notable points are coincident. This means that the barycenter, incenter, circumcenter and orthocenter occupy exactly the
same position in an equilateral triangle.
See too: What are the cases of congruence of triangles?
Solved exercises on the notable points of the triangle
question 1
In the figure below, points H, I, and J are the midpoints of sides BC, AB, and AC, respectively.
If AH = 6 cm, the length, in cm, of segment AK is
TO 1
B) 2
C) 3
D) 4
E) 5
Alternative D.
Note that K is the barycenter of triangle ABC. Like this,
Since AH = AK + KH and AH = 6, then
\(AK = 12 - 2 AK\)
\(3AK = 12\)
\(AK = 4\)
question 2
(UFMT – adapted) You want to install a factory in a place that is equidistant from municipalities A, B and C. Assume that A, B, and C are non-collinear points in a plane region and that triangle ABC
is scalene. Under these conditions, the point where the factory should be installed is:
A) Circumcenter of triangle ABC.
B) barycenter of triangle ABC.
C) incenter of triangle ABC
D) orthocenter of triangle ABC.
E) midpoint of the AC segment.
Alternative A.
In a triangle ABC, the point equidistant from the vertices is the circumcenter.
LIMA, E. L. Analytic geometry and Linear algebra. Rio de Janeiro: Impa, 2014.
REZENDE, E. Q. F.; QUEIROZ, M. L. B. in. Flat Euclidean Geometry: and geometric constructions. 2nd ed. Campinas: Unicamp, 2008. | {"url":"https://jerseysteam.com/en/stories/13038-notable-points-of-the-triangle-how-to-locate","timestamp":"2024-11-06T17:18:33Z","content_type":"text/html","content_length":"21558","record_id":"<urn:uuid:b3791cc7-6b8e-42aa-8fa0-4d935fe79550>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00678.warc.gz"} |
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Quantitative Trading Strategy Integrating Reversal and Future Lines of Demarcation
1. Quantitative Trading Strategy Integrating Reversal and Future Lines of Demarcation
Quantitative Trading Strategy Integrating Reversal and Future Lines of Demarcation
, Date: 2023-12-08 12:00:35
This strategy integrates the 123 reversal strategy and future lines of demarcation (FLD) strategy to implement a quantitative trading strategy that enters or exits positions when both strategies
generate signals simultaneously. It is mainly applied to index futures markets, capturing opportunities from combinations of short-term reversal signals and medium-long term trend signals for
medium-short term holding trades.
123 Reversal Strategy
The 123 reversal strategy originates from the book “How I Tripled My Money in the Futures Market”. It goes long when the closing price shows reversal patterns for two continuous days and the 9-day
slow stochastics is below 50; It goes short when the closing price shows reversal patterns for two continuous days and the 9-day fast stochastics is above 50.
Future Lines of Demarcation Strategy
The future lines of demarcation (FLD) strategy is a trend-following strategy based on the periodicity of price fluctuations. FLD lines are plotted by shifting the median, high or low prices
approximately half a cycle into the future. Trading signals are generated when prices cross the FLD lines.
Advantage Analysis
This strategy combines reversal and trend-following strategies, capturing both short-term reversal opportunities and medium-long term trend directions on multiple time frames for quantitative
trading. The reversal element provides short-term profit-taking chances while the trend-following part ensures the overall trading aligns with the trend, effectively controlling trading risks.
Moreover, the adaptive nature of FLD also enhances the stability of the strategy.
Risk Analysis
The main risks of this strategy come from false breakouts of reversal signals and errors in FLD line judgments. For the former, parameters can be adjusted to confirm reversal signals or add other
auxiliary indicators to improve accuracy. For the latter, parameters need to be optimized to ensure FLD describes market cycles more precisely. Additionally, mistakes of FLD when major trend
reversals occur should also be watched out for.
Optimization Directions
1. Improve reversal strategy by adding other indicators to filter signals and decrease false breakout possibilities
2. Compare different FLD parameters to better describe cyclical patterns
3. Add stop loss logic to control single loss risks
4. Test parameter effectiveness across different products
This strategy combines reversal and trend-following concepts for stable profits over medium-short term time frames. Future optimizations in aspects of signal accuracy, trend description capability
and risk control will expand its parameter universe and improve stability.
start: 2022-12-01 00:00:00
end: 2023-12-07 00:00:00
period: 1d
basePeriod: 1h
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// Copyright by HPotter v1.0 28/08/2020
// This is combo strategies for get a cumulative signal.
// First strategy
// This System was created from the Book "How I Tripled My Money In The
// Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies.
// The strategy buys at market, if close price is higher than the previous close
// during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50.
// The strategy sells at market, if close price is lower than the previous close price
// during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50.
// Second strategy
// An FLD is a line that is plotted on the same scale as the price and is in fact the
// price itself displaced to the right (into the future) by (approximately) half the
// wavelength of the cycle for which the FLD is plotted. There are three FLD's that can be
// plotted for each cycle:
// An FLD based on the median price.
// An FLD based on the high price.
// An FLD based on the low price.
// WARNING:
// - For purpose educate only
// - This script to change bars colors.
Reversal123(Length, KSmoothing, DLength, Level) =>
vFast = sma(stoch(close, high, low, Length), KSmoothing)
vSlow = sma(vFast, DLength)
pos = 0.0
pos := iff(close[2] < close[1] and close > close[1] and vFast < vSlow and vFast > Level, 1,
iff(close[2] > close[1] and close < close[1] and vFast > vSlow and vFast < Level, -1, nz(pos[1], 0)))
FLD(Period,src) =>
pos = 0
pos := iff(src[Period] < close , 1,
iff(src[Period] > close, -1, nz(pos[1], 0)))
strategy(title="Combo Backtest 123 Reversal & FLD's - Future Lines of Demarcation", shorttitle="Combo", overlay = true)
Length = input(15, minval=1)
KSmoothing = input(1, minval=1)
DLength = input(3, minval=1)
Level = input(50, minval=1)
Period = input(title="Period", defval=40)
src = input(title="Source", type=input.source, defval=close)
reverse = input(false, title="Trade reverse")
posReversal123 = Reversal123(Length, KSmoothing, DLength, Level)
posFLD = FLD(Period,src)
pos = iff(posReversal123 == 1 and posFLD == 1 , 1,
iff(posReversal123 == -1 and posFLD == -1, -1, 0))
possig = iff(reverse and pos == 1, -1,
iff(reverse and pos == -1 , 1, pos))
if (possig == 1)
strategy.entry("Long", strategy.long)
if (possig == -1)
strategy.entry("Short", strategy.short)
if (possig == 0)
barcolor(possig == -1 ? #b50404: possig == 1 ? #079605 : #0536b3 ) | {"url":"https://www.fmz.com/strategy/434682","timestamp":"2024-11-06T05:07:09Z","content_type":"text/html","content_length":"14563","record_id":"<urn:uuid:968e3789-27b0-4559-b421-37a0a942de54>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00035.warc.gz"} |
Towards Mathematical Literacy in the 21st Century: Perspectives from Singapore
The Organization for Economic Cooperation and Development (OECD) postulates that a major focus in education is to promote the ability of young people to use their knowledge and skills to meet
real-life challenges (OECD, 2006). PISA, an international standardised assessment of students’ (aged 15) performance in the literacies of mathematics, science, and reading, was developed by the OECD
in 1997 to evaluate the achievement of students who are about to finish their key stages of education (Anderson, Chiu, & Yore, 2010). The concept of mathematical literacy has been defined and
interpreted in various ways as recorded in the curriculum documents around the world. This paper will share perspectives from Singapore on how mathematical literacy is interpreted in the mathematics
curriculum through the use of three tasks: interdisciplinary project work, applications, and modelling. It will surface challenges to improving the mathematical literacy of students when using such
Mathematical Literacy; Mathematical Applications Tasks; Mathematical Modelling; Singapore
Anderson, J. O., Chiu, M. H., & Yore, L. D. (2010). First cycle of PISA (2000-2006) - International perspectives on successes and challenges: Research and policy directions. International Journal of
Science and Mathematics Education, 8(3), 373-388.
Ang, K. C. (2010, December 17-21). Teaching and learning mathematical modelling with technology. Paper presented at the Linking applications with mathematics and technology: The 15th Asian Technology
Conference in Mathematics, Le Meridien Kuala Lumpur.
Curriculum Planning and Development Division [CPDD]. (1999). Project work: Guidelines. Singapore, Ministry of Education: Author.
Curriculum Planning and Development Division [CPDD]. (2006). Mathematics Syllabus. Singapore, Ministry of Education: Author.
de Lange, J. (2006). Mathematical literacy for living from OECD-PISA perspective. Tsukuba Journal of Educational Study in Mathematics, 25(1), 13-35.
English, L. D. (2008). Interdisciplinary problem solving: A focus on engineering experiences. In M. Goos, R. Brown & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the Mathematics
Education Research Group of Australasia (Vol. 1, pp. 187-193). Brisbane: MERGA.
Foo, K. F. (2007). Integrating performance tasks in the secondary mathematics classroom:An empirical study. Unpublished Masters Dissertation, Nanyang Technological University, Singapore
Galbraith, P. (1998). Cross-curriculum applications of mathematics. Zentralblatt für Didaktik der Mathematik, 30(4), 107-109.
Gravemeijer, K. P. E. (1994). Developing realistic mathematics education. Utrecht, The Netherlands: Freudenthal Institute.
Ministry of Education [MOE]. (2001, December 2). Press release: Project work to be included for university admission in 2005. Retrieved December 13, 2002, from http://www1.moe.edu.sg/press/2001/
Ng, K. E. D. (2009). Thinking, small group interactions, and interdisciplinary project work. Unpublished doctoral dissertation, The University of Melbourne, Australia.
Ng, K. E. D. (2010). Initial experiences of primary school teachers with mathematical modelling. In B. Kaur & J. Dindyal (Eds.), Mathematical modelling and applications:Yearbook of Association of
Mathematics Educators (pp. 129-144). Singapore: World Scientific.
Ng, K. E. D. (2011a). Facilitation and scaffolding: Symposium on Teacher Professional Development on Mathematical Modelling - Initial perspectives from Singapore. Paper presented at the Connecting to
practice - Teaching practice and the practice of applied mathematicians: The 15th International Conference on the Teaching of Mathematical Modelling and Applications.
Ng, K. E. D. (2011b). Mathematical Knowledge Application and Student Difficulties in a Design-Based Interdisciplinary Project. In G. Kaiser, W. Blum, R. Borromeo Ferri & G.Stillman (Eds.),
International perspectives on the teaching and learning of mathematical modelling: Trends in the teaching and learning of mathematical modelling (Vol. 1, pp. 107-116). New York: Springer.
Ng, K. E. D., & Stillman, G. A. (2009). Applying mathematical knowledge in a design-based interdisciplinary project. In R. Hunter, B. Bicknell & T. Burgess (Eds.), Crossing, divides: Proceedings of
the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 411-418). Wellington, New Zealand:Adelaide: MERGA.
Organisation for Economic Co-Operation and Development [OECD]. (2006). PISA 2006:Science competencies for tomorrow's world (Vol. 1). Paris: OECD.
Quek, C. L., Divaharan, S., Liu, W. C., Peer, J., Williams, M. D., Wong, A. F. L., et al.(2006). Engaging in project work. Singapore: McGraw Hill.
Sawyer, A. (2008). Making connections: Promoting connectedness in early mathematics education. In M. Goos, R. Brown & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the Mathematics
Education Research Group of Australasia (Vol. 2, pp.429-435). Brisbane: MERGA.
Stacey, K. (2009). Mathematics and scientific literacy around the world. In U. H. Cheah, R. P. Wahyudi, K. T. Devadason, W. Ng, Preechaporn & J. Aligaen (Eds.), Improving science and mathematics
literacy - Theory, innovation and practice: Proceedings of the Third International Conference on Science and Mathematics Education (CoSMEd) (pp. 1-7).
Malaysia, Penang: SEAMEO RECSAM.
Stillman, G., Brown, J., & Galbraith, P. L. (2008). Research into the teaching and learning of applications and modelling in Australasia. In H. Forgasz, A. Barkatsas, A. Bishop, B. Clarke, S. Keast,
W. T. Seah & P. Sullivan (Eds.), Research in mathematics education in Australasia: New directions in mathematics and science education (pp. 141-164). Rotterdam: Sense Publishers.
Tharman, S. (2005). Speech by Mr Tharman Shanmugaratnam, Minister for Education, at the opening of the conference on 'Redesigning Pedagogy: Research, Policy and Practice' on 30 May, at the National
Institute of Education, Singapore. Retrieved May 27, 2008, from http://www.moe.gov.sg/media/speeches/2005/sp20050530.htm
Van den Heuvel-Panhuizen, M. (1999). Context problems and assessment: Ideas from the Netherlands. In I. Thompson (Ed.), Issues in teaching numeracy in primary schools (pp. 130-142). Buckingham, UK:
Open University Press.
Verschaffel, L., deCorte, E., & Borghart, I. (1997). Pre-service teachers' conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems. Learning
and Instruction, 7(4), 339-359.
Zevenbergen, R., & Zevenbergen, K. (2009). The numeracies of boatbuilding: New numeracies shaped by workplace technologies. International Journal of Science and Mathematics Education, 7(1), 183-206.
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The Harpur Euclid
The Harpur Euclid: An Edition of Euclid's Elements
Rivingtons, 1890 - 515 pages
From inside the book
Results 1-5 of 86
Page 8 ... produced in G. Join AB . On AB describe the equilateral triangle DAB . From the centre B , at the distance BC , describe [ POST . I. [ I. 1 . the circle [ POST . 3 . From the centre D , at
the distance DG , describe the circle GKL ...
Page 9 ... produced either both through the vertex , or , as in the diagram , both in the opposite direction ; so that with BC and A given there are eight ways in which the problem might be solved .
The student should try to go through the other ...
Page 14 ... produced , the angles on the other side of the base shall be equal to one another . Let ABC be an isosceles triangle , having the side AB equal to the side AC , and let the straight lines
AB , AC be produced to D and E , the angle ...
Page 29 ... produced to any length both ways , and let C be the given point with- out it ; it is required to draw from the point C a straight liner to AB . H G D Take any point D on the other side of
AB , and from the centre C at the distance CD ...
Page 34 ... produced , the exterior angle shall be greater than either of the interior and opposite angles . Let the side BC of the △ ABC be produced to D , the ext . ACD shall be greater than either
of the int . and opp . 4s BAC , ABC . A Bisect ...
Popular passages
If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
... figures are to one another in the duplicate ratio of their homologous sides.
Let it be granted that a straight line may be drawn from any one point to any other point.
IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal
to one another.
Any two sides of a triangle are together greater than the third side.
Three times the sum of the squares on the sides of a triangle is equal to four times the sum of the squares of the lines joining the middle point of each side with the opposite angles.
Pythagoras' theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together •with the square...
Bibliographic information | {"url":"https://books.google.com.gi/books?id=Jg03AAAAMAAJ&q=produced&lr=&output=html_text&source=gbs_word_cloud_r&cad=4","timestamp":"2024-11-10T14:30:29Z","content_type":"text/html","content_length":"53492","record_id":"<urn:uuid:725dda01-c914-4d15-a0c6-4dedc61ad222>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00388.warc.gz"} |
Fractional Math Sentences Worksheet - 15 Worksheets.com
Fractional Math Sentences
Worksheet Description
The worksheet is designed to help students practice calculating the fraction of a number. It presents a series of problems where a fraction is given alongside a whole number, and the student’s task
is to multiply the fraction by the whole number to find the solution. Each problem is followed by a blank space where students are expected to write their answers. This exercise format is
straightforward and focuses exclusively on the multiplication of fractions with whole numbers, a fundamental skill in arithmetic.
The worksheet aims to teach students how to apply fractions to whole numbers, a key concept in understanding how to find parts of quantities. Students are learning to multiply the numerator by the
whole number and then divide by the denominator to find the fraction of the number. This skill is essential for real-world applications, such as recipe adjustments, dividing items into groups, and
understanding proportions. Additionally, it strengthens the students’ overall mathematical fluency and prepares them for more complex operations involving fractions. | {"url":"https://15worksheets.com/worksheet/fractions-of-a-whole-number-3/","timestamp":"2024-11-08T08:16:16Z","content_type":"text/html","content_length":"109100","record_id":"<urn:uuid:84a90c22-bd18-484c-84a9-3f146c7bd825>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00536.warc.gz"} |
Excel DVARP Function - Free Excel Tutorial
This post will guide you how to use Excel DVARP function with syntax and examples in Microsoft excel.
The Excel DVARP Function will get the variance of a population based on the entire population of numbers in a column in a list or database based on a given criteria. And the DVARP function can be
used to evaluate text values and logical values in references in Excel.
The DVARP function is a build-in function in Microsoft Excel and it is categorized as a Database Function.
The DVARP function is available in Excel 2016, Excel 2013, Excel 2010, Excel 2007, Excel 2003, Excel XP, Excel 2000, Excel 2011 for Mac.
The syntax of the DVARP function is as below:
= DVARP(database, field, criteria)
Where the DVARP function arguments are:
• Database -This is a required argument. The range of cells that containing the database.
• Field – This is a required argument. The column with database that you want the minimum of.
• Criteria – The range of cells that contains the conditions that you specify.
Excel DVARP Function Examples
The below examples will show you how to use Excel DVARP Function to get the variance of a population based on a sample by using the numbers in a column in a list or database that match a given
#1 to get the variance in the cost of excel project in a range A1:C11, using the following formula:
=DVARP (A1:C11,C1,E1:E2)
Related Functions
• Excel DVAR Function
The Excel DVAR Function will get the variance of a population based on a sample of numbers in a column in a list or database based on a given criteria.The syntax of the DVAR function is as below:
= DVAR (database, field, criteria)… | {"url":"https://www.excelhow.net/excel-dvarp-function.html","timestamp":"2024-11-12T22:45:16Z","content_type":"text/html","content_length":"85735","record_id":"<urn:uuid:cf2b7d39-25fb-4027-9cd6-4234cbe93cb7>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00323.warc.gz"} |
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Weak magnetic field corrections to light vector or axial mesons mixings and vector meson dominance
Nenhuma Miniatura disponível
Weak magnetic-field-induced corrections to effective coupling constants describing light vector meson mixing and vector meson dominance (VMD) are derived. The magnetic field must be weak with respect
to an effective quark mass M∗ such that: eB0/M∗2 < 1 or eB0/M∗2 1. On that basis, a flavor SU(2) quark–quark interaction due to a non-perturbative one gluon exchange is considered. By means of
methods usually applied to the Nambu—JonaLasinio and global color models, the equations for leading light vector/axial meson coupling to a background electromagnetic field are derived. The
corresponding effective coupling constants are resolved in structureless mesons and long-wavelength limits. Some of the resulting coupling constants are redefined, so as to become
magnetic-field-induced corrections to vector or axial meson couplings. Due to the approximated chiral symmetry of the model, light axial meson mixing induced by the magnetic field is also obtained.
Numerical estimates are presented for the coupling constants and some of the corresponding momentum-dependent form factors. The contributions of the induced VMD and vector meson mixing couplings for
the low-momWeak magnetic-field-induced corrections to effective coupling constants describing light vector meson mixing and vector meson dominance (VMD) are derived. The magnetic field must be weak
with respect to an effective quark mass M∗ such that: eB0/M∗2 < 1 or eB0/M∗2 1. On that basis, a flavor SU(2) quark–quark interaction due to a non-perturbative one gluon exchange is considered. By
means of methods usually applied to the Nambu—JonaLasinio and global color models, the equations for leading light vector/axial meson coupling to a background electromagnetic field are derived. The
corresponding effective coupling constants are resolved in structureless mesons and long-wavelength limits. Some of the resulting coupling constants are redefined, so as to become
magnetic-field-induced corrections to vector or axial meson couplings. Due to the approximated chiral symmetry of the model, light axial meson mixing induced by the magnetic field is also obtained.
Numerical estimates are presented for the coupling constants and some of the corresponding momentum-dependent form factors. The contributions of the induced VMD and vector meson mixing couplings for
the low-momentum pion electromagnetic form factor and for the (off-shell) charge symmetry violation potential at the constituent quark level are estimated. The relative overall weak
magneticfield-induced anisotropic corrections are of the order of (eB0/M∗2 ) n, where n = 2 or n = 1, respectivelyentum pion electromagnetic form factor and for the (off-shell) charge symmetry
violation potential at the constituent quark level are estimated. The relative overall weak magneticfield-induced anisotropic corrections are of the order of (eB0/M∗2 ) n, where n = 2 or n = 1,
Chiral Lagrangians, Sea quark determinant, Meson mixing, Vector meson dominance, Magnetic field, rho–pion coupling
BRAGHIN, F. L. Weak magnetic field corrections to light vector or axial mesons mixings and vector meson dominance. Journal of Physics G: nuclear and particle physics, Bristol, v. 47, e115102, 2020.
DOI: 10.1088/1361-6471/aba7c9. Disponível em: https://iopscience.iop.org/article/10.1088/1361-6471/aba7c9. Acesso em: 14 set. 2023. | {"url":"https://repositorio.bc.ufg.br/items/5862d253-8d24-491d-b21c-d829c72bcbda","timestamp":"2024-11-05T09:54:33Z","content_type":"text/html","content_length":"443586","record_id":"<urn:uuid:7bb742bb-62fa-4a8b-aa21-bbc98b66817e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00504.warc.gz"} |
Case Study: Clipboard Health Lyft Pricing Exercise
Here is my answer to the Clipboard Health product case study. I didn't get the job.
Update Feb 29, 2024.
I told myself I wouldn't put any more time into this, but
someone was wrong on the internet
and it was me (there was a typo in my model). So I have updated this post with a correction which changes the final result from about $9 to about $5.
Clipboard Health is a Y-combinator funded startup building a platform to match medical professionals to healthcare facilities. Sounds like a cool company! They're hiring.
Good news: If you apply for a job at Clipboard Health, you'll probably get a response.
Bad news: You'll be asked to complete a lengthy case study. You're almost guaranteed to fail. And you'll receive zero feedback.
I probably should have listened to the hundreds of negative reviews on Glassdoor. But I didn't. I got nerd sniped and I spent several hours completing the case study. I've published my work below to
help others solve the case study faster, and because I have no idea how I failed. Do you see my mistake? If so, let me know!
The Challenge
Identify the optimal take rate (how much Lyft makes per ride) to maximize net revenue (the difference between amount riders pay and the amount Lyft pays to drivers) for the next 12 months on a given
route in Toledo.
We are told that current price for the route is $25 and take is $6, so drivers currently earn $19 per trip, which results in a 60% match rate – 60 out of 100 riders find a ride at this rate.
And when we run a pricing experiment reducing Lyft's take from $6 to $3, we see that match rate rises from 60% to 93%. In other words, as Lyft takes less, drivers earn more, and they're more likely
to accept a ride.
We are given additional information to help us build a model. The most important is churn. Riders who don't fail to find a driver churn at 10% monthly, while riders who do fail to find a driver churn
at 33%. I like to think positively, so I will use retention instead of churn.
We're given other inputs as well, such as CAC for both drivers and riders, but this information shouldn't be required to produce a result given that net revenue, as defined in the exercise, is
independent of cost of acquisition. So we should be able to ignore CAC for now.
My Result
Is this wrong? It might be! Let me know what you think by leaving a comment or finding me on LinkedIn.
The optimal take rate for the business is $4 to $5 per ride.
HOW I DID IT
First, we build a model from the two pricing experiments, and then we use that model to project retention (churn), and total rides in 12 months, which we can multiply by our take rate to calculate
net revenue.
First, let's look at what a whole year might look like under the current $6 take rate...
See the google sheet for all the work
We can do the same for the $3 experiment, where match rate increases to .93, shifting retention up to .88.
From this, we can build a simple linear model to project match rate for all $1 increments.
See the google sheet for all the work
We see that for every extra dollar that we take from drivers, our match rate drops by 11 percentage points (our slope).
Match rate is the determining factor for rider retention, and we see that retention drops 2-3 percentage points as match rate drops at every extra dollar we take.
At $5, our net revenue peaks, and then it starts to drop as rider churn outweighs the money that’s coming in per ride.
Double check the work
Let's double check the result we arrived at above by adding another layer to our model: market penetration.
It stands to reason that there will be diminishing returns the longer we are in market. So let's cut new riders by about 10% for every subsequent monthly cohort and see how this changes the result.
See the google sheet for all the work
With this new model revenue peaks at $4. So it is fair to say that the optimal take rate for the business is between $4 and $5.
Caveats apply
This is a super simplified linear model to project match rate. In fact, match rate might drop off a lot more steeply once we get past a $6 take rate, which would result in an optimal take rate closer
to $6.
That's it!
Nope, that's not it.
The morning after submitting my response I received a generic rejection letter from the hiring team. I asked for feedback, and did not receive any. I am clearly not alone.
What do you think? How did I do?
Other articles on this website:
Histograms with SQL
Even if all you do is sit around and send histograms through Slack, you’ll be a decent enough product manager.
How to use BigQuery
Follow these instructions to run your first query.
Best BigQuery Datasets
BigQuery is a tremendously powerful tool. Not just because you can use it as a SQL client for your own data (for instance, you can use it to query a CSV that you upload), but also because it’s a
tremendous tool for teaching. BigQuery includes hundreds of public datasets that | {"url":"https://www.samthebrand.com/clipboard-health-product-case-study/","timestamp":"2024-11-02T07:52:49Z","content_type":"text/html","content_length":"38326","record_id":"<urn:uuid:0c0e39bb-9f35-4909-8c08-3a2d0295b27f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00089.warc.gz"} |
Decibel Scale¶
An important property of music concerns the dynamics, a general term that is used to refer to the volume of a sound as well as to the musical symbols that indicate the volume. From a physical point
of view, sound power expresses how much energy per unit time is emitted by a sound source passing in all directions through the air. The term sound intensity is then used to denote the sound power
per unit area. In practice, sound power and sound intensity can show extremely small values that are still relevant for human listeners. For example, the threshold of hearing (TOH), which is the
minimum sound intensity of a pure tone a human can hear, is as small as
Furthermore, the range of intensities a human can perceive is extremely large with $I_\mathrm{TOP}:=10~\mathrm{W}/\mathrm{m}^2$ being the threshold of pain (TOP). For practical reasons, one switches
to a logarithmic scale to express power and intensity. More precisely, one uses a decibel (dB) scale, which is a logarithmic unit expressing the ratio between two values. Typically, one of the values
serves as a reference, such as $I_\mathrm{TOH}$ in the case of sound intensity. Then the intensity measured in $\mathrm{dB}$ is defined as
$$\mathrm{dB}(I) := 10\cdot \log_{10}\left(\frac{I}{I_\mathrm{TOH}}\right).$$
From this definition, one obtains $\mathrm{dB}(I_\mathrm{TOH})=0$, and a doubling of the intensity results in an increase of roughly $3~\mathrm{dB}$:
$$\mathrm{dB}(2\cdot I) = 10\cdot \log_{10}(2) + \mathrm{dB}(I) \approx 3 + \mathrm{dB}(I).$$
When specifying intensity values in terms of decibels, one also speaks of intensity levels. The following table shows some typical intensity values given in $\mathrm{W}/\mathrm{m}^2$ as well as in
decibels for some sound sources and dynamics indicators. | {"url":"https://www.audiolabs-erlangen.de/resources/MIR/FMP/C1/C1S3_Dynamics.html","timestamp":"2024-11-06T14:03:18Z","content_type":"text/html","content_length":"1049050","record_id":"<urn:uuid:09b0945b-3c2b-49c6-a5be-8f82980a208b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00224.warc.gz"} |
wwojciech/stratallo: Optimum Sample Allocation in Stratified Sampling version 2.2.1 from GitHub
Functions in this package provide solution to classical problem in survey methodology - an optimum sample allocation in stratified sampling. In this context, the optimum allocation is in the
classical Tschuprow-Neyman's sense and it satisfies additional lower or upper bounds restrictions imposed on sample sizes in strata. There are few different algorithms available to use, and one them
is based on popular sample allocation method that applies Neyman allocation to recursively reduced set of strata. This package also provides the function that computes a solution to the minimum cost
allocation problem, which is a minor modification of the classical optimum sample allocation. This problem lies in the determination of a vector of strata sample sizes that minimizes total cost of
the survey, under assumed fixed level of the stratified estimator's variance. As in the case of the classical optimum allocation, the problem of minimum cost allocation can be complemented by
imposing upper-bounds constraints on sample sizes in strata.
License GPL-2
Version 2.2.1
URL https://github.com/wwojciech/stratallo
Package repository View on GitHub
Install the latest version of this package by entering the following in R:
wwojciech/stratallo documentation
built on Oct. 31, 2024, 3:46 a.m. | {"url":"https://rdrr.io/github/wwojciech/stratallo/","timestamp":"2024-11-04T11:48:04Z","content_type":"text/html","content_length":"24983","record_id":"<urn:uuid:c818afbd-42c4-46f8-919f-eff1926172b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00558.warc.gz"} |
CaCert is a Certification Authority that works with a web of trust: people meet and assure (similar to keysigning) eachother. If you’ve been assured by enough people you’ll be able to let your ssl
server key be certified by cacert. It’s a lot more secure than other CA’s who just give anyone a certificate who pays enough.
Still a hierarchical system with a CA is flawed. When the CA is compromised, the whole system fails. PGP’s web of trust hasn’t got this weakness.
(Got a nice shiny cacert certified ssl certificate on my webserver now)
“Nothing to hide”
In this short essay, written for a symposium in the San Diego Law Review, Professor Daniel Solove examines the “nothing to hide” argument. When asked about government surveillance and data
mining, many people respond by declaring: “I’ve got nothing to hide.” According to the “nothing to hide” argument, there is no threat to privacy unless the government uncovers unlawful activity,
in which case a person has no legitimate justification to claim that it remain private. The “nothing to hide” argument and its variants are quite prevalent, and thus are worth addressing. In this
essay, Solove critiques the “nothing to hide” argument and exposes its faulty underpinnings.
“I’ve Got Nothing to Hide” and Other Misunderstandings of Privacy
Not only is its subject very relevant, the Essay is very well written and a pleasure to read.
Don’t use md5(microtime()). You might think it’s more secure than md5(rand()), but it isn’t.
With a decent amount of tries and a method of syncing (like a clock on your website) one can predict the result of microtime() to the millisecond. This only leaves about a 1000 different possible
return values for microtime() to be guessed. That isn’t safe.
Just stick with md5(rand()), and if you’re lucky and rand() is backed by /dev/random you won’t even need the md5(). In both cases it will be quite a lot more secure than using microtime().
Simple Branch Prediction Analysis
This paper outlines simple branch prediction analysis attack against the RSA decryption algorithm.
At the core of RSA decryption is a loop over all bits of the secret key number d. When the bit 1 there is other code executed than when the bit is 0. The CPU branches on a different bit.
A spy process can be run on the CPU which measures the branch cache of the CPU by flooding the cache with branches and measuring the time it takes. When the sequentially running secret process doing
RSA decryption makes a different branch (1 instead of 0) it can be noticed in a change of execution time on the spy process’s branches.
In this way quite a lot of secret bits can be derived.
There are some clear buts:
• You must be able to insert a spy process on the computer itself and it should know exactly when the RSA process runs.
• To attain clear readings, there shouldn’t be other processes claiming too much CPU time.
• The spy and CPU process should run on the same physical processor and preferably at the same time (dual core)
An easy fix would be to allocate a whole processor for the RSA decryption time, so no process can spy. Another option would be to add noise in the Branch Prediction Buffer, but that would result in a
performance loss.
DDOS on Hash Tables (Self Balancing Hash Tables)
Hash Tables are widely used in server software. A malicious user can easily forge keys in the communication with the server that will result in hashes from the keys so that they will end up in the
same bucket. This will force the hashtable to grow each time this bucket is filled.
In a worst case implementation, this will result in an enormous amount of empty buckets and only one full bucket. Even if the implementation is smart and will therefore only will grow the targetted
bucket, the hash table will fail its use. The hash table is meant to make lookups for a key fast by distributing the pairs in buckets, this effect is gone when all pairs are located in one bucket,
which will cause linear time look-ups.
One malicious user won’t be able to crash a server, but a DDOS attack with several users would be tremendously more efficient when targeting this weakness in hash tables.
A simple solution would be to randomly pick a salt for the hash function on each new instance of a hash table. Basically we prefix each key with a random salt before we hash it. The malicious user
might know the hash function, but he has to guess the salt, which neglects the effect.
Additionally logic could be added to detect an unbalanced hash table and switch the salt on when the table is unbalanced.
This could also be usefull to balance hash tables when there is no malicious user, but the keys are possibly unfortunate.
Rainbow Tables: Coverage
A rainbow table is generated by creating (m) chains using randomly picked starting keys. The reduction functions result (or ought to result at least) in evenly distributed new keys. Their is only a
probability that a certain key/cipher pair is in the table.
I’ll gradually explain how the coverage can be calculated.
Take m chains, each 1 long to cover a keyspace of N keys. The chance that you can find the key is:
[tex]\displaystyle P = \frac{m}{N}[/tex]
When we have 1/2N chains, we’ll subsequently have a P of 1/2, a 50% coverage. Obvious.
When each chain would be 2 in length it gets trickier.
[tex]P = \frac{m}{N}[/tex] still is the chance that we can find the key in the first column (first item of each chain of the table) of the table. The second column isn’t as straight forward. We’ve
blindly assumed that each key in the first column is unique, which is true for the first column, but certainly isn’t for the second column. A few chains can merge. The second column is ‘random’.
The chance that a certain key is in a specific chain of the second column is [tex]\frac{1}{N}[/tex]. The chance that a certain key is in the second column is [tex]1 – \left(1 – \frac{1}{N} \right)^m
[/tex] (One minus the chance that there isn’t a certain key in a certain chain multiplied by itself the-amount-of-chains times).
The amount of keys covered by this column is the chance that a certain key is in the column times the amount of keys: [tex]N \left(1 – \left(1 – \frac{1}{N} \right)^m \right)[/tex].
The chance a certain key is in any of the two columns would be:
[tex]\left( \frac{m}{N} \right) \cdot N \left(1 – \left(1 – \frac{1}{N} \right)^m \right) \over N[/tex]
The third column’s coverage can be calculated by taking the unique keys in the second column as m was taken for the amount of unique keys for the first column. With each chain t in length this
formula applies:
[tex]\displaystyle m_1=m \\ m_{n+1}=N \left( 1 – \left( 1 – \frac {1}{N} \right) \right) \\ P=1-\prod^m_{i=1}1-\frac{m_i}{N}[/tex]
Reversing CRC
Cyclic Redundancy Code
CRC is a hash which is frequently used as a checksum for data in for instance archives. Who hasn’t had bad CRC errors once when opening corrupted zips. CRC is a very old algorithm and over time it
changed a lot from the original idea.
The original idea behind CRC was representing the data that you wanted the hash from as a big number and dividing it by a certain number called the polynomial and taking the remainder of the division
as the hash. For instance: 23 % 3 = 2 (% is the modulus operator, which is the remainder of a division)
The initial problem was that dividing is a rather intensive operation. They wanted to simplify CRC to make it easier to implement in hardware and make it faster. They did this by getting rid of the
carry used in the substraction of the division:
Normal substraction (binary): 10101 - 01100 = 01001
Without carry: 10101 - 01100 = 11001
Substraction without a carry is basicly a eXclusive bitwise OR (XOR), which returns only 1 when one of the operands is 1 and the other 0.
The algorithm required was faster bit still worked bit by bit, which isn’t really what a computer likes. A computer works best with one to four bytes. To make CRC faster they cached 8 operations at
the time by precomputing the results for a certain start value and put it in a table called a XOR Table.
The required code for the CRC calculation itself now became very simple:
hash = (hash >> 8 ) ^ table[data ^ (0xff & hash)]
They changed the CRC algorithm once more by making it reflected. This means that the input data is reversed bitwise: 011101011 <-> 110101110. This was done because most of the hardware chips at the
time reversed data bitwise. For it was too much work to reflect each byte of incoming data they changed the algorithm that generates the Crc table to create a table which has the effect of reflected
This is by the way not totally correct; the result still was different for a reflected than a non-reflected algorithm for they wouldn’t cache the whole piece of data to reverse it but did it per byte
at calculation.
At this moment CRC barely resembles the original idea of a modulus.
Reversing CRC
First off, credits for the original idea of CRC patching go to anarchriz.
CRC is a cryptographicly very weak algorithm. It can be easily reversed for it has got the property that with 4 bytes you append to the current computed hash you can get every required hash. You can
change the whole message and add 4 patch bytes to patch the hash to the one you like.
The ability to patch a CRC also makes it possible to very efficiently generate all possible source data of a checksum. Although it still is a bruteforce method you got 4 bytes freely and patching is
faster than calculating.
Patching is basicly going back the way CRC works. Crc basicly takes the hash, moves it 1 byte to the right (dropping one byte) and xor-ring it with the table entry. The nice thing about normal CRC is
that the first byte of a table entry is unique for that entry.
For the first byte of the entry is unique for that entry and it is put in the hash xor-red with 0 for that is what is shifted in from the right you can work back the whole entry used.
For instance:
My is: 0x012345678, this means that it was xorred with the entry in the CRC table that starts with 0x12. When you xor the hash with that full entry the only thing that the next byte was xorred with
was the start of a table entry too.
When reversing the current hash you know what will be xorred on the patch you’ll give. Xorring this with your wanted hash is enough.
The resulting algorithm is suprisingly simple:
– Put the current hash byte wise reversed at the start of a buffer. Put the wanted hash byte wise reversed at the end of the current hash in the same buffer.
– Look up the entry in the table that starts with byte 7 in the buffer. Xor this value of position 4, and Xor the entry number on position 3. Repeat this 4 times with the positions each time one
decremented. (thus 7,6,5,4 and 4,3,2,1 and 3,2,1,0)
When you’ve done this the required patch bytes are the first 4 bytes in the buffer.
I’ve made a simple python script to work with crc32 and to patch a hash.
You can download it Here. And there is a C implementation here.
Fixed a bug in the example.
Andrea Pannitti wrote an implementation in java.
I found a great article by Stigge, Pl�tz, M�ller and Redlich about reversing CRC. They pretty much nailed it by bringing it down to elementary algebra. They explain an algorithm which is able to
patch data at any given point to adjust the CRC to any desired value.
Linux Mount Security
With the linux Set UID Attribute you can let the owner of the file be the one the execute it when another user executes the file. This feature has traditionaly be used for system tools in linux which
require root access to run but also must be able to be run my users.
It came to mind that a floppy with the ext2 filesystem could contain files of the root user with this Set UID Attribute set. Which theoraticly would allow anyone who is allowed to mount floppy’s or
other media with an filesystem that supports this attribute to gain root access for a program.
On my system I got this entry in my /mnt/fstab, which allows an user to mount the floppy:
/dev/floppy/0 /mnt/floppy auto noauto,user 0 0
I made a simple C program which would show the contents of /etc/shadow, which contains the password hashes of the users, and chmodded it accordingly. (chmod = showshadow; chmod go+x showshadow; chmod
u+rs showshadow)
I ran my program, and it seemed to work! The contents of the /etc/shadow file was streaming on my console.
Euforicly I went to another linux computer and tried the same trick.
darkshines@darkshines-one /mnt/floppy $ ./showshadow
bash: ./showshadow: Permission denied
Dissapointed but releived it seemed that linux had already some precaution against a root Set UID-ed executable.
I copied the contents of the folder whilest preserving permissions to another folder outside the /mnt/floppy and it all seemed to work again, although I couldn’t do it with a normal user account for
I can’t preserve the owner when copying a file as a normal user.
I wondered how linux would secure it and tried to run the program while it was unchmodded.
darkshines@darkshines-one /mnt/floppy $ ./showshadow.unchmodded
bash: ./showshadow.unchmodded: Permission denied
The warning is from bash which can’t seem to execute the program. (note that it isn’t the program that can’t acces shadow) . After recompiling it on the floppy itself it seems that linux prevents any
program to be executed in an user mounted folder.
I recon that that security precaution is a bit too strict. Although copying the file from the medium to a normal folder and then executing is still possible, I find it a bit strange that nothing of
the user itself can be executed.
This could result in trouble when dealing with netmounts where one user can has a share on a server where he puts his files and can access only that mount for space on a terminal, when dealing with
an user mount which would be required for security.
Safe web authentication
The major problem with security of web applications is that the client sends the login name and password in plain text if https isn’t available. A nasty person with access to the network could use
ARP poisening alongside packet sniffing to acquire the login, which wouldn’t really be desirable.
I stumbled accross a very interesting piece javascript which implements the md5 hash algorithm: http://pajhome.org.uk/crypt/md5/.
Using a hash makes it impossible to reverse engineer a password and makes authentication safer. An issue with this is that you only require the hash, not the password to get in. To prevent this the
password should be salted before hashed.
Basicly a secure authentication via http would look like this:
Client sends request for login to server.
Server sends the login form which includes a login id and salt to the client.
Server stores the login id and salt it sent to the client.
Client sends the hash of the filled out password and received hash alongside the login id from the server to the server.
Server checks whether the hash of the password in the database and the received hash combined with the login id are valid.
Server sends whether authentication was a success.
Maybe I’ll implement an example application :-). In any case I hope that this will be employed.
Update, most authentication system used by webbased software are still vulnerable and would almost neglect the use of this by being able to hijack a session by just getting the session key. The
client however could also implement javascript to use a similar method with a salt to protect the session key. The problem still is that it is extra overhead on the client and that not every client
has got javascript enabled.
Copy protecting
From software, audio to video are being illegaly copied and everytime the major brands try to implement some kind of protection. They always claim their protection to be perfect, and yet it is always
broken, for it is quite simple:
As long as the intended user has the platform on which he`ll run it in his own possession he can always adapt it in someway to extract the data. Even the best video protection can’t beat making a
bypass in your monitor to acquire the image on your screen.
Even protecting something like a DVD is almost impossible. The dvd player hardware and software must be able to read what’s on the cd, and a protection must be able to be read to. Also there must be
dvd writers to write a protection. Now all major brands can say they’ll put a protection in their DVD burners to protect from writing to the DVD protection section, but then another brand creates
their DVD burners that can write to it and everyone will buy those which the big brands won’t let happen. And even if they got the disk itself truly protected someone can emulate the DVD using
software or even hardware.
Also there are things that allow itself to be copied, but the original copier can be tracked; this by putting in every video/song/software a unique signature which can be tracked back to the store
which then can track it back to the person who copied it. Sounds great, would be impossible to forge when they use strong RSA like cryptography, just one problem, when inserting random trash instead
of the signature someone can know the piece is illegal but cannot track someone, hopeless.
The only, and only way, to stop illegal copying is making buying legaly less of an effort than acquiring illegaly. I hope they will relize this sooner or later for honoustly I`m becoming sick of all
those ‘magic’ protections.
PHP Security Consortium
The PHPSC is a site managing a lot of resources on PHP security.
For all those starting or sometimes using PHP this is a must read.
Also I’d advice for people who want to know whether there site is safe enough is to try to play the other site by trying out hacking yourself: hackthissite.org. It is easier than you might have | {"url":"https://blog.affien.com/archives/tag/security/","timestamp":"2024-11-11T08:18:48Z","content_type":"text/html","content_length":"85625","record_id":"<urn:uuid:b4997b50-001b-4725-a649-904dfbed5537>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00759.warc.gz"} |
Work Problems
Y'all know the drill.
Person A does a job in A hours, Person B does the same job in B hours. How long does it take them to do the job together?
Teaching the work problem can turn into a debacle fast. Really fast. Most teachers boil these down to an algorithm they themselves might not understand.
"Here's the formula, kids. Learn it. Know it. Use it. Now let's move on to something else."
That's never set well and a couple of years ago, it dawned on me that a work problem is nothing more than a complex rate problem.
Person A can do 1/A of the job per hour and person person B can do 1/B of the job per hour. How much of the job do they complete per hour when working together?
This has been a game changer. Find the common denominator and add 'em up. Now we have a common rate. From there it's pretty easy to determine how long it takes to complete the job. We give no
formulas and kids understand
to find an answer and
it works.
We had kids derive two different algorithms for work problems.
x = (AB)/(A+B)
1/A + 1/B = 1/X
Sure, kids. Use it. G'ahead.
But then we added the dreaded twist.
Person A can do the job in A hours. If person B helps, they can do the job in X hours. How long would it take person B to do the job when working alone?
I give you,
Clinton's Theorem
I kid you not, this kid came up with this all himself. By "integer portion of X" he means, everything to the left of the decimal. He's truncated the number to make things "simpler."
As he was presenting this to us before we left for break, I was baffled. It works, but why? So today, I sat down next to him and had him explain the formula to me again, I took notes and then
simplified the expression. I won't deprive you the joy of doing so yourself.
Seriously, have at it.
4 comments:
Chris Okasaki said...
Well that's just silly. X_T is all wrong. You don't just truncate it to the integer part. You also need to delete all the prime digits from the integer part, as well as the vowels when written in
hexadecimal. And then add pi at the end, 'cause you don't want it to be, you know, zero.
David Cox said...
Do you want to tell him, or should I?
A X_T + X_T is just a red herring.
Divide by it top and bottom and you get the obvious answer.
David Cox said...
Of course it's a red herring. An 8th grade kid came up with this. Overcomplicated? Absolutely. His own creation? You bet.
I'll take it. | {"url":"http://coxmath.blogspot.com/2011/01/work-problems.html","timestamp":"2024-11-06T07:57:02Z","content_type":"text/html","content_length":"73365","record_id":"<urn:uuid:36cea4e6-b680-4bab-b6f0-c988484ce443>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00311.warc.gz"} |
multiple discriminant analysis
That is, using coefficients a, b, c, and d, the function is: D = a * climate + b * urban + c * population + d * gross domestic product per capita. However, multiple regression can sometimes be
preferred to the discriminant analysis because it requires less restrictive assumptions to be met to be valid (Warner, 2013). With developments and improvements in the techniques in discriminant
analysis, it has been adapted into a form that can provide solutions to modern-day problems. It minimizes the dissimilarity between many variables, and organize them into large groups, … MULTIPLE
DISCRIMINANT ANALYSIS A. DEFINISI Menurut Cramer, Multiple Discriminant Analysis merupakan teknik parametrik yang digunakan untuk menentukan bobot dari prediktor yg paling baik untuk membedakan dua
atau lebih kelompok kasus, yang tidak terjadi secara kebetulan. Chapter 7 Multiple Discriminant Analysis and Logistic Regression 335 What Are Discriminant Analysis and Logistic Regression? A
classifier with a linear decision boundary, generated by … In addition, discriminant analysis is used to determine the minimum number of … The program will do this automatically, even if only the
Canonical option is selected. The book presents the theory and applications of Discriminant analysis, one of the most important areas of multivariate statistical analysis. Web Extension 25A Multiple
Discriminant Analysis 25WA-3 Z 5 0.2 Companies that lie to the left of the line (and also have Z , 0) are unlikely to go bankrupt; those that lie to the right (and have Z . Linear Discriminant
Analysis. However, SAS PROC DISCRIM does not perform Multiple discriminant analysis, it only works on a single dependent variable. A Linear Discriminant Analysis should be performed before a
Canonical one. An appendix provides mathematical derivations and computation procedures for the techniques applied. In this example, we specify in the groups subcommand that we are interested in the
variable job, and we list in parenthesis the minimum and maximum values seen in job . On the other hand, in the case of multiple discriminant analysis, more than one discriminant function can be
computed. If, on the contrary, it is assumed that the covariance matrices differ in at least two groups, then the quadratic discriminant analysis should be preferred . 0) are likely to go bankrupt. 3
0. Discriminant analysis is used to predict the probability of belonging to a given class (or category) based on one or multiple predictor variables. 339 Discriminant Analysis 340 Logistic Regression
341 Analogy with Regression and MANOVA 341 Hypothetical Example of Discriminant Analysis 342 A Two-Group Discriminant Analysis: Purchasers Versus Nonpurchasers 342 Multiple discriminant analysis
(MDA) is a statistical measure that financial planners use to ascertain the prospective investments when a lot of variables need to be considered. Meaning of Multiple-discriminant analysis … The
major distinction to the types of discriminant analysis is that for a two group, it is possible to derive only one discriminant function. Group of cases used in estimating the discriminant function
(s). Nowhere is it more active than in the area of bankruptcy prediction and the use of statistical models and accounting ratios in an effort to predict company failure for up to four years in
advance.' Two models of Discriminant Analysis are used depending on a basic assumption: if the covariance matrices are assumed to be identical, linear discriminant analysis is used. STUDY. 648 Z
Score range ±0. DISCRIMINANT ANALYSIS I n the previous chapter, multiple regression was presented as a flexible technique for analyzing the relationships between multiple independent variables and a
single dependent variable. The three empirical models used in the study are recursive partitioning, logistic regression, and multiple discriminant analysis. Linear and Canonical discriminant analyses
can be performed with or without stepwise selection of variables. The model was developed basing on empirical studies, to predict the sickness of a … S B generalization to multiple classes in not
obvious. Multiple-discriminant analysis (MDA) Statistical technique for distinguishing between two groups on the basis of their observed characteristics. 8.2.1. It works with continuous and/or
categorical predictor variables. Multiple discriminant analysis may be considered as a principal component analysis (chapter 31) in which the principal axes of between-groups variation are determined
after within-groups variation has been taken as a yardstick (sections 33.3 and 33.12). Edward I. Altman (1968) developed Z score model in order to detect the financial health of industrial units with
a view to prevent the industrial sickness. Multiple Discriminant Analysis • c-class problem • Natural generalization of Fisher’s Linear Discriminant function involves c-1 discriminant functions •
Projection is from a d-dimensional space to a c-1 dimensional space. 583 –0. Findings: The results indicate that idea, efficiency, adventure, and gratification shopping motivations are significant
determinants of mobile shoppers, implying that those shopping motivations are push factors of mobile shopping. Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant
function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or
more classes of objects or events. Furthermore, the logistic regression may be chosen over the discriminant analysis when needed to estimate the probability of a particular outcome given a change in
scores on predictor variables. Multiple discriminant analysis was used to analyze the data. For K-class problem, Fisher Discriminant Analysis involves (K 1) discriminant functions. Discriminant
analysis is a multivariate statistical tool that generates a discriminant function to predict about the group membership of sampled experimental data. Make W d (K 1) where each column describes a
discriminant. There are many examples that can explain when discriminant analysis fits. Chapter 6 of Multivariate Methods for Market and Survey Research The chapter provides a brief discussion of
marketing research uses of multiple discriminant analysis. Multivariate Data Analysis Hair et al. If you really have multiple dependent class variables, you could combine them into a single class
variable encompassing all of the multiple class variables, or perhaps something like PROC PLS will work (or maybe it won't, I haven't really tried). Included are test hypotheses regarding group
means, classification, and perceptual mapping. Linear discriminant function analysis (i.e., discriminant analysis) performs a multivariate test of differences between groups. multiple discriminant
analysis. (ii) Quadratic Discriminant Analysis (QDA) In Quadratic Discriminant Analysis, each class uses its own estimate of variance when there is a single input variable. BY MEANS OF MULTIPLE
DISCRIMINANT ANALYSIS PAULBARNES+ INTRODUCTION The use of accounfing data (either explicitly or implicitly) for predictive purposes is at the heart of financial decision making. 4 Web Extension 22A:
Multiple Discriminant Analysis FIGURE 22A-2 Probability Distributions of Z Scores Probability Density Nonbankrupt Zone of Ignorance Bankrupt –0. Discriminant Analysis Discriminant Function Canonical
Correlation Water Resource Research Kind Permission These keywords were added by machine and not by the authors. (iii) Regularized Discriminant Analysis (RDA) Discriminant Analysis and Applications
comprises the proceedings of the NATO Advanced Study Institute on Discriminant Analysis and Applications held in Kifissia, Athens, Greece in June 1972. It can be seen from the graph that one X
(indicating a failing company) lies to the left 3 0 +0. The remainder of the paper is designed as follows: Section 1 provides a detailed survey of prior research and the related literature. In case
of multiple input variables, each class uses its own estimate of covariance. 7th edition. Definition of Multiple-discriminant analysis in the Financial Dictionary - by Free online English dictionary
and encyclopedia. Version info: Code for this page was tested in IBM SPSS 20. Discriminant analysis uses OLS to estimate the values of the parameters (a) and Wk that minimize the Within Group SS An
Example of Discriminant Analysis with a Binary Dependent Variable Predicting whether a felony offender will receive a probated or prison sentence as … Analysis sample. Here, m is the number of
classes, is the overall sample mean, and is the number of samples in the k-th class. sklearn.discriminant_analysis.LinearDiscriminantAnalysis¶ class
sklearn.discriminant_analysis.LinearDiscriminantAnalysis (solver = 'svd', shrinkage = None, priors = None, n_components = None, store_covariance = False, tol = 0.0001, covariance_estimator = None)
[source] ¶. The discriminant command in SPSS performs canonical linear discriminant analysis which is the classical form of discriminant analysis. Discriminant analysis allows you to estimate
coefficients of the linear discriminant function, which looks like the right side of a multiple linear regression equation. Abstract: In many real-world applications, an object can be described from
multiple views or styles, leading to the emerging multi-view analysis. When the criterion variable has two categories, the technique is known as two-group discriminant analysis. CSE 555: Srihari 22
Mapping from d-dimensional space to c-dimensional space Previously, we have described the logistic regression for two-class classification problems, that is when the outcome variable has two possible
values (0/1, no/yes, negative/positive). To overcome this difficulty multiple discriminant analysis is used. For discussions of multiple discriminant analysis and logit, which have been used
extensively in previous insolvency studies, see BarNiv and Hershbarger (1990). Box's M. Statistical test for the equality of the covariance matrices of the independent variables across the groups of
… This process is experimental and the keywords may be updated as the learning algorithm improves. It should be noted that nonlinear discriminant functions may be used, and we could also use more
dependent variables. Then, multi-class LDA can be formulated as an optimization problem to find a set of linear combinations (with coefficients ) that maximizes the ratio of the between-class
scattering to the within-class scattering, as Multiple Discriminant Analysis. Multiple discriminant analysis • Discriminant analysis techniques are described by the number of categories possessed by
the criterion variable. Multiple Discriminant Analysis. Much of its flexibility is due to the way in which all … What is Multiple-discriminant analysis? PLAY. So now, we have to update the two
notions we have de ned for a 2-class problem, S B and S W. S W = XK i=1 i. Incremental DA is a wonderful way of using multiple discriminant analysis to solve the current challenges. Case of multiple
discriminant analysis, more than one discriminant function analysis (,... More than one discriminant function analysis ( i.e., discriminant analysis involves ( K 1 ) discriminant.. Wonderful way of
using multiple discriminant analysis FIGURE 22A-2 Probability Distributions of Scores. Many real-world applications, an object can be described from multiple views or styles, leading to emerging.
Perform multiple discriminant analysis is used Free online English Dictionary and encyclopedia only on. Classification, and perceptual mapping process is experimental and the related literature
estimate of covariance this,! Survey of prior research and the related literature analysis to solve the challenges! Group of cases used in estimating the discriminant function ( s ) analysis involves
K. Variable has two categories, the technique is known as two-group discriminant analysis was used analyze! Of cases used in estimating the discriminant command in SPSS performs Canonical linear
discriminant analysis single dependent variable of most. May be updated as the learning algorithm improves possessed by the number of categories possessed multiple discriminant analysis criterion...
Multiple discriminant analysis fits Free online English Dictionary and encyclopedia difficulty multiple discriminant analysis the techniques applied known as discriminant... Analyses can be performed
before a Canonical one analysis fits used in estimating the command! S ) involves ( K 1 ) discriminant functions where each column describes a discriminant of Z Scores Density. Sas PROC DISCRIM does
not perform multiple discriminant analysis fits the discriminant command in SPSS performs linear..., in the case of multiple discriminant analysis involves ( K 1 ) each... Analysis was used to
analyze the data linear discriminant analysis is used Canonical linear discriminant analysis to solve the challenges. Canonical linear discriminant analysis techniques are described by the number of
categories possessed by number. Multivariate test of differences between groups Fisher discriminant analysis related literature wonderful way of using multiple discriminant analysis are... Selection
of variables, each class uses its own estimate of covariance FIGURE 22A-2 Probability Distributions of Scores. A linear discriminant analysis to solve the current challenges definition of
Multiple-discriminant analysis in the of... Analyze the data of discriminant analysis should be performed before a Canonical one analysis FIGURE Probability! The Financial Dictionary - by Free online
English Dictionary and encyclopedia estimate of covariance stepwise selection of variables ) each. Dictionary - by Free online English Dictionary and encyclopedia analysis techniques are described
the... Not perform multiple discriminant analysis fits group of cases used in estimating the discriminant (. Can explain when discriminant analysis FIGURE 22A-2 Probability Distributions of Z Scores
Probability Density Nonbankrupt Zone Ignorance!: Section 1 provides a detailed survey of prior research and the keywords may be updated as the algorithm! Analyses can be computed of covariance used
in estimating the discriminant function can be described multiple. Spss performs Canonical linear discriminant function can be described from multiple views styles! Classes in not obvious, and
perceptual mapping styles, leading to multiple discriminant analysis emerging multi-view analysis estimating discriminant. This difficulty multiple discriminant analysis Code for this page was tested
in SPSS. A linear discriminant function ( s ) on the other hand, the. In estimating the discriminant command in SPSS performs Canonical linear discriminant analysis was to. ) discriminant functions
multiple views or styles, leading to the emerging analysis! The technique is known as two-group discriminant analysis, it only works on a dependent... In case of multiple input variables, each class
uses its own estimate of covariance described by the number categories! Provides mathematical derivations and computation procedures for the techniques applied provides a detailed survey of prior
research and the may... Automatically, even if only the Canonical option is selected s B generalization to multiple classes in not.! A multivariate test of differences between groups class uses its
own estimate of covariance single! Designed as follows: Section 1 provides a detailed survey of prior and... Leading to the emerging multi-view analysis many examples that can explain when
discriminant analysis fits in many applications... Is known as two-group discriminant analysis • discriminant analysis, it only works on a dependent... Works on a single dependent variable wonderful
way of using multiple discriminant analysis FIGURE Probability! S ) other hand, in the Financial Dictionary - by Free online English Dictionary encyclopedia... Analyze the data perform multiple
discriminant analysis techniques are described by the variable., more than one discriminant function ( s ) there are many that! Prior research and the related literature techniques are described by
the criterion variable has two categories, the is! The number of categories possessed by the number of categories possessed by the number of categories possessed by the of! Da is a wonderful way of
using multiple discriminant analysis on a single dependent variable test hypotheses regarding group,. Discrim does not perform multiple discriminant analysis two-group discriminant analysis, it only
works on a single dependent.! By the criterion variable keywords may be updated as the learning algorithm improves ) discriminant functions it works... Discrim does not perform multiple discriminant
analysis, more than one discriminant function can be performed with or stepwise... Multiple classes in not obvious, in the Financial Dictionary - by Free online English Dictionary and.! Group of
cases used in estimating the discriminant function analysis ( i.e., discriminant analysis • analysis. | {"url":"http://www.cirro.dk/39wcdgou/75078e-multiple-discriminant-analysis","timestamp":"2024-11-03T12:55:27Z","content_type":"text/html","content_length":"27266","record_id":"<urn:uuid:06982e39-1c1a-49e6-98b0-99be0cc1ad76>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00337.warc.gz"} |
Infrared signature of the superconducting state in Pr<sub>2-x</sub>Ce <sub>x</sub>CuO<sub>4</sub>
We measured the far infrared reflectivity of two superconducting Pr [2-x]Ce[x]CuCO[4] films above and below T [c]. The reflectivity in the superconducting state increases and the optical conductivity
drops at low energies, in agreement with the opening of a (possibly) anisotropic superconducting gap. The maximum energy of the gap scales roughly with T[c] as 2Δ[max]/k[B]T [c]≈4.7. We determined
absolute values of the penetration depth at 5 K as λ[ab]=(3300±700) Å for x=0.15 and λ[ab]=(2000±300) Å for x=0.17. A spectral weight analysis shows that the Ferrell-Glover-Tinkham sum rule is
satisfied at conventional low energy scales -4Δ[max].
Funders Funder number
National Science Foundation
Directorate for Mathematical and Physical Sciences 0102350
Directorate for Mathematical and Physical Sciences
Dive into the research topics of 'Infrared signature of the superconducting state in Pr[2-x]Ce [x]CuO[4]'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/infrared-signature-of-the-superconducting-state-in-prsub2-xsubce-","timestamp":"2024-11-06T17:50:33Z","content_type":"text/html","content_length":"49660","record_id":"<urn:uuid:b3e90eb6-1e6e-44f0-a4af-67a9243a8a79>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00736.warc.gz"} |
Lesson 5
Splitting Triangle Sides with Dilation, Part 1
• Let’s draw segments connecting midpoints of the sides of triangles.
5.1: Notice and Wonder: Midpoints
Here’s a triangle \(ABC\) with midpoints \(L, M\), and \(N\).
What do you notice? What do you wonder?
5.2: Dilation or Violation?
Here’s a triangle \(ABC\). Points \(M\) and \(N\) are the midpoints of 2 sides.
1. Convince yourself triangle \(ABC\) is a dilation of triangle \(AMN\). What is the center of the dilation? What is the scale factor?
2. Convince your partner that triangle \(ABC\) is a dilation of triangle \(AMN\), with the center and scale factor you found.
3. With your partner, check the definition of dilation on your reference chart and make sure both of you could convince a skeptic that \(ABC\) definitely fits the definition of dilation.
4. Convince your partner that segment \(BC\) is twice as long as segment \(MN\).
5. Prove that \(BC=2MN\). Convince a skeptic.
5.3: A Little Bit Farther Now
Here’s a triangle \(ABC\). \(M\) is \(\frac23\) of the way from \(A\) to \(B\). \(N\) is \(\frac23\) of the way from \(A\) to \(C\).
What can you say about segment \(MN\), compared to segment \(BC\)? Provide a reason for each of your conjectures.
1. Dilate triangle \(DEF\) using a scale factor of -1 and center \(F\).
2. How does \(DF\) compare to \(D'F'\)?
3. Are \(E\), \(F\), and \(E'\) collinear? Explain or show your reasoning.
Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If \(D\) is the midpoint of segment \(BC\) and \(E\) is the midpoint of segment \(BA\), then what can we say
about \(ED\) and triangle \(ABC\)?
Segment \(ED\) is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if \(AC=10\), then \(ED=5\). This happens because the entire triangle
\(EBD\) is a dilation of triangle \(ABC\) with a scale factor of \(\frac12\).
In triangle \(ABC\), segment \(FG\) divides segments \(AB\) and \(CB\) proportionally. In other words, \(\frac{BG}{GA}\)=\(\frac{BF}{FC}\). Again, there is a dilation that takes triangle \(ABC\) to
triangle \(GBF\), so \(FG\) is parallel to \(AC\) and we can calculate its length using the same scale factor.
\(\overleftrightarrow{FG} \parallel \overleftrightarrow{AC}\)
• dilation
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
• scale factor
The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2,
then you have a scaled copy with a scale factor of 2. | {"url":"https://im-beta.kendallhunt.com/HS/students/2/3/5/index.html","timestamp":"2024-11-03T06:10:19Z","content_type":"text/html","content_length":"95105","record_id":"<urn:uuid:6bf43c5c-306d-4ec8-82e8-5a84ad923f8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00050.warc.gz"} |
EViews Help: Nonlinear Equation Solution Methods
Nonlinear Equation Solution Methods
When solving a nonlinear equation system, EViews first analyzes the system to determine if the system can be separated into two or more blocks of equations which can be solved sequentially rather
than simultaneously. Technically, this is done by using a graph representation of the equation system where each variable is a vertex and each equation provides a set of edges. A well known algorithm
from graph theory is then used to find the strongly connected components of the directed graph.
Once the blocks have been determined, each block is solved for in turn. If the block contains no simultaneity, each equation in the block is simply evaluated once to obtain values for each of the
If a block contains simultaneity, the equations in that block are solved by either a Gauss-Seidel or Newton method, depending on how the solver options have been set.
By default, EViews uses the Gauss-Seidel method when solving systems of nonlinear equations. Suppose the system of equations is given by:
The problem is to find a fixed point such that
to find the solution. At each iteration, EViews solves the equations in the order that they appear in the model. If an endogenous variable that has already been solved for in that iteration appears
later in some other equation, EViews uses the value as solved in that iteration. For example, the k-th variable in the i-th iteration is solved by:
The performance of the Gauss-Seidel method can be affected be reordering of the equations. If the Gauss-Seidel method converges slowly or fails to converge, you should try moving the equations with
relatively few and unimportant right-hand side endogenous variables so that they appear early in the model.
Newton's Method
Newton’s method for solving a system of nonlinear equations consists of repeatedly solving a local linear approximation to the system.
Consider the system of equations written in implicit form:
In Newton’s method, we take a linear approximation to the system around some values
and then use this approximation to construct an iterative procedure for updating our current guess for
where raising to the power of -1 denotes matrix inversion.
The procedure is repeated until the changes in
Note that in contrast to Gauss-Seidel, the ordering of equations under Newton does not affect the rate of convergence of the algorithm.
Broyden's Method
Broyden's Method is a modification of Newton's method which tries to decrease the calculational cost of each iteration by using an approximation to the derivatives of the equation system rather than
the true derivatives of the equation system when calculating the Newton step. That is, at each iteration, Broyden's method takes a step:
As well as updating the value of
In particular, Broyden's method uses the following equation to update
In EViews, the Jacobian approximation is initialized by taking the true derivatives of the equation system at the starting values of
Broyden's method shares many of the properties of Newton's method including the fact that it is not dependent on the ordering of equations in the system and that it will generally converge quickly in
the vicinity of a solution. In comparison to Newton's method, Broyden's method will typically take less time to perform each iteration, but may take more iterations to converge to a solution. In most
cases Broyden's method will take less overall time to solve a system than Newton's method, but the relative performance will depend on the structure of the derivatives of the equation system. | {"url":"https://help.eviews.com/content/optimize-Nonlinear_Equation_Solution_Methods.html","timestamp":"2024-11-07T03:12:20Z","content_type":"application/xhtml+xml","content_length":"19409","record_id":"<urn:uuid:fa01124e-c8d0-45c7-833b-7dd2cd032fd2>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00591.warc.gz"} |
Categories: WIN rds-hosted Mathematics
Permutations 962K
Author: Fred Ransom Date: Apr 10 2013
Permutations: This presents a Permutations namespace for n items taken m (<= n) at a time. It creates and delivers one permutation with each call to Permutations:next(). It also keeps track of
parity, thus making it useful for some things in linear algebra. Determinants, matrix inverse and solving simultaneous equations are illustrated. All routines were chosen to illustrate use of
permutations rather than speed. Although the field of real numbers was assumed here, these routines will work for any field, such as complex numbers or the integers mod 7 when you define appropriate
arithmetic operations. EU 4, Judith's IDE and Win32Lib. Permutations.e is general. | {"url":"http://phix.x10.mx/pmwiki/pmwiki.php?n=Main.Permutations","timestamp":"2024-11-04T22:04:32Z","content_type":"application/xhtml+xml","content_length":"9735","record_id":"<urn:uuid:49c89ea3-1fd6-4a93-86b5-c441201525ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00754.warc.gz"} |
Half planes covering the plane - Problem
This problem was posted to sci.math on 4 Oct 1998 by Norman Grégoire (normand@contact.net)
│Here is a problem I'd like to prove, but I don't know how to proceed : │
│ │
│We have a set H of n half-planes (n>3), covering R^2. │
│Prove that some subset of three half-planes of H is enough to cover R^2.│
This page Feedback by email
last changed: or Web form
17 Jan 2000
[Validate HTML] | {"url":"http://f2.org/maths/halfplane-prob.html","timestamp":"2024-11-15T02:56:39Z","content_type":"text/html","content_length":"3987","record_id":"<urn:uuid:e1f8d3ea-3f1c-47ae-acdf-05495ef9628b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00584.warc.gz"} |
Slant stack and Fourier transform
Next: Inverse slant stack Up: SLANT STACK Previous: Interface velocity from head
Let u(x,t) be a wavefield. The slant stack
The integral across x in (14) is done at constant x,t)-plane.
Slant stack is readily expressed in Fourier space. The definition of the two-dimensional Fourier transformation of the wavefield u(x,t) is
Recall the definition of Snell's parameter in Fourier space k from the 2-D Fourier transform (15). Change the integration variable from t to Insert the definition (14) into (17). Think of
The inverse Fourier transform of (18) is
The result (19) states that a slant stack can be created by Fourier-domain operations. First you transform u(x,t) to p.
Both (14) and (19) are used in practice. In (14) you have better control of truncation and aliasing. For large datasets, (19) is much faster.
Next: Inverse slant stack Up: SLANT STACK Previous: Interface velocity from head Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/iei/slnt/paper_html/node14.html","timestamp":"2024-11-11T08:20:32Z","content_type":"text/html","content_length":"9327","record_id":"<urn:uuid:36315d44-f403-4d57-9ce2-9e754a5ff9d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00166.warc.gz"} |
Displaying 61 - 72 of 409
• This course on quantum mechanics is divided in two parts:
The aim of the first part is to review the basis of quantum mechanics. The course aims to provide an overview of the perturbation theory to handle perturbations in quantum systems. Time evolution
of quantum systems using the Schrodinger, Heisenberg and interaction pictures will be covered. Basics of quantum statistical mechanics for distinguishable particles, bosons, and fermions will be
covered. A brief overview of density matrix approach and quantum systems interacting with the environment will be given.
The second part of the course is an introduction to scalar quantum field theory. The Feynman diagram technique for perturbation theory is developed and applied to the scattering of relativistic
particles. Renormalization is briefly discussed.
• This class is an introduction to cosmology. We'll cover expansion history of the universe, thermal history, dark matter models, and as much cosmological perturbation theory as time permits.
• Topics will include (but are not limited to): - Quantum error correction in quantum gravity and condensed matter - Quantum information scrambling and black hole information - Physics of random
tensor networks and random unitary circuits
• This course is designed to introduce modern machine learning techniques for studying classical and quantum many-body problems encountered in condensed matter, quantum information, and related
fields of physics. Lectures will focus on introducing machine learning algorithms and discussing how they can be applied to solve problem in statistical physics. Tutorials and homework
assignments will concentrate on developing programming skills to study the problems presented in lecture.
• This course will introduce some advanced topics in general relativity related to describing gravity in the strong field and dynamical regime. Topics covered include properties of spinning black
holes, black hole thermodynamics and energy extraction, how to define horizons in a dynamical setting, formulations of the Einstein equations as constraint and evolution equations, and
gravitational waves and how they are sourced.
• Topics will include (but are not limited to): Canonical formulation of constrained systems, The Dirac program, First order formalism of gravity, Loop Quantum Gravity, Spinfoam models, Research at
PI and other approaches to quantum gravity.
• Chaos, popularly known as the butterfly effect, is a ubiquitous phenomenon that renders a system's evolution unpredictable due to extreme sensitivity to initial conditions. Within the context of
classical physics, it often occurs in nonintegrable Hamiltonian systems and is characterized by positive Lyapunov exponents. On the other hand, the notion of nonintegrability and chaos in quantum
physics is still not well-understood and is an area of active research. Several signatures have been studied in the literature to identify quantum chaos but all of them fall short in some way or
the other. In this course, we will first discuss the notions of classical integrability, and classical chaos and its characterization with Lyapunov exponents. Then, we will discuss a few
well-studied signatures of quantum chaos and the subtleties associated with them.
• The aim of this course is to introduce concepts in topology and geometry for applications in theoretical physics. The topics will be chosen depending on time availability from the following list:
topological manifolds and smooth manifolds, differential forms and integration on manifolds, Lie groups and Lie algebras, and Riemann surfaces, cohomology and the fundamental group.
• We will review the notion of entanglement in quantum mechanics form the point of view of information theory, and how to quantify it and distinguish it from classical correlations. We will derive
Bell inequalities and discuss their importance, and how quantum information protocols can use entanglement as a resource. Then we will analyze measurement theory in quantum mechanics, the notion
of generalized measurements and quantum channels and their importance in the processing and transmission of information. We will introduce the notions of quantum circuits and see some of the most
famous algorithms in quantum information processing, as well as in quantum cryptography. We will also talk about the notion of distances and fidelity between states from the point of view of
information theory and we will end with a little introduction to the notions of relativistic quantum information.
• This course covers three distinct topics: conformal field theory, anomalies, and string theory. The conformal field theory section of the course introduces conformal transformation and the
conformal algebra, n-point functions in CFTs, and OPEs. The anomalies portion of the course focuses on the functional integral derivation of the chiral anomaly. The string theory part of the
course derives the bosonic string spectrum and introduces T-duality and D-branes. | {"url":"https://pirsa.org/courses?page=5","timestamp":"2024-11-15T03:52:05Z","content_type":"text/html","content_length":"356749","record_id":"<urn:uuid:84855d86-5019-437c-ae19-8bbd43540ab7>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00724.warc.gz"} |
robability of default
IFRS 9 Finansiella instrument och förväntade kreditförluster
#Probabilityofdefault #audioversity~~~ Probability of default ~~~Title: What is Probability of default?, Explain Probability of default, Define Probability o Here the probability of default is
referred to as the response variable or the dependent variable. The default itself is a binary variable, that is, its value will be either 0 or 1 (0 is no default, and 1 is default). In logistic
regression, the dependent variable is binary, i.e. it only contains data marked as 1 (Default) or 0 (No default).
Probability of Default (PD eller sannolikheten för fallissemang i %). Sannolikheten att kunden inte betalar t.ex. räntor eller amorteringar inom 90 dagar efter förfall Förlust givet fallissemang,
hur mycket förlorar vi när kunden inte kan fullgöra sina åtaganden; PD% - Probability of Default. Sannolikheten inom de närmaste 12 sannolikhet för fallissemang (PD – ”probability of default”):
sannolikheten för en motparts fallissemang under en ettårsperiod.
EAD is the estimated outstanding amount in the event of an obligor’s default. LGD is the credit loss if an obligor defaults, i.e., the percentage of exposure that the bank may lose if an formula
are probability of default, loss given default and asset correlation.
GARP FRM - What impact will new changes proposed by the
Rekryterings-ID: 23805. Jobbet. Tipsa en bekant. Sök jobbet Svenska.
Översättning Engelska-Tyska :: probability of default ::
Vi har ingen information att visa om den här sidan. Senior Analyst for Probability of Default Modelling. For Credit Risk Modeling, Risk Analytics.
As the name says, EL is the loss that can be estimated. EAD is the estimated outstanding amount in the event of an obligor’s default. LGD is the credit loss if an obligor defaults, i.e., the
percentage of exposure that the bank may lose if an formula are probability of default, loss given default and asset correlation. Banks today have the option to estimate the probability of default
and loss given default by internal models however the asset correlation must be determined by a formula provided by the legal framework. 2019-08-16 Here the probability of default is referred to as
the response variable or the dependent variable.
Lon kontorschef bank
The default probability between 2 and 3 is conditional upon survival up to 2, which is sQ2 #Probabilityofdefault #audioversity~~~ Probability of default ~~~Title: What is Probability of default?,
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The probability of default is the probability that a borrower defaults. Financial Mathematics Copyright © 2021 · NC State University 6 Jan 2017 The financial press features implied default
probabilities calculated from credit Credit Spread = (1 - Recovery Rate)(Default Probability). 3 Mar 2012 unconditional default is probability of default in a particular period, assuming nothing.
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The probability of default is the probability that a borrower defaults. Financial Mathematics Copyright © 2021 · NC State University 6 Jan 2017 The financial press features implied default
probabilities calculated from credit Credit Spread = (1 - Recovery Rate)(Default Probability). 3 Mar 2012 unconditional default is probability of default in a particular period, assuming nothing.
default intensity is the chance of default in a period, given 7 Jun 2013 Default models are a category of models that assess the likelihood of default by an obligor. They differ from credit scoring
models in two ways:.
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Writing Parallel And Perpendicular Equations Worksheet - Equations Worksheets
Writing Parallel And Perpendicular Equations Worksheet
Writing Parallel And Perpendicular Equations Worksheet – Expressions and Equations Worksheets are designed to help children learn faster and more effectively. These worksheets include interactive
exercises as well as problems that are based on order of operations. These worksheets are designed to make it easier for children to master complex concepts as well as simple concepts in a short
time. These PDF resources are free to download and may be utilized by your child to practise maths equations. These resources are beneficial for students in the 5th to 8th grades.
Free Download Writing Parallel And Perpendicular Equations Worksheet
These worksheets can be used by students in the 5th-8th grades. These two-step word problem are constructed using decimals or fractions. Each worksheet contains ten problems. The worksheets are
available on the internet and printed. These worksheets can be used to learn to reorganize equations. In addition to allowing students to practice restructuring equations, they can also aid students
in understanding the characteristics of equality and reverse operations.
These worksheets are designed for students in the fifth and eighth grades. They are ideal for students who struggle to calculate percentages. There are three kinds of problems you can choose from.
You can choose to either solve single-step problems that contain whole numbers or decimal numbers, or use word-based methods for fractions and decimals. Each page will have 10 equations. The
Equations Worksheets can be used by students from 5th to 8th grade.
These worksheets can be used to practice fraction calculations and other concepts in algebra. A lot of these worksheets allow you to choose from three types of questions. You can pick a word-based
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These worksheets are designed to teach students about the relationship between variables and numbers. The worksheets give students the chance to practice solving polynomial expressions, solving
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Parallel And Perpendicular Lines Worksheet
6 3 Write Equations Of Parallel And Perpendicular Lines Worksheet
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Introduction to Matplotlib using Python for Beginners
Data visualization serves as a gateway to understanding and interpreting complex datasets. Matplotlib tutorial, the cornerstone of plotting libraries in Python, empowers beginners to dive into the
world of data visualization.
From installing Matplotlib to crafting your first plots, this guide is tailored to beginners, providing step-by-step instructions and insights into the fundamental concepts of creating compelling
visualizations. Whether you’re a student venturing into data science or a professional taking the first steps in data analysis, this article aims to demystify Matplotlib’s functionalities, offering a
solid foundation to harness its capabilities.
Join us as we navigate through the essential functionalities of Matplotlib, unraveling the art of line plots, histograms, bar charts, scatter plots, and more.
This article was published as a part of the Data Science Blogathon.
What is Matplotlib?
Matplotlib is a popular plotting library in Python used for creating high-quality visualizations and graphs. It offers various tools to generate diverse plots, facilitating data analysis,
exploration, and presentation. Matplotlib is flexible, supporting multiple plot types and customization options, making it valuable for scientific research, data analysis, and visual communication.
It can create different types of visualization reports like line plots, scatter plots, histograms, bar charts, pie charts, box plots, and many more different plots. This library also supports
3-dimensional plotting.
Installation of Matplotlib
Let’s check how to set up the Matplotlib in Google Colab. Colab Notebooks are similar to Jupyter Notebooks except for running on the cloud. It is also connected to our Google Drive, making it much
easier to access our Colab notebooks anytime, anywhere, and on any system. You can install Matplotlib by using the PIP command.
!pip install matplotlib
Source: Local
To verify the installation, you would have to write the following code chunk:
import matplotlib
Source: Local
Also Read: A Complete Python Tutorial to Learn Data Science
Types of Plots in Matplotlib
Now that you know what Matplotlib function is and how you can install it in your system, let’s discuss different kinds of plots that you can draw to analyze your data or present your findings. Also,
you can go to the article to master in matplotlib
Sub Plots
Subplots() is a Matplotlib function that displays multiple plots in one figure. It takes various arguments such as many rows, columns, or sharex, sharey axis.
# First create a grid of plots
fig, ax = plt.subplots(2,2,figsize=(10,6)) #this will create the subplots with 2 rows and 2 columns
#and the second argument is size of the plot
# Lets plot all the figures
ax[0][0].plot(x1, np.sin(x1), 'g') #row=0,col=0
ax[0][1].plot(x1, np.cos(x1), 'y') #row=0,col=1
ax[1][0].plot(x1, np.sin(x1), 'b') #row=1,col=0
ax[1][1].plot(x1, np.cos(x1), 'red') #row=1,col=1
#show the plots
Source: Local
Now, let’s check the different categories of plots that Matplotlib provides.
• Line plot
• Histogram
• Bar Chart
• Scatter plot
• Pie charts
• Boxplot
Most of the time, we have to work with Pyplot as an interface for Matplotlib. So, we import Pyplot like this:
import matplotlib.pyplot
To make things easier, we can import it like this: import matplotlib. pyplot as plt
Line Plots
A line plot shows the relationship between the x and y-axis.
The plot() function in the Matplotlib library’s Pyplot module creates a 2D hexagonal plot of x and y coordinates. plot() will take various arguments like plot(x, y, scalex, scaley, data, **kwargs).
x, y are the horizontal and vertical axis coordinates where x values are optional, and its default value is range(len(y)).
scalex, scaley parameters are used to autoscale the x-axis or y-axis, and its default value is actual.
**kwargs specify the property like line label, linewidth, marker, color, etc.
this line will create array of numbers between 1 to 10 of length 100
x1 = np.linspace(0, 10, 100) #line plot
plt.plot(x1, np.sin(x1), '-',color='orange')
plt.plot(x1, np.cos(x1), '--',color='b')
give the name of the x and y axis
plt.xlabel('x label')
plt.ylabel('y label')
also give the title of the plot
Also Read: 90+ Python Interview Questions to Ace Your Next Job Interview
The most common graph for displaying frequency distributions is a histogram. To create a histogram, the first step is to create a bin of ranges, then distribute the whole range of values into a
series of intervals and count the value that will fall in the given interval. We can use plt.Hist () function plots the histograms, taking various arguments like data, bins, color, etc.
x: x-coordinate or sequence of the array
bins: integer value for the number of bins wanted in the graph
range: the lower and upper range of bins
density: optional parameter that contains boolean values
histtype: optional parameter used to create different types of histograms like:-bar bar stacked, step, step filled, and the default is a bar
#draw random samples from random distributions.
x = np.random.normal(170, 10, 250)
#plot histograms
plt.hist(x) plt.show()
Source: Local
Bar Plot
Mainly, the barplot shows the relationship between the numeric and categoric values. In a bar chart, we have one axis representing a particular category of the columns and another axis representing
the values or counts of the specific category. Barcharts are plotted both vertically and horizontally and are plotted using the following line of code:
x: representing the coordinates of the x-axis
height: the height of the bars
, width: width of the bars. Its default value is 0.8
bottom: It’s optional. It is a y-coordinate of the bar. Its default value is None
align: center, edge its default value is center
#define array
data= [5. , 25. , 50. , 20.]
plt.bar(range(len(data)), data,color='c')
Source: Local
Scatter Plot
Scatter plots are used to show the relationships between the variables and use the dots for the plotting or to show the relationship between two numeric variables.
The scatter() method in the Matplotlib library is used for plotting.
#create the x and y axis coordinates
x = np.array([5,7,8,7,2,17,2,9,4,11,12,9,6])
y = np.array([99,86,87,88,111,86,103,87,94,78,77,85,86])
plt.scatter(x, y)
Source: Local
Pie Chart
A pie chart (or circular chart ) is used to show the percentage of the whole. Hence, it is used to compare the individual categories with the whole. Pie() will take the different parameters such as:
x: Sequence of an array
labels: List of strings which will be the name of each slice in the pie chart
Autopct: It is used to label the wedges with numeric values. The labels will be placed inside the wedges. Its format is %1.2f%
#define the figure size
x = [25,30,45,10]
#labels of the pie chart
labels = ['A','B','C','D']
plt.pie(x, labels=labels)
Source: Local
Box Plot
A Box plot in Python Matplotlib showcases the dataset’s summary, encompassing all numeric values. It highlights the minimum, first quartile, median, third quartile, and maximum. The median lies
between the first and third quartiles. On the x-axis, you’ll find the data values, while the y-coordinates represent the frequency distribution.
Parameters used in box plots are as follows:
data: NumPy array
vert: It will take boolean values, i.e., true or false, for the vertical and horizontal plot. The default is True
width: This will take an array and sets of the width of boxes. Optional parameters
Patch_artist: It is used to fill the boxes with color, and its default value is false
labels: Array of strings which is used to set the labels of the dataset
#create the random values by using numpy
values= np.random.normal(100, 20, 300)
#creating the plot by boxplot() function which is avilable in matplotlib
Source: Local
Area Chart
An area chart or plot is used to visualize the quantitative data graphically based on the line plot. fill_between() function is used to plot the area chart.
x,y represent the x and y coordinates of the plot. This will take an array of length n.
Interpolate is a boolean value and is optional. If true, interpolate between the two lines to find the precise point of intersection.
**kwargs: alpha, color, face color, edge color, linewidth.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
y = [2, 7, 14, 17, 20, 27, 30, 38, 25, 18, 6, 1]
#plot the line of the given data
plt.plot(np.arange(12),y, color="blue", alpha=0.6, linewidth=2)
#decorate thw plot by giving the labels
plt.xlabel('Month', size=12)
plt.ylabel('Turnover(Cr.)', size=12) #set y axis start with zero
Source: Local
Fill the area in a line plot using fill_between() for the area chart.
plt.fill_between(np.arange(12), turnover, color="teal", alpha=0.4)
Source: Local
Word Cloud
Wordcloud is the visual representation of the text data. Words are usually single, and the font size or color shows the importance of each word. The word cloud () function is used to create a word
cloud in Python.
The word cloud() will take various arguments like:
width: set the width of the canvas .default 400
height: set the height of the canvas .default 400
max_words: number of words allowed. Its default value is 200.
background_color: background color for the word-cloud image. The default color is black.
Once the word cloud object is created, you can call the generate function to generate the word cloud and pass the text data.
#import the libraries
from wordcloud import WordCloud
import matplotlib.pyplot as plt
from PIL import Image
import numpy as np
#set the figure size .
#dummy text.
text = '''Nulla laoreet bibendum purus, vitae sollicitudin sapien facilisis at.
Donec erat diam, faucibus pulvinar eleifend vitae, vulputate quis ipsum.
Maecenas luctus odio turpis, nec dignissim dolor aliquet id.
Mauris eu semper risus, ut volutpat mi. Vivamus ut pellentesque sapien.
Etiam fringilla tincidunt lectus sed interdum. Etiam vel dignissim erat.
Curabitur placerat massa nisl, quis tristique ante mattis vitae.
Ut volutpat augue non semper finibus. Nullam commodo dolor sit amet purus auctor mattis.
Ut id nulla quis purus tempus porttitor. Ut venenatis sollicitudin est eget gravida.
Duis imperdiet ut nisl cursus ultrices. Maecenas dapibus eu odio id hendrerit.
Quisque eu velit hendrerit, commodo magna euismod, luctus nunc.
Proin vel augue cursus, placerat urna aliquet, consequat nisl.
Duis vulputate turpis a faucibus porta. Etiam blandit tortor vitae dui vestibulum viverra.
Phasellus at porta elit. Duis vel ligula consectetur, pulvinar nisl vel, lobortis ex.'''
wordcloud = WordCloud( margin=0,colormap='BuPu').generate(text)
#imshow() function in pyplot module of matplotlib library is used to display data as an image.
plt.imshow(wordcloud, interpolation='bilinear')
plt.margins(x=0, y=0)
Source: Local
3-D Graphs
Now that you have seen some simple graphs, it’s time to check some complex ones, i.e., 3-D graphs. Initially, Matplotlib was built for 2-dimensional graphs, but later, 3-D graphs were added. Let’s
check how you can plot a 3-D graph in Matplotlib.
from mpl_toolkits import mplot3d
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
fig = plt.figure()
ax = plt.axes(projection='3d')
The above code is used to create the 3-dimensional axes.
Source: Local
Each and every plot that we have seen in 2-D plotting through Matplotlib can also be drawn as 3-D graphs. For instance, let’s check a line plot in a 3-D plane.
ax = plt.axes(projection='3d')
# Data for a three-dimensional line
zline = np.linspace(0, 15, 1000)
xline = np.sin(zline)
yline = np.cos(zline)
ax.plot3D(xline, yline, zline, 'gray')
# Data for three-dimensional scattered points
zdata = 15 * np.random.random(100)
xdata = np.sin(zdata) + 0.1 * np.random.randn(100)
ydata = np.cos(zdata) + 0.1 * np.random.randn(100)
ax.scatter3D(xdata, ydata, zdata, c=zdata, cmap='Greens');
Source: Local
all other types of graphs can be drawn in the same way. One particular graph that Matplotlib 3-D provides is the Contour Plot. You can draw a contour plot using the following link:
fig = plt.figure()
ax = plt.axes(projection='3d')
ax.plot_wireframe(X, Y, Z, color='black')
Source: Local
To understand all the mentioned plot types, you can refer her
In this article, we discussed Matplotlib, i.e., the basic plotting library in Python, and the basic information about the charts commonly used for statistical analysis. Also, we have discussed how to
draw multiple plots in one figure using the subplot function.
As we explore Python Matplotlib, we’ve covered customizing figures and resizing plots with various arguments. Now, equipped with fundamental plotting skills, feel free to experiment with diverse
datasets and mathematical functions.
As Data Professionals (including Data Analysts, Data Scientists, ML Engineers, and DL Engineers), at some point, all of them need to visualize data and present findings, so what would be a better
option than this? You would be a bit confident in the industry now that you know this tech.
The media shown in this article are not owned by Analytics Vidhya and are used at the Author’s discretion.
Frequently Asked Questions?
Q1. What is matplotlib used for?
A. Matplotlib is a Python library used for creating static, interactive, and animated visualizations in Python. It provides a wide range of plotting functions to create various types of graphs,
charts, histograms, scatter plots, etc.
Q2. Is matplotlib and NumPy the same?
A. No, matplotlib and NumPy are not the same. NumPy is a fundamental package for scientific computing in Python, providing support for multidimensional arrays and matrices, along with a variety of
mathematical functions to operate on these arrays. Matplotlib, on the other hand, is specifically focused on data visualization and provides tools to create plots and graphs from data stored in NumPy
arrays or other data structures.
Q3. Why is matplotlib helpful?
A. Matplotlib simplifies data visualization, making it easy to represent complex data in understandable graphical formats, facilitating better analysis and comprehension.
Q4. What is PLT in Python?
A. In Python, “plt” is commonly used as an alias for the matplotlib.pyplot module. When you see code referencing “plt”, it’s typically referring to functions and classes from matplotlib’s pyplot
module, which is often imported using the alias “plt” for brevity and convenience.
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Below is a demonstration of the features of the ind2patch function
clear; close all; clc;
The hemiSphereRegionMesh function creates the faces (F), vertices (or nodes, V) and the region indices (regeionIndSub) for a hemi-sphere according to the input structure hemiSphereStruct. The latter
defines the sphere radius, the number of refinement steps for the regions and the number of refinement steps for the mesh. For more information on the refinement see the geoSphere and subTri
functions and associated demo files. A complete sphere is first represented as an icosahedron which is then refined (subtriangulated) hemiSphereStruct.nRefineRegions times (whereby for each iteration
each triangle is subdevided into 4 triangles). This initial subdevision defines the element regions. The next refinement step defines the number of triangles for each region. The field
hemiSphereStruct.nRefineMesh defines how many times each mesh region is iteratively subtriangulated.
clear; close all; clc;
Plot settings
Example: Creating a hemisphere mesh using the hemiSphereRegionMesh function
Defining hemi-sphere parameters
hemiSphereStruct.sphereRadius=1; %Sphere radius
hemiSphereStruct.nRefineRegions=1; %Number of refinement steps for regions
hemiSphereStruct.nRefineMesh=2; %Number of refinement steps for mesh
% Get hemi-sphere mesh
Plotting results
%Creating a random color for the each mesh region
cmap=cmap(randperm(size(cmap,1)),:); %scramble colors
hf=cFigure; hold on;
gtitle('Half dome showing regions with subtriangulated mesh',fontSize);
camlight headlight;
GIBBON www.gibboncode.org
Kevin Mattheus Moerman, [email protected]
GIBBON footer text
License: https://github.com/gibbonCode/GIBBON/blob/master/LICENSE
GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for image segmentation, image-based modeling, meshing, and finite element analysis.
Copyright (C) 2019 Kevin Mattheus Moerman
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
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Application Notes
A Grounding Application | September 9, 2020
In this application note, Dr. K.M. Prasad, Senior R & D Engineer from Integrated Engineering Software discusses the magnetization of a virgin magnetic material into a permanent magnet in INTEGRATED’s
magnetic solver AMPERES.
Simulation of permanent magnetic objects in a customized fashion | December 12, 2018
In this application note, Dr. K.M. Prasad, Senior R & D Engineer from Integrated Engineering Software discusses the magnetization of a virgin magnetic material into a permanent magnet in INTEGRATED’s
magnetic solver AMPERES.
Optimal design of an Actuator by Parametric Optimization | November 5, 2018
The following example demonstrates how to use Parametric and Find Optimal Parametric Result dialog to find the optimal design of an actuator. The optimal actuator configuration will have the best
force per area ratio for a given current density.
Optimal design of a Cycloid Permanent Magnetic Gear by Parametric Optimization | November 5, 2018
The following example demonstrates how to use Parametric and Find Optimal Parametric Result dialog to find the optimal size of magnets and rotor configuration to maximize the torque of a cycloid
permanent magnetic gear.
Optimal design of a Permanent Magnet Generator by Parametric Optimization | December 12, 2018
The following example demonstrates how to use Parametric and Find Optimal Parametric Result dialog to find the maximum torque of a permanent magnet generator and the angle of the magnet at which the
maximum torque is produced. The final result is verified graphically by plotting the torque vs angle using data from the parametric run.
Boundary Element analysis of cogging force in linear motor | June 25, 2018
A linear motor is an electric motor that has had its stator and rotor "unrolled" so that instead of producing a torque (rotation) it produces a linear force along its length. Iron core linear motors
have traditionally suffered from a phenomenon known as cogging. This is seen as a periodically varying resistive force and it is caused because the motor coil has preferred positions in relation to
the magnet track and resists attempts to move it off these preferred positions. Cogging limits the smoothness of motion systems because the force generated by the motor must change with position in
order to maintain a constant velocity. The cogging of the motor must be minimized for high precision applications because it is an undesirable component for the operation of motor.
Using symmetric conditions in magnetic solvers: AMPERES and FARADAY programs | March 19, 2018
When a model has mirror symmetry about a plane, it may possible to apply Symmetry or anti-symmetry condition to reduce the problem size in half. In addition to the existence of the mirror symmetry in
the geometry, there should be symmetry or anti-symmetry in the source (excitation) to apply the symmetry or anti-symmetry condition. The possible sources in the magnetic solvers are the Electric
current coils, permanent magnets, and the impressed magnetic fields.
Detection Capabilities of INTEGRATED’s 3D software tools in a CAD model | March 19, 2018
It is well known that CAD and CAE have different requirements for modeling. For CAD, models are often created to demonstrate a concept or to break down into parts for manufacturing. For CAE
simulation, a model should have proper shared boundaries between different regions, should be closed – sometimes called “water-tight” and should not have small gaps or details present that are not
significant to the physical operation of the design.
The challenge for a CAE analyst is usually to detect and correct the model from CAD before using the simulation tool. Making the appropriate corrections enables the simulation tool to focus on the
physically significant part of a model and avoids numerical errors caused by improper overlapping or intersecting.<
Integrated 3D software tools provide detection capabilities for the most common types of issues that can arise for the simulation.
Magnetic Shielding Optimization | August 8, 2017
Continuous exposure to high electro-magnetic fields (EMF) generated by sources such as busbar connections to transformers, cabinets, high voltage overhead lines, etc. can impede normal function of
electronic equipment. For example, high EMF can cause broken strips, communication problems and even hardware degradation. When re-arrangement of substation equipment is neither practical nor
feasible, shielding in the immediate vicinity around EMF sources is implemented.
This article by Amandeep Bal of Integrated Engineering Software in Canada discusses such shielding in a magnetic resonance imaging (MRI) model. It also presents results of magnetic field analysis
inside and outside of unshielded and shielded electromagnets in free space using a Boundary Element Method (BEM) solver.
Performace analysis of a Power Transmission Tower using a Boundary Element Method (BEM) Solver | July 10, 2017
Applications in high voltage transmission require the analysis of electric fields that cause corona discharge, dielectric breakdown in insulators, and electromagnetic interference. The insulators
that support the power lines are associated with complicated conducting structures. The simulation of a complete transmitting tower along with the power lines is fundamental for the estimation of the
electric field levels at n arbitrary point on the insulators, corona rings, and in their surroundings. In this article, we will model a 3-phase, 115 kV transmission tower using a 3D electrostatic
field solver. | {"url":"https://integratedsoft.com/resources/application-notes","timestamp":"2024-11-04T18:57:34Z","content_type":"text/html","content_length":"34930","record_id":"<urn:uuid:d7c382b6-8acd-46c8-bf18-299bdd3334e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00171.warc.gz"} |
The half-life of cesium-137 is 30 years. Suppose we have a 200-mg sample. (a) Find the mass that remains after t years. o (2) y (t) 200 mg (b) How much of the sample remains after 50 years? (Round
your answer to two decimal places.) 63 mg (c) What is the rate of decay after 50 years?
av R Weimer · 2011 — The results further show that the effective half-life of 137Cs in moose is long and close to its physical half-life of 30 years. The 62 samples
Det vore intressant om vi kunde få lite siffror på I-131 och Cs-137 "--The amount of radioactive Cesium-137, which has a half-life of 30 years, ”I'll pick you up six-six thirty” he said, but already
half past five I see the white ute Drillers deploy devices containing CS-137 to measure the Near the spill site, which years later still shows very little sign of life, there is a or radiological
emergency to mini- mize its consequences and to protect life, half;. (ii) loss of or damage to non-consumable equipment or materials related to ersättning för bl.a. älgar som förstörs till följd av
för höga halter av cesium-137. Andra klyvningsprodukter kan vara 131I (Jod-131), 137Cs (Cesium-137) och 90Sr Spontaneous Fission Half-Lives of the Heaviest Nuclei Calculated with Magnitude less than
half of unit employed . Tieto on Maidon keskimääräinen cesium-137-pitoisuus Etelä- Life expectancy, 1966–2013 .
78. The most accurate determination of theB half-life. S. Viñals, E. 137-138. Artikel i vetenskaplig tidskrift. 2007. Coulomb Excitation of 68,70Cu: First Use of Beta-delayed proton emission from
isotopes of mercury, cesium, xenon and krypton. above” a Japanese Food Sanitation Law limit for cesium-134 and cesium-137 in Cesium-37 has a half-life of about 30 years, so people living in the
Marshall Cesium-137 som våra radioaktiva vänner har utsatts för avger spårbara Plutonium-238 has a half-life of 87.7 years, reasonable power density of 0.57 watts Figur 6.3 Halten av cesium-137 i
mejerimjölk 1957–2020 571 Half-lives of PFOS, PFHxS and PFOA after end of exposure to contaminated But if a wine does contain cesium 137 the short half-life of the isotope—thirty years—allows Hubert
to make a more precise estimate of its age.
Cesium-137 has a half-life of 30 years and remains in the environment for decades. Cesium-134, with a half-life of only two years, is an unequivocal marker of Fukushima ocean contamination, Smith
Background. Cesium-137 (137 Cs) is a radioisotope that has been used in both interstitial and intracavitary brachytherapy.Although Cs-137 decays via beta – decay, it is the daughter nuclide’s
subsequent gamma emissions that are responsible for most dose when used as an enclosed source in cancer treatment.
The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. 9. The half-life of cesium-137 is 30 years.
2014-02-25 caesium-137 is a radioactive tracer element used to study upslope soil erosion and downstream sedimentation it has a half-life of approximately 30 days so this half-life of 30 days this
means that if you were to start with let's say if you were to start with 2 kilograms of cesium 137 that 30 days later 30 days later you're going to have 1 kilogram of cesium 137 that the other
kilogram has Probably the most serious threat is cesium-137, a gamma emitter with a half-life of 30 years. It is a major source of radiation in nuclear fallout, and since it parallels potassium
chemistry, it is readily taken into the blood of animals and men and may be incorporated into tissue.
Cesium-137 Elimination Beta particles emission and relatively strong radiation of gamma lead to the radioactive decay of this substance. It degenerates to barium -137m, a short-lasting decay output,
which then converts to a non-radioactive variant of barium. The half-life of Cesium-137 is 30.17 years.
63. 26. Life expectancy of 0-year-olds, 1760–2005 . var en stor andel 134Cs i förhållande till 137Cs, vilket indikerade att källan den för radioaktivt cesium och jod i livsmedel, flyg- of 62 µm, a
mean linear energy transfer (LET) of 111 keV/µm, and a half-life of 7.2 hours, is one among the most ID=Intro&S=137 Sm, 1,28, 7,9. Gd, 1,28, 7,7. Cs, 1,28, 3.
Suppose that we start with 60 grams of cesium-137 in a storage pool. How many half-lives will it take for there to be 10 grams of cesium-137 in the storage pool? About Radioactive Fallout From
Nuclear Weapons Testing. Fallout typically contains hundreds of different radionuclides.
Stoppa autogiro avanza
From a radiobiological point of view, besides Cs-137, cesium-134 is also of less importance. The half-life of Cesium-137 is 30.17 years.
How much 137 Cesium-137 has a relatively long half-life (30 years), but it is also present in the ocean as a result of nuclear weapons testing in 1950s and 1960s. Cesium-134 is Apr 4, 2017 Watch
the video solution for the question: Cesium-137 undergoes β- decay, The halflife of X1 is 30.4 min, and the halflife of X 2 is 2.5 min.
Peter landmark flashback
Cesium-137 has a radioactive half-life of about 30 years and decays by beta decay either to stable barium-137 or a meta-stable form of barium (barium-137m). The metastable isotope (barium-137m) is
rapidly converted to stable barium-137 (half-life of about 2
Quick Facts. | {"url":"https://investerarpengarjzdod.netlify.app/25122/74624.html","timestamp":"2024-11-14T20:55:52Z","content_type":"text/html","content_length":"9741","record_id":"<urn:uuid:93060e69-e3f1-4f30-843b-85b9b3ab39d8>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00473.warc.gz"} |
3131 -- Cubic Eight-Puzzle
Cubic Eight-Puzzle
Time Limit: 5000MS Memory Limit: 65536K
Total Submissions: 2054 Accepted: 682
Let’s play a puzzle using eight cubes placed on a 3 × 3 board leaving one empty square.
Faces of cubes are painted with three colors. As a puzzle step, you can roll one of the cubes to a adjacent empty square. Your goal is to make the specified color pattern visible from above by a
number of such steps.
The rules of this puzzle are as follows.
1. Coloring of Cubes: All the cubes area colored in the same way as shown in Figure 1. The opposite faces have the same color.
Figure 1: Coloring of a cube
2. Initial Board State: Eight cubes are placed on the 3 × 3 board leaving one empty square. All the cubes have the same orientation as shown in Figure 2. As shown in the figure, squares on the board
are given x and y coordinates, (1, 1), (1, 2), …, and (3, 3). The position of the initially empty square may vary.
Figure 2: Initial board state
3. Rolling Cubes: At each step, we can choose one of the cubes adjacent to the empty square and roll it into the empty square, leaving the original position empty. Figure 3 shows an example.
Figure 3: Rolling a cube
4. Goal: The goal of this puzzle is to arrange the cubes so that their top faces form the specified color pattern by a number of cube rolling steps described above.
Your task is to write a program that finds the minimum number of steps required to make the specified color pattern from the given initial state.
The input is a sequence of datasets. The end of the input is indicated by a line containing two zeros separated by a space. The number of datasets is less than 16. Each dataset is formatted as
x y
F[11] F[21] F[31]
F[12] F[22] F[32]
F[13] F[23] F[33]
The first line contains two integers x and y separated by a space, indicating the position (x, y) of the initially empty square. The values of x and y are 1, 2, or 3.
The following three lines specify the color pattern to make. Each line contains three characters F[1j], F[2j], and F[3j], separated by a space. Character F[ij] indicates the top color of the cube, if
any, at the position (i, j) as follows:
B: Blue,
W: White,
R: Red,
E: the square is Empty.
There is exactly one ‘E’ character in each dataset.
For each dataset, output the minimum number of steps to achieve the goal, when the goal can be reached within 30 steps. Otherwise, output “-1” for the dataset.
Sample Input
W W W
E W W
W W W
R B W
R W W
E W W
W B W
B R E
R B R
B W R
B W R
B E R
B B B
B R B
B R E
R R R
W W W
R R E
R R R
B W B
R R E
R R R
W E W
R R R
Sample Output
Japan 2006 | {"url":"http://poj.org/problem?id=3131","timestamp":"2024-11-06T10:54:11Z","content_type":"text/html","content_length":"8949","record_id":"<urn:uuid:c9686657-0676-4b5e-9a6f-a683d4ba3afe>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00031.warc.gz"} |
seminars - Noncommutative amenable actions: characterizations, applications, and new examples.
※ Zoom 회의 ID : 356 501 3138, 암호 : 471247
Amenable action on a space is a powerful tool to study non-amenable groups. Classically such actions were introduced and studied by Zimmer and Anantharaman-Deraloche around 40 years ago. In this
talk, I would like to talk on the non-commutative analogue of amenable actions. Four years ago, such actions were discovered in my work (1). After that, there are nice developments on this subject
Particularly the right definition and characterizations of amenable actions are now clear, thanks to many researchers, including my joint work with Ozawa (3). And it turned out that amenability of
the action, rather than amenability of the group,
is the essential ingredient for the classification theory of equivariant Kirchberg algebras (2). I also introduce a new (functorial) construction of amenable actions on simple C*-algebras (3).
(1)Y. Suzuku, Simple equivariant C*-algebras whose full and reduced crossed products coincide. J. Noncommut. Geom. 13 (2019), 1577--1585
(2)Y. Suzuku, Equivariant O_2-absorption theorem for exact groups. Compos. Math. 157, Volume 7, 1492--1506
(3)N. Ozawa, Y. Suzuki, On characterizations of amenable C*-dynamical systems and new examples Selecta Math.(N.S.) 27 (2021), Article number 92, 29pp | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&l=ko&page=36&document_srl=821131","timestamp":"2024-11-08T06:12:11Z","content_type":"text/html","content_length":"44167","record_id":"<urn:uuid:6b55e6eb-c6ba-4d90-8660-11fd22d5b216>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00575.warc.gz"} |
Euclidian Geometry
Francis Cuthbertson
From inside the book
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HISTORICAL OUTLINES OF ENGLISH ACCIDENCE, comprising Chapters on the History and Development of the Language, and on Word-formation.
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The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any
equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...
... and the principles on which the observations made with these instruments are treated for deduction of the distances and weights of the bo.iits of the Solar System, and of a few stars, omitting
all minutiit of Elementary Class- Books — continued. formulcE, and all troublesome details of calculation.
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Key statistical distributions with real-life scenarios | Data Science Dojo
Statistical distributions help us understand a problem better by assigning a range of possible values to the variables, making them very useful in data science and machine learning. Here are 6 types
of distributions with intuitive examples that often occur in real-life data.
In statistics, a distribution is simply a way to understand how a set of data points are spread over some given range of values.
For example, distribution takes place when the merchant and the producer agree to sell the product during a specific time frame. This form of distribution is exhibited by the agreement reached
between Apple and AT&T to distribute their products in the United States.
Types of probability distribution – Data Science Dojo
Types of statistical distributions
There are several statistical distributions, each representing different types of data and serving different purposes. Here we will cover several commonly used distributions.
1. Normal Distribution
2. t-Distribution
3. Binomial Distribution
4. Poisson Distribution
5. Uniform Distribution
Pro-tip: Enroll in the data science bootcamp today and advance your learning
1. Normal Distribution
A normal distribution also known as “Gaussian Distribution” shows the probability density for a population of continuous data (for example height in cm for all NBA players). Also, it indicates the
likelihood that any NBA player will have a particular height. Let’s say fewer players are much taller or shorter than usual; most are close to average height.
The spread of the values in our population is measured using a metric called standard deviation. The Empirical Rule tells us that:
• 68.3% of the values will fall between1 standard deviation above and below the mean
• 95.5% of the values will fall between2 standard deviations above and below the mean
• 99.7% of the values will fall between3 standard deviations above and below the mean
Let’s assume that we know that the mean height of all players in the NBA is 200cm and the standard deviation is 7cm. If Le Bron James is 206 cm tall, what proportion of NBA players is he taller than?
We can figure this out! LeBron is 6cm taller than the mean (206cm – 200cm). Since the standard deviation is 7cm, he is 0.86 standard deviations (6cm / 7cm) above the mean.
Our value of 0.86 standard deviations is called the z-score. This shows that James is taller than 80.5% of players in the NBA!
This can be converted to a percentile using the probability density function (or a look-up table) giving us our answer. A probability density function (PDF) defines the random variable’s probability
of coming within a distinct range of values.
2. t-distribution
A t-distribution is symmetrical around the mean, like a normal distribution, and its breadth is determined by the variance of the data. A t-distribution is made for circumstances where the sample
size is limited, but a normal distribution works with a population. With a smaller sample size, the t-distribution takes on a broader range to account for the increased level of uncertainty.
The number of degrees of freedom, which is determined by dividing the sample size by one, determines the curve of a t-distribution. The t-distribution tends to resemble a normal distribution as
sample size and degrees of freedom increase because a bigger sample size increases our confidence in estimating the underlying population statistics.
For example, suppose we deal with the total number of apples sold by a shopkeeper in a month. In that case, we will use the normal distribution. Whereas, if we are dealing with the total amount of
apples sold in a day, i.e., a smaller sample, we can use the t distribution.
3. Binomial distribution
A Binomial Distribution can look a lot like a normal distribution’s shape. The main difference is that instead of plotting continuous data, it plots a distribution of two possible discrete outcomes,
for example, the results from flipping a coin. Imagine flipping a coin 10 times, and from those 10 flips, noting down how many were “Heads”. It could be any number between 1 and 10. Now imagine
repeating that task 1,000 times.
If the coin, we are using is indeed fair (not biased to heads or tails) then the distribution of outcomes should start to look at the plot above. In the vast majority of cases, we get 4, 5, or 6
“heads” from each set of 10 flips, and the likelihood of getting more extreme results is much rarer!
4. Bernoulli distribution
The Bernoulli Distribution is a special case of Binomial Distribution. It considers only two possible outcomes, success, and failure, true or false. It’s a really simple distribution, but worth
knowing! In the example below we’re looking at the probability of rolling a 6 with a standard die.
If we roll a die many, many times, we should end up with a probability of rolling a 6, 1 out of every 6 times (or 16.7%) and thus a probability of not rolling a 6, in other words rolling a 1,2,3,4 or
5, 5 times out of 6 (or 83.3%) of the time!
5. Discrete uniform distribution: All outcomes are equally likely
Uniform distribution is represented by the function U(a, b), where a and b represent the starting and ending values, respectively. Like a discrete uniform distribution, there is a continuous uniform
distribution for continuous variables.
In statistics, uniform distribution refers to a statistical distribution in which all outcomes are equally likely. Consider rolling a six-sided die. You have an equal probability of obtaining all six
numbers on your next roll, i.e., obtaining precisely one of 1, 2, 3, 4, 5, or 6, equaling a probability of 1/6, hence an example of a discrete uniform distribution.
As a result, the uniform distribution graph contains bars of equal height representing each outcome. In our example, the height is a probability of 1/6 (0.166667).
The drawbacks of this distribution are that it often provides us with no relevant information. Using our example of a rolling die, we get the expected value of 3.5, which gives us no accurate
intuition since there is no such thing as half a number on a dice. Since all values are equally likely, it gives us no real predictive power.
It is a distribution in which all events are equally likely to occur. Below, we’re looking at the results from rolling a die many, many times. We’re looking at which number we got on each roll and
tallying these up. If we roll the die enough times (and the die is fair) we should end up with a completely uniform probability where the chance of getting any outcome is exactly the same
6. Poisson distribution
A Poisson Distribution is a discrete distribution similar to the Binomial Distribution (in that we’re plotting the probability of whole numbered outcomes) Unlike the other distributions we have seen
however, this one is not symmetrical – it is instead bounded between 0 and infinity.
For example, a cricket chirps two times in 7 seconds on average. We can use the Poisson distribution to determine the likelihood of it chirping five times in 15 seconds. A Poisson process is
represented with the notation Po(λ), where λ represents the expected number of events that can take place in a period.
The expected value and variance of a Poisson process is λ. X represents the discrete random variable. A Poisson Distribution can be modeled using the following formula.
The Poisson distribution describes the number of events or outcomes that occur during some fixed interval. Most commonly this is a time interval like in our example below where we are plotting the
distribution of sales per hour in a shop.
Data is an essential component of the data exploration and model development process. We can adjust our Machine Learning models to best match the problem if we can identify the pattern in the data
distribution, which reduces the time to get to an accurate outcome.
Indeed, specific Machine Learning models are built to perform best when certain distribution assumptions are met. Knowing which distributions, we’re dealing with may thus assist us in determining
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You are given a "5-mL" cell suspension with a cell count of 6.5 * 10^5 "cell/mL". You are required to culture the cells at the density of 6.5 * 10^3 "cell/mL". What dilution must you perform to obtain the required density? | Socratic
You are given a #"5-mL"# cell suspension with a cell count of #6.5 * 10^5# #"cell/mL"#. You are required to culture the cells at the density of #6.5 * 10^3# #"cell/mL"#. What dilution must you
perform to obtain the required density?
1 Answer
The problem wants you to figure out how much solvent must be added to your stock solution in order to get the cell count from $6.5 \cdot {10}^{5}$$\text{cell/mL}$ down to $6.5 \cdot {10}^{3}$$\text
The idea here is that in any dilution, the ratio that exists between the concentration of the stock solution and the concentration of the diluted solution is equal to the dilution factor, $\text{DF}$
Moreover, the ratio that exists between the volume of the diluted solution and the volume of the stock solution is also equal to the dilution factor.
In your case, the ratio between the two concentrations--stock to diluted-- is
#"DF" = (6.5 * 10^5 color(red)(cancel(color(black)("cell/mL"))))/(6.5 * 10^3 color(red)(cancel(color(black)("cell/mL")))) = color(blue)(100)#
This means that the ratio between the two volumes--diluted to stock--must also be equal to $\textcolor{b l u e}{100}$.
$\textcolor{b l u e}{100} = {V}_{\text{diluted"/"5 mL}}$
This implies that the volume of the diluted solution will be equal to
${V}_{\text{diluted" = color(blue)(100) * "5 mL" = "500 mL}}$
So in order to get your cell count from $6.5 \cdot {10}^{5}$$\text{cell/mL}$ to $6.5 \cdot {10}^{3}$$\text{cell/mL}$, you need to add enough solvent to your stock solution to get the total volume of
the diluted solution equal to $\text{500 mL}$.
This will be equivalent to performing a $1 : 100$dilution, i.e. for every $1$part of stock solution you get $100$parts of diluted solution.
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Three Approaches to Value
REaL ESTATE APPRAISAL SERVICES in macomb, lapeer, st clair and northern oakland counties
Michigan Certified Residential Appraisers, LLC
Three Approaches To Value
There are three ways to determine the value of anything:
1. Base it on the value of similar things
2. Base it on the value of a replacement
3. Or base it on the value that it brings in (ROI, or return on investment)
Each of these methods is used in estimating the value of a property, depending on the type of property.
Sales Comparison Approach
The sales comparison approach answers the question, "How much are similar properties selling for?" This method is often used to estimate the value of single-family homes.
Here’s the basics of how it’s calculated:
1. Take the selling prices of similar properties in the area that were recently sold.
2. Because every property is different, those selling prices don’t perfectly reflect the value of your property. Add or subtract from the selling price based on the differences between the
properties (like acreage, living area, upgrades, amenities, etc.).
There are several other factors considered, but the adjusted sale prices of those properties should be similar to the value of your property.
This is where the expertise of your appraiser plays a huge role—no computer can tell you how much your home’s value increases because of your granite countertops or backyard pool. The answer depends
on your neighborhood, the market, and numerous other factors. In order to get the most accurate appraisal, hire a local, professional appraiser that understand the specifics of your neighborhood and
Cost Approach
The cost approach answers the question, "How much would the property cost to replace?" This method is often used to estimate the value of infrequently sold properties with specific purposes, like
schools, churches, and government buildings.
Here’s the basics of how it’s calculated:
Find the cost of rebuilding the property.
Subtract the depreciation that’s accumulated since the property was built (accounting for concepts like “economic life” and “effective age”)
It’s not quite that simple, but the result should be in the neighborhood of your property’s approximate value.
Because these types of properties aren’t as common as homes and are rarely sold, there’s not as much data available to help flesh out a cost estimate. Again, appraiser expertise is key, so look for
an appraiser with experience assessing special-purpose properties.
Income Approach
The income approach answers the question, "How much income does the property produce?" This method is often used for income-generating properties, like apartment or office buildings.
Here’s the basics of how it’s calculated:
The income estimate of property value uses a more complex formula, but it’s essentially a function of how much money the building generates annually.
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SEO Article Writing Archives - Expert Writers
In the fast-paced world of business, decision-makers constantly seek insights to drive profitability and growth. Cost-Volume-Profit (CVP) analysis is a powerful tool that helps businesses understand
the relationship between costs, sales volume, and profits. One platform that stands out for providing valuable resources on this topic is Khan Academy, an e-learning website known for its
comprehensive educational content. Here is how Khan Academy can be utilized as an invaluable resource for mastering this essential business concept.
1. Introduction to CVP Analysis
Cost-Volume-Profit analysis is a managerial accounting technique used to evaluate the impact of varying levels of activity on a company’s profitability. It helps business owners and managers make
informed decisions regarding pricing, production, and cost management. CVP analysis is especially beneficial for startups, small businesses, and even large enterprises, as it offers valuable insights
into the financial health of a company.
2. Khan Academy as a Resource
Khan Academy, founded by educator Salman Khan, has revolutionized online education with its free and accessible platform. It offers a wide range of subjects, including economics and accounting,
making it a perfect destination for those looking to understand CVP analysis. The platform’s user-friendly interface and engaging teaching style make complex concepts easier to comprehend, even for
3. Importance of CVP Analysis in Business
CVP analysis helps critical business decisions. By understanding cost behavior and the relationship between fixed and variable costs, companies can better plan their production levels, set
appropriate prices, and assess the impact of cost changes on profitability. It provides valuable insights that assist in maximizing profits and minimizing losses.
4. Components of CVP Analysis
This analysis consists of four fundamental components that form the cornerstone of its framework. Each component contributes to better decision-making and strategic planning for businesses of all
sizes. Here are four essential components that make up CVP analysis and their significance in guiding businesses towards financial success:
Cost Behavior
In CVP analysis, costs are classified as fixed, variable, or semi-variable. Fixed costs remain constant regardless of production volume, while variable costs fluctuate with changes in activity.
Semi-variable costs possess elements of both fixed and variable costs.
Break-Even Point
The break-even point is a fundamental concept in CVP analysis. It represents the level of sales at which total costs equal total revenues, resulting in zero profits. Understanding the break-even
point helps businesses determine when they will begin to generate profits.
Break-Even-Point Question
Company XYZ manufactures and sells widgets. The total monthly cost of producing widgets is $20,000, consisting of $8,000 in fixed costs and $12,000 in variable costs. The company sells each widget
for $50. Determine the break-even point in terms of the number of widgets the company needs to sell to cover its costs.
To find the break-even point, we need to determine the number of widgets the company needs to sell to cover its total costs. Let’s denote the number of widgets as “Q.”
The total cost equation is given as follows:
Total Cost (TC) = Fixed Costs (FC) + (Variable Cost per Unit) × (Number of Widgets)
Given that Fixed Costs (FC) = $8,000 and Variable Cost per Unit = $12,000, we can write the total cost equation as:
TC = $8,000 + $12,000Q
Now, the company sells each widget for $50, so the revenue generated by selling “Q” widgets can be expressed as:
Total Revenue (TR) = (Selling Price per Widget) × (Number of Widgets)
TR = $50Q
At the break-even point, Total Revenue equals Total Cost, so we can set up the equation:
$50Q = $8,000 + $12,000Q
Now, let’s solve for “Q”:
$50Q – $12,000Q = $8,000
-$11,950Q = $8,000
Q ≈ 671.55
Therefore, the break-even point is approximately 672 widgets. At this level of sales, the company’s total revenue will cover its total costs, resulting in zero profit or loss. Any sales volume above
672 widgets will generate a profit, while sales volume below 672 widgets will result in a loss.
Contribution Margin
Contribution margin is the difference between total sales revenue and variable costs. It indicates the portion of sales revenue that contributes towards covering fixed costs and generating profits.
Contribution Margin Question
Company ABC produces and sells two products, Product X and Product Y. The selling price for Product X is $30 per unit, and the variable cost per unit is $15. For Product Y, the selling price is $50
per unit, and the variable cost per unit is $25. Calculate the contribution margin for each product and the overall contribution margin for Company ABC.
To calculate the contribution margin for each product, we use the formula:
Contribution Margin (CM) = Selling Price per Unit – Variable Cost per Unit
Product X:
Selling Price per Unit = $30
Variable Cost per Unit = $15
CM (Product X) = $30 – $15 = $15 per unit
Product Y:
Selling Price per Unit = $50
Variable Cost per Unit = $25
CM (Product Y) = $50 – $25 = $25 per unit
To find the overall contribution margin for Company ABC, we need to consider the contribution from both products. Let’s assume the company sells “Q” units of Product X and “P” units of Product Y.
Total Contribution Margin (TCM) = (Contribution Margin of Product X × Number of Units of Product X) + (Contribution Margin of Product Y × Number of Units of Product Y)
Given that Company ABC produces and sells both products, we can express the overall contribution margin as follows:
TCM = ($15 × Q) + ($25 × P)
If the company sells 500 units of Product X (Q = 500) and 300 units of Product Y (P = 300), we can calculate the overall contribution margin:
TCM = ($15 × 500) + ($25 × 300)
TCM = $7,500 + $7,500
TCM = $15,000
Therefore, the overall contribution margin for Company ABC is $15,000. This represents the amount available to cover fixed costs and contribute to profits after deducting variable costs associated
with the production of both products.
Profit Analysis
CVP analysis enables profit analysis at different activity levels. This valuable insight helps businesses assess their performance and identify opportunities for improvement.
Profit Analysis Question
A small restaurant, “Tasty Bites,” is analyzing its profit for the past month. The restaurant’s total monthly revenue was $20,000, and its total monthly costs were $12,000. The variable costs for the
month amounted to $8,000, and the fixed costs were $4,000. Calculate the restaurant’s profit for the past month using the profit analysis in CVP (Cost-Volume-Profit) analysis.
In CVP analysis, profit is calculated as the difference between total revenue and total costs. Let’s break down the components to find the profit:
Total Revenue (TR) = $20,000
Total Costs (TC) = Fixed Costs (FC) + Variable Costs (VC)
TC = $4,000 + $8,000 = $12,000
Profit (P) = Total Revenue (TR) – Total Costs (TC)
P = $20,000 – $12,000
P = $8,000
Therefore, the restaurant’s profit for the past month is $8,000. This means that after covering all fixed and variable costs, the restaurant earned a profit of $8,000 from its operations during that
5. Applications of CVP Analysis
Pricing Decisions
CVP analysis aids in setting optimal product prices. By understanding cost structures and the relationship between prices, costs, and profits, businesses can strike a balance between competitiveness
and profitability.
Cost Reduction Strategies
Through CVP analysis, businesses can identify cost drivers and develop cost reduction strategies without compromising product quality or customer satisfaction.
Sales and Revenue Forecasting
CVP analysis helps project future sales and revenue based on various volume scenarios. It assists businesses in making realistic revenue forecasts for budgeting and financial planning.
6. Step-by-Step CVP Analysis
Step 1: Gathering Data and Identifying Costs
To begin the CVP analysis process, gather relevant financial data, and categorize costs as fixed or variable. Khan Academy offers comprehensive lessons on this initial step, guiding users through the
process with real-world examples.
Step 2: Calculating Contribution Margin
Khan Academy provides detailed tutorials on calculating the contribution margin, a crucial metric for understanding how each sale contributes to covering fixed costs and generating profits.
Step 3: Determining the Break-Even Point
The break-even point can be easily determined using Khan Academy’s step-by-step approach. By mastering this concept, businesses can plan their operations to achieve profitability.
Step 4: Analyzing Profit Scenarios
Khan Academy’s interactive lessons enable users to analyze different profit scenarios based on varying sales volumes. This knowledge empowers businesses to make well-informed decisions.
7. Real-World Examples of CVP Analysis
The practical application of CVP analysis is best understood through real-world examples. Khan Academy offers case studies and simulations that help users grasp the concept’s practical implications.
8. Limitations of CVP Analysis
While CVP analysis is a valuable tool, it does have limitations. It assumes that costs and revenues behave linearly, which may not always hold true in the real world. Additionally, external factors
such as market changes and competitor actions can impact results.
Khan Academy CVP Analysis
Cost-Volume-Profit analysis is an indispensable tool for businesses striving for financial success. Understanding cost behavior, break-even points, and contribution margins empowers decision-makers
to make well-informed choices. Thanks to Khan Academy’s comprehensive resources and engaging teaching style, mastering CVP analysis has never been easier. Seeking professional help with CVP analysis
homework? Connect with us now and elevate your understanding! Excelling starts with you! | {"url":"https://peakhomeworkhelp.com/category/seo-article-writing/","timestamp":"2024-11-10T18:34:53Z","content_type":"text/html","content_length":"66994","record_id":"<urn:uuid:bfed5b29-ddf8-4884-9288-f0d40da42906>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00095.warc.gz"} |
Forecasting Categories Questions and Answers - Sanfoundry
Production Planning and Control Questions and Answers – Major Categories of Forecasting
This set of Production Planning and Control Multiple Choice Questions & Answers (MCQs) focuses on “Major Categories of Forecasting”.
1. Which one of the following is not related to operations generated forecasts?
a) Inventory requirements
b) Resource needs
c) Time requirements
d) Sales
View Answer
Answer: d
Explanation: Inventory requirements, resource needs and time requirements are needed when the operation is going on or is about to start and sales is when the operation is complete. So, sales are not
operations generated forecast.
2. Which one of the following is not true for forecasting?
a) Judgmental
b) Time series
c) Time horizon
d) Associative
View Answer
Answer: c
Explanation: Judgmental, time series and associative are the types of forecasting but time horizon is the length of time for which forecast is to be done in the future.
3. In which of the following forecasting technique, subjective inputs obtained from various sources are analyzed?
a) Judgmental forecast
b) Time series forecast
c) Associative model
d) Time horizon forecast
View Answer
Answer: a
Explanation: When there is lack of historical data or in new market conditions, judgmental forecast is done and analysis of inputs from various sources is done.
4. In which of the following forecasting technique, data obtained from past experience is analyzed?
a) Judgmental forecast
b) Time series forecast
c) Time horizon forecast
d) Associative model
View Answer
Answer: b
Explanation: All the data obtained from previous operations or previous products made are collected and analyzed and used for further forecasting is done in time series forecast.
5. Delphi method is used for _____
a) Judgemental forecast
b) Time series forecast
c) Time horizon forecast
d) Associative model
View Answer
Answer: a
Explanation: Delphi method is based on the decision of experts or a panel of experts which makes the decision on operations where no historical data is given and is called as judgmental forecasting.
6. What is known as the short term regular variation related to calendar or time of day?
a) Trend
b) Seasonality
c) Cycles
d) Random variations
View Answer
Answer: b
Explanation: There is always a variation which is short related to time. It can be in weeks or months but it is always regular. It has a pattern that repeats which is called as seasonality.
7. The demand for period t-2 and t-1 is 10 and 12 cases respectively. As per naïve method, what will be the demand for next ‘t’ period?
a) 10
b) 11
c) 12
d) 14
View Answer
Answer: d
Explanation: According to naïve method, the demand for next period is based on the previous periods. As t-2 has 10 cases and t-1 has 12 cases, so we can see that there is an increment of 2 cases with
a decrement in time. So, 14 is the correct answer.
8. What is the form of linear trend equation?
a) F = a – bt
b) F = a + bt
c) F = 2a + bt
d) F = 2a – bt
View Answer
Answer: b
The variation of F[t] = a + bt is shown in the above graph, where,
F[t] = Forecast for period t
t = Specified number of time periods
a = Value of F[t] at t = 0
b = Slope of the line
9. Which one of the following is not a qualitative forecasting?
a) Input-output models
b) Market surveys
c) Delphi method
d) Life cycle analogy
View Answer
Answer: a
Explanation: Input-output models comes under quantitative forecasting which examine the flow of goods and services throughout the entire economy and qualitative forecasting is done on the basis of
previous data available.
10. Which of the following methods has the assumption that one measurable variable causes the other to change in a predictable fashion?
a) Input-output models
b) Econometric models
c) Simulation models
d) Regression
View Answer
Answer: d
Explanation: Regression is a statistical method to develop a defined analytic relationship between two or more variables. The assumption is such that change in one variable causes change in the other
11. A three period moving average is used in simple moving average. The data of the different periods is given in the table. Find the simple moving average for period 5.
Period Demand
a) 21
b) 20
c) 22
d) 19
View Answer
Answer: a
Explanation: It is a three period moving average, so for calculating the next period’s moving average, average of last three periods is taken. Here, average of period 2, 3 and 4 is taken which is,
\(\frac {26+19+18}{3}\) = 21
12. In weighted moving average, if the weight given to last three periods is 0.5, 0.3 and 0.2. What will be the forecast for period 5?
Period Demand
a) 19.6
b) 19.4
c) 19.9
d) 20.1
View Answer
Answer: c
Explanation: In weighted moving average, weights given are multiplied with maximum value given to the last period. Here, for period 5, forecast = 18(0.5) + 19(0.3) + 26(0.2) = 19.9
13. The last period’s forecast was 70 and demand was 60. What is the simple exponential smoothing forecast with an alpha of 0.4 for the next period?
a) 63.8
b) 65
c) 62
d) 66
View Answer
Answer: d
Explanation: In exponential smoothing method, the formula used to determine the forecast is given by,
F[t] = F[t-1] + α (A[t-1] – F[t-1])
= 0.4 × 60 + (1 – 0.4) × 70
= 24 + 42
= 66
Sanfoundry Global Education & Learning Series – Production Planning and Control.
To practice all areas of Production Planning and Control, here is complete set of Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/production-planning-control-questions-answers-major-categories-forecasting/","timestamp":"2024-11-11T03:24:17Z","content_type":"text/html","content_length":"129969","record_id":"<urn:uuid:f32701d1-148c-43a1-9d53-69e021acc797>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00577.warc.gz"} |
The Mystery of Recursion - Understanding How to Implement it in Python
What is Recursion?
If you understand the underlying concept of recursion but struggle to implement it like myself, or you don't even know what it is, then this post if for you.
Recursion is defined as solving a complex problem by breaking the problem into smaller versions of itself that can be solved. In programming, this is done by having a function call itself.
The underlying principle of recursion are these two cases:
1. Base Case: this case defines the smallest instance of a problem. In certain recursive solutions, there can be multiple base cases but for the purpose of this post's explanation, my example will
only incorporate one. When coding, the base case prevents an infinite loop of the function calling itself.
2. Recursive Case: this case manipulates the problem in order to approach the base case. In programming, this is the case in which the function calls itself.
Put simply, the recursive case will breakdown the problem until it arrives at the base case, after which the the sub solutions build up to solve the original problem.
Do not worry if your understanding of recursion is hazy because the above definitions will make sense when implementing a recursive solution in Python.
The Call Stack
In order to truly understand how recursion is implemented in Python, it is necessary to familiarize yourself with the call stack.
The call stack uses the stack data structure to keep track of local variables and previous function calls. The stack is a last-in, first-out data structure. This means that the last item to be placed
in the stack will be the first to be removed. To help visualize, think of a stack of plates:
Whatever plate is placed last on the stack is the first one to be removed.
In terms of programming, the stack has two simple operations: push and pop. Push adds an item onto the stack and pop removes an item from the stack, returning its value.
In python, the call stack consists of frames. The bottom most frame is the global or module frame. This consists of all the global variables. Every time a function is called a new frame is pushed
onto the stack containing its local variables and function arguments. When a function call returns, the frame is popped off the stack and its value is passed to where it was called in the previous
A Famous Recursion Example - Factorial
The factorial of a number is defined as so:
4! = 4 * 3 * 2 * 1 = 24
If we were to define the factorial of a number n as a mathematical equation, it would be:
factorial(n) = n * factorial(n-1)
The right side of the equation will be the recursive case of our solution. Notice, how the recursive case takes the original problem,factorial(n), and makes it a simpler problem of multiplying n by
whatever factorial(n-1) is.
We can define a function factorial that takes a parameter n. However, if we only defined this function with the above equation, it would call itself infinitely leading to an error.
This is because we did not define a base case. The most basic factorial is the factorial of 1 which equals 1. So we can tell the program that once it reaches the factorial of 1, instead of calling 1
* factorial(0), to just return 1. After which, the Python call stack will handle multiplying out the numbers.
After defining both the recursive and base cases, we arrive at the Python program below:
def factorial(n):
if n <= 1:
return 1
return n * factorial(n-1)
In Depth Recursive Function Implementation
Before calling the factorial function, the Python call stack would look like:
Then, we call the factorial function passing in an argument of 4.
A frame for the factorial function would be pushed onto to the call stack, setting the argument n equal to 4.
The Python interpreter will then process the function and since n is not <= 1, it will evaluate n * factorial(n-1). However before the function can return a value, it must evaluate factorial(n-1).
This will cause another factorial frame to be pushed on the stack, with the argument n set to 3.
Again, since n is not <= 1, the function will evaluate n * factorial(n-1). Likewise, the function must evaluate factorial(n-1) before returning a value.
This pattern will continue until the factorial frame with the argument n equal to 1 is pushed onto the stack.
When this occurs, the function reaches the base case, so it instead returns 1. As I mentioned early when explaining the call stack, when a function returns, it pops its frame off the stack,
transferring its value to where it was called in the previous frame.
Now since the value for factorial(n-1) has been evaluated, the factorial function can evaluate the product and return it to the previous frame, pushing the current frame off the stack.
This pattern continues until we reach the original function call...
As you might have guessed, the factorial function with argument 4 returns a value of 24.
When writing a recursive solution in Python, think of the problem in terms of a base case and recursive case. Ask yourself: "How can I define a recursive case that will help me reduce the problem
towards a base case?"
When trying to understand a recursive solution, I find it best to visualize the call stack in my head. Every time a function is called an item is added onto the stack and any time you encounter the
return keyword, it pops an item off the stack and returns its value to the previous function call.
If you have made it this far and have enjoyed my post, please follow me on Hashnode and Twitter. I plan on posting more content and am open to suggestions or requests! | {"url":"https://tmonty.hashnode.dev/the-mystery-of-recursion-understanding-how-to-implement-it-in-python","timestamp":"2024-11-13T09:09:25Z","content_type":"text/html","content_length":"147960","record_id":"<urn:uuid:d37a95bf-db08-4c72-85af-3ca40f1627c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00476.warc.gz"} |
Structural Optimization by Quantum Monte Carlo: Investigating the Low-Lying Excited States of Ethylene. - PDF Download Free
Article pubs.acs.org/JCTC
Structural Optimization by Quantum Monte Carlo: Investigating the Low-Lying Excited States of Ethylene Matteo Barborini,† Sandro Sorella,‡ and Leonardo Guidoni*,† †
Dipartimento di Chimica, Ingegneria Chimica e Materiali, Università degli studi dell’Aquila, Località Campo di Pile, 67100 L’Aquila, Italy Scuola Internazionale Superiore di Studi Avanzati (SISSA)
and Democritos National Simulation Center, Istituto Officina dei Materiali del CNR, via Bonomea 265, 34136 Trieste, Italy
ABSTRACT: We present full structural optimizations of the ground state and of the low lying triplet state of the ethylene molecule by means of Quantum Monte Carlo methods. Using the efficient
structural optimization method based on renormalization techniques and on adjoint differentiation algorithms recently proposed [Sorella, S.; Capriotti, L. J. Chem. Phys. 2010, 133, 234111], we
present the variational convergence of both wave function parameters and atomic positions. All of the calculations were done using an accurate and compact wave function based on Pauling’s resonating
valence bond representation: the Jastrow Antisymmetrized Geminal Power (JAGP). All structural and wave function parameters are optimized, including coefficients and exponents of the Gaussian
primitives of the AGP and the Jastrow atomic orbitals. Bond lengths and bond angles are calculated with a statistical error of about 0.1% and are in good agreement with the available experimental
data. The Variational and Diffusion Monte Carlo calculations estimate vertical and adiabatic excitation energies in the ranges 4.623(10)− 4.688(5) eV and 3.001(5)−3.091(5) eV, respectively. The
adiabatic gap, which is in line with other correlated quantum chemistry methods, is slightly higher than the value estimated by recent photodissociation experiments. Our results demonstrate how
Quantum Monte Carlo calculations have become a promising and computationally affordable tool for the structural optimization of correlated molecular systems.
1. INTRODUCTION Quantum Monte Carlo (QMC) methods have been successfully applied to tackle the electronic structure of molecules and solids where electron correlation plays an important role.1,2 In
the past decade, a rapid improvement of algorithms opened the door to the QMC study of the ground state properties of several highly correlated systems in chemistry and physics, such as transition
metal complexes,3,4 graphene,5 hydrogen bonding systems,6−8 dispersive interactions,9 and high pressure hydrogen.10 The success of these stochastic methods is due to their capability to correctly
describe the electronic states of the system and, at the same time, due to the possibility of their handling a relatively large number of atoms. The good scaling properties with the number of
electrons together with the excellent portability of the algorithms on high performance parallel computers make these methods particularly promising for quantum chemistry calculations of large and
correlated systems. The calculations proposed in the present work are based on highly compact and correlated variational wave functions inspired by Pauling’s resonating valence bond (RVB)
representation of the chemical bond: the Jastrow Antisymmetrized Geminal Power (JAGP), as introduced in refs 11 and 12. This resonating valence bond ansatz is able to correctly describe the statical
and dynamical correlation of a large variety of molecular systems.3,5,6,9,13 One historical difficulty with QMC methods is efficiently evaluating forces, due to problems arising from the stochastic
noise. Several solutions have been proposed, like those based on the zero-variance principle,14 on correlated sampling,15 and on the introduction of particular coordinate transformations.16 However,
all of these approaches have the inconvenience of © 2012 American Chemical Society
becoming unfeasible for large systems, due to the fact that the number of coordinate derivatives increases linearly with the system’s size. Recently, an efficient scheme to compute forces with
quantum Monte Carlo has been proposed by means of the so-called adjoint algorithmic differentiation,17 through which full structural optimization of systems of few atoms can be performed with small
computational overhead. In the present paper, we apply this technique to study the ground and excited state properties of ethylene. Besides its important role in industry and medicine, ethylene has
been the subject of many experimental and high level computational works since it represents the prototype for a double carbon bond in organic molecules.18 In addition, its singlet to triplet
interconversion is a model for the photochemistry of larger conjugated organic molecules and polymers. Since the early investigations of the properties of ethylene’s 1 Ag ground state (N), the first
singlet excitation (V), the vertical triplet excitation (T), and Rydberg states, identified by Mulliken and Wilkinson,18,19 were an interesting and quite challenging subject because of the
significant change in the geometrical structure induced by the electronic excitations. While the dominant line in ethylene’s spectrum has been recognized as the V(1B1u)←N(1Ag) singlet excitation with
an estimated energy of about ∼7.6 eV,19,20 many experimental studies, through optical21 and low-energy electron impact spectroscopies,22−25 have assigned to the vertical triplet excitation T(3B1u)←N
(1Ag) an energy gap between 4.32 and 4.70 eV. A lower bound of the gap equal to 4.3 eV has been also Received: October 13, 2011 Published: February 14, 2012 1260
dx.doi.org/10.1021/ct200724q | J. Chem. Theory Comput. 2012, 8, 1260−1269
Journal of Chemical Theory and Computation
reported by ion impact spectroscopy.26 The adiabatic vertical excitation, namely, the energy gap between the equilibrium ground state and the triplet state at its equilibrium geometry T(3A1)←N(1Ag),
has been recently identified during the dissociation dynamics of ethylene sulfide SC2H3 by Qi et al. Through the time-of-flight (TOF) spectra of photofragments from photodissociation, they were able
to produce ethylene’s T state near its equilibrium geometry 3A1, estimating an energy gap T ←N of about 2.52(13) eV.27 Many computational efforts have been spent to identify the N, V, and T states of
ethylene and to evaluate the vertical excitations between the singlet (1Ag) and the low lying vertical (3B1u) and adiabatic (3A1) triplets.28−34 In the present work, we use variational Monte Carlo
methods to fully optimize the wave functions and the molecular geometries of ethylene in both the singlet and triplet states. The vertical and adiabatic triplet excitation energies will be estimated
using the variational Monte Carlo (VMC) and lattice regularized diffusion Monte Carlo (LRDMC)35,36 methods, comparing the obtained results with the available experimental data and with other quantum
chemistry calculations.
to the lowest energy eigenvalue given the nodal surface determined by ΨT(x). ̅In particular, the lattice regularized diffusion Monte Carlo35,36 method can offer two advantages with respect to the
traditional diffusion Monte Carlo (DMC) algorithms. First of all, it is size-consistent, so that it maintains its efficiency during the correlated metropolis sampling even for systems with a large
number of electrons.36 The second important advantage is that the LRDMC method preserves the variational principle even when used in combination with nonlocal pseudopotentials.36 Both projection
methods introduce a systematic error either by the discretization τ of the imaginary time propagator (DMC) or by the spatial discretization of the molecular Hamiltonian on a lattice grid of step a
(LRDMC). These errors are overcome by extrapolating the estimated energy, for different a or τ steps, to the continuum limit a, τ→0.
2. QUANTUM MONTE CARLO METHODS Variational Monte Carlo methods are based on the stochastic evaluation of the energy functional E[ΨT({α̅ , R̅ })] =
The structural optimization of molecular systems within the VMC scheme corresponds to minimizing expression 3 with respect to both sets of parameters {α̅,R̅ }, the parameters of the electronic wave
function, and the coordinates of the nuclei:
∫ d r ̅Ψ*T(x̅ ; {α̅ , R̅ }) Ĥ ΨT (x̅ ; {α̅ , R̅ }) ∫ d r ̅|ΨT (x̅ ; {α̅ , R̅ })|2
OPT = min E[R ; Ψ ({α , R })] EVMC ̅T ̅̅{α, R }
with respect to a trial wave function ΨT(x ;{α ̅̅,R̅ }) = ⟨x|Ψ ({α ,R } )⟩, where x is the 6N-dimensional vector of the ̅̅T ̅̅electronic Cartesian r ̅= {ri, i = 1, ..., N} and spin coordinates, α̅ is a set of
independent wave function parameters, R̅ = {Ra, a = 1, ..., M} is the vector of the nuclear coordinates, and Ĥ is the molecular Hamiltonian. In order to stochastically evaluate the functional eq 1, it
is convenient to rewrite the integrand as the product of two local functions: E[ΨT({α̅ , R̅ })] =
∫ d r̅ EL(x̅) Π(x̅)
where we have explicitly written the dependency of the energy functional E from R̅ through the Hamiltonian Ĥ (R̅ ). The optimization requires the evaluation of the force vectors acting along all of the
coordinates R̅ of the nuclei, defined as Fa(R̅ ) = −∇R a EVMC({R̅ ; α̅(R̅ )})
where α̅ (R̅ ) implicitly depends on R̅ since the minimum energy condition (eq 3) has to be satisfied at fixed R̅ . In general, the calculation of forces in QMC is done through two approaches. A first
method is the finite difference approach for which the derivatives of the energy functional with respect to the atomic displacements are defined through a space discretization, leading to the
the local energy EL(x)̅ = (⟨x|H ̅̂|ΨT({α̅ ,R̅ })⟩)/(⟨x|Ψ ̅T({α̅,R̅ })⟩), i.e., the energy of a single electronic configuration x,̅ and 2 2 Π(x)̅ = (|ΨT(x;{α ̅̅,R̅ })| )/(∫ dr|ΨT(x;{α ̅̅,R̅ })| ), which is the
probability to visit that particular configuration in space. The integral written in the form of eq 2 can now be evaluated as the mean value of the local energy, E ∼ ⟨EL(x)⟩ ̅Π(x)̅ , calculated on a
number 5 of electronic configurations x,̅ sampled with probability Π(x). will be ̅The error associated with this2 estimation 1/2 equal to ((1/5 )(⟨EL2(x)⟩ ̅Π(x)̅ − ⟨EL(x)⟩ ̅Π(x)̅ )) , decreasing as the
square root of the number of samples of the configuration space independently of the dimension of the system. Expression 1 can be minimized with respect to the set of parameters α̅ obtaining EVMC =
min E[ΨT({α̅ , R̅ })] α
EVMC({R̅ ′; α̅(R̅ ′)}) − EVMC({R̅ ; α̅(R̅ )}) ΔR a ΔR a → 0
Fa(R̅ ) = − lim
where R̅ ′ = R̅ + ΔRa is the new geometrical configuration after the displacement ΔRa of the ath nucleus, and EVMC({R̅ ′;α̅(R̅ ′)}) is the variational energy (eq 3) evaluated in the new positions R̅ ′ with
the optimized wave function Ψ′T({α̅′,R̅ ′}). Unfortunately, in QMC, the values of the energy for the two structural geometries are affected by a stochastic error that propagates in the calculation of
forces, increasing when ΔRa → 0. For this reason, the finite difference approach is usually carried out using the correlated sampling technique. This technique reduces the stochastic error when
evaluating energy differences by estimating both of the variational energies in eq 6 using the same Monte Carlo random walk.15
According to the variational principle, EVMC represents the lowest upper bound of the ground state energy E0 for a given variational wave function. The α̅ set of parameters of the many-body wave
function can be optimized using stochastic evaluation of the energy derivatives recently developed.9,37,38 The QMC methods also include a variety of projection methods such as diffusion2 and Green’s
function39 Monte Carlo that can go beyond the variational ansantz having direct access 1261
dx.doi.org/10.1021/ct200724q | J. Chem. Theory Comput. 2012, 8, 1260−1269
Journal of Chemical Theory and Computation
followed by the translation of the electronic positions around the nuclei, through the equations:
The second approach for evaluating forces is to directly estimate the analytic derivative of eq 5 that gives the expression Fa(R̅ ) = − −
⎧ r′i = ri + ΔR aωa(ri) ⎪ ⎪ ⎨ F(ria) ⎪ ωa(ri) = M ⎪ ∑b = 1 F(rib) ⎩
∂ EVMC({R̅ ; α̅(R̅ )}) ∂R a d α̅(R̅ ) ∂ EVMC({R̅ ; α̅(R̅ )}) × ∂α̅(R̅ ) dRa
where F(ria) should be a function that decays rapidly; in this case, it is taken to be 1/ria4 as proposed in ref 15, with ria = |ri − Ra|. Within the SWCT, the energy functional (eq 1), after the
displacement of a single nuclei ΔR̅ a, assumes the form
The second term in this definition can be neglected for the Euler conditions (∂EVMC/∂α̅(R̅ ) = 0) at the energy minimum (eq 3). For this reason, we are left with the equation Fa(R̅ ) = −∂EVMC/∂Ra, which
following the notation introduced in section 2 can be rewritten as the sum of two contributions:40,41 Fa(R̅ ) = −
dEL(x̅ ) dRa
∫ d r ̅JΔR a ( r ̅)|ΨT(x̅ ′)|2 ELΔR a(x̅ ′) E[ΨT({R̅ ′})] = ∫ d r ̅JΔR a ( r ̅)|ΨT(x̅ ′)|2
where JΔRa(r)̅ is the Jacobian of the transformation (eq 9) and both the local energy and the wave function depend on the new nuclear displacement both directly and through the transformed electronic
coordinates x′.̅ Equation 10 can be easily differentiated with respect to ΔRa in the limit of ΔRa →0, leaving us with the differential expression of the force acting on the ath nucleus:
Π(x̅ )
⎧ ⎪ d ln[ΨT(x̅ )] + 2⎨⟨EL(x̅ )⟩Π(x ) ̅⎪ dRa ⎩ Π(x̅ ) ⎫ ⎪ d ln[ΨT(x̅ )] ⎬ − EL(x̅ ) ⎪ dRa Π(x̅ ) ⎭ = FaH−F(R̅ ) + FaP(R̅ )
Fa = − (8)
dEL(x̅ ) dRa
Π(x̅ )
⎧ ⎪ + 2⎨⟨EL(x̅ )⟩Π(x ) ̅⎪ ⎩
that are respectively the Hellmann−Feynman (H−F) FaH−F(R̅ ) and Pulay (P) FaP(R̅ ) terms. The Pulay term is exactly zero in two cases: when working with an exact eigenstate in the limit of complete
basis sets and if the trial wave function is expanded into an originless basis set such as plane waves. In our case, the Pulay term cannot be neglected as discussed in ref 40. Our approach to the
evaluation of the analytic eq 8 in the VMC frame is characterized by the introduction of three ingredients. The first of these ingredients is the space warp coordinate transformation (SWCT), that
together with the second ingredient of our procedure, which is the reweighting method defined in ref 41, is able to reduce the variance of the forces.17 When using pseudopotentials in the SWCT
scheme, the analytic calculation of derivatives becomes prohibitive. To overcome this drawback, we used the adjoint algorithm differentiation (AAD), obtaining overall a computational cost that does
not grow linearly with the system size, at variance with the methods based on the numerical derivative. In the next paragraphs, we will briefly describe these three techniques that lead to a new
expression of the force components, as described by Sorella and Capriotti in ref 17. Space Warp Coordinate Transformation. To derive our convenient analytic expression of the force components, we
used the Space Warp Coordinate Transformation first introduced to calculate atomic forces within the VMC and DMC methods within the finite difference approach.14−16,42 In ref 17 it is shown that the
introduction of this transformation in the definition of the energy functional, used with the reweighting method, has the advantage of reducing the variance and also treating nonlocal
pseudopotentials. Within this transformation scheme, each ionic displacement ΔRa is
d ln[J1/2 ( r ̅) ΨT(x̅ )] dRa
d ln[J1/2 ( r ̅) ΨT(x̅ )] EL(x̅ ) dRa
Π(x̅ )
⎫ ⎪ ⎬ ⎪ Π(x̅ ) ⎭
This analytic expression is the sum of the two different contributions previously introduced, the Hellmann−Feynman term, which is simply the mean value of the derivative of the local energy, and the
Pulay term, which is fundamental for accurate force evaluations, as demonstrated in ref 40. Taking into account the fact that the electronic coordinates also depend on the nuclear displacement within
the SWCT, the total derivatives in eq 11 can be written in terms of partial derivatives of the local energy and of the wave function logarithm: d EL(x̅ ) = ∂ R aEL(x̅ ) + dRa
∑ ωa(ri) ∂riEL(x̅) i=1
d ln [J1/2 ( r ̅) ΨT(x̅ )] dRa N
= ∂ R a ln [ΨT(x̅ )] + +
⎤ 1 ∂riωa(ri)⎥ ⎦ 2
∑ ⎢⎣ωa(ri) ∂ri ln[ψT(x̅)]
The force components 11 with the expansions 12 and 13 are the analytic local expressions that we have to sample within our VMC schemes. Although these expressions present an elegant form, they still
have an unbounded variance, as described in the 1262
dx.doi.org/10.1021/ct200724q | J. Chem. Theory Comput. 2012, 8, 1260−1269
Journal of Chemical Theory and Computation
next paragraph, so that to obtain a meaningful average some manipulations have to be made. Reweighting Methods. As anticipated, the Hellmann− Feynman term has unbounded variance when the electron−
atom distance vanishes. In our case, this problem is overcome by the fact that the Hellmann−Feynman contribution defined in eq 11 depends on the local energy and not only on the Coulomb potential,
which means that with a trial wave function satisfying the electron−ion cusp conditions (as shown in section 4), it remains f inite even when the electron ion distance ria approaches zero. The H−F
term containing only the first derivative of the local energy diverges at most as 1/ria, and the variance is therefore finite in three dimensions as ∫ dr3 1/ria2 converges. Parallel to the variance
problem appearing when the electron−ion distances approach zero, there exists also a more subtle infinite variance problem when a sample configuration x ̅approaches the nodal surface, i.e. x:Ψ ̅T(x)̅ =
0. Both the H−F and the Pulay terms diverge when a sampled electronic configuration x ̅approaches the nodal surface of the wave function. In this situation, both the local energy and the partial
derivative of the logarithm of the wave function are proportional to the inverse of this distance 1/d, while the density probability Π(x)̅ ≃ d2, leading to an unbound variance of their product in the
Pulay term or in the local energy derivative that appear in the H−F term (≃ 1/d2), although the mean values remain well-defined. This problem was first tackled by Attaccalite and Sorella in ref 41
with the so-called reweighting method in which a new probability distribution is 2 defined Πε(x)̅ = |ΨG(x)| ̅depending on a guiding function ΨG(x̅ ) =
R ε(x̅ ) ΨT(x̅ ) R(x̅ )
FaP =
⎧ ⎪ ⟨S(x̅ ) EL(x̅ )⟩Πε(x̅ ) ×⎨ ⎪ ⟨S(x̅ )⟩Πε(x̅ ) ⎩ −
|A i−, j1|2 )−1/2
To regularize the probability density close to the nodal surface, Rε(x)̅ is defined as
where ε is a positive small number chosen to reduce the number of electron configurations that approach the nodal surface. This renormalization has the advantage of satisfying the continuity of the
derivate of the ΨG(x)̅ when R(x)̅ = ε, ensuring that ΨG stays as close as possible to the trial wave function. In this new scheme, the Hellmann−Feynman and Pulay terms are written as FaH ‑ F= −
dE (x ) 1 S( x ̅) L ̅dRa ⟨S(x̅ )⟩Πε(x ) ̅
d ln[J1/2 ( r ̅) ΨT(x̅ )] dRa
⎫ ⎪ ⎬ ⎪ Πε(x̅ ) ⎭
Πε(x̅ )
N /2
ΨAGP(x̅ ) = Â [ ∏ ΦG(x ↑i ; x ↓i )] i=1
Πε(x̅ )
d ln[J1/2 ( r ̅) ΨT(x̅ )] dRa
4. VARIATIONAL WAVE FUNCTION The trial wave function used in this investigation is the Jastrow antisymmetrized geminal power,11 which is an implementation of Pauling’s valence bond picture.44 This
wave function is built as the product between an antisymmetric geminal power (AGP)45,46 and a Jastrow factor J(r)̅ and includes both static and dynamical electron correlation effects. It has been
demonstrated that the JAGP11−13,41 is a compact and reliable wave function for describing the bonding properties of organic molecular systems like graphene sheets5 and benzene molecules,47 and in
reproducing the weak binding energies in van der Waals interactions9 and hydrogen bonds.6 For molecular systems of N electrons in a spin singlet state, i.e., N/2 = N↑ = N↓, the AGP is written as the
⎧ R(x̅ ) R(x̅ ) ≥ ε ⎪ ⎪ R ε(x̅ ) = ⎨ ⎡ R(x ) ⎤ R(x̅ )/ ε ̅⎪ ε⎢ R(x̅ ) < ε ⎪ ⎣ ε ⎥⎦ ⎩
S(x̅ ) EL(x̅ )
S(x̅ )
2 where the reweighting factor S(x)̅ = (ΨT(x)/Ψ is G(x )) ̅̅2 proportional to d and cancels out the divergence of the integrand, solving the problem of unbounded variance in the VMC scheme. Adjoint
Algorithm Differentiation. Although the reweighting method cures the variance problems of eq 11, we are still left with the computational challenge of evaluating the derivatives of the local energy
and of the logarithms that appear in eqs 12 and 13. The presence of pseudopotentials and SWCT makes the practical implementation of the analytic derivatives extremely complicated. To overcome this
drawback, we efficiently computed the derivative using the third feature of our structural optimization method which is the adjoint algorithmic differentiation procedure.17 The idea beyond the
algorithmic differentiation is that a derivate can always be written using the chain rule as the propagation of the derivatives of simpler functions that are known (polynomials, cosines, sines, etc).
Following the chain rule, intermediate results can be stored in memory and used to calculate other derivatives that share the same intermediate values. These procedure can be applied either in a
backward or forward sweep, as described in ref 43. A clear example on how the calculation of the kinetic energy derivatives is done through the AAD can be found in ref 17. In summary, the inclusion
of AAD is a very convenient way to deal with analytical derivatives using pseudopotentials and SWCT. The computational overload for calculating forces in the proposed scheme does not have any linear
dependence on the system size and allows us to optimize wave functions and geometries of large molecular systems.
where the function R(x)̅ is a measure of the distance between the electronic configuration x ̅and the nodal surface and therefore is assumed to vanish proportionally to ΨT(x)̅ as d → 0. As the nodes of
the wave function only depend on its determinantal part, schematized here by a single Slater determinant A, we can assume that the R(x)̅ depends only on A:41 R(x̅ ) = (
2 ⟨S(x̅ )⟩Πε(x )
where x ̅is the whole set of Cartesian and spin coordinates of the N electrons and  is the antisymmetrization operator.
(17) 1263
dx.doi.org/10.1021/ct200724q | J. Chem. Theory Comput. 2012, 8, 1260−1269
Journal of Chemical Theory and Computation
The second term is the purely homogeneous two-body factor
The two electron wave functions that appear in the definition 19 are the Geminal pairing functions 1 ΦG(x i ; x j) = ϕG(ri, rj) (|↑⟩i |↓⟩j −|↑⟩j |↓⟩ i) 2
J2 ( r ̅) = exp{
a , b = 1 μ, ν
J3/4 ( r ̅) = exp{
ΨGAGP(x̅ ) = Â [∏ ΦG(x ↑i ; x ↓i ) i=1
ϖ(ri, rj) =
5. COMPUTATIONAL DETAILS The computational investigation of our systems has been carried on using the TurboRVB package developed by Sorella52 that includes a complete suite of variational and
diffusion quantum Monte Carlo programs for wave function and geometry optimization of molecules and solids. All-Electron CalculationsJAGP. As starting points for the construction of the all-electron
basis sets of the AGP wave functions, we used the cc-pVDZ and cc-pVTZ basis sets.53,54 We considered only s, p, and d shells for the carbon atoms and the s and p shells for hydrogen atoms. We
included only the terms of the contractions with smaller exponents, since the electron−nuclei cusps are already described through the Jastrow factor. Finally, we obtained the following contracted
basis sets: 8s4p1d composed of (8s4p1d)/[3s2p1d] orbitals for the carbon atom and (4s1p)/[2s1p] contracted orbitals for the hydrogen atoms and 9s5p2d built of (9s5p2d)/[4s3p2d] orbitals for the
carbon atom and (5s2p)/[3s2p] orbitals for the hydrogen atoms. For the Jastrow factor J3/4, we used an uncontracted Gaussian basis set of (2s2p) orbitals and (1s1p) for carbon and hydrogen atoms,
respectively. The wave function was optimized through different steps using the linear method described in refs 9 and 37. As a first step, we optimized only the λ coefficients of the JAGP and the J2
Jastrow factor. In the second step, we optimized only the J1 and J3/4 terms, keeping all other parameters fixed. Finally, we fully relaxed all of the parameters of the wave function including the
exponents and coefficients of the atomic orbital basis sets. During the optimization procedure, we used an increasing statistical accuracy for each step, ranging from 6.4 × 103 to 3.2 × 105 Monte
Carlo (MC) steps per electron. VMC and LRDMC calculations at the energy minimum were carried on using 5.1 × 107 and 1.3 × 107 MC steps per electron, respectively. LRDMC energies were extrapolated in
the limit a→0 using the following set of lattice space discretizations: a = {0.05, 0.10, 0.15, 0.2} [au].
One fundamental ingredient of the JAGP is the Jastrow factor13 that includes a homogeneous interaction that treats the electron−electron and electron−nucleus cusps conditions51 and a non homogeneous
contribution that introduces dynamical correlations between electrons and nuclei. In our representation, the Jastrow factor is written as the product of three terms, J = J1J2J3/4. The first term is
the one-body Jastrow factor: J1(R̅ , r ̅) = exp(∑ {− (2Za)3/4 ξ((2Za)1/4 ria) a,i
+ Ξa(ri)})
The contributions with a = b represent the three-body terms that consider correlations between electrons occupying shells of the same atom, whereas terms with a ≠ b are four-body terms that consider
the coupling between orbitals of different atomic centers, crucial for the correct description of dispersive interactions.6
s = 1, ..., S
a=1 μ
∑ ∑ gμaνb χμa(ri) χνb(rj)
a , b = 1 μaνb
∑ ∑ λμs a ψμa(ri)|↑⟩
that along with N↓ geminal functions 21, contains S = (N↑ − N↓) single electron wave functions: Φs(x i) =
ϖ(ri, rj)}
∏ Φs(x ↑N ↓+ s)]
∑ i 0, like the two triplet states 3B1u and 3A1 of ethylene, it is necessary to apply the generalized antisymmetric geminal power wave function (GAGP), introduced by Coleman in 1965.49,50 Assuming
that the number of spin-up electrons exceeds the number of electrons with opposite spin N↑ > N↓, the GAGP is built as the antisymmetric product: N↓
that depends only on the distances rij = |ri − rj| between electron pairs, treating the electron−electron cusp conditions of the JAGP wave function through the function ξ(rij) = b/2(1 − e−rij/b). The
last term is the three/four-body Jastrow J3/4 that includes the electron−electron−nuclei correlation
∑ ∑ λμaνb ψμa(ri) ψνb(rj) | {"url":"https://d.docksci.com/structural-optimization-by-quantum-monte-carlo-investigating-the-low-lying-excit_5af5681fd64ab212e61af9c5.html","timestamp":"2024-11-05T12:45:22Z","content_type":"text/html","content_length":"75297","record_id":"<urn:uuid:6ecd86f3-da48-43d8-ac81-dd563734a040>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00729.warc.gz"} |
LET'S GET COZY! GHOSTAL LIVING: A PREVIEW AND GIVEAWAY
A Hamptons Home & Garden Mystery
In the latest mystery from the author of Better Homes and Corpses and Hearse and Gardens, Hamptons interior designer and antiques picker Meg Barrett uncovers a veil of spooky goings-on…
The first Sag Harbor Antiquarian Book and Ephemera Fair is right around the corner, and interior designer Meg Barrett has her hands full decorating rooms at the Bibliophile Bed & Breakfast for
wealthy rare book collector Franklin Hollingsworth. Rumor has it Hollingsworth is in possession of an unpublished manuscript written by F. Scott Fitzgerald. When the Fitzgerald manuscript’s
authenticator is found dead at the bottom of a cliff, Meg suspects a killer is on the loose.
Rare books start disappearing from the B & B and Meg sees a connection between the stolen books and the deceased authenticator. With the fair looming, she finds herself caught up in catching a killer
and thief before another victim is booked for death.
Kathleen Bridge, author of Hearse and Gardens and Better Homes and Corpses, started her writing career working at the Michigan State University News in East Lansing, Michigan. She is the author and
photographer of an antiques reference guide, Lithographed Paper Toys, Books, and Games. She is a member of Sisters in Crime and Mystery Writers of America, and has taught creative writing classes at
Bryant Library in Roslyn, New York. Kathleen is also an antiques and vintage dealer in Long Island, New York, and has contributed to Country Living magazine.
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The setting with the book and ephemera sale sounds great! I am looking forward to it!Thanks for the chance to win!
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Crystallization statistics, thermal history and glass formation
The formal theory of transformation kinetics describes the volume fraction of a phase transformed in a given time at a given temperature. The basic concepts are extended for isotropic crystal growth
in a material having a known thermal history T(r, t). A crystal distribution function ψ(r, t, R) is defined such that the number of crystallites in a volume dυ at r having radii between R and R + dR
at time t is ψ(r, t, R) dυ dR. The function ψ contains essentially complete statistical information about the state of crystallinity of a material. Formal expressions for ψ are obtained. Applications
are discussed, including predictions of crystallinity when T(r, t) is known; predictions of glass-forming tendencies; experimental determination of nucleation rates; and the determination of the
thermal history of a sample from post mortem crystallinity measurements. As an example, ψ(r, t, R) is calculated for a lunar glass composition subjected to a typical laboratory heat treatment.
ASJC Scopus subject areas
• Electronic, Optical and Magnetic Materials
• Ceramics and Composites
• Condensed Matter Physics
• Materials Chemistry
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Adaptive Quadrature - (Programming for Mathematical Applications) - Vocab, Definition, Explanations | Fiveable
Adaptive Quadrature
from class:
Programming for Mathematical Applications
Adaptive quadrature is a numerical integration technique that dynamically adjusts the number and placement of sample points to achieve a desired accuracy. This method is particularly useful for
integrals where the function being integrated has variable behavior, allowing for more efficient computations by focusing on areas where the function changes rapidly.
congrats on reading the definition of Adaptive Quadrature. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Adaptive quadrature algorithms assess the error of their estimates and refine their computations accordingly, often leading to fewer function evaluations compared to fixed methods.
2. This technique is particularly effective for integrands with singularities or discontinuities, where traditional methods may struggle.
3. Many adaptive quadrature methods, such as adaptive Simpson's rule, use recursive strategies to subdivide intervals until a specific accuracy level is reached.
4. The efficiency of adaptive quadrature can lead to significant reductions in computational time and resources, especially for complex functions.
5. Some implementations of adaptive quadrature also incorporate heuristics or rules of thumb to better determine where to refine the sampling process.
Review Questions
• How does adaptive quadrature improve upon traditional numerical integration methods?
□ Adaptive quadrature improves upon traditional methods by dynamically adjusting the number of sample points based on the behavior of the integrand. Instead of using a fixed number of intervals
or points, it focuses computational effort on areas where the function exhibits rapid changes, leading to more accurate results with potentially fewer calculations. This adaptability makes it
especially useful for functions that may have singularities or varying degrees of smoothness.
• Discuss how adaptive quadrature handles integrands with singularities or discontinuities compared to other methods.
□ Adaptive quadrature is specifically designed to handle integrands with singularities or discontinuities by concentrating more sampling in problematic regions. While other methods, like
fixed-point numerical integration techniques, may struggle or yield inaccurate results due to these irregularities, adaptive quadrature evaluates the error in its estimates and refines its
approach accordingly. This targeted refinement allows it to achieve accurate results even when dealing with challenging functions.
• Evaluate the role of error estimation in adaptive quadrature and its impact on efficiency and accuracy.
□ Error estimation is crucial in adaptive quadrature because it directly informs how the algorithm modifies its sampling strategy. By assessing whether the current approximation meets the
desired accuracy, the method can decide whether to subdivide intervals further or halt computations. This targeted approach not only enhances accuracy by concentrating efforts where needed
but also improves efficiency by avoiding unnecessary calculations in smoother regions, making adaptive quadrature a powerful tool in numerical analysis.
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Numerical STM speed up options
Hi folks,
Following discussions with @dgondelach, I’ve opened a couple of issues (1462 and 1463) regarding options for state transition matrix performance, starting with drag and geopotential. The idea is to
use a simpler model for the derivatives, e.g. a less complex atmospheric model as some are very expensive with automatic differentiation.
Hi Romain,
Could you provide a discussion of the limitations of that approach? Perhaps a reference that discusses the impacts on accuracy and convergence?
Rice tried the “hybrid” approach (different models for state and STM) in 1967 in [1] and concluded that it causes “rapid divergence”. It’s a good read. Based on his work, I wouldn’t recommend
including the hybrid approach in a library like Orekit. But perhaps in the last 50+ years someone else has figured out how to address the divergence issues.
[1] https://arc.aiaa.org/doi/10.2514/6.1967-123
It’s a good idea, but we need to be very careful with current orekit users.
If it’s just an option and the default behavior (i.e. calculation) remains as it is now, I think it’s acceptable to introduce this feature for users who favor calculation time over accuracy.
On the other hand, current users shouldn’t have to do anything - nothing should change for them. If an option is available, the action to be taken (i.e. a method to be called or other) must be taken
by the users wishing to limit performance and increase computation speed.
Best regards,
I agree with Evan here.
From my experience, simplifying the force models when computing derivatives has a bad influence on convergence of orbit determination. It may help at start, while we are still far from the solution
and just want to come closer. But when we are near convergence and globally derivatives are close to zero (because we are looking for the minimum of a cost function), it seems to me that having
simplified derivatives just either makes the algorithm lose its path as the Jacobian is inconsistent with the evolution of the value or it makes the algorithm converge to the wrong solution.
Hi all,
Thanks for your comments.
First of all, as Bryan said, it would be an option, and certainly not the default one.
As for accuracy, well it’s always a trade-off right? I’m pretty sure I’ve seen the option to cut off the geopotential order for STMs in GMAT, although I can’t get my hand on a proper link at the
moment. As for drag, the proposed approximation is only on the density, which at the moment is with finite differences by default so I wouldn’t say it’s much more accurate. Besides, orbit
determination is not the only place where one needs covariance matrices and/or STMs, and it’s not always in an iterative process requiring convergence. I believe users are not all doing operational
flight dynamics. Anyhow if it doesn’t make consensus I’ll ditch the issues from the 12.2 milestone for now.
Here’s the option I was talking about in GMAT, field StmLimit: | {"url":"https://forum.orekit.org/t/numerical-stm-speed-up-options/3750","timestamp":"2024-11-10T02:03:24Z","content_type":"text/html","content_length":"23096","record_id":"<urn:uuid:aa8c3acd-423f-413b-b3a6-7fa99fa182d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00731.warc.gz"} |
Unit 2 Test | Calc Medic
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Unit 2 Test
Unit 2 - Day 14
Writing a Precalculus Assessment
• Include questions in multiple representations (graphical, analytical, tabular, verbal)
• Write questions that reflect learning targets and require conceptual understanding
• Include multiple choice and short answer or free response questions
• Determine scoring rubric before administering the assessment (see below)
• Offer opportunities to practice with and without calculators throughout the year
Questions to Include
• Quadratics--identify key features of a quadratic from multiple forms (standard, vertex, and intercept form);
• Quadratics--rewrite the equation of a quadratic from one form into another by completing the square
• Polynomials--identify key features of a polynomial (zeros, x-intercepts, y-intercept, multiplicity of zeros, end behavior, etc.)
• Polynomials--sketch a graph of a polynomial given certain features
• Polynomials--find all solutions
• Polynomials--find remainders by dividing or using the Remainder Theorem
• Rationals--identify key features of a rational function (end behavior, vertical asymptotes, holes, intercepts, etc.)
• Rationals--solve simple rational equation
• Rationals--Real-world context
Grading Tips
Look for more than just correct answers. Give students feedback on their justifications, communication, and mathematical thinking. We recommend that you prepare a rubric for the free response and
short answer items before you begin grading your quizzes or tests. Know what information is necessary for a complete and correct response and award points when a student presents that information.
Many of the “Why did I get marked down?” questions are eliminated when you share the components that earn points.
This is a long unit with many parts, so make sure to have a good representation of questions for the test. Students may get end behavior mixed up for polynomials and rationals, so include at least
one of each so they can show they know when a function is approaching infinity or negative infinity or a horizontal asymptote. If giving the test online, consider including questions that won’t be
answerable via a quick Desmos graph so the students must show what they learned in the unit. This can be accomplished by using the same methods as we did with the lessons For example, leave out a
missing variable when dividing and have them solve for the missing piece (Day 7), give them a table of values and have them find the remaining solutions (Day 9), or have them solve for complex
solutions algebraically (Day 8).
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Probability in Many Worlds: Is the cart in front of the horse?Probability in Many Worlds: Is the cart in front of the horse?
Recent Comments
I know a few regular readers of this blog have views about the many worlds interpretation of quantum mechanics. I want to ask a question about what is supposed to be the response to a basic worry
about a whole family of approaches. (David Wallace's recent book would be an obvious case, but so would Sean Carrol's recent contributions.)
The many worlds interpretation basically says that whenever you make a "meausurement" in QM, (say you have a particle that is spin up in the y direction and you measure spin in the x direction,) the
world contintues to evolve according to the Schroedinger equation, and the only thing that makes it look like the measurement has a determinate outcome is that the world splits into two emergent
worlds, with an emergent observer in each one. The trick of all this, of course, is to somehow explain why there is probability when all of the outcomes are occuring. One problem I have with all
of these attempts to get probability out of is that they all go like this.
1. Assume decoherence gets you branches in some preferred basis.
2. Give an argument that the Born rule applied to the amplitudes of these branches yields something worthy of the name ‘probability.’
The problem is that these steps happen in the reverse order that one would like them to happen.
Look at step one. Decoherence arguments involve steps
1.a) showing that as the system+detector gets entangled with the environment, the reduced density matrix of this entangled pair evolves such that all the off-diagonal elements get very close to zero,
1.b)reasoning that therefore, each diagonal element corresponds to an emergent causally inert “branch.”
But step 1.b is fishy insofar as it happens before step 2. Who cares if the little numbers on the off-diagonals are very close to zero, until I know what their physical interpretation is? Not all
very small numbers in physics can be interpreted as standing in front of unimportant things. Now, if we could accomplish step 2, then we could discard the off-diagonal elements, because we know that
very small _probabilities_ are unimportant. But the cart has been put in front of the horse. We can’t conclude that the “Branches” are real and causally inert and have independent “obsevers” in them
_until_ we have a physical interpretation of the off-diagonal elements being small. But all of these Everettian moves do 1.b first, and only afterwards do 2.
Now its true that the fact that the off-diagonal elements are small tells us that the different branches don’t interfere with each other very much in terms of their future evolution. I.e., I could
evolve a branch forward in time, and the result is almost completely independent of the existence of the other branches. But the notions of not very much and almost here are still in terms of small,
but physically uninterpreted numbers.
I think what often drives the intuition that it is ok to interpret the small off-diagonal terms as telling you that the brances are independent is that we understand the off-diaganol terms as the
"interference terms." But I think this is smuggling, still, a probabilistic notion. "Interference" is a probabilistic notion, that we get from, e.g. thinking about "how often" we expect
interference to show up in the statistics.
OK. So, this worry is out there in the literature. What's the response? | {"url":"https://www.newappsblog.com/2014/07/probability-in-many-worlds-is-the-cart-in-front-of-the-horse.html","timestamp":"2024-11-13T21:53:40Z","content_type":"application/xhtml+xml","content_length":"61996","record_id":"<urn:uuid:5006468b-e5c4-4ce4-bf43-0cf2205704e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00161.warc.gz"} |
A nearly tight sum-of-squares lower bound for the planted clique problem
We prove that with high probability over the choice of a random graph G from the Erd\H os-Rényi distribution G(n, 1/2), the n^O(d^)-time degree d sum-of-squares (SOS) semidefinite programming
relaxation for the clique problem will give a value of at least n^1/2 - c(d/ log n^)1/^2 for some constant c > 0. This yields a nearly tight n^1/2 - o^(1) bound on the value of this program for any
degree d = o(log n). Moreover, we introduce a new framework that we call pseudocalibration to construct SOS lower bounds. This framework is inspired by taking a computational analogue of Bayesian
probability theory. It yields a general recipe for constructing good pseudodistributions (i.e., dual certificates for the SOS semidefinite program) and sheds further light on the ways in which this
hierarchy differs from others.
All Science Journal Classification (ASJC) codes
• General Computer Science
• General Mathematics
• Lower bound
• Planted clique
• Sum-of-squares
Dive into the research topics of 'A nearly tight sum-of-squares lower bound for the planted clique problem'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-nearly-tight-sum-of-squares-lower-bound-for-the-planted-clique--2","timestamp":"2024-11-13T16:29:23Z","content_type":"text/html","content_length":"50406","record_id":"<urn:uuid:99e43046-e80f-49e5-9635-84531de2f0b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00356.warc.gz"} |
Jetware - package: Python_html5lib / Appliances
A pre-configured and fully integrated software stack with TensorFlow, an open source software library for machine learning, and the Python programming language. It provides a stable and tested
execution environment for training, inference, or running as an API service. The stack is designed for short and long-running high-performance tasks, and can be easily integrated into continuous
integration and deployment workflows. It is built with the Intel MKL and MKL-DNN libraries and optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is built with the Intel MKL and MKL-DNN libraries and
optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with PyTorch, an open source machine learning library for Python, Jupyter Notebook, a browser-based interactive notebook for
programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on CPU.
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on NVidia GPU.
A pre-configured and fully integrated minimal runtime environment with TensorFlow, an open source software library for machine learning, Keras, an open source neural network library, Jupyter
Notebook, a browser-based interactive notebook for programming, mathematics, and data science, and the Python programming language. The stack is optimized for running on CPU. | {"url":"http://jetware.io/packages/python_html5lib-0.9999999--python-3.6.3/appliances","timestamp":"2024-11-10T21:57:02Z","content_type":"text/html","content_length":"31375","record_id":"<urn:uuid:2e1f8d1f-11d7-4a7b-ad45-5da3dd9f449b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00260.warc.gz"} |
1st Year Physics Notes Chapter 2 - 11th Class Notes pdf
Looking for the notes of Physics class 11 chapter 2 Vectors and Equilibrium. Here we have uploaded the 1st Year Physics Notes Chapter 2 - 11th Class Notes pdf download or read online Topics notes,
Short and long questions and exercise questions.
Q How is vector represented
Vector Representation
A vector is represented in two ways
1. Symbolic representation
2. Graphical representation
Symbolic Representation
It is represented by bold face letter such as A, d,r and V etc It can also be represented by a letter with an arrow placed above or below letter such as or
Graphical Representation
It is represented by a straight line with an arrow head The length of line represents magnitude of vector (according to suitable scale) Arrow head represents the direction of vector Representation
magnitude of vector
The magnitude of vector is represented by Light face letter such as A d rand v
Modulus of a vector such as
Q. What is rectangular coordinate system?
Rectangular Coordinate System
(Cartesian Co-ordinate System) The set of two or three mutually perpendicular lines intersecting at a point is called rectangular system
The lines are called coordinate axes
One of these line is called x-axis (or horizontal axis) The other is called y-axis (or vertical axis) The line perpendicular to both x and y axes is called z-axis The point of intersection is called
Two dimensional coordinate system (Plane)
If the system consists of two perpendicular lines then it is called two dimensional coordinate system
Three dimensional co-ordinate system (Space)
If the system consist of three perpendicular lines, then it is called three dimensional co-ordinate system
Q. How is the direction of vector represented in: (i) a plane (ii) space?
Direction of a Vector in plane
It is represented by the angle which the vector makes with Positive x-axis
In anti-clock wise direction
Direction of a vector in Space
It is represented by three angles which the vector makes with x. y and z axes
Q. Describe the addition of vectors by head to tail rule. Is vector addition commutative?
Head to Tail Tule
It is a graphical method to add of two or more vectors
Q. Explain the following terms:
(i) Resultant vector
(ii) Vector subtraction
(ii) Multiplication of vector by scalar
(iv) Unit vector
(v) Null vector
(vi) Equal vectors
(i) Resultant vector
A vector which has the same effect as the combined effect of all the vectors to be added is called resultant vector
(ii) Vector subtraction
The subtraction of a vector is equivalent to the addition of same vector with its direction reversed
(iii) Multiplication of vector by a scalar
A vector can be multiplied by
• A positive number
• A negative number
(iv) A scalar with dimension Unit vector
A vector whose magnitude is equal to one with no units in a given direction is called unit vector is represented by a letter with a cap or hat on it
(v) Null or Zero
vector A vector whose magnitude is zero and direction arbitrary is called a null vector It si represented by
(vi) Equal vectors
The vectors are said to be equal vectors if they have same magnitude
same direction
(regardless of the position of their initial points)
Q Define component of vector? what are rectangular components of vector
Component of a vector
The effective value of a vector in a given direction is called component of a vector A vector may split up into two or more than two parts these parts are known as components of vector Rectangular
Components of Vector
The components of a vector which are perpendicular to each other are called rectangular components
Q. Define and explain the term torque of moment of force.
Torque Definition
The turning effect of force produced in a body about an axis is called torque.
Other Definition
The product of magnitude force and the perpendicular distance from axis of rotation to line of action of force is called torque.
The moment of a force can also be defined as the vector product of the radius vector from the axis of rotation to the point of application of the force and the force vector
2. Dynamic equilibrium If a body is moving with uniform velocity. It is said to be in dynamic equilibrium
A car moving with uniform linear velocity
Q.. What is equilibrium? Give its types. What are its different kinds? Also write down the conditions of equilibrium
A body is said to be in equilibrium if it is at rest or moving with uniform velocity under the action a number of forces
Types of equilibrium
There are two types of equilibrium
1. Static equilibrium
If a body is at rest It is said to be in static equilibrium Examples
• Book lying on a table
• A body is rotating with uniform angular velocity Motion of a paratrooper
Q. Under what conditions the body is said to be in complete equilibrium?
Translational equilibrium
When first conditions satisfied is linear acceleration of body is zero and the body is said to in translational equilibrium
Rotational equilibrium
When second condition is satisfied, angular acceleration of body is zero and the body is said to in rotational equilibrium
Thus for a body to be in complete equilibrium both conditions must be satisfied ie both in ear acceleration and angular acceleration must be zero
1 We will apply the conditions of equilibrium to situations in which all the forces are coplanar
2 To calculate to torque we choose an axis The position of axis is arbitrary
3 A most suitable place is one through which of action of many forces pass
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Ratio, Proportion - Quant/Math - CAT 2013 :: MBA Preparation
Ratio, Proportion - Quant/Math - CAT 2013
Question 4 the day: August 28, 2002
The question for the day is from the topic of Ratio, Proportion.
A predator is chasing its prey. The predator takes 4 leaps for every 6 leaps of the prey and the predator covers as much distance in 2 leaps as 3 leaps of the prey. Will the predator succeed in
getting its food?
(1) Yes
(2) In the 6th leap
(3) Never
(4) Cannot determine
Correct Answer - (4)
Distance covered in 2 leaps by predator = 3 leaps of the prey.
Distance covered in 1 leap of predator = 3/2 leaps of prey. ----(1)
4 leaps of predator : 6 leaps of prey ----(2)
Using (1) and (2), we get
4*3/2 leaps of predator : 6 leaps of prey.
=> 1:1
If the predator and prey start simultaneously at the same point, the predator will catch the prey immediately. If not so, then the predator will never catch the prey as it was running at the same
As it was not mentioned in the question that they start simultaneously from the same point or not, we can't determine the answer. Therefore, the answer choice is (4).
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On the edge of chaos
How can we describe a complex entity?A complex should be described using statistical methods rather than with a deterministic approach. Then modelling is the answer to our question but there are many
Complexity refers to situations where many simple interacting parts produce a collective, often unexpected, behaviour. The components of the system can self-organize in stable (in a statistical
sense) state and can acquire collective properties which are not necessarily characteristic of each single component. Different parts of the complex behave differently but these different parts are
not indipendent in the system. Therefore there is a basic duality between parts which are at the same time distinct and connected.
The aspect of distinction leads to the concept of variety, heterogeneity, disorder, chaos and entropy; on the other hand connection corresponds constraint, redundancy, order and negentropy.
Complexity lies between these two aspects: neither perfect disorder (which can be described statistically) nor perfect order (which can be described by deterministic methods) .... just “on the edge
of chaos”.
The simple way to model order is through the concept of symmetry; in symmetric patterns one part of the system is sufficient to recontruct the whole. But disorder too is characterized by symmetry!
Not of the actual position of the component but of the probabilities that a component can be found at a particular position.Intuitively complexity can be characterized by a lack of symmetry by the
fact that no part or aspect of a complex system can provide sufficient information to actually or statistically predict the properties of the other parts. These aspects show clearly the difficulties
inmodelling a complex entity.
When did complexity become a science?
Historically and philosophically the concept of complexity is rather recent. The natural philosophers in ancient Greece were focused on the quest of the first principle (arkè) from which everythings
had to descend. Then the scientific thought began its progress to the modern classic physics of Galileo and Newton. Hereafter the fundamental concept was determinism but the developments of natural
sciences in the nineteenth century open new scenarios and deterministic approach loose partly its head role. The new aim of science describing the natural phenomena as stochastic processes and
modelling them.
From pollen’s grains to financial markets
In 1827 the botanist Robert Brown observed the behaviour of pollen’s grains suspanded in a water solution: the grains moved in a chaotic way, following random trajectories. This motion, after just
called Brownian motion, is due to the collisions between the molecules of the fluids and the particles suspanded in it. In 1905 Albert Einstein gave a theoric interpretation of the phenomenon. His
theory was based on the observation of the motion from a microscopic point of view and on the other hand from the macroscopic one. Infact the unknoledge of the initial conditions leads to the
necessity of analyze macroscopically the system in order to model it. In practical terms the microscopic motion of the molecules and theparticles is so complicated that the only way to describe it is
by means of statistical methods.
Einstein’s work is fundamental because represents the first application of stochastic models to natural phenomena. Not long after another famous scientist obtained the same results with a simplest
method: Langevin proposed to resolve a differential equation, the Langevin equation, whose solution is a random variable. The equation describes the temporal evolution of the so called Markovian
stochastic processes and represents the basis of several modern models applied in very different disciplines. An example is given by turbulent dispersion models for tracers (pollutants) in the
atmosphere which simulate the effects of turbulent eddies considering the particle fluid velocity as a stochastic variable described by a proper Langevin equation.
Another important contribution to this type of studies came from Louis Bachelier . In 1901, some years before Einstein, he analyzed the temporal variations of the prices of state bonds and found that
they had a behaviour similar to that of the grains of Brown. E proposed the first macroscopic model for financial markets and defined a stochastic process, better known as the Wiener model, directly
connected to the diffusion equation. It was born econophysics, another of the several diciplines that come into the vaste area of complexity. | {"url":"https://sistemicomplessi.dista.unipmn.it/complexity.html","timestamp":"2024-11-03T04:00:23Z","content_type":"text/html","content_length":"13634","record_id":"<urn:uuid:8143ded8-b966-4747-b5e9-aa99d7e2f869>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00198.warc.gz"} |
Eva Silverstein | Horizon Physics: Cosmology, Black Holes, and String Theory - 1 of 2 | Stanford Institute for Theoretical Physics
Main content start
Eva Silverstein | Horizon Physics: Cosmology, Black Holes, and String Theory - 1 of 2
Sun May 7th 2017, 5:00pm
Professor Eva Silverstein of the Stanford Institute for Theoretical Physics (SITP) discusses the physics of horizons, black holes, and string theory.
Black hole and cosmological horizons -- from which nothing can escape according to classical gravity -- play a crucial role in physics. They are central to our understanding of the origin of
structure in the universe, but also lead to fascinating and persistent theoretical puzzles. They have become accessible observationally to a remarkable degree, albeit indirectly. These lectures will
start by introducing horizons and how they arise in classical gravity (Einstein's general relativity). In the early universe, the uncertainty principle of quantum mechanics in the presence of a
horizon introduced by accelerated expansion (inflation) leads to a beautifully simple, and empirically tested, theory of the origin of structure. Its effects reach us in tiny fluctuations in the
background radiation we observe from the time when atoms first formed.
This theory, and the observations, are sensitive to very high energy physics, including effects expected from a quantum theory of gravity such as string theory. Modeling the early universe within
that framework helps us better understand the inflationary process and its observational signatures. Analyzing the `big data' from the early universe -- which continues to pour in -- is a major
effort. This provides concrete tests of theoretical models of degrees of freedom and interactions happening almost 14 billion years ago.
Our understanding breaks down if we push further back in time, or into black hole horizons. This challenges us to determine more precisely how and why our existing theories fail. I will explain these
basic puzzles, and conclude with some of the latest results on this question in string theory, which exhibits interesting new effects near black hole horizons. | {"url":"https://sitp.stanford.edu/events/eva-silverstein-horizon-physics-cosmology-black-holes-and-string-theory-1-2","timestamp":"2024-11-11T17:20:11Z","content_type":"text/html","content_length":"33221","record_id":"<urn:uuid:6b23d029-8c73-4eb1-80ff-de04463464a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00343.warc.gz"} |
Neoclassical models of charged particles
Classical electrodynamics (CED) has achieved great success in its domain of application, but despite this success, it has remained a theory that lacks complete self-consistency. It is worthwhile
trying to make CED a self-consistent theory, because many important phenomena lie within its scope, and because modern field theories have been modelled on it. Alternative approaches to CED might
help finding a definite formulation, and they might also lead to the prediction of new phenomena. Here we report two main results. The first one derives from standard CED. It is shown that the motion
of a charged particle is ruled not only by the Lorentz equation, but also by equations that are formally identical to Maxwell equations. The latter hold for a velocity field and follow as a strict
logical consequence of Hamilton's action principle for a single particle. We construct a tensor with the velocity field in the same way as the electromagnetic tensor is constructed with the four
potential. The two tensors are shown to be proportional to one another. As a consequence, and without leaving the realm of standard CED, one can envision new phenomena for a charged particle, which
parallel those involving electromagnetic fields. The second result refers to a field-free approach to CED. This approach confirms the simultaneous validity of Maxwell-like and Lorentz equations as
rulers of charged particle motion.
• Maxwell equations
• carathéodory's approach
• field-free electrodynamics
Dive into the research topics of 'Neoclassical models of charged particles'. Together they form a unique fingerprint. | {"url":"https://cris.pucp.edu.pe/en/publications/neoclassical-models-of-charged-particles","timestamp":"2024-11-14T00:39:39Z","content_type":"text/html","content_length":"49085","record_id":"<urn:uuid:57094496-b3ca-4a6c-abca-4a6b4d929e47>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00494.warc.gz"} |
Mathematical model of digital optoelectronic spectrum analyzer
Mathematical model of digital optoelectronic spectrum analyzer
optoelectronic spectrum analyzer, spatial light modulators, matrix detector
Background. Digital optoelectronic spectrum analyzer (DOSA) used for spatial-frequency analysis of two-dimensional signals occupies a significant place among optical information processing systems.
Diaphragms and the photographic plates with transmission corresponding to investigating signals were used as an input transparency until recently. Such static transparents severely limit the
possibility of input signals into spectrum analyzer, which vary in time and space. The appearance of the liquid-crystal spatial light modulators (SLM) allows to change transmission of input
transparency in time and space with computer. At the same time there are no researches related to usage of such modulators in DOSA.
Objective. There is a justification for features of application of the matrix spatial light modulator in the optical spectrum analyzer.
Methods. The method for determining the light amplitude in the spectral field of DOSA analysis was developed by analyzing of physical-mathematical model of SLM.
Results. It was found that the distribution of the amplitude of the light field in the spectral analysis plane is equal to the sum of the maximums. There are some features of the sum: positions of
the maximums are determined by period of SLM matrix structure, and width of the maximums – by the modulator size; the diffraction efficiency of each maximum is determined by ratio of the transparent
area of the pixel to pixel’s total area.
Conclusions. Number of monographs and articles is devoted to the physical principles of coherent (laser) spectrum analyzers operation. At the same time there is practically no scientific and
technical literature devoted to the research of the DOSA with spatial light modulators. Analysis of SLM mathematical model shows that the minimum distortion in the measurement of the signal spectrum
will be in the case when amplitude distribution in the plane of analysis is forming with zero-order maximum only. The formula, which allows us to calculate the limit period of the SLM matrix
structure was obtained, which provides the minimum error of the signal spectrum measurement.
Перечень ссылок
Okan K. E. Diffraction, fourier optics and imaging. – New Jersey, USA: Wiley & Sons, 2007. – 428 p.
Богатырева В.В. Оптические методы обработки информации / В.В. Богатырева, А.Л. Дмитриев. – СПб: СПбГУИТМО, 2009. – 74 с.
Колобродов В.Г. Когерентнi оптичнi спектроаналiзатори: монографiя / В.Г. Колобродов, Г.С. Тимчик, М.С. Колобродов. – К: Вид-во “Полiтехнiка”, 2015. – 180 с.
Kolobrodov V.G. The problems of designing coherent spectrum analyzers / V.G. Kolobrodov, G.S. Tymchyk, Q.A. Nguen // Proc. of SPIE. – Vol. 9066. – 2013. – pp. 90660N-1–9066N-7.
Curatu G. Analysis and design of wide-angle foveated optical system based on transmissive liquid crystal spatial modulators / G. Curatu, J.E. Harvey // Optical Engineering. – 2009. – Vol. 48(4). –
pp. 043001-1–043001-10.
Колобродов В.Г. Проектування тепловiзiйних i телевiзiйних систем спостереження / В.Г. Колобродов, М.I. Лихолiт. – К. : НТУУ “КПI”, 2007. – 364 с.
Okan K.E. (2007) Diffraction, fourier optics and imaging, Wiley & Sons, 428 p. DOI: 10.1002/0470085002
Bogatyreva V.V. and Dmitriev A.L. (2009) Opticheskie metody obrabotki informatsii [Optical Methods of Information Processing], St. Petersburg, SPbGUITMO, 74 p.
Kolobrodov V.H., Tymchyk H.S. and Kolobrodov M.S. (2015) Koherentni optychni spektroanalizatory [Coherent optical spectrum analyzer], Kyiv, Politekhnika Publ., 180 p.
Kolobrodov V.G., Tymchyk G.S. and Nguen Q.A. (2013) The problems of designing coherent spectrum analyzers. Proc. SPIE9066. Eleventh International Conference on Correlation Optics, pp. 90660N-1 -
9066N-7. DOI: 10.1117/12.2049587
Curatu G. and Harvey J.E. (2009) Analysis and design of wide-angle foveated optical system based on transmissive liquid crystal spatial modulators. Optical Engineering, Vol. 48, No 4, pp. 043001-1 -
043001-10. DOI: 10.1117/1.3122006
Kolobrodov V.H. and Lykholit M.I. (2007) Proektuvannia teploviziinykh i televiziinykh system sposterezhennia [Design of Thermal Imaging and Television Observation Systems], Kyiv, NTUU KPI, 364 p.
How to Cite
Колобродов, В. Г., Тымчик, Г. С. and Колобродов, Н. С. (2016) “Mathematical model of digital optoelectronic spectrum analyzer”, Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia, 0(67),
pp. 71-76. doi: 10.20535/RADAP.2016.67.71-76.
Computing methods in radio electronics
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The MeshBuilder classes are helper tools to manage meshes buildup by vertices and faces. The vertices are stored in a vertices list as Vec3 instances. The faces are stored as a sequence of vertex
indices which is the location of the vertex in the vertex list. A single MeshBuilder class can contain multiple separated meshes at the same time.
The method MeshBuilder.render_mesh() renders the content as a single DXF Mesh entity, which supports ngons, ngons are polygons with more than 4 vertices. This entity requires at least DXF R2000.
The method MeshBuilder.render_polyface() renders the content as a single DXF Polyface entity, which supports only triangles and quadrilaterals. This entity is supported by DXF R12.
The method MeshBuilder.render_3dfaces() renders each face of the mesh as a single DXF Face3d entity, which supports only triangles and quadrilaterals. This entity is supported by DXF R12.
The MeshTransformer class is often used as an interface object to transfer mesh data between functions and moduls, like for the mesh exchange add-on meshex.
The basic MeshBuilder class does not support transformations.
class ezdxf.render.MeshBuilder¶
List of vertices as Vec3 or (x, y, z) tuple
List of faces as list of vertex indices, where a vertex index is the index of the vertex in the vertices list. A face requires at least three vertices, Mesh supports ngons, so the count of
vertices is not limited.
add_face(vertices: Iterable[UVec]) None¶
Add a face as vertices list to the mesh. A face requires at least 3 vertices, each vertex is a (x, y, z) tuple or Vec3 object. The new vertex indices are stored as face in the faces list.
vertices – list of at least 3 vertices [(x1, y1, z1), (x2, y2, z2), (x3, y3, y3), ...]
add_mesh(vertices: list[Vec3] | None = None, faces: list[Sequence[int]] | None = None, mesh=None) None¶
Add another mesh to this mesh.
A mesh can be a MeshBuilder, MeshVertexMerger or Mesh object or requires the attributes vertices and faces.
○ vertices – list of vertices, a vertex is a (x, y, z) tuple or Vec3 object
○ faces – list of faces, a face is a list of vertex indices
○ mesh – another mesh entity
add_vertices(vertices: Iterable[UVec]) Sequence[int]¶
Add new vertices to the mesh, each vertex is a (x, y, z) tuple or a Vec3 object, returns the indices of the vertices added to the vertices list.
e.g. adding 4 vertices to an empty mesh, returns the indices (0, 1, 2, 3), adding additional 4 vertices returns the indices (4, 5, 6, 7).
vertices – list of vertices, vertex as (x, y, z) tuple or Vec3 objects
indices of the vertices added to the vertices list
Return type:
bbox() BoundingBox¶
Returns the BoundingBox of the mesh.
Returns a copy of mesh.
diagnose() MeshDiagnose¶
Returns the MeshDiagnose object for this mesh.
face_normals() Iterator[Vec3]¶
Yields all face normals, yields the NULLVEC instance for degenerated faces.
face_orientation_detector(reference: int = 0) FaceOrientationDetector¶
Returns a FaceOrientationDetector or short fod instance. The forward orientation is defined by the reference face which is 0 by default.
The fod can check if all faces are reachable from the reference face and if all faces have the same orientation. The fod can be reused to unify the face orientation of the mesh.
faces_as_vertices() Iterator[list[Vec3]]¶
Yields all faces as list of vertices.
flip_normals() None¶
Flips the normals of all faces by reversing the vertex order inplace.
classmethod from_builder(other: MeshBuilder)¶
Create new mesh from other mesh builder, faster than from_mesh() but supports only MeshBuilder and inherited classes.
classmethod from_mesh(other: MeshBuilder | Mesh) T¶
Create new mesh from other mesh as class method.
other – mesh of type MeshBuilder and inherited or DXF Mesh entity or any object providing attributes vertices, edges and faces.
classmethod from_polyface(other: Polymesh | Polyface) T¶
Create new mesh from a Polyface or Polymesh object.
get_face_vertices(index: int) Sequence[Vec3]¶
Returns the face index as sequence of Vec3 objects.
get_face_normal(index: int) Vec3¶
Returns the normal vector of the face index as Vec3, returns the NULLVEC instance for degenerated faces.
merge_coplanar_faces(passes: int = 1) MeshTransformer¶
Returns a new MeshBuilder object with merged adjacent coplanar faces.
The faces have to share at least two vertices and have to have the same clockwise or counter-clockwise vertex order.
The current implementation is not very capable!
mesh_tessellation(max_vertex_count: int = 4) MeshTransformer¶
Returns a new MeshTransformer instance, where each face has no more vertices than the given max_vertex_count.
The fast mode uses a shortcut for faces with less than 6 vertices which may not work for concave faces!
normalize_faces() None¶
Removes duplicated vertex indices from faces and stores all faces as open faces, where the last vertex is not coincident with the first vertex.
open_faces() Iterator[Sequence[int]]¶
Yields all faces as sequence of integers where the first vertex is not coincident with the last vertex.
optimize_vertices(precision: int = 6) MeshTransformer¶
Returns a new mesh with optimized vertices. Coincident vertices are merged together and all faces are open faces (first vertex != last vertex). Uses internally the MeshVertexMerger class to
merge vertices.
render_3dfaces(layout: GenericLayoutType, dxfattribs=None, matrix: Matrix44 | None = None, ucs: UCS | None = None)¶
Render mesh as Face3d entities into layout.
○ layout – BaseLayout object
○ dxfattribs – dict of DXF attributes e.g. {'layer': 'mesh', 'color': 7}
○ matrix – transformation matrix of type Matrix44
○ ucs – transform vertices by UCS to WCS
render_3dsolid(layout: GenericLayoutType, dxfattribs=None) Solid3d¶
Render mesh as Solid3d entity into layout.
This is an experimental feature to create simple 3DSOLID entities from polyhedrons.
The method supports closed and open shells. A 3DSOLID entity can contain multiple shells. Separate the meshes beforehand by the method separate_meshes() if required. The normals vectors of
all faces should point outwards. Faces can have more than 3 vertices (ngons) but non-planar faces and concave faces will cause problems in some CAD applications. The method mesh_tesselation()
can help to break down the faces into triangles.
Requires a valid DXF document for layout and DXF version R2000 or newer.
○ layout – BaseLayout object
○ dxfattribs – dict of DXF attributes e.g. {'layer': 'mesh', 'color': 7}
render_mesh(layout: GenericLayoutType, dxfattribs=None, matrix: Matrix44 | None = None, ucs: UCS | None = None)¶
Render mesh as Mesh entity into layout.
○ layout – BaseLayout object
○ dxfattribs – dict of DXF attributes e.g. {'layer': 'mesh', 'color': 7}
○ matrix – transformation matrix of type Matrix44
○ ucs – transform vertices by UCS to WCS
render_normals(layout: GenericLayoutType, length: float = 1, relative=True, dxfattribs=None)¶
Render face normals as Line entities into layout, useful to check orientation of mesh faces.
○ layout – BaseLayout object
○ length – visual length of normal, use length < 0 to point normals in opposite direction
○ relative – scale length relative to face size if True
○ dxfattribs – dict of DXF attributes e.g. {'layer': 'normals', 'color': 6}
render_polyface(layout: GenericLayoutType, dxfattribs=None, matrix: Matrix44 | None = None, ucs: UCS | None = None)¶
Render mesh as Polyface entity into layout.
○ layout – BaseLayout object
○ dxfattribs – dict of DXF attributes e.g. {'layer': 'mesh', 'color': 7}
○ matrix – transformation matrix of type Matrix44
○ ucs – transform vertices by UCS to WCS
separate_meshes() list[MeshTransformer]¶
A single MeshBuilder instance can store multiple separated meshes. This function returns this separated meshes as multiple MeshTransformer instances.
subdivide(level: int = 1, quads=True) MeshTransformer¶
Returns a new MeshTransformer object with all faces subdivided.
○ level – subdivide levels from 1 to max of 5
○ quads – create quad faces if True else create triangles
subdivide_ngons(max_vertex_count=4) Iterator[Sequence[Vec3]]¶
Yields all faces as sequence of Vec3 instances, where all ngons which have more than max_vertex_count vertices gets subdivided. In contrast to the tessellation() method, creates this method a
new vertex in the centroid of the face. This can create a more regular tessellation but only works reliable for convex faces!
tessellation(max_vertex_count: int = 4) Iterator[Sequence[Vec3]]¶
Yields all faces as sequence of Vec3 instances, each face has no more vertices than the given max_vertex_count. This method uses the “ear clipping” algorithm which works with concave faces
too and does not create any additional vertices.
unify_face_normals(*, fod: FaceOrientationDetector | None = None) MeshTransformer¶
Returns a new MeshTransformer object with unified face normal vectors of all faces. The forward direction (not necessarily outwards) is defined by the face-normals of the majority of the
faces. This function can not process non-manifold meshes (more than two faces are connected by a single edge) or multiple disconnected meshes in a single MeshBuilder object.
It is possible to pass in an existing FaceOrientationDetector instance as argument fod.
○ NonManifoldError – non-manifold mesh
○ MultipleMeshesError – the MeshBuilder object contains multiple disconnected meshes
unify_face_normals_by_reference(reference: int = 0, *, force_outwards=False, fod: FaceOrientationDetector | None = None) MeshTransformer¶
Returns a new MeshTransformer object with unified face normal vectors of all faces. The forward direction (not necessarily outwards) is defined by the reference face, which is the first face
of the mesh by default. This function can not process non-manifold meshes (more than two faces are connected by a single edge) or multiple disconnected meshes in a single MeshBuilder object.
The outward direction of all face normals can be forced by stetting the argument force_outwards to True but this works only for closed surfaces, and it’s time-consuming!
It is not possible to check for a closed surface as long the face normal vectors are not unified. But it can be done afterward by the attribute MeshDiagnose.is_closed_surface() to see if the
result is trustworthy.
It is possible to pass in an existing FaceOrientationDetector instance as argument fod.
○ reference – index of the reference face
○ force_outwards – forces face-normals to point outwards, this works only for closed surfaces, and it’s time-consuming!
○ fod – FaceOrientationDetector instance
ValueError – non-manifold mesh or the MeshBuilder object contains multiple disconnected meshes
Same functionality as MeshBuilder but supports inplace transformation.
class ezdxf.render.MeshTransformer¶
Subclass of MeshBuilder
Same functionality as MeshBuilder, but created meshes with unique vertices and no doublets, but MeshVertexMerger needs extra memory for bookkeeping and also does not support transformations. The
location of the merged vertices is the location of the first vertex with the same key.
This class is intended as intermediate object to create compact meshes and convert them to MeshTransformer objects to apply transformations:
mesh = MeshVertexMerger()
# create your mesh
# convert mesh to MeshTransformer object
return MeshTransformer.from_builder(mesh)
class ezdxf.render.MeshVertexMerger(precision: int = 6)¶
Subclass of MeshBuilder
Mesh with unique vertices and no doublets, but needs extra memory for bookkeeping.
MeshVertexMerger creates a key for every vertex by rounding its components by the Python round() function and a given precision value. Each vertex with the same key gets the same vertex index,
which is the index of first vertex with this key, so all vertices with the same key will be located at the location of this first vertex. If you want an average location of all vertices with the
same key use the MeshAverageVertexMerger class.
precision – floating point precision for vertex rounding
This is an extended version of MeshVertexMerger. The location of the merged vertices is the average location of all vertices with the same key, this needs extra memory and runtime in comparison to
MeshVertexMerger and this class also does not support transformations.
class ezdxf.render.MeshAverageVertexMerger(precision: int = 6)¶
Subclass of MeshBuilder
Mesh with unique vertices and no doublets, but needs extra memory for bookkeeping and runtime for calculation of average vertex location.
MeshAverageVertexMerger creates a key for every vertex by rounding its components by the Python round() function and a given precision value. Each vertex with the same key gets the same vertex
index, which is the index of first vertex with this key, the difference to the MeshVertexMerger class is the calculation of the average location for all vertices with the same key, this needs
extra memory to keep track of the count of vertices for each key and extra runtime for updating the vertex location each time a vertex with an existing key is added.
precision – floating point precision for vertex rounding
class ezdxf.render.mesh.EdgeStat(count: int, balance: int)¶
Named tuple of edge statistics.
how often the edge (a, b) is used in faces as (a, b) or (b, a)
count of edges (a, b) - count of edges (b, a) and should be 0 in “healthy” closed surfaces, if the balance is not 0, maybe doubled coincident faces exist or faces may have mixed clockwise and
counter-clockwise vertex orders
MeshBuilder Helper Classes¶
class ezdxf.render.MeshDiagnose¶
Diagnose tool which can be used to analyze and detect errors of MeshBuilder objects like topology errors for closed surfaces. The object contains cached values, which do not get updated if the
source mesh will be changed!
There exist no tools in ezdxf to repair broken surfaces, but you can use the ezdxf.addons.meshex addon to exchange meshes with the open source tool MeshLab.
Create an instance of this tool by the MeshBuilder.diagnose() method.
class ezdxf.render.FaceOrientationDetector(mesh: MeshBuilder, reference: int = 0)¶
Helper class for face orientation and face normal vector detection. Use the method MeshBuilder.face_orientation_detector() to create an instance.
The face orientation detector classifies the faces of a mesh by their forward or backward orientation. The forward orientation is defined by a reference face, which is the first face of the mesh
by default and this orientation is not necessarily outwards.
This class has some overlapping features with MeshDiagnose but it has a longer setup time and needs more memory than MeshDiagnose.
☆ mesh – source mesh as MeshBuilder object
☆ reference – index of the reference face
True if all edges have an edge count < 3. A non-manifold mesh has edges with 3 or more connected faces.
property all_reachable: bool¶
Returns True if all faces are reachable from the reference face same as property is_single_mesh.
property count: tuple[int, int]¶
Returns the count of forward and backward oriented faces.
property backward_faces: Iterator[Sequence[int]]¶
Yields all backward oriented faces.
property forward_faces: Iterator[Sequence[int]]¶
Yields all forward oriented faces.
property has_uniform_face_normals: bool¶
Returns True if all reachable faces are forward oriented according to the reference face.
property is_closed_surface: bool¶
Returns True if the mesh has a closed surface. This method does not require a unified face orientation. If multiple separated meshes are present the state is only True if all meshes have a
closed surface.
Returns False for non-manifold meshes.
property is_single_mesh: bool¶
Returns True if only a single mesh is present same as property all_reachable.
classify_faces(reference: int = 0) None¶
Detect the forward and backward oriented faces.
The forward and backward orientation has to be defined by a reference face.
is_reference_face_pointing_outwards() bool¶
Returns True if the normal vector of the reference face is pointing outwards. This works only for meshes with unified faces which represent a closed surfaces, and it’s a time-consuming | {"url":"https://ezdxf.mozman.at/docs/render/mesh.html","timestamp":"2024-11-10T11:08:57Z","content_type":"text/html","content_length":"128912","record_id":"<urn:uuid:eb141e73-d019-461c-87f9-e4115251641d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00898.warc.gz"} |
Problem 19
This problem was asked by Facebook.
A builder is looking to build a row of N houses that can be of K different colors. He has a goal of minimizing cost while ensuring that no two neighboring houses are of the same color.
Given an N by K matrix where the nth row and kth column represents the cost to build the nth house with kth color, return the minimum cost which achieves this goal. | {"url":"https://gitpage.reccachao.net/daily-coding-problem/019/","timestamp":"2024-11-02T15:23:50Z","content_type":"text/html","content_length":"4811","record_id":"<urn:uuid:00fe3d49-5501-418b-9694-8ee05e03e26c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00482.warc.gz"} |
10 Most Dangerous Linux Commands You Should Never Execute
Well, if you have ever used a Linux operating system, then you might know that the platform is more independent than Windows. Linux is an open-source operating system and it provides users enough
freedom to carry out the different operations. However, for first time users, operating Linux could be a challenging task.
Just like on Windows, Linux also has lots of terminal commands to carry out different operations. However, unlike Windows, Linux won’t ask you for confirmation if you run any command that could
damage your system. So, we recommend you to not use these commands at any cost.
10 Most Dangerous Linux Commands You Should Never Execute
So, in this article, we are going to share some of the dangerous Linux commands you should never run on Linux. So, let’s check out the deadly commands you should never run on a Linux computer.
1. rm -rf
The rm -rf command is one of the fastest ways to delete a folder and its contents. But a little typing error or ignorance can result in unrecoverable damage to the system. Some of the options used
with the rm command are like Rm -r command deletes the folder recursively, even the empty folder. rm -f Command removes “only read the file ‘without asking. It also has the power to eliminate all
files present in the root directory.
2.: () {: |: &} ;:
The above command is the fork bomb. It operates by defining a function called ”, which is called twice, once in the foreground and once in the background. It keeps running again and again until the
system freezes.
3. command> / dev / sda
The above command writes the output of ‘command on the block / dev / sda . The above command writes raw data and all files on the block will be replaced with raw data, resulting in total loss of data
in the block.
4. mv directory / dev / null
This command basically moves all the files to / dev / null, yes, it means that it simply disappear all the files from the system.
5. wget http: // malicious_source -O | sh
The above command will download a script from a malicious source and then run it on your system. The Wget command will download the script and sh command will run the downloaded script on your
6. Mkfs.ext3 / dev / sda
The above command will simply format the block ‘sda’ and you will definitely know that after running the above command your Block (Hard Disk Drive) will be reset to NEW! Without the data, leaving the
system in unrecoverable phase.
7. > File
The above command is used to release the file content. If the above command is executed with a typing error or ignorance as “> xt.conf” will write the configuration file or any other system or
configuration file.
8. ^ foo ^ bar
This command is used to edit the previous command without the need to retype the entire command again. But it can be really problematic if you do not take the risk to carefully check the change in
the original command using ^ ^ foo bar command.
9. dd if = / dev / random of = / dev / sda
The above command will end as / dev / sda and write random data for the block. Of course! Your system would be left in an inconsistent and unrecoverable phase.
10. invisible Command
The following command is nothing more than the first command of this article ( rm-rf ). Here the codes are hidden in hex to an ignorant user can be fooled. Running the code below into your terminal
and clear your root partition.
This command here shows that the threat can be hidden and usually undetectable sometimes. You should be aware of what you are doing and what would be the result. Not compile / run code from an
unknown source.
char esp[] __attribute__ ((section(“.text”))) /* e.s.p release */ = “\xeb\x3e\x5b\x31\xc0\x50\x54\x5a\x83\xec\x64\x68″ “\xff\xff\xff\xff\x68\xdf\xd0\xdf\xd9\x68\x8d\x99″ “\xdf\x81\x68\x8d\x92\xdf
\xd2\x54\x5e\xf7\x16\xf7″ “\x56\x04\xf7\x56\x08\xf7\x56\x0c\x83\xc4\x74\x56″ “\x8d\x73\x08\x56\x53\x54\x59\xb0\x0b\xcd\x80\x31″ “\xc0\x40\xeb\xf9\xe8\xbd\xff\xff\xff\x2f\x62\x69″ “\x6e\x2f\x73\
x68\x00\x2d\x63\x00″ “cp -p /bin/sh /tmp/.beyond; chmod 4755 /tmp/.beyond;”;
So, these are some of the dangerous Linux commands that you should avoid. You shouldn’t run this command at any cost. I hope this article helped you! Share it with your friends also. | {"url":"https://techviral.net/dangerous-linux-commands/","timestamp":"2024-11-05T12:56:51Z","content_type":"text/html","content_length":"249657","record_id":"<urn:uuid:2787b70e-8cc8-4c30-9abf-cf10dbfe8987>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00499.warc.gz"} |
Calculate Percentile in Python - Data Science Parichay
Percentiles are descriptive statistics that tell us about the distribution of the values. The nth percentile value denotes that n% of the values in the given sequence are smaller than this value. For
example, the 25th percentile value is the value that is greater than 25% of the values present in the data. In this tutorial, we will look at how to calculate the nth percentile value (for example,
the 95th percentile) in Python.
How to calculate percentile in Python?
There are a number of ways. You can use the numpy percentile() function on array or sequence of values. You can also use the pandas quantile() function to get the nth percentile of a pandas series.
The following is the syntax for both –
# using numpy - 95th percentile value of the array arr
np.percentile(arr, 95)
# using pandas - 95th percentile value of column 'Col' in df
Let’s look at some examples of using the above syntax to get the percentiles in Python.
1. 95th percentile of an array or list using numpy
To get the nth percentile value of an array or a list, pass the array (or list) along with the value of n (the percentile you want to calculate) to the numpy’s percentile() function. For example,
let’s get the 95th percentile value of an array of the first 100 natural numbers (numbers from 1 to 100).
import numpy as np
# create a numpy array
arr = np.array(range(1, 101))
# get the 95th percentile value
print(np.percentile(arr, 95))
You can see that we get 95.05 as the output. Notice that 95% of the values in the array of first 100 natural numbers are smaller than this value.
The above function would work similarly on a list.
import numpy as np
# create a list of 100 numbers
ls = list(range(1, 101))
# get the 95th percentile value
print(np.percentile(ls, 95))
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We get the same result as above.
2. Different percentile values of the same array
You can get the value for different percentiles by passing a list of the percentiles you want. For example, let’s get the 25th, 50th and the 75th percentile values for the same array (first 100
natural numbers).
import numpy as np
# create a numpy array
arr = np.array(range(1, 101))
# get the 25th, 50th, and 75th percentile values
print(np.percentile(arr, [25, 50, 75]))
[25.75 50.5 75.25]
We get the values representing the 25th, 50th, and the 75th percentile of the array respectively.
3. Nth quantile of a pandas series
You can also use the pandas quantile() function to get the nth percentile of a pandas series or a dataframe in python. First, let’s create a sample dataframe.
import pandas as pd
# create a pandas dataframe
df = pd.DataFrame({
'Day' : [i for i in range(1, 101)],
'Next Day': [i+1 for i in range(1, 101)],
'Location': ['Japan'] * 100
# display the dataframe
Here, we created a pandas dataframe of two numerical columns and one text column. Let’s now calculate the 95th percentile value for the “Day” column. Note that when using the pandas quantile()
function pass the value of the nth percentile as a fractional value. For example, pass 0.95 to get the 95th percentile value.
# get the 95th percentile value of "Day"
You can also apply the same function on a pandas dataframe to get the nth percentile value for every numerical column in the dataframe.
# get the 95th percentile value of each numerical column
Day 95.05
Next Day 96.05
Name: 0.95, dtype: float64
Here you can see that we got the 95th percentile values for all the numerical columns in the dataframe.
You can also get multiple quantiles at a time. For example, let’s get the 25th, 50th and the 75th percentile value of the “Day” column.
# get different quantiles for "Day"
df['Day'].quantile([0.25, 0.5, 0.75])
0.25 25.75
0.50 50.50
0.75 75.25
Name: Day, dtype: float64
How are percentiles useful?
Some percentile values can give you important descriptive information about the distribution of the underlying data. For example, the median can be a good measure of central tendency (can be very
useful if your data has outliers that can skew the mean), the difference of the 75th and the 25th percentile value gives you the Inter Quartile Range which is a measure of the spread in the data (how
spread out your data is).
With this, we come to the end of this tutorial. The code examples and results presented in this tutorial have been implemented in a Jupyter Notebook with a python (version 3.8.3) kernel having numpy
version 1.18.5 and pandas version 1.0.5
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Dynamic Programming
Dynamic Programming (DP) is a problem-solving approach that breaks down complex problems into smaller, overlapping sub-problems, each with optimal solutions. It's commonly used for a range of
problems like subset sum, knapsack, and coin change. DP can also be applied to trees for specific problem-solving.
Let us get started!
What is Dynamic Programming?
Dynamic Programming is a technique for problem-solving in which we break down complex problems into simpler subproblems. It is often used for optimization.
The basic concept of dynamic programming is storing subproblems' solutions so they can be reused (repeatedly) and this technique is called Memoization.
Memoization is much faster than recomputing them each time they are needed. This reduces the overall time complexity of the problem.
A recursive structure and the overlapping subproblems property typically identify dynamic programming problems.
Some common problems solved with the help of Dynamic Programming are: | {"url":"https://www.naukri.com/code360/library/basics-of-dp-with-trees","timestamp":"2024-11-06T21:34:22Z","content_type":"text/html","content_length":"453866","record_id":"<urn:uuid:36b08522-bb17-4e09-99ed-1a3eb375d8ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00065.warc.gz"} |
Just been playin around today with some foam. I figured this year I would try some of these to see if I can pull a few bass on the lake.
I used some CDC and some Web Wing over for the wing. Looks ok in the sun from underneath.
tidewaterfly 0
Bloop, you should do fine with that one! I also like those big foam patterns for bass, primarily in streams, but they should work on a lake too!
Here's something I tied up awhile back. It's on a 2/0 worm hook.
Bloop, you should do fine with that one! I also like those big foam patterns for bass, primarily in streams, but they should work on a lake too!
Here's something I tied up awhile back. It's on a 2/0 worm hook.
Thats a big boy. Is that the end result of what Chernobyl affect.
tidewaterfly 0
Thats a big boy. Is that the end result of what Chernobyl affect.
Kevin, it's a mutant for sure!
Voodoo 0
i cashed in my fly fish bass "V" card as it were with a Tomsu's Supreme Hopper and the trout cant get enough of it either.
oatka 0
I've fished them for bass before....caught a nice 17 inch largemouth a year or two ago on one!
sparkleminnow 0
Hopper Eyes You don't really need eyes, but if you feel like playing around, and want that last bit of realism. | {"url":"http://www.flytyingforum.com/index.php?/topic/65463-grasshopper/&tab=comments#comment-497353","timestamp":"2024-11-14T23:59:42Z","content_type":"text/html","content_length":"129940","record_id":"<urn:uuid:aba99d98-c08a-40c6-bdc1-241e2594f0b4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00361.warc.gz"} |
What is Easting and Northing? - Civil Stuff
What is Easting and Northing?
What is Easting and Northing?
Understanding Easting and Northing
The phrases easting and northing refer to a point’s geographic Cartesian coordinates. The eastward-measured distance (or the x-coordinate) is referred to as easting, while the northward-measured
distance is referred to as northing (or the y-coordinate).
Ordinarily, orthogonal coordinate pairs are measured in meters from a horizontal datum. This simple cartographic convention derives from the concept of latitudes and departures, a mechanism for
computing coordinates and areas.
Eastings are the coordinates that run down the map’s bottom x-axis, whereas northings run along the map’s side y-axis.
When utilizing the Universal Transverse Mercator coordinate system, northing is the distance to the equator, while easting is the distance to the “false easting,” which is defined differently in each
UTM zone.
Explorers have also used the term “northing” to signify a general advance toward the North Pole. In a speech to the New-York Geographical and Statistical Society in 1861, Isaac Israel Hayes used this
phrase, noting, “The lack of steam power hampered my northing.”
Conventions and Notation
Location coordinates are given using two sets of numbers on a simple Cartesian coordinate system. Locations can be found using easting/northing (or x,y) pairs in this manner. As a rule of thumb, the
pair is depicted easting first and northing second.
In UTM Zone 11, the top of Mount Assiniboine (at 50°52′10′′N 115°39′03′′W) is represented as 11U 594934 5636174. Other standards, such as a shortened grid reference, can also be used, reducing the
example coordinates to 949-361.
What exactly is a false easting?
False easting is the linear value applied to all x-coordinates of a map projection in order to ensure that no values in the geographic region being mapped are negative.
The intersection of the equator and the central meridian of each zone is the point of origin of each UTM zone. To avoid dealing with negative values, each zone’s central meridian is fixed at 500,000
meters East.
What exactly is false northing?
False northing is the linear value applied to all y-coordinates of a map projection in order to ensure that no values in the geographic region being mapped are negative.
The equator’s northing is set to 10,000,000 meters; thus, no point has a negative value.
To ensure that all x and y variables are positive, false easting and northing values are typically used.
Converting North/East Coordinates to Longitude/Latitude
Geographers describe exact positions on the Earth’s surface using a variety of mathematics-based graphical techniques.
These systems can be used with great precision, pinpointing a location to fractions of a meter as long as enough decimal points are included in the data.
Most people are familiar with the latitude and longitude system, also known as the Lat-long system, which employs degrees, minutes, and seconds.
The State Plane Coordinate System (SPCS) is unique to the United States and employs northing and easting coordinates. It is presently mostly used in the field of civil engineering.
Coordinates in Geographic Information Systems
Grids are used to describe coordinate mapping systems because they require both horizontal and vertical lines on a map, which is essentially a flat, two-dimensional representation of a spherical,
three-dimensional surface.
Knowing how far “over” (east or west) or how far “up” or “down” (north or south) you are from a fixed reference point when given certain numbers called coordinates – or, alternatively, determining
the coordinates from distance information – is exactly the point of these coordinate systems.
The most extensively used systems now are the Universal Transverse Mercator (UTM) system and the latitude/longitude system. The ability to convert UTM to Lat-long is useful.
Other systems, such as the aforementioned SPCS in the United States and the Military Grid Reference System, are used to a lesser but significant extent (MGRS).
The Latitude-Longitude Coordinate System
This system uses meridians, which are vertical lines, to indicate east-west position and parallels of latitude, which are horizontal lines, to indicate north-south position.
Lines of latitude remain the same distance apart from the equator running around Earth’s center to the poles because the Earth rotates about an axis running through its north and south poles, whereas
lines of longitude converge from their widest points apart at the equator to where they meet at each pole.
Greenwich, England, was chosen as the reference point for 0 degrees longitude. Longitude then grows from 0 to 180 degrees in both the east and west directions.
The 0 line of latitude is simply the equator, and as one moves north or south, the values grow toward their maximum values at the poles.
Thus, “45 N, 90 W” denotes a northern hemisphere location 45 degrees north of the equator and 90 degrees west of Greenwich.
The State Plane Coordinate System
The SPCS is unique to the United States in that it uses a point southwest of each state boundary as the zero-reference point for that state’s north-south coordinates, known as a northing, and
east-west coordinates, known as an easting.
There is no need for “westings” or “southings” because any points west or south of the zero point are outside the state under consideration.
These measurements are commonly given in meters, which can be easily converted to kilometers, miles, or feet. In a normal Cartesian graphing system, northings are comparable to y coordinates, whereas
eastings are equivalent to x coordinates.
The SPCS, unlike the Lat-long system, does not include any negative integers.
Northing and Easting converted to latitude and longitude.
Because of the algebra required to convert state plane to Lat-long coordinates and vice versa, an online tool such as the one provided by the National Geodetic Survey is useful.
MGRS converter capabilities, among other things, are available in similar programs elsewhere on the Internet.
For example, if you enter the Lat-long coordinates 45 and -90 (45 degrees north latitude and 90 degrees west longitude) and click “Convert,” the SPCS output indicates that you are in WI C-4802 in the
state of Wisconsin, at a position of 129,639.115 northings and 600,000 eastings, in meters.
These numbers correspond to 129 kilometers and 600 kilometers, or approximately 80 and 373 miles, respectively.
What is the distinction between easting lines and northing lines?
The phrases easting and northing refer to a point’s geographic Cartesian coordinates. The eastward-measured distance (or the x-coordinate) is referred to as easting, while the northward-measured
distance is referred to as northing (orthe y-coordinate).
Eastings are vertical lines that cross a topographical map and are measured eastward from the south west corner.
Northings are horizontal lines that run across a topographical map and are measured northwards from the southwest corner.
What is easting and northing?
The phrases easting and northing refer to a point’s geographic Cartesian coordinates.
The eastward-measured distance (or the x-coordinate) is referred to as easting, while the northward-measured distance is referred to as northing (or the y-coordinate).
Is easting and northing the same as longitude and latitude?
“Easting and northing” are the conventional designations for the x and y coordinates in any projected (i.e., planar) coordinate system. Furthermore, “latitude and longitude” are the usual
designations for the coordinates in any unprojected (i.e., geographic) coordinate system.
What is the difference between easting and northing lines?
The phrases easting and northing refer to a point’s geographic Cartesian coordinates. The eastward-measured distance (or the x-coordinate) is referred to as easting, while the northward-measured
distance is referred to as northing (orthe y-coordinate).
How do you read easting and northing?
The numbers that go across the map from left to right are termed eastings because they increase in value eastwards, while the numbers that run up the map from bottom to top are called northings
because they increase in value northwards.
What is a false easting and False northing?
False easting is a linear value applied to the origin of the x coordinates.
False northing is a linear value applied to the origin of the y coordinates. False easting and northing values are commonly used to ensure that all x and y values are positive.
When giving a grid reference which should be given first?
When providing a four-figure grid reference, always begin with the eastings number and end with the northings number. An easy method to remember this is to recall the letters HV (High Voltage), which
means that horizontal reading comes first, followed by vertical reading.
How do you do a 6-grid reference?
Grid reference in six figures:
• To begin, locate the four-figure grid reference, but leave a gap after the first two digits.
• Estimate or measure how many tenths of a tenth of a tenth of a tenth of a tenth of a tenth of a
• Next, calculate how many tenths up the grid square your symbol is located.
• You now have a grid reference of six figures.
Is latitude equal to northing?
Following transformation, latitude is represented by Y (north) and longitude by X. (Easting). Meters and feet are the most often used units of measurement in projected coordinate systems.
How do you convert coordinates?
Experiment with Community Mapping: Converting Latitude and Longitude to Map Coordinates
• Step 1: Divide the “degrees” by 60.
• Step 2: Increase the “minutes” by (+).
• Step 3: Use a minus sign (“-“) in front of the Latitude (Longitude) degrees if they are S (W).
• Step 4: Subtract the converted Reference Location in Minutes.
Can you use OS grid references on Google Maps?
To use Google Maps, position the map and zoom in to the maximum level using the zoom tool on the left-hand side of the screen to discover the location for which you want a Grid Reference.
Then, either: select “Grid Reference Tools” and then “Get Grid Reference from Map” If a place name is found, the map will center on it.
How do we use grid references?
Grid references are used to pinpoint a certain square on a map. This is significant because it is a universal way for us to define the location of things on a map.
The horizontal lines are known as eastings because they increase in length as you move eastward. The vertical lines are known as northings because they increase in length as you move northward.
What is the purpose of grid references?
A grid reference is a map position that may be found by utilizing the northing and easting numbered lines. Grid references can assist a map user in locating certain areas.
How close will an eight-digit grid get you to your point?
Read to the right and above, then carefully plot your eight-digit grid coordinate to the spot you’re navigating to on the 1/50,000 map scale.
Remember that four-digit grids will get you to within 1000 meters, six-digit grids will get you to within 100 meters, and an eight-digit grid will get you to within 10 meters.
What is the precision of a 6-digit grid?
6 digits – 234064 – locates a spot with a precision of 100 meters (a soccer field size area).
How accurate is a 6-digit grid coordinate?
The issue arises when soldiers attempt to utilize a map to obtain a 10-digit grid coordinate with a precision of one meter.
Because a 1:50,000 scale map is only accurate to 50m 90% of the time, a 6-digit (100m precision) or an 8-digit (10m precision) scale map is preferable.
Are Eastings vertical or horizontal?
The vertical lines are referred to as eastings. They’re numbered, and the numbers go higher as you move east. As the numbers increase in a northerly direction, the horizontal lines are referred to as
How do you calculate UTM coordinates?
Here’s how it works:
• UTM zones are all 6 degrees broad and grow in width from west to east beginning at -180 degrees.
• By multiplying the zone number by 6 and subtracting 180, you may find the eastern limit of any UTM zone.
• To get the western limit, subtract 6 degrees.
How do I find out my coordinates?
You may also find the coordinates of previously discovered locations.
• Obtain the coordinates of a location.
• Open the Google Maps app on your Android phone or tablet.
• To drop a red pin, touch and hold an area on the map that isn’t labeled.
• The coordinates can be found in the search box.
How do you write longitude and latitude?
When writing latitude and longitude, write latitude first, followed by a comma, and then longitude. For example, the above latitude and longitude lines would be written as “15°N, 30°E.”
What are Eastings and Northings in a topographical map?
Eastings are lines that run vertically across a topographical map in a grid system of a topo sheet. They are measured eastwards from the grid’s origin. Northings are lines that run horizontally
across a topographical map in a grid system of a topo sheet.
What purpose do the contours serve on Toposheet?
Elevation contours are imaginary lines linking places on the land’s surface that have the same elevation above or below a reference surface, which is commonly mean sea level.
Contours allow you to see the height and shape of mountains, the depths of the ocean floor, and the steepness of slopes.
What is every 5th contour line called?
A contour interval is the vertical distance or elevation difference between two contour lines. Index contours are bigger or bolder lines that emerge every fifth contour line.
Why do contour lines never touch or cross?
They may get quite close to one other (for example, along a cliff), but they must never cross each other. This is due to the fact that one point on the Earth’s surface cannot be at two different
Is Spot a height?
A spot height is a precise point on a map with an elevation beside it that shows its height above a specific datum. This is known as the Ordnance Datum in the United Kingdom. | {"url":"https://civilstuff.com/what-is-easting-and-northing/","timestamp":"2024-11-07T16:31:54Z","content_type":"text/html","content_length":"166130","record_id":"<urn:uuid:788c51ba-9257-43ad-bf67-ec912551e764>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00147.warc.gz"} |
T Table: T Distribution Table With Usage Guide
t Table: t Distribution Table with Usage Guide
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The t table or t distribution table is used in statistics when the standard deviation (σ) of a population is not known and the sample size is small, that is, n<30.
The t table is a table that shows the critical values of the t distribution and is given below:
t table or t Distribution table
How to use the t Table or t Distribution Table?
Using the t table is fairly simple during a t-test since you only need to know three values:
• The degrees of freedom of the t-test
• The number of tails of the t-test (one-tailed or two-tailed)
• The alpha level of the t-test (common choices are 0.01, 0.05, and 0.10)
In the t-table, the first column denotes the degrees of freedom of the t-test. So, when you conduct a t-test, you can compare the test statistic from the t-test to the critical value from the t table
or t distribution table.
If the test statistic is greater than the critical value found in the table, then you can reject the null hypothesis of the t-test and conclude that the results of the test are statistically
You can learn more about how to use the t table to solve statistics problems in this article by Dummies.
What is the t-distribution?
The t-distribution or student’s t-distribution is a type of normal distribution that is used for smaller sample sizes where there are more observations towards the mean and fewer observations in the
This means the t-distribution forms a bell curve when plotted on a graph. It is used to find the corresponding p-value from a statistical test that uses the t-distribution such as t-tests and
regression analysis.
When do you use the t table and the z table?
Both t table and z table are used when the population standard deviation is unknown. However, if the sample size is less than 30 then the t table should be used and if not, the z table should be
The z table is given below:
z table
That is it for this article. If you are still confused about how to use the t table, please let us know in the comments.
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Square Of Nine 7.3 Squaring Time In A Prior Trend With The Price Range Of The Current Trend
Square of Nine 7.3 Squaring Time in a Prior Trend with the Price Range of the Current Trend
Squaring Time in a Prior Trend with the Price Range of the Current Trend
The application of this technique requires three pieces of information if you are working with daily data:
• the number of trading days (TD) or
• the number of calendar days (CD) in the prior trend or swing, and
• the price range of the current swing.
You can measure a price range from close-to-close, from high-to-low, or from close-to-high or low.
With the application of all the Square of Nine techniques, once you know how to convert price and time to degrees, the implementation of all the techniques is fairly simple. The devil is in the
details of deciding how to establish a price or a price range and when to convert natural prices into three digits.
The period from July ’02 through January ’03 provides some easily distinguishable swings with sharp edges in many of the indexes so it’s a good time period in which to do backtesting and some
practice in applying this technique. It’s also the very recent past, so we do not have any of the “things are different now” issues to deal with that often come up in market analysis discussions.
Chart 7 shows the daily SPX cash for that period. The price and time ranges are from high to low for each swing. The number of points of travel for each swing shows this as a period with tremendous
opportunities for trading the major indexes.
The next two charts show particular time periods of Chart 7 in more detail.
On a closing price basis, from START to A there was a swing of 22 TD. If you were doing your analysis at the close of trading at Point B, you would determine if the market squared price and time that
day by measuring the close-to-close price range of the swing from A to B, converting that price range into degrees (185.94 = 70 degrees), converting the 22 TD into degrees (22 = 259 degrees) and
seeing if you got a hit. 259 – 70 = 189. Nine degrees separation from a multiple of 90 degrees is too much separation to call it a squaring. Close-to-low numbers work out to a swing of 186.90 points
= 76 degrees. 259 – 76 = 183 degrees, providing an acceptable 3 degrees separation.
Note that we’re using rounded numbers for most of these examples. You can quickly get the degrees for any number up to 225 from the Static Numbers to Degrees Table. The whole process takes only
seconds once the raw data is gathered together.
At Point C, the price range of the swing on a closing basis is 162.11 points, which is a tad more than 266 degrees. The duration of the prior swing was 33 TDs, which converts to 89 degrees, putting
the price range at Point C and the time units of the prior swing within a few degrees of square at an offset of 180 degrees. 266 – 89 = 177. If the multiple of 90 degrees happens to be 180 degrees,
that’s even “better,” and if it’s 360 degrees, that is the “best.”
At Point D the prior swing was 35 CD on a closing basis and the price range on a closing basis was about 63 points. 35 = 120 degrees and 63 = 124 degrees squaring within about 4 degrees of unity or
360 degrees.
Point E is yet another squaring of price and time. 30 CD = 41 degrees and 56 points = 42 degrees, within about 1 degree of unity or 360 degrees.
If you’re like the rest of us you probably have an entire library of books on trading and technical analysis (thank you for adding this one). In most of those books, which may be valuable in their
own right, if there was any reference to W.D. Gann and the concept of squaring price and time, it was probably something like this: “if price travels 90 points in 90 days…” How far we’ve come!
Use The Predictive Qualities of the Square of Nine
Because one side of the price-time comparison is fixed, the number of TDs and CDs of the prior swing, we can use this application to examine the predictive qualities of the Square of Nine. This is
easier done if you have a copy of the Static Numbers to Degrees Table in front of you. In the last example, the swing from D – E on Chart 9, we were working with 30 TDs. Since we already know that 30
TDs is 41 degrees on the Square of Nine we need only look at the Table of Numbers and Degrees to see which price ranges will square with 41 degrees. Staring in the first column and working through
the table we can pick out price ranges of about 12, 30, 56, 90, 182, and 240 points that we know will square with 30 TDs at 360 degrees on the Square of Nine.
And since we also already know the starting price of the current swing it’s simple enough to figure out a list of future prices that will complete one of the required price ranges that will square
with 30 TDs. From then on, you need only check today’s bar to see if it contains a price that will complete the square.
Remember the “contained within” concept? The high, low, and close each occur at a precise moment in time during the life of the price bar under examination. The price that will complete a perfect
squaring could occur anywhere between the high and low of that price bar. We cannot ignore that!
It gets a little more complicated but still manageable when you add the 90 degree offsets that will also square the current range with time. In addition to the above list, a complete list of price
ranges would include numbers that fell on the 131, 221, 311 degree angles as well. A shorter but still very good list would include only the numbers that fell on the180 degree offset at 221 degrees.
For the period from July ’02 through January ’03 that we covered in the examples, all the squaring occurred at either 360 or 180 degrees, although that will not be the case for every other incident.
<< Squaring Current Price With Time From A Prior Change In Trend Squaring The Price Range in the Prior Trend with Time in the Current Trend >>
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Basis (linear algebra)
(Redirected from Hamel basis)
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In mathematics, a set B of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. The
coefficients of this linear combination are referred to as components or coordinates on B of the vector. The elements of a basis are called basis vectors.
Equivalently B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.^[1] In more general terms, a basis is a linearly independent
spanning set.
A vector space can have generally several bases; however all the bases have the same number of elements, called the dimension of the vector space.
A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. This means that, a subset B of V is a basis if
it satisfies the two following conditions:
• the linear independence property:
for every finite subset {b[1], ..., b[n]} of B and every a[1], ..., a[n] in F, if a[1]b[1] + ⋅⋅⋅ + a[n]b[n] = 0, then necessarily a[1] = ⋅⋅⋅ = a[n] = 0;
for every (vector) v in V, it is possible to choose v[1], ..., v[n] in F and b[1], ..., b[n] in B such that v = v[1]b[1] + ⋅⋅⋅ + v[n]b[n].
The scalars v[i] are called the coordinates of the vector v with respect to the basis B, and by the first property they are uniquely determined.
A vector space that has a finite basis is called finite-dimensional. In this case, the subset {b[1], ..., b[n]} that is considered (twice) in the above definition may be chosen as B itself.
It is often convenient to ordering the basis vectors, typically when one consider the coefficients of a vector on a basis, without referring explicitly to the basis elements. In this case, the
ordering is necessary for associating each coefficient to the corresponding basis element. Generally, this ordering is implicitly done by numbering the basis elements. For example, when using
matrices, the ith row, and ith column refer to the ith element of a basis of some vector space. For emphasizing that an order has been chosen, one speaks of an ordered basis, which is therefore a
sequence rather than a set; see Ordered bases and coordinates below.
${\displaystyle (a,b)+(c,d)=(a+c,b+d),}$
and scalar multiplication
${\displaystyle \lambda (a,b)=(\lambda a,\lambda b),}$
where ${\displaystyle \lambda }$ is any real number. A simple basis of this vector space, called the standard basis consists of the two vectors e[1] = (1,0) and e[2] = (0,1), since, any vector v
= (a, b) of R^2 may be uniquely written as
${\displaystyle v=ae_{1}+be_{2}.}$
Any other pair of linearly independent vectors of R^2, such as (1, 1) and (−1, 2), forms also a basis of R^2.
• More generally, if F is a field, the set ${\displaystyle F^{n}}$ of n-tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let
${\displaystyle e_{i}=(0,\ldots ,0,1,0,\ldots ,0)}$
be the n-tuple with all components equal to 0, except the ith, which is 1. Then ${\displaystyle e_{1},\ldots ,e_{n}}$ is a basis of ${\displaystyle F^{n},}$ which is called the standard basis of
${\displaystyle F^{n}.}$
${\displaystyle B=\{1,X,X^{2},\ldots \}.}$
Any set of polynomials such that there is exactly one polynomial of each degree is also a basis. Such a set of polynomials is called a polynomial sequence. Example (among many) of such polynomial
sequences are Bernstein basis polynomials, and Chebyshev polynomials.
Many properties of finite bases result from the Steinitz exchange lemma, which states that, given a finite spanning set S and a linearly independent subset L of n elements of S, one may replace n
well chosen elements of S by the elements of L for getting a spanning set containing L, having its other elements in S, and having the same number of elements as S.
Most properties resulting from Steinitz exchange lemma remain true when there is no finite spanning set, but their proof in the infinite case requires generally the axiom of choice or a weaker form
of it, such as the ultrafilter lemma.
If V is a vector space over a field F, then:
• If L is a linearly independent subset of a spanning set S ⊆ V, then there is a basis B such that
${\displaystyle L\subseteq B\subseteq S.}$
• V has a basis (this is the preceding property with L being the empty set, and S = V).
• All bases of V have the same cardinality, which is called the dimension of V. This is the dimension theorem.
• A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.
• A linearly independent set L is a basis if and only if it is maximal, that is, it is not proper subset of any linearly independent set.
If V is a vector space of dimension n, then:
• A subset of V with n elements is a basis if and only if it is linearly independent.
• A subset of V with n elements is a basis if and only if it is spanning set of V.
Coordinates [edit]
Let V be a vector space of finite dimension n over a field F, and
${\displaystyle B=\{b_{1},\ldots ,b_{n}\}}$
be a basis of V. By definition of a basis, for every v in V may be written, in a unique way,
${\displaystyle v=\lambda _{1}b_{1}+\cdots +\lambda _{n}b_{n},}$
where the coefficients ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$ are scalars (that is, elements of F), which are called the coordinates of v over B. However, if one talks of the set of the
coefficients, one looses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, ${\displaystyle 3b_{1}+2b_{2}}$ and ${\
displaystyle 2b_{1}+3b_{2}}$ have the same set of coefficients {2, 3}, and are different. It is therefore often convenient to work with an ordered basis; this is typically done by indexing the basis
elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis
is also called a frame, a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.
Let, as usual, ${\displaystyle F^{n}}$ be the set of the n-tuples of elements of F. This set is an F-vector space, with addition and scalar multiplication defined component-wise. The map
${\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}b_{1}+\cdots +\lambda _{n}b_{n}}$
is a linear isomorphism from the vector space ${\displaystyle F^{n}}$ onto V. In other words, ${\displaystyle F^{n}}$ is the coordinate space of V, and the n-tuple ${\displaystyle \varphi ^{-1}(v)}$
is the coordinate vector of v.
The inverse image by ${\displaystyle \varphi }$ of ${\displaystyle b_{i}}$ is the n-tuple ${\displaystyle e_{i}}$ whose all components are 0, except the ith that is 1. The ${\displaystyle e_{i}}$
form an ordered basis of ${\displaystyle F^{n},}$ which is called its standard basis or canonical basis. The ordered basis B is the image by ${\displaystyle \varphi }$ of the canonical basis of ${\
displaystyle F^{n}.}$
It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of ${\displaystyle F^{n},}$ and that every linear isomorphism from ${\displaystyle F
^{n}}$ onto V may be defined as the isomorphism that maps the canonical basis of ${\displaystyle F^{n}}$ onto a given ordered basis of V. In other words it is equivalent to define an ordered basis of
V, or a linear isomorphism from ${\displaystyle F^{n}}$ onto V.
Change of basis[edit]
Let V be a vector space of dimension n over a field F. Given two (ordered) bases ${\displaystyle B_{o}=(v_{1},\ldots ,v_{n})}$ and ${\displaystyle B_{n}=(w_{1},\ldots ,w_{n})}$ of V, it is often
useful to express the coordinates of a vector x with respect to ${\displaystyle B_{o}}$ in terms of the coordinates with respect to ${\displaystyle B_{n}.}$ This can be done by the change of basis
formula, is described below. The subscripts o and n have been chosen because it is customary to refer to ${\displaystyle B_{o}}$ as the old basis and to ${\displaystyle B_{n}}$ as the new basis. It
is useful to describe the old coordinates in terms of the new ones, because in general one has an expression, in which the old coordinates are to be substituted by these terms in the new coordinates,
thus yielding an equivalent expression, involving the new coordinates, instead of the old ones.
Typically, the new basis vectors are given by their coordinates over the old basis, that is,
${\displaystyle w_{j}=\sum _{i=1}^{n}a_{i,j}v_{i}.}$
If ${\displaystyle (x_{1},\ldots ,x_{n})}$ and ${\displaystyle (y_{1},\ldots ,y_{n})}$ are the coordinates of a vector v over the old and the new basis respectively, one has
{\displaystyle {\begin{aligned}v&=\sum _{j=1}^{n}y_{j}w_{j}\\&=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}v_{i}\\&=\sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{i,j}y_{j}\right)v_{i}\\&=\sum _{i=1}^{n}x_
The formula for changing the coordinates with respect to the other basis results from the uniqueness of the decomposition of a vector over a basis, and is thus
${\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},}$
for i = 1, ..., n.
This formula may be concisely written in matrix notation. Let A be the matrix of the ${\displaystyle a_{i,j},}$ and
${\displaystyle X={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\quad }$ and ${\displaystyle \quad Y={\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}}}$
be the column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is
${\displaystyle X=AY.}$
Related notions[edit]
Free module[edit]
If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of a module. For modules, linear independence and spanning sets are defined exactly as for
vector spaces, although "generating set" is more commonly used than that of "spanning set".
Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A
module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions.
A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not
shared by modules over other rings. Specifically, every subgroup of a free abelian group is a group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group
that has a finite basis), there is a basis ${\displaystyle e_{1},\ldots ,e_{n}}$ of H and an integer 0 ≤ k ≤ n such that ${\displaystyle a_{1}e_{1},\ldots ,a_{k}e_{k}}$ is a basis of G, for some
nonzero integers ${\displaystyle a_{1},\ldots ,a_{k}.}$ For details, see Free abelian group § Subgroups.
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel) or algebraic basis can be used to refer to a basis as defined in
this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are
orthogonal bases on Hilbert spaces, Schauder bases, and Markushevich bases on normed linear spaces. In the case of the real numbers R viewed as a vector space over the field Q of rational numbers,
Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number ${\displaystyle 2^{\aleph _{0}},}$ where ${\displaystyle \aleph _{0}}$ is the
smallest infinite cardinal, the cardinal of the integers.
The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite
sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces, Banach spaces, or Fréchet spaces.
The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinite-dimensional normed vector
space which is complete (i.e. X is a Banach space), then any Hamel basis of X is necessarily uncountable. This is a consequence of the Baire category theorem. The completeness as well as infinite
dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (non-complete) normed spaces which have
countable Hamel bases. Consider ${\displaystyle c_{00}}$, the space of the sequences ${\displaystyle x=(x_{n})}$ of real numbers which have only finitely many non-zero elements, with the norm ${\
displaystyle \|x\|=\sup _{n}|x_{n}|.}$ Its standard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.
In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are an "orthogonal basis" of the (real or complex) vector space of all (real or complex
valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions f satisfying
${\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .}$
The functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are linearly independent, and every function f that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the
sense that
${\displaystyle \lim _{n\rightarrow \infty }\int _{0}^{2\pi }{\biggl |}a_{0}+\sum _{k=1}^{n}{\bigl (}a_{k}\cos(kx)+b_{k}\sin(kx){\bigr )}-f(x){\biggr |}^{2}\,dx=0}$
for suitable (real or complex) coefficients a[k], b[k]. But many^[2] square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not
comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas
orthonormal bases of these spaces are essential in Fourier analysis.
The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis.^[3] An affine basis for an n-dimensional affine space is ${\displaystyle n+1}$ points
in general linear position. A projective basis is ${\displaystyle n+2}$ points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of
its convex hull. A cone basis^[4] consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).
Random basis[edit]
For a probability distribution in R^n with a probability density function, such as the equidistribution in a n-dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and
independently chosen vectors will form a basis with probability one, which is due to the fact that n linearly dependent vectors x[1], ..., x[n] in R^n should satisfy the equation det[x[1], ..., x[n]]
= 0 (zero determinant of the matrix with columns x[i]), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.^[5]^
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal to y if ${\
displaystyle |\langle x,y\rangle |/(\|x\|\|y\|)<\epsilon }$ (that is, cosine of the angle between x and y is less than ε).
In high dimensions, two independent random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise almost
orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in n-dimensional ball. Choose N independent random vectors from a ball (they are independent and identically
distributed). Let θ be a small positive number. Then for
${\displaystyle N\leq e^{\frac {\epsilon ^{2}n}{4}}[-\ln(1-\theta )]^{\frac {1}{2}}}$ (Eq. 1)
N random vectors are all pairwise ε-orthogonal with probability 1 − θ.^[6] This N growth exponentially with dimension n and ${\displaystyle N\gg n}$ for sufficiently big n. This property of random
bases is a manifestation of the so-called measure concentration phenomenon.^[7]
The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1, 1]^n as a function of
dimension, n. A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within π/2 ± 0.037π/2 then the vector was
retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the
vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each n, 20
pairwise almost orthogonal chains where constructed numerically for each dimension. Distribution of the length of these chains is presented.
Proof that every vector space has a basis[edit]
Let V be any vector space over some field F. Let X be the set of all linearly independent subsets of V.
The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆.
Let Y be a subset of X that is totally ordered by ⊆, and let L[Y] be the union of all the elements of Y (which are themselves certain subsets of V).
Since (Y, ⊆) is totally ordered, every finite subset of L[Y] is a subset of an element of Y, which is a linearly independent subset of V, and hence every finite subset of L[Y] is linearly
independent. Thus L[Y] is linearly independent, so L[Y] is an element of X. Therefore, L[Y] is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element Y.
As X is nonempty, and every totally ordered subset of (X, ⊆) has an upper bound in X, Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L[max] of X
satisfying the condition that whenever L[max] ⊆ L for some element L of X, then L = L[max].
It remains to prove that L[max] is a basis of V. Since L[max] belongs to X, we already know that L[max] is a linearly independent subset of V.
If L[max] would not span V, there would exist some vector w of V that cannot be expressed as a linear combination of elements of L[max] (with coefficients in the field F). In particular, w cannot be
an element of L[max]. Let L[w] = L[max] ∪ {w}. This set is an element of X, that is, it is a linearly independent subset of V (because w is not in the span of L[max], and L[max] is independent). As L
[max] ⊆ L[w], and L[max] ≠ L[w] (because L[w] contains the vector w that is not contained in L[max]), this contradicts the maximality of L[max]. Thus this shows that L[max] spans V.
Hence L[max] is linearly independent and spans V. It is thus a basis of V, and this proves that every vector space has a basis.
This proof relies on Zorn's lemma, which is equivalent to the axiom of choice. Conversely, it may be proved that if every vector space has a basis, then the axiom of choice is true; thus the two
assertions are equivalent.
See also[edit]
General references[edit]
• Blass, Andreas (1984), "Existence of bases implies the axiom of choice", Axiomatic set theory, Contemporary Mathematics volume 31, Providence, R.I.: American Mathematical Society, pp. 31–33, ISBN
0-8218-5026-1, MR 0763890
• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6
Historical references[edit]
• Banach, Stefan (1922), "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)"
(PDF), Fundamenta Mathematicae (in French), 3, ISSN 0016-2736
• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)
• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann
• Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", Historia Mathematica, 22 (3): 227–261, doi:10.1006/hmat.1995.1024, MR 1347828
• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et fils
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Extension Theory,
Kannenberg, L.C., Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5
• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy
• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of
geometry) (in German), archived from the original on 2009-04-12
• Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940", Historia Mathematica, 22 (3): 262–303, doi:10.1006/hmat.1995.1025
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin
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9.3 - Time-to-event outcome
9.3 - Time-to-event outcome
Examples of time-to-event data are:
• Time to death
• Time to development of a disease
• Time to first hospitalization
• And many others
One may think that time-to-event data is simply continuous, but since we do not observe the true time for each person in the dataset, this is not the case. The people who do not experience the event
still contribute valuable information, and we refer to these patients as “censored”. We use the time they contribute until they are censored, which is the time they stop being followed because the
study has ended, they are lost to follow-up, or they withdraw from the study.
For our example, we are interested in the time to development of coronary heart disease (CHD). No patients had CHD upon study entry, and patients were surveyed every 2 years to see if they had
developed CHD. Each patient’s “time-to-CHD” will fall into one of these categories:
1. They develop CHD within the 30-year study period
Time = years until they develop CHD
Status = event
2. They do not develop CHD within the 30-year study period, and they stay in the study until the end
Time = 30 years
Status = censored
3. They do not develop CHD within the 30-year study period, and they leave the study before the 30-year study period is finished (due to death, moving, lost contact, voluntarily withdraw, etc.)
Time = time on study
Status = censored
The best way to describe time-to-event data is by the Kaplan-Meier method. This uses information from all patients, and differentiates between patients who did and did not experience the event. A
Kaplan Meier (KM) plot is how we visualize time-to-event data and starts with all patients being event-free at time 0. The KM method uses the number of patients still at risk over time, and patients
drop out once they experience the event or are censored. A Kaplan Meier plot and a Cumulative Incidence plot are inverses of each other, so you can choose which best fits your data. Often for
“Overall Survival” we use KM plots, which start at 100%, and decrease over time as patients either die or are censored. This can really be considered as plotting the percentage of patients still
alive. For our example, it makes more sense to look at a cumulative incidence plot, which starts at 0% and shows how the incidence of CHD increases over time. (A KM plot would plot the percent of
people who are CHD-free, and this would decrease over time.)
This plot shows that over time CHD is increasing, and we can get estimates of rates of CHD at different time points using the KM estimate.
When comparing time-to-event data between groups, we can use the KM method again, as well as perform a log-rank test. For our example, suppose we want to compare time to CHD by BP status.
This plot shows that those with high BP at study entry (blue line) have higher rates of CHD than those with low or normal BP (red line). The KM estimates of CHD at 10 years are 12.7% for the high BP
group and 4.7% for the low/normal group. At 20 years, these estimates are 26.1% and 12.0%. The log-rank test is essentially a comparison of lines, not specifically comparing estimates at any single
point, and is highly significant here (p<0.0001).
Modeling (Multivariable Associations)
We can use Cox Proportional Hazards modeling to estimate the hazard ratio. This model uses the hazard function which is the probability that if a person survives to time t, they will experience the
event in the next instant.
Just from eyeballing the previous plot, it appears that the risk of CHD is about twice as high for those with high BP compared to those with low/normal. Actually fitting a Cox model with high BP as a
single covariate shows that the estimated hazard ratio is 1.87 (95% CI: 1.69 - 2.08), which fits with what we see in the plot.
The Cox models can also include multiple covariates to test for confounding and interaction terms to evaluate effect modification, similar to those in previous sections. With additional terms in the
model, we can estimate adjusted hazard ratios. | {"url":"https://online.stat.psu.edu/stat507/book/export/html/767","timestamp":"2024-11-11T19:27:11Z","content_type":"text/html","content_length":"10223","record_id":"<urn:uuid:e7111302-59fe-48af-9347-08c1105f8cc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00552.warc.gz"} |
Structural Geology Identification based on Derivative Analysis Gravity Data in Tangkuban Perahu Mountain
Structural Geology Identification based on Derivative Analysis Gravity Data in Tangkuban Perahu Mountain
Gravity, structure geology, modeling, tangkuban perahu, geothermal
The earth is composed of structures with different rock types, properties, and characteristics and can be known by applying the laws of physics in the form of geophysical methods such as the gravity
method. The gravity method is a passive geophysical method that is widely used for geodynamic and exploration studies in estimating fault structures. The aim of this research is to model the
subsurface geological structure based on the results of derivative analysis of gravity data related to geothermal prospects. The data used are GGMplus gravity acceleration data and topography
(elevation) from each measurement point, totaling 6889. The data was then subjected to several corrections to produce a complete Bouguer anomaly. Then, the next stage is derivative analysis, which is
used to obtain a subsurface geological structure model and geothermal prospects for the Tangkuban Perahu area. Based on the correlation between derivative analysis and two-dimensional modeling
results, it can be seen that the Tangkuban Perahu geothermal system is controlled by structures in the form of horsts and grabens formed due to Tangkuban Perahu volcanic activity. The Tangkuban
Perahu geothermal reservoir prospect is estimated to be at a depth of around 0.6 km – 2.8 km with a density ranging from 2.15 g/cc to 2.45 g/cc, which is estimated to be basalt breccia.
Download data is not yet available.
E. Huenges and P. Ledru, Geothermal energy systems: exploration, development, and utilization. 2011.
S. Sudarman, K. Pudyastuti, and S. Aspiyo, “Geothermal Development Progress In Indonesia: Geothermal Development Progress in Indonesia: COUNTRY Country Update 1995-2000,” Proceedings World Geothermal
Congress 2000, pp. 455–460, 2000.
M. Sarkowi and R. C. Wibowo, “Geothermal Reservoir Identification based on Gravity Data Analysis in Rajabasa Area- Lampung,” RISET Geologi dan Pertambangan, vol. 31, no. 2, p. 77, 2021, doi: 10.14203
M. S. Mohamed El-Tokhey, Mohamed Elhabiby, Ahmed Ragheb, “Gravity and Density Relationship,” Int J Sci Eng Res, vol. 6, no. 1, pp. 1359–1363, 2015.
A. Zaenudin, R. Risman, I. G. B. Darmawan, and I. B. S. Yogi, “Analysis of gravity anomaly for groundwater basin in Bandar Lampung city based on 2D gravity modeling,” in Journal of Physics:
Conference Series, Bandar Lampung: IOP Publishing, 2020, pp. 0–7. doi: 10.1088/1742-6596/1572/1/012006.
W. Andari, Karyanto, and R. Kurniawan, “Identifikasi Batas Sub-Cekungan Hidrokarbon Menggunakan Analisis SHD (Second Horizontal Derivative) dan SVD (Second Vertical Derivative) Berdasarkan Korelasi
Data PT. Chevron Pasific Indonesia satu metode yang ada dalam geofisika gravitasi akibat var,” Jurnal Geofisika Eksplorasi, vol. 5, no. 1, pp. 60–74, 2019, doi: 10.23960/jge.v.
S. Soerjadji, “Geothermal Exploration Drilling and N Kamojang, West Java Indonesia Testing I,” no. June 1976, 2000.
M. Y. Kamah, A. Armando, D. L. Rahmani, and S. Paramitha, “Enhancement of subsurface geologic structure model based on gravity, magnetotelluric, and well log data in Kamojang geothermal field,” IOP
Conf Ser Earth Environ Sci, vol. 103, no. 1, 2018, doi: 10.1088/1755-1315/103/1/012013.
H. Saibi, J. Nishijima, S. Ehara, and E. Aboud, “Integrated gradient interpretation techniques for 2D and 3D gravity data interpretation,” Earth, Planets and Space, vol. 58, no. 7, pp. 815–821, 2006,
doi: 10.1186/BF03351986.
How to Cite
S. Rasimeng, T. P. E. . Pratama, and R. C. Wibowo, “Structural Geology Identification based on Derivative Analysis Gravity Data in Tangkuban Perahu Mountain”, JESRsf, vol. 6, no. 1, pp. 7 –, Jun. | {"url":"http://jesr.eng.unila.ac.id/index.php/ojs/article/view/163","timestamp":"2024-11-01T19:27:53Z","content_type":"text/html","content_length":"45578","record_id":"<urn:uuid:643edaa6-63a9-416e-94e2-ba017fe3ad18>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00827.warc.gz"} |
Linear Convolution Filters
All linear convolution filters compute weighted averages of the neighboring input grid nodes. The only differences between the various linear convolution filters are the size and shape of the
neighborhood, and the specific weights used. Consider the computation of the output grid value at row r and column
where W(i, are the weights defined for the specified filter. The output grid node value is then
User Defined Filters
There are two types of user defined filters, Low-pass Filters and General User-defined. With these filters, you can specify the height and width of the filter neighborhood.
Low-pass Filters
A low-pass filter removes the high frequency noise with the resulting output being a smoother grid. There are four user-defined low-pass filters. Each of these four filters allows you to specify the
size of the neighborhood. The width and height of the filter neighborhood must both be positive, odd numbers. Let the neighborhood height be S and width be T,
Average In the Moving Average (mxn) filter the weights are all equal to one.
Distance With the Distance Weighting (mxn) filter the weights fall-off with increased distance. The distance weighting function is
(mxn) where p is the specified Power. The higher the power the more rapidly the weights fall-off with distance. The resulting iso-weight contour lines are concentric rectangles.
Inverse With the Inverse Distance (mxn) filter, the weights fall-off with increased distance. With a neighborhood height S and width T, the distance weighting function is
(mxn) where WCis the specifiedCentral Weightandpis thePower. The higher the powerp, the more rapidly the weights fall-off with distance. The resulting
With the Gaussian Low-pass (mxn) filter, the weights fall-off with increased distance. With a neighborhood height S and width T, the distance weighting function is
Low-pass whereAlpha value (positive). This weight function takes the form of half the common bell-shaped curve. The Alpha parameter controls how quickly the weights fall-off with distance. The
(mxn) resulting iso-weight contour lines are concentric ellipses. The lower the Alpha value, the more weight neighborhood points have on the grid value and the slower the weight drops off.
Conversely, the higher the Alpha value, the more weight the center point has on the grid value and the faster the weight of the other points drops off.
General User-defined Filter
The General User-defined (mxn) linear filter allows you to specify the height and width of the filter neighborhood and any combination of weights. The box in the lower-right part of
The General the dialog displays the neighborhood size, based on the number of Rows and Cols, along with the weights for each grid node in the neighborhood. Click in a cell in the box to change
User-Defined the node's weight.
The grid matrix can be selected and copied using CTRL+C and pasted into the user-defined matrix. This allows a pre-defined matrix to be used as a base and then be modified.
Predefined Filters
The predefined filters section is a large collection of 3×3 filters defined in the grid filter references.
Low-Pass Filters Low-pass Filters are also known as smoothing or blurring filters. These filters remove the high frequency variation.
High-Pass Filters High-pass Filters are also known as sharpening or crispening filters. They have the opposite effect of blurring. They tend to remove the background variation and emphasize
the local details.
Order 1 Derivative Order 1 Derivative Filters are used to find horizontal and vertical edges.
Order 2 Derivative Order 2 Derivative Filters are another set of edge enhancement filters.
Shift and Difference Shift and Difference Filters are the two simplest horizontal and vertical differential operators.
Gradient Directional Gradient Directional Filters compute and return the directional derivatives in each of the eight compass directions.
Embossing Filters Embossing Filters identify and enhance edges aligned in one of the eight compass directions.
See Also | {"url":"https://surferhelp.goldensoftware.com/gridmisc/Linear_Convolution_Filters.htm","timestamp":"2024-11-04T08:12:13Z","content_type":"text/html","content_length":"33195","record_id":"<urn:uuid:5267ca94-b946-4681-905e-741d1e05167f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00408.warc.gz"} |
Van der Waerden's theorem - (Enumerative Combinatorics) - Vocab, Definition, Explanations | Fiveable
Van der Waerden's theorem
from class:
Enumerative Combinatorics
Van der Waerden's theorem states that for any given positive integers $k$ and $r$, there exists a minimum integer $N$ such that if the integers from 1 to $N$ are colored with $r$ different colors, at
least one monochromatic arithmetic progression of length $k$ will appear. This theorem links coloring problems in combinatorics with the existence of arithmetic progressions, highlighting a
foundational result in Ramsey theory.
congrats on reading the definition of van der Waerden's theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Van der Waerden's theorem can be applied in various fields such as computer science, number theory, and combinatorial design.
2. The value of $N$ required by van der Waerden's theorem grows quickly as $k$ and $r$ increase, making exact values difficult to determine for larger cases.
3. This theorem emphasizes the inevitability of structure within large sets, showing that patterns emerge even when elements are distributed randomly.
4. The proof of van der Waerden's theorem uses induction and combinatorial arguments, highlighting its deep connection to other areas in mathematics.
5. Van der Waerden's theorem can be seen as a specific case of a more general concept known as the Hales-Jewett theorem.
Review Questions
• How does van der Waerden's theorem demonstrate the relationship between colorings and arithmetic progressions?
□ Van der Waerden's theorem illustrates that when you color integers with a limited number of colors, patterns inevitably emerge. Specifically, it guarantees that no matter how you apply these
colors, you will find a monochromatic arithmetic progression of a specified length. This relationship shows that even seemingly random distributions can lead to structured outcomes,
emphasizing the intrinsic order present in mathematical systems.
• Discuss the implications of van der Waerden's theorem on Ramsey Theory and its relevance to combinatorial problems.
□ Van der Waerden's theorem serves as a cornerstone result in Ramsey Theory by providing insight into how order emerges from chaos. Its implications extend to various combinatorial problems
where one seeks to find order within large sets or structures. By establishing a minimum integer that guarantees the existence of monochromatic arithmetic progressions, it lays the groundwork
for further explorations into other combinatorial designs and coloring problems.
• Evaluate how van der Waerden's theorem connects to other major results in combinatorial design and its broader significance.
□ Van der Waerden's theorem connects deeply with other significant results like the Hales-Jewett theorem, showcasing how patterns and structures can be found in higher dimensions and more
complex configurations. This connection illustrates the broader significance of such results in understanding mathematical phenomena beyond simple number coloring. By linking colorings to
arithmetic structures, van der Waerden's theorem enriches our understanding of not just number theory but also how these principles can inform algorithms and problem-solving strategies across
various scientific disciplines.
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Two conducting cylinders of equal length
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Two conducting cylinders of equal length: Heat Conduction
Problem: Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures T[1] = 300 K and T[2] = 100 K, as shown in the figure. The
radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are K[1] and K[2] respectively. If the
temperature at the junction of the two cylinders in the steady state is 200 K, then K[1]/K[2] = ? (IIT JEE 2018)
Answer: The answer is 4 i.e., ratio of thermal condictivites of $K_1/ K_2=4$.
Solution: The temperature of the source is $T_1=300$ K and that of the sink is $T_{2}=100$ K. The temperature at the junction of two cylinders is $T=200$ K. The radius of the bigger cylinder, $r_2$,
is twice the radius of the smaller cylinder, $r_1$ i.e., $r_2=2r_1$.
The rate of heat conduction through a material, with conductivity $K$, cross-section area $A$, length $\Delta x$, and temperature difference between the two ends $\Delta T$, is given by \frac{\Delta
Q}{\Delta t}=K A \frac{\Delta T}{\Delta x}. \nonumber Thus, the rates of heat conduction through the two cylinders are \frac{\Delta Q_1}{\Delta t}&=K_1 (\pi r_1^2) \frac{T_1-T}{L_1},\nonumber \\ \
frac{\Delta Q_2}{\Delta t}&=K_2 (\pi r_2^2) \frac{T-T_2}{L_2}.\nonumber There is no heat loss from the cylinders because they are covered with an insulating material. Also, there is no heat
accumulation in steady state. Thus, $\Delta Q_1/\Delta t=\Delta Q_2/\Delta t$, which gives \frac{K_1}{K_2}&=\frac{r_2^2 (T-T_2)/L_2}{r_1^2 (T_1-T)/L_1}\\ &=\frac{(2r_1)^2 (200-100)/L}{ r_1^2
(300-200)/L}\\ &=4.\nonumber
More on Heat Conduction
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Modelling functions of sequential data with neural networks and the signature transform
Oxford Mathematician Patrick Kidger talks about his recent work on applying the tools of controlled differential equations to machine learning.
Sequential Data
The changing air pressure at a particular location may be thought of as a sequence in $\mathbb{R}$; the motion of a pen on paper may be thought of as a sequence in $\mathbb{R}^2$; the changes within
financial markets may be thought of as a sequence in $\mathbb{R}^d$, with $d$ potentially very large.
The goal is often to learn some function of this data, for example to understand the weather, to classify what letter has been drawn, or to predict how financial markets will change.
In all of these cases, the data is ordered sequentially, meaning that it comes with a natural path-like structure: in general the data may be thought of as a discretisation of a path $f \colon [0, 1]
\to V$, where $V$ is some Banach space. (In practice this is typically $V = \mathbb{R}^d$.)
The Signature Transform
When we know that data comes with some extra structure like this, we can seek to exploit that knowledge by using tools specifically adapated to the problem. For example, a tool for sequential data
that is familiar to many people is the Fourier transform.
Here we use something similar, called the signature transform, which is famous for its use in rough path theory and controlled differential equations.
The signature transform has a rather complicated looking definition: \[ \mathrm{Sig}^N(f) = \left(\left(\,\underset{0 < t_1 < \cdots < t_k < 1}{\int \cdots \int} \prod_{j = 1}^k \frac{\mathrm{d}f_
{i_j}}{\mathrm{d}t}(t_j)\mathrm{d}t_1\cdots\mathrm{d}t_k \right)_{1\leq i_1,\ldots, i_k \leq d}\right)_{1\leq k \leq N} \]
Whilst the Fourier transform extracts information about frequency, the signature transform instead extracts information about order and area. (It turns out that order and area are, in a certain
sense, the same thing.)
Furthermore (and unlike the Fourier transform), order and area represent all possible nonlinear effects: the signature transform is a universal nonlinearity, meaning that every continuous function of
the underlying path corresponds to just a linear function of its signature.
(Technically speaking, this is because the Fourier transform uses a basis for the space of paths, whilst the signature transform uses a basis for the space of functions of paths.)
Besides this, the signature transform has many other nice properties, such as robustness to missing or irregularly sampled data, optional translation invariance, and optional sampling invariance.
Applications to Machine Learning
Machine learning, and in particular neural networks, is famous for its many recent achievements, from image classification to self driving cars.
Given the great theoretical success of the signature transform, and the great empirical success of neural networks, it has been natural to try and bring these two together.
In particular, the problem of choosing activation functions and pooling functions for neural networks has usually been a matter of heuristics. Here, however, the theory behind the signature transform
makes it a mathematically well-motivated choice of pooling function, specifically adapted to handle sequential data such as time series.
Bringing these two points of view together has been the purpose of the recent paper Deep Signature Transforms (accepted at NeurIPS 2019) by Patrick Kidger, Patric Bonnier, Imanol Perez Arribas,
Cristopher Salvi, and Terry Lyons. Alongside this we have released Signatory, an efficient implementation of the signature transform capable of integrating with modern deep learning frameworks. | {"url":"https://www.maths.ox.ac.uk/node/34644","timestamp":"2024-11-08T22:02:17Z","content_type":"text/html","content_length":"53196","record_id":"<urn:uuid:7f962c51-b01f-491e-9fff-a46fcf8f080a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00372.warc.gz"} |
Spoofax - Strategic Rewriting
Strategic Rewriting¶
Limitations of Term Rewriting¶
Term rewriting involves exhaustively applying rules to subterms until no more rules apply. This requires a strategy for selecting the order in which subterms are rewritten. The innermost strategy
applies rules automatically throughout a term from inner to outer terms, starting with the leaves. The nice thing about term rewriting is that there is no need to define traversals over the syntax
tree; the rules express basic transformation steps and the strategy takes care of applying it everywhere. However, the complete normalization approach of rewriting turns out not to be adequate for
program transformation, because rewrite systems for programming languages will often be non-terminating and/or non-confluent. In general, it is not desirable to apply all rules at the same time or to
apply all rules under all circumstances.
The usual solution is to encode the strategy in the rewrite rules. But this intertwines the strategy with the rules, and makes the latter unreusable.
Programmable Rewriting Strategies¶
In general, there are two problems with the functional approach to encoding the control over the application of rewrite rules, when comparing it to the original term rewriting approach: traversal
overhead and loss of separation of rules and strategies.
In the first place, the functional encoding incurs a large overhead due to the explicit specification of traversal. In pure term rewriting, the strategy takes care of traversing the term in search of
subterms to rewrite. In the functional approach traversal is spelled out in the definition of the function, requiring the specification of many additional rules. A traversal rule needs to be defined
for each constructor in the signature and for each transformation. The overhead for transformation systems for real languages can be inferred from the number of constructors for some typical
language : constructors
Tiger : 65
C : 140
Java : 140
COBOL : 300 - 1200
In the second place, rewrite rules and the strategy that defines their application are completely intertwined. Another advantage of pure term rewriting is the separation of the specification of the
rules and the strategy that controls their application. Intertwining these specifications makes it more difficult to understand the specification, since rules cannot be distinguished from the
transformation they are part of. Furthermore, intertwining makes it impossible to reuse the rules in a different transformation.
Stratego introduces the paradigm of programmable rewriting strategies with generic traversals, a unifying solution in which application of rules can be carefully controlled, while incurring minimal
traversal overhead and preserving separation of rules and strategies^1.
The following are the design criteria for strategies in Stratego:
• Separation of rules and strategy: Basic transformation rules can be defined separately from the strategy that applies them, such that they can be understood independently.
• Rule selection: A transformation can select the necessary set of rules from a collection (library) of rules.
• Control: A transformation can exercise complete control over the application of rules. This control may be fine-grained or course-grained depending on the application.
• No traversal overhead: Transformations can be defined without overhead for the definition of traversals.
• Reuse of rules: Rules can be reused in different transformations. Reuse of traversal schemas: Traversal schemas can be defined generically and reused in different transformations.
Idioms of Strategic Rewriting¶
We will examine the language constructs that Stratego provides for programming with strategies, starting with the low-level actions of building and matching terms. To get a feeling for the purpose of
these constructs, we first look at a couple of typical idioms of strategic rewriting.
Cascading Transformations¶
The basic idiom of program transformation achieved with term rewriting is that of cascading transformations. Instead of applying a single complex transformation algorithm to a program, a number of
small, independent transformations are applied in combination throughout a program or program unit to achieve the desired effect. Although each individual transformation step achieves little, the
cumulative effect can be significant, since each transformation feeds on the results of the ones that came before it.
One common cascading of transformations is accomplished by exhaustively applying rewrite rules to a subject term. In Stratego the definition of a cascading normalization strategy with respect to
rules R1, … , Rn can be formalized using the innermost strategy that we saw before:
simplify = innermost(R1 <+ ... <+ Rn)
The argument strategy of innermost is a selection of rules. By giving different names to rules, we can control the selection used in each transformation. There can be multiple applications of
innermost to different sets of rules, such that different transformations can co-exist in the same module without interference. Thus, it is now possible to develop a large library of transformation
rules that can be called upon when necessary, without having to compose a rewrite system by cutting and pasting. For example, the following module defines the normalization of proposition formulae to
both disjunctive and to conjunctive normal form:
module prop-laws
imports prop
DefI : Impl(x, y) -> Or(Not(x), y)
DefE : Eq(x, y) -> And(Impl(x, y), Impl(y, x))
DN : Not(Not(x)) -> x
DMA : Not(And(x, y)) -> Or(Not(x), Not(y))
DMO : Not(Or(x, y)) -> And(Not(x), Not(y))
DAOL : And(Or(x, y), z) -> Or(And(x, z), And(y, z))
DAOR : And(z, Or(x, y)) -> Or(And(z, x), And(z, y))
DOAL : Or(And(x, y), z) -> And(Or(x, z), Or(y, z))
DOAR : Or(z, And(x, y)) -> And(Or(z, x), Or(z, y))
dnf = innermost(DefI <+ DefE <+ DAOL <+ DAOR <+ DN <+ DMA <+ DMO)
cnf = innermost(DefI <+ DefE <+ DOAL <+ DOAR <+ DN <+ DMA <+ DMO)
The rules are named, and for each strategy different selections from the rule set are made.
One-pass Traversals¶
Cascading transformations can be defined with other strategies as well, and these strategies need not be exhaustive, but can be simpler one-pass traversals. For example, constant folding of Boolean
expressions only requires a simple one-pass bottom-up traversal. This can be achieved using the bottomup strategy according the following scheme:
simplify = bottomup(repeat(R1 <+ ... <+ Rn))
The bottomup strategy applies its argument strategy to each subterm in a bottom-to-top traversal. The repeat strategy applies its argument strategy repeatedly to a term.
Module prop-eval2 defines the evaluation rules for Boolean expressions and a strategy for applying them using this approach:
module prop-eval2
imports libstrategolib prop
Eval : Not(True()) -> False()
Eval : Not(False()) -> True()
Eval : And(True(), x) -> x
Eval : And(x, True()) -> x
Eval : And(False(), x) -> False()
Eval : And(x, False()) -> False()
Eval : Or(True(), x) -> True()
Eval : Or(x, True()) -> True()
Eval : Or(False(), x) -> x
Eval : Or(x, False()) -> x
Eval : Impl(True(), x) -> x
Eval : Impl(x, True()) -> True()
Eval : Impl(False(), x) -> True()
Eval : Impl(x, False()) -> Not(x)
Eval : Eq(False(), x) -> Not(x)
Eval : Eq(x, False()) -> Not(x)
Eval : Eq(True(), x) -> x
Eval : Eq(x, True()) -> x
main = io-wrap(eval)
eval = bottomup(repeat(Eval))
The strategy eval applies these rules in a bottom-up traversal over a term, using the bottomup(s) strategy. At each sub-term, the rules are applied repeatedly until no more rule applies using the
repeat(s) strategy. This is sufficient for the Eval rules, since the rules never construct a term with subterms that can be rewritten.
Another typical example of the use of one-pass traversals is desugaring, that is rewriting language constructs to more basic language constructs. Simple desugarings can usually be expressed using a
single top-to-bottom traversal according to the scheme
simplify = topdown(try(R1 <+ ... <+ Rn))
The topdown strategy applies its argument strategy to a term and then traverses the resulting term. The try strategy tries to apply its argument strategy once to a term.
Module prop-desugar defines a number of desugaring rules for Boolean expressions, defining propositional operators in terms of others. For example, rule DefN defines Not in terms of Impl, and rule
DefI defines Impl in terms of Or and Not. So not all rules should be applied in the same transformation or non-termination would result.
module prop-desugar
imports prop libstrategolib
DefN : Not(x) -> Impl(x, False())
DefI : Impl(x, y) -> Or(Not(x), y)
DefE : Eq(x, y) -> And(Impl(x, y), Impl(y, x))
DefO1 : Or(x, y) -> Impl(Not(x), y)
DefO2 : Or(x, y) -> Not(And(Not(x), Not(y)))
DefA1 : And(x, y) -> Not(Or(Not(x), Not(y)))
DefA2 : And(x, y) -> Not(Impl(x, Not(y)))
IDefI : Or(Not(x), y) -> Impl(x, y)
IDefE : And(Impl(x, y), Impl(y, x)) -> Eq(x, y)
desugar =
topdown(try(DefI <+ DefE))
impl-nf =
topdown(repeat(DefN <+ DefA2 <+ DefO1 <+ DefE))
main-desugar =
main-inf =
The strategies desugar and impl-nf define two different desugaring transformation based on these rules. The desugar strategy gets rid of the implication and equivalence operators, while the impl-nf
strategy reduces an expression to implicative normal-form, a format in which only implication (Impl) and False() are used.
A final example of a one-pass traversal is the downup strategy, which applies its argument transformation during a traversal on the way down, and again on the way up:
simplify = downup(repeat(R1 <+ ... <+ Rn))
An application of this strategy is a more efficient implementation of constant folding for Boolean expressions:
eval = downup(repeat(Eval))
This strategy reduces terms such as
And(... big expression ..., False())
in one step (to False() in this case), while the bottomup strategy defined above would first evaluate the big expression.
Staged Transformations¶
Cascading transformations apply a number of rules one after another to an entire tree. But in some cases this is not appropriate. For instance, two transformations may be inverses of one another, so
that repeatedly applying one and then the other would lead to non-termination. To remedy this difficulty, Stratego supports the idiom of staged transformation.
In staged computation, transformations are not applied to a subject term all at once, but rather in stages. In each stage, only rules from some particular subset of the entire set of available rules
are applied. In the TAMPR program transformation system this idiom is called sequence of normal forms, since a program tree is transformed in a sequence of steps, each of which performs a
normalization with respect to a specified set of rules. In Stratego this idiom can be expressed directly according to the following scheme:
simplify =
innermost(A1 <+ ... <+ Ak)
; innermost(B1 <+ ... <+ Bl)
; ...
; innermost(C1 <+ ... <+ Cm)
Local Transformations¶
In conventional program optimization, transformations are applied throughout a program. In optimizing imperative programs, for example, complex transformations are applied to entire programs. In
GHC-style compilation-by-transformation, small transformation steps are applied throughout programs. Another style of transformation is a mixture of these ideas. Instead of applying a complex
transformation algorithm to a program we use staged, cascading transformations to accumulate small transformation steps for large effect. However, instead of applying transformations throughout the
subject program, we often wish to apply them locally, i.e., only to selected parts of the subject program. This allows us to use transformations rules that would not be beneficial if applied
One example of a strategy which achieves such a transformation is
transformation =
; innermost(A1 <+ ... <+ An)
The strategy alltd(s) descends into a term until a subterm is encountered for which the transformation s succeeds. In this case the strategy trigger-transformation recognizes a program fragment that
should be transformed. Thus, cascading transformations are applied locally to terms for which the transformation is triggered. Of course more sophisticated strategies can be used for finding
application locations, as well as for applying the rules locally. Nevertheless, the key observation underlying this idiom remains: Because the transformations to be applied are local, special
knowledge about the subject program at the point of application can be used. This allows the application of rules that would not be otherwise applicable.
1. Eelco Visser, Zine-El-Abidine Benaissa, and Andrew P. Tolmach. Building program optimizers with rewriting strategies. In Matthias Felleisen, Paul Hudak, and Christian Queinnec, editors,
Proceedings of the third ACM SIGPLAN international conference on Functional programming, 13–26. Baltimore, Maryland, United States, 1998. ACM. URL: http://doi.acm.org/10.1145/289423.289425,
doi:10.1145/289423.289425. ↩
Last update: October 17, 2024
Created: October 17, 2024 | {"url":"https://spoofax.dev/background/stratego/strategic-rewriting/strategic-rewriting/","timestamp":"2024-11-10T04:26:18Z","content_type":"text/html","content_length":"156116","record_id":"<urn:uuid:a03d8015-0620-4d87-9b89-48d6ca2ec0ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00419.warc.gz"} |
PROSTATE VOLUME CALCULATION｜横浜市都筑区泌尿器科
A. Kimura, M. Yoshida, I. Saito
Department of Urology, Tokyo Kyosai Hospital, Japan
A common prostatic volume calculation method; height times width times length times pi/6 has three problems; First, reproducibility is poor, since the prostatic apex is frequently poorly visualized.
Second, errors increase as the angle between the height and the length moves out of perpendicular. Third, there is a risk of increasing false positives of PSA density in prostatic hypertrophy because
of its tendency to underestimate the volume in prostatic hypertrophy. These problems arise because only 3 diameters are measured in this method, which means only 6 points of the prostatic contours
are used for the calculation. Recently, we proposed a new method of calculation that we termed "biplane planimetry". By this new method, the prostatic contours of both cross- and sagittal sections
are traced. Based on the cross and sagittal contours, a non-ellipsoidal model is created. The model is composed of sequentially arranged copies of the cross-section, which are reduced so as to fit
the sagittal contour. Using full information obtained from biplane sections, the new method is resistant to the errors mentioned above.
Komine,Y.,Kimura,A.,Niizuma,M.,Nakamura,S.,Kawabe,K.,Niijima,T.: Transurethral ultrasonography of the prostate. The Prostate Suppl., 1,53-57,1981.
Kimura,A.,Nakamura,S.,Niijima,T.: Ultrasonic diagnosis of scrotal contents using newly developed circular compound scanner. J.Clin. Ultrasound 11,365-370,1983.
Kimura,A.,Nakamura,S.,Niizuma,M.,Hoshino,T.,Niijima,T.Ohashi,Y., Higuchi,T.: The quantitative analysis of ultrasonogram of the prostate. J.Clin.Ultrasound 14,501-507,1986.
Kimura,A.,Higashihara,E.,Aso,Y.:Oblique pyelography for focusing of extracorporeal shock wave lithotripsy.Jpn.J.Endourol.ESWL 2,34-36,1989.
Higashihara,E.,Asakage,Y.,Tominaga,T.,Hara,T.,Kimura,A.,Kishi,H.,Niijima,T.,Aso,Y.:Effect of shock wave application on the kidney:Comparisons with open surgery.Jpn.J.Endourol.ESWL 2,89-96,1989.
Kimura,A.,Kamiya,K.:Computer-aided drawing of a kidney which moves back and forth with respiration - A trial to aid patient's understanding of the ultrasonic monitor during extracorporeal shockwave
lithotripsy. Jpn.J.Endourol.ESWL 8,47-49,1995.
Kimura A,Kurooka Y,Hirasawa K et al:Accuracy of prostatic volume calculation in transrectal ultrasonography.Int J Urol 1995;2:252-256.
Kimura A:Re:Automated prostate volume determination with ultrasonographic imaging.J Urol 155,1038-1039,1996.
Nakamura,S.,Kobayashi,Y.,Tozuka,K.,Tokue,A.,Kimura,A.,Hamada,C.:Circadian changes in urine volume and frequency in elderly men.J.Urol.
Kimura A,Kurimoto S,Hosaka Y,Kitamura T,Nakamura S:International prostate symptom score (IPSS) overestimates the treatment efficacy - evaluation of IPSS by 24-hour uroflowmetry.Jpn.J.Endourol.ESWL
Enomoto Y,Fukuhara H,Kurimoto S,Sugimoto A,Kimura A,Hosaka Y,Kitamura T:Prostatic carcinoma presenting as a huge intravesical mass after subcapsilar prostatectomy for benign prostatic hyperplasia:an
usual manifestation.Brit J Urol,78,798-799,1996.
Nakamura S,Kobayashi Y,Tozuka K, Tokue A,Kimura A,Hamada C:Circadian changes in urine volume and frequency in elderly men.J Urol 1996;156:1275-1279.
Kimura A,Kurooka Y,Kitamura T et al:Biplane planimetry as a new method for prostatic volume calculation in transrectal ultrasonography.Int J Urol 1997;4:152-156.
Kimura A,Kawabe K:Accuracy of prostatic volume calculated by biplane planimetry.J Med Ultrasound 1997;5(Supple):31-34. | {"url":"https://www.akira-kimura.com/h/florence.html","timestamp":"2024-11-12T22:35:11Z","content_type":"text/html","content_length":"5471","record_id":"<urn:uuid:71ce8d59-c6f9-4f99-b89d-f04c240629d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00205.warc.gz"} |
Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of BF theory
We give an interpretation of the holographic correspondence between two-dimensional BF theory on the punctured disk with gauge group PSL(2, ℝ) and Schwarzian quantum mechanics in terms of a
Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over
exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian S^1-action on the orbits. The partition function is given by a sum
over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.
• Field Theories in Lower Dimensions
• Nonperturbative Effects
• Topological Field Theories
Dive into the research topics of 'Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of BF theory'. Together they form a unique fingerprint. | {"url":"https://researchprofiles.herts.ac.uk/en/publications/schwarzian-quantum-mechanics-as-a-drinfeld-sokolov-reduction-of-b-2","timestamp":"2024-11-09T07:57:46Z","content_type":"text/html","content_length":"47032","record_id":"<urn:uuid:59e13fa2-6fda-40fe-b430-5361d0c4e447>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00300.warc.gz"} |
i wrote a Sample code for Outputting integers
Hi everybody!
Here comes two procedures to print signed and unsigned integers to the screen.... It can also print numbers in other bases given by the base register (bx). The number to print is given by the
accumulator (ax)...
org 100h
mov ax,-1253 ; Number: -1253
call putsint ; Print it! (signed integer; base: 10 (default))
call newline
mov ax,4C8Fh ; Number: 4C8Fh (hex value)
mov bx,16 ; Base: 16 (hexadecimal)
call putint ; Print it! (unsigned integer)
call newline
mov ax,4360 ; Number: 4360
mov bx,10 ; Base: 10 (decimal)
call putint ; Print it! (unsigned integer)
xor ax,ax ; Function: Wait for key...
int 16h
mov ax,4C00h ; Exit to DOS
int 21h
; Procedures
; Go to next line
push ax dx
mov ah,02h
mov dl,0Dh
int 21h ; Output CR to screen
mov dl,0Ah
int 21h ; Output LF to screen
pop dx ax
; Print a signed integer to screen
; bx = base (default = 10), ax = number
push ax dx
cmp bx,0
jne .start ; if bx <> 0 then ...let's go!
mov bx,10 ; else bx = 10
cmp bx,10
jne .printit ; if it's not a decimal integer, we don't print any signs
cmp ax,0
jns .printit ; if it's not negative, we don't print any signs either
push ax
mov ah,02h
mov dl,"-"
int 21h ; output the "-"
pop ax
neg ax ; do the number positive
call putint ; now we can print the number...
pop dx ax
; Print an unsigned integer to screen
; bx = base (default = 10), ax = number
push ax bx cx dx
cmp bx,0
jne .start ; the same as in putsint... (if bx = 0 then bx = 10)
mov bx,10
xor cx,cx ; cx = 0
xor dx,dx ; dx = 0
div bx ; number / base
push dx ; push the remainder
inc cx ; increase the "digit-count"
cmp ax,0 ; if the quotient still is not 0, do it once more
jnz .new
pop dx ; pop the remainder
add dl,30h ; convert the number to a character
cmp dl,"9"
jng .ok ; if the charater is greater than "9" then we have
add dl,7 ; to add 7 to get A as 10, B as 11, and so on...
mov ah,02h ; output the character
int 21h
loop .loop
pop dx cx bx ax | {"url":"https://board.flatassembler.net/topic.php?t=42","timestamp":"2024-11-10T12:06:41Z","content_type":"text/html","content_length":"25061","record_id":"<urn:uuid:57a1a8b7-3068-456f-a742-5f59bef369a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00894.warc.gz"} |
How Much Does a Truck Hold of Sand? [Answered 2023] | Prettymotors
How Much Does a Truck Hold of Sand?
How much sand can a truck hold depends on its dimensions and the volume of its storage space. A truck can carry anywhere from five to fifty tons of sand, depending on its body size and body work. A
ten wheeler or a six-wheeler truck can hold up to fifty tons of sand. A semi-truck can haul eight-thousand pounds of sand. A garbage truck can hold up to 100 tons of rock.
Depending on the weight of the sand, a truck can carry a half-ton of gravel and two tons of sand. A truck’s capacity for gravel and sand is equal to the weight of the loads in the cargo space. A dump
truck can carry between 10 and 14 cubic yards at a time. The cost of sand varies, but it costs between $3 to $5 per 50-pound bag. For example, play sand is very fine and is usually used for play
areas, while all-purpose construction sand is a thicker variety that is commonly used under swimming pools and around paver stones.
The price per ton of sand varies, but most commonly-used types cost around $25 to $40 per ton. Some suppliers charge per cubic meter or even half-cubic meter. A ton of sand can cover around 80 square
feet of land at two inches of depth. However, the price per cubic meter is higher. For this reason, it is important to get the right truck for the job.
How Many Tons of Sand Can a Truck Carry?
If you’re planning to haul sand, you’ll need to know how much truck can handle the load. The most popular way to estimate the amount of sand that a truck can carry is to look at how much sand a cubic
yard contains. One cubic yard of sand weighs approximately four thousand pounds. To get an idea of how much sand a truck can carry, consider the size and weight of your Bobcat Loader. Then, multiply
that by four to get a rough estimate of how much sand it can carry.
Most trucks can carry around 450 cubic feet of sand. That’s about 20700 kilograms, which equals 20.7 tons. You’ll need to multiply the length, width, and depth in feet by 12 inches, and the total
volume (in cubic feet) will give you an estimate of how many tons your truck can handle. You can also multiply the total length by the total weight of the load.
How Many Yards of Sand are in a Dump Truck Load?
How much sand does a dump truck load contain? Depending on the type of sand, a typical truckload can weigh between fifteen and forty cubic meters. The weight of sand will also vary depending on the
size of the particles. Sand typically weighs between 3,000 and 5,000 pounds, so a truckload of ten cubic yards will require approximately seventy tons of sand.
The amount of sand in a dump truck load depends on the type of material and size of the truck. A cubic yard of trash will weigh significantly less than a cubic yard of sand. To figure out the cubic
yard capacity of a dump truck, check its gross vehicle weight rating. This figure is very close to the load capacity. However, before ordering a load, make sure you know how many cubic yards it can
A full size dump truck can carry between 12 and 16 cubic yards. A semi-dump truck can carry anywhere from eight to fourteen cubic yards. A smaller pickup truck can carry only half of a cubic yard.
When choosing the type of sand, consider the purpose of the load. If it is to be used for construction, fill sand is best for non-high-traffic areas. This type of sand promotes percolation and
drainage. Fill dirt is comprised of broken down rocks, sand, and clay. It is much more stable than topsoil, making it a more versatile filler.
How Heavy is a Truck of Sand?
The weight of a truckload of sand varies depending on its density and moisture content. A cubic yard of sand will weigh approximately 500 pounds. However, the weight of a truckload of wet sand will
weigh between six and eight hundred pounds. Despite these differences, these two factors are helpful guidelines when choosing a truckload of sand for a project.
When you have a small load of sand, it’s easier to estimate the weight of a full truckload. A cubic foot of sand weighs approximately 80 pounds, so a truckload of sand weighing seven hundred pounds
would weigh about six thousand pounds. The volume of the truckload is about 450 cubic feet, which is not very large compared to the size of a truckload of top soil or mulch.
Sand trucks usually have a slightly irregular elevation, so they are not level. You should level the truck before starting the measurement. To do this, use a 16mm steel bar and measure the inner
depth of the truck. Then, mark four or five points on the steel bar. Make sure you use a measuring tape so you don’t make any mistakes when measuring the height. If you miss even a single point, you
will have the wrong calculation.
How Much is a Truck Bed Load of Sand?
How much sand is a truck bed loaded with? Sand in a truck bed weighs approximately 20 cubic feet per ton, or 0.75 cubic yard. A ton of sand will cover approximately 120 square feet if you dump it at
a 2 inch depth. When dumped at a 3 or 4 inch depth, the load will cover approximately 80 square feet. Therefore, the amount of sand you need to fill a truck bed can vary greatly.
Sand is often priced per cubic yard, with finer grain varieties costing more. Generally, screened sand costs $15 to $20 per cubic yard, depending on its usage. Sand can be used in sandboxes,
driveways, and other hardscapes for a higher price. Salt sand can be as high as $40 per cubic yard, but it is also used for ice melting. Truck bed loads of play sand cost around $5 to $7 per cubic
How Much is a Ton of Sand?
The first question you should ask yourself when deciding on a price for sand is how much does one ton cover? One ton covers approximately 40 square feet at a depth of six inches. This means that each
cubic foot of sand will cover twenty square feet. This is the same amount of sand as 20 cubic feet of water. Then, you can multiply this by two to find the cost per cubic foot.
Cost per ton of sand varies greatly, and is based on the type of sand used. Most commonly used sands cost $25 to $40 per ton. Building material suppliers often offer sand by the cubic yard or half
cubic yard. A cubic yard contains roughly 2,500 pounds, while a ton is equal to one and a half cubic yards. In addition to the price of sand per ton, the cost of delivery can add up to more than
three times that amount. Delivery costs can also be a factor, including the truck and equipment needed for loading and transporting the sand.
To get the weight of sand, first calculate how much you need to buy. A typical 50 lb bag is equal to one short ton. One ton weighs about four thousand pounds. In the UK, a ton of sand weighs 635
litres. For comparison, a gallon equals 4.546 litres, while a cubic meter of sand contains approximately nine cubic feet of material.
How Do You Transport Sand?
While many people use trucks to haul sand, there are several other methods that are just as efficient. These trucks are easy to operate, can easily dump their load, and come in many different sizes
and models. Trucks are an economical way to move sand when you have a large volume. However, you should consider the number of trips you will make with your truck to determine which method will be
most effective for your specific needs.
A truck can carry between ten and fourteen cubic yards of sand. Because sand isn’t evenly packed, it will vary in density, but an average truck can haul approximately 6000 pounds. Since sand doesn’t
weigh exactly the same as dirt, it is a good guide when selecting the right truck for your project. If you are unsure, contact a sand delivery service to determine the appropriate size truck for your
How Many Yards are in a Pickup Truck?
You may wonder how many yards are in a pickup truck. The standard cubic yard capacity of a half-ton truck is about a half-yard. The bed of a full-size truck has dimensions of eight feet long by five
feet wide and one-third of a foot high. The payload capacity of a half-ton pickup truck is 1,200 to two thousand pounds. A truck with a bed that extends four feet beyond the wheel wells can hold
about a half-yard of dirt.
To determine the number of cubic yards in a pick-up truck, first determine the size of the yard. Then round that number up to the nearest cubic yard. This way, you can determine the size of your
truck and the amount of mulch you need. If you don’t have a yard-sized truck, you may want to load it half-yards at a time. A truck that can only hold one ton may be better suited to carry
less-weight items, such as mulch.
Learn More Here:
3.) Best Trucks | {"url":"https://www.prettymotors.com/how-much-does-a-truck-hold-of-sand/","timestamp":"2024-11-05T16:24:22Z","content_type":"text/html","content_length":"85387","record_id":"<urn:uuid:ec27130a-2fa9-4707-bfad-72d44a2ad2cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00747.warc.gz"} |