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# Grover Iteration

I have two questions regarding a search algorithm based on the Grover Iteration.

Lets assume I would have a classical database:

Binary Index   Value
00             3
01             1
10             0
11             2

1. In the examples I have seen a register of lets say 2 qubits + 1 ancilla qubit (for the orcale) are used. Next, a uniform superposition of the 2 qubits is generated, followed by $$r$$-iterations of the oracle and diffusion operator. In the examples, people are using an oracle e.g. for the state "11" and ideally they finally measure "11" with $$P = 1$$. But this is not realy what we want in a database (?). How could I load the database shown above and search within it (e.g. to find the number "3") using the algorithm. In the end, I would be interested in the Index of an element I am searching for, right?

Imaginge I would have the following database:

Binary Index   Value
00             5
01             1
10             0
11             2

I would now like to search for "3" within this database. I would expect that the Grover algorithm would return $$P = 0$$ in this case.

1. There is another question regading the efficiency $$\sqrt{N}$$ of a $$N$$-dimensional database. Since we measure only the states "00", "01", ... "11" by a single shot, we would need to measure $$i$$-times to get a certain probability of our search element, e.g. "11". Would this not affect the actual efficiency of the algorithm? I understand when $$N$$ is a large number, the angle of rotation per iteration becomes small, so that after the rotations the database and search item are almost perfectly aligned. In this case we do not need to measure many times because the probability for "11" would be almost $$P = 1$$?

1. I'm not exactly sure what you're asking. Grover's algorithm works when you have an oracle operator $$O_f$$ such that $$O_f|x\rangle |0\rangle = |x\rangle |f(x)\rangle,$$ where $$f(x)\in\{0,1\}$$. For lots of applications (like your example) we have a function $$g:\{0,1\}^n\rightarrow\{0,1\}^m$$, i.e., the output is not a single bit, and we want to find the pre-image of some output. In your example, $$g$$ represents a look-up in the table, so $$g(0)=3$$, $$g(1)=1$$, etc. To translate this to the function for a Grover operator, we assume we have $$O_g:|x\rangle |0^m\rangle = |x\rangle |g(x)\rangle$$ (the second register has $$m$$ qubits). Then we create some equality operator for the specific point we're searching for. In your example, if we're looking for $$3$$, we would create an operator $$O_{=3}$$ such that $$O_{=3}|x\rangle |0\rangle = |x\rangle |0\rangle$$ if $$x\neq 3$$ and $$O_{=3}|x\rangle|0\rangle = |x\rangle | 1\rangle$$ if $$x=3$$. Then $$O_{=3}O_g$$ has the desired effect (if we apply $$O_{=3}$$ t the output register of $$O_g$$, and if we uncompute the output of $$O_g$$ after).

If the Grover search then returns an index $$x$$, and you want to find the value associated to $$x$$, then because we implement $$O_g$$ in this way, we can just run $$O_g |x\rangle |0\rangle$$ and measure the second register (of course, in your example, that will just give us 3, which we already know because that was the goal of our search!).

If you're actually searching a database, maybe there is more data associated to that index. For example, a database of students might have "student ID" and "major", and you want to find the major of the student with ID=3. So you find the index in the database of the student with ID=3, then look up that index to find the major.

1(b): Asking how you load the database: well, Grover's search isn't actually well suited for database search. How do you actually implement $$O_g$$? This relies on something called "QRAM" (sometimes called "QROM" or "QRACM"). This is a bit controversial and while there are some dubious proposals for a physical device that would efficiently do this, building it out of traditional quantum gates is quite expensive: essentially you iterate over the entire table, and write the corresponding value if the input matches.

But, if you iterate over the entire table, this costs $$O(N)$$. For that cost, you might as well search the database classically! So unless you believe in cheap QRAM (I don't) then Grover's algorithm is actually unhelpful for database search. It is a continuous annoyance to myself and others that Grover's algorithm is often described as a "database search" when it is really best described as a "pre-image search".

1(c): If you run a Grover search and the specific element is not in the "table" (i.e., there is no $$x$$ such that $$f(x)=1$$), then you end up with a uniform distribution over output values. Thus, when you measure, you'll get a uniformly random $$x$$. You would need to check the value of $$x$$ by computing $$f$$ (either classically or quantumly) to check whether the search succeeded or not.

1. You're basically right: the point of Grover's algorithm is amplify the probability of the correct answer (in your example, "11") so that when you measure at the end, the probability of measuring that answer is almost 1. So if you do the right number of Grover iterations, there is no extra complexity to repeat the measurements. (Though the probability won't be exactly 1, so you repeat maybe $$O(1)$$ times).

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# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

All Issues

## Discrete & Continuous Dynamical Systems - B

March 2013 , Volume 18 , Issue 2

Select all articles

Export/Reference:

2013, 18(2): i-iii doi: 10.3934/dcdsb.2013.18.2i +[Abstract](1402) +[PDF](1512.2KB)
Abstract:
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

2013, 18(2): 283-293 doi: 10.3934/dcdsb.2013.18.283 +[Abstract](1386) +[PDF](332.3KB)
Abstract:
Motivated by the fact that various experimental results yield strong confirmatory support for the hypothesis that the mixing of a wheat-flour dough is essentially a rate-independent process'', this paper examines how the mixing can be modelled using the rigorous mathematical framework developed to model an incremental time evolving deformation of an elasto-plastic material. Initially, for the time evolution of a rate-independent elastic process, the concept is introduced of an "energetic solution'' [24] as the characterization for the rate-independent deformations occurring. The framework in which it is defined is formulated in terms of a polyconvex stored energy density and a multiplicative decomposition of large deformations into elastic and nonelastic (plastic or viscous) components. The mixing of a dough to peak dough development is then modelled as a sequence of incremental elasto-nonelastic deformations. For such incremental processes, the existence of Sobolev solutions is guaranteed. Finally, the limit passage to vanishing time increment leads to the existence of an energetic solution to our problem.
2013, 18(2): 295-312 doi: 10.3934/dcdsb.2013.18.295 +[Abstract](2035) +[PDF](413.2KB)
Abstract:
In this paper we consider Galerkin finite element discretizations of semilinear elliptic differential inclusions that satisfy a relaxed one-sided Lipschitz condition. The properties of the set-valued Nemytskii operators are discussed, and it is shown that the solution sets of both, the continuous and the discrete system, are nonempty, closed, bounded, and connected sets in $H^1$-norm. Moreover, the solution sets of the Galerkin inclusion converge with respect to the Hausdorff distance measured in $L^p$-spaces.
2013, 18(2): 313-329 doi: 10.3934/dcdsb.2013.18.313 +[Abstract](1448) +[PDF](425.4KB)
Abstract:
One of the main paradigms of the theory of weakly interacting chaotic systems is the absence of phase transitions in generic situation. We propose a new type of multicomponent systems demonstrating in the weak interaction limit both collective and independent behavior of local components depending on fine properties of the interaction. The model under consideration is related to dynamical networks and sheds a new light to the problem of synchronization under weak interactions.
2013, 18(2): 331-348 doi: 10.3934/dcdsb.2013.18.331 +[Abstract](2215) +[PDF](424.1KB)
Abstract:
This paper is concerned with an optimal control problem for a system of ordinary differential equations with rate independent hysteresis modelled as a rate independent evolution variational inequality with a closed convex constraint $Z\subset \mathbb{R}^m$. We prove existence of optimal solutions as well as necessary optimality conditions of first order. In particular, under certain regularity assumptions we completely characterize the jump behaviour of the adjoint.
2013, 18(2): 349-376 doi: 10.3934/dcdsb.2013.18.349 +[Abstract](2016) +[PDF](450.1KB)
Abstract:
If financial markets displayed the informational efficiency postulated in the efficient markets hypothesis (EMH), arbitrage operations would be self-extinguishing. The present paper considers arbitrage sequences in foreign exchange (FX) markets, in which trading platforms and information are fragmented. In [18,9] it was shown that sequences of triangular arbitrage operations in FX markets containing $4$ currencies and trader-arbitrageurs tend to display periodicity or grow exponentially rather than being self-extinguishing. This paper extends the analysis to $5$ or higher-order currency worlds. The key findings are that in a $5$-currency world arbitrage sequences may also follow an exponential law as well as display periodicity, but that in higher-order currency worlds a double exponential law may additionally apply. There is an inheritance of instability'' in the higher-order currency worlds. Profitable arbitrage operations are thus endemic rather that displaying the self-extinguishing properties implied by the EMH.
2013, 18(2): 377-401 doi: 10.3934/dcdsb.2013.18.377 +[Abstract](1842) +[PDF](1557.1KB)
Abstract:
The new consensus'' DSGE(dynamic stochastic general equilibrium) macroeconomic model has microfoundations provided by a single representative agent. In this model shocks to the economic environment do not have any lasting effects. In reality adjustments at the micro level are made by heterogeneous agents, and the aggregation problem cannot be assumed away. In this paper we show that the discontinuous adjustments made by heterogeneous agents at the micro level mean that shocks have lasting effects, aggregate variables containing a selective, erasable memory of the shocks experienced. This hysteresis framework provides foundations for the post-Walrasian analysis of macroeconomic systems.
2013, 18(2): 403-415 doi: 10.3934/dcdsb.2013.18.403 +[Abstract](1884) +[PDF](436.0KB)
Abstract:
Following the approach of [22], we derive a system of Fokker-Planck equations to model a stock-market in which hysteretic agents can take long and short positions. We show numerically that the resulting mesoscopic model has rich behaviour, being hysteretic at the mesoscale and displaying bubbles and volatility clustering in particular.
2013, 18(2): 417-435 doi: 10.3934/dcdsb.2013.18.417 +[Abstract](2118) +[PDF](1263.7KB)
Abstract:
Identification of biological models is often complicated by the fact that the available experimental data from field measurements is noisy or incomplete. Moreover, models may be complex, and contain a large number of correlated parameters. As a result, the parameters are poorly identified by the data, and the reliability of the model predictions is questionable. We consider a general scheme for reduction and identification of dynamic models using two modern approaches, Markov chain Monte Carlo sampling methods together with asymptotic model reduction techniques. The ideas are illustrated using a simple example related to bio-medical applications: a model of a generic receptor. In this paper we want to point out what the researchers working in biological, medical, etc., fields should look for in order to identify such problematic situations in modelling, and how to overcome these problems.
2013, 18(2): 437-452 doi: 10.3934/dcdsb.2013.18.437 +[Abstract](1461) +[PDF](377.3KB)
Abstract:
This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.
2013, 18(2): 453-465 doi: 10.3934/dcdsb.2013.18.453 +[Abstract](1924) +[PDF](403.3KB)
Abstract:
Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path $\omega$ and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.
2013, 18(2): 467-482 doi: 10.3934/dcdsb.2013.18.467 +[Abstract](1854) +[PDF](561.9KB)
Abstract:
We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.
2013, 18(2): 483-493 doi: 10.3934/dcdsb.2013.18.483 +[Abstract](1563) +[PDF](383.9KB)
Abstract:
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.
2013, 18(2): 495-512 doi: 10.3934/dcdsb.2013.18.495 +[Abstract](1712) +[PDF](917.1KB)
Abstract:
The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.
2013, 18(2): 513-521 doi: 10.3934/dcdsb.2013.18.513 +[Abstract](1493) +[PDF](858.3KB)
Abstract:
The existence of canard cascades is studied in the paper as a problem of gluing of stable and unstable one-dimensional slow invariant manifolds at turning points. This way of looking is made feasible to establish the existence of canard cascades, that can be considered as a generalization of canards. A further development of this approach, with applications to the van der Pol equation and a problem of population dynamics, is contained in the paper.
2013, 18(2): 523-549 doi: 10.3934/dcdsb.2013.18.523 +[Abstract](1705) +[PDF](441.2KB)
Abstract:
We establish a large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one, including diffusion.
2013, 18(2): 551-563 doi: 10.3934/dcdsb.2013.18.551 +[Abstract](1876) +[PDF](390.1KB)
Abstract:
Electromagnetic processes in a ferromagnetic conductor (e.g., an electric transformer) are here described by coupling the Maxwell equations with nonlinear constitutive laws of the form $$\vec B \in \mu_0\vec H + {\mathcal M}(x) \vec H/|\vec H|, \qquad \vec J = \sigma(x) \big( \vec E + \vec E_a(x,t) + h(x)\vec J \!\times\! \vec B \big).$$ Here $\vec E_a$ stands for an applied electromotive force; the saturation ${\mathcal M}(x)$, the conductivity $\sigma(x)$ and the Hall coefficient $h(x)$ are also prescribed. The first relation accounts for hysteresis-free ferromagnetism, the second one for the Ohm law and the Hall effect.
This model leads to the formulation of an initial-value problem for a doubly-nonlinear parabolic-hyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by time-discretization, derivation of a priori estimates, and passage to the limit. This final step rests upon a time-dependent extension of the Murat and Tartar div-curl lemma, and on compactness by strict convexity.
2013, 18(2): 565-573 doi: 10.3934/dcdsb.2013.18.565 +[Abstract](1669) +[PDF](309.0KB)
Abstract:
We prove that the sweeping process on a "regular" class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.
2013, 18(2): 575-600 doi: 10.3934/dcdsb.2013.18.575 +[Abstract](1853) +[PDF](743.5KB)
Abstract:
We consider an "elastic'' version of the statistical mechanical monomer-dimer problem on the $n$-dimensional integer lattice. Our setting includes the classical "rigid'' formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan'' EDM systems where the dimer potential is a weighted $l_1$-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.

2018  Impact Factor: 1.008


text:
# Thread: Cauchy sequence and limitpoint

1. ## Cauchy sequence and limitpoint

Hello,

I am trying to prove that if a sequence $\displaystyle \{x_{n}\}$ in a set $\displaystyle M\in X$ ($\displaystyle X$ is a metric space) is a Cauchy sequence and the sequence has a limit point $\displaystyle x_{0}$ then $\displaystyle x_{n} \rightarrow x_{0}$. I wish to understand the details in such a proof whence I will try to explain my thoughts.

I know that the sequence $\displaystyle \{x_{n}\}_{N\in \mathbb{N}}$ is a Cauchy sequence which means that for all $\displaystyle \epsilon >0$ I can find an $\displaystyle N \in \mathbb{N}$ beyond which the distance between any to elements $\displaystyle x_{n}$ and $\displaystyle x_{m}$ of the sequence will be smaller than $\displaystyle \epsilon$. This means that I can make the distance between any two element of the sequence arbitrarily small.

Furthermore we know that $\displaystyle x_{0}$ is a limit point which means that for every $\displaystyle \epsilon>0$ there is a point $\displaystyle m \in M$ such that $\displaystyle m \in B(x_{0},\epsilon)$.

I wish to argue as follows:

1) I would like to argue that $\displaystyle x_{m}=m \in B(x_{0},\frac{\epsilon}{2})$ (I can freely choose $\displaystyle r=\frac{\epsilon}{2}$ since $\displaystyle x_{0}$ is a limit point).

2) Since $\displaystyle \{x_{n}\}$ is a Cauchy sequence $\displaystyle d(x_{n},x_{m})$ will be smaller than any $\displaystyle \epsilon$ I choose, so I choose $\displaystyle \frac{\epsilon}{2}$. Therefore $\displaystyle d(x_{n},x_{m})<\frac{\epsilon}{2}$.

3) Since $\displaystyle x_{m}=m \in B(x_{0},\frac{\epsilon}{2})$ then $\displaystyle d(x_{m},x_{0})<\frac{\epsilon}{2}$.

4) Using the triangle inequality we have $\displaystyle d(x_{n},x_{0}) \leq d(x_{n},x_{m}) + d(x_{m},x_{0}) \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$

5) We can conclude that $\displaystyle x_{n} \rightarrow x_{0}$

Can someone please comment on the proof and the arguments presented?
How can I convince myself of 1) ?
How do I convince myself of the choice of epsilon (I start my choice of epsilon from the fact that $\displaystyle x_{0}$ is a limit point and from this I choose my epsilon for the distance between $\displaystyle x_{n}$ and $\displaystyle x_{m}$!!).

Thanks

2. ## Re: Cauchy sequence and limitpoint

Originally Posted by detalosi
I am trying to prove that if a sequence $\displaystyle \{x_{n}\}$ in a set $\displaystyle M\in X$ ($\displaystyle X$ is a metric space) is a Cauchy sequence and the sequence has a limit point $\displaystyle x_{0}$ then $\displaystyle x_{n} \rightarrow x_{0}$. I wish to understand the details in such a proof whence I will try to explain my thoughts.
I know that the sequence $\displaystyle \{x_{n}\}_{N\in \mathbb{N}}$ is a Cauchy sequence which means that for all $\displaystyle \epsilon >0$ I can find an $\displaystyle N \in \mathbb{N}$ beyond which the distance between any to elements $\displaystyle x_{n}$ and $\displaystyle x_{m}$ of the sequence will be smaller than $\displaystyle \epsilon$. This means that I can make the distance between any two element of the sequence arbitrarily small.
Furthermore we know that $\displaystyle x_{0}$ is a limit point which means that for every $\displaystyle \epsilon>0$ there is a point $\displaystyle m \in M$ such that $\displaystyle m \in B(x_{0},\epsilon)$.

1) I would like to argue that $\displaystyle x_{m}=m \in B(x_{0},\frac{\epsilon}{2})$ (I can freely choose $\displaystyle r=\frac{\epsilon}{2}$ since $\displaystyle x_{0}$ is a limit point).

2) Since $\displaystyle \{x_{n}\}$ is a Cauchy sequence $\displaystyle d(x_{n},x_{m})$ will be smaller than any $\displaystyle \epsilon$ I choose, so I choose $\displaystyle \frac{\epsilon}{2}$. Therefore $\displaystyle d(x_{n},x_{m})<\frac{\epsilon}{2}$.

3) Since $\displaystyle x_{m}=m \in B(x_{0},\frac{\epsilon}{2})$ then $\displaystyle d(x_{m},x_{0})<\frac{\epsilon}{2}$.

4) Using the triangle inequality we have $\displaystyle d(x_{n},x_{0}) \leq d(x_{n},x_{m}) + d(x_{m},x_{0}) \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$

5) We can conclude that $\displaystyle x_{n} \rightarrow x_{0}$
Can someone please comment on the proof and the arguments presented?
How do I convince myself of the choice of epsilon (I start my choice of epsilon from the fact that $\displaystyle x_{0}$ is a limit point and from this I choose my epsilon for the distance between $\displaystyle x_{n}$ and $\displaystyle x_{m}$!!).
The good news is that I would say that the proof is basically correct BUT difficult to follow.
You should know that $\{x_n\}$ is a Cauchy sequence in $M\subseteq X$ a metric metric space.
Also, you are given that $x_0$ is a limit point of the sequence.
You are to prove that $(x_n)\to x_0$.
Being a limit point of the sequence, every open set containing $x_0$ contains some point of the sequence different from $x_0$
One property of Cauchy sequences us that for $\forall\varepsilon>0$ the ball $\mathscr{B}_{\varepsilon }(x_0)$ will contain almost all terms of the sequence.
Now, almost all has a definite meaning: it means all but an finite collection.
I think you have that idea. Can you tighten the argument?


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Error mitigation with Pennylane on IBMQ backends#

In this tutorial we will cover how to use Mitiq in conjunction with PennyLane, and further how to run error-mitigated circuits on IBMQ backends.

Setup: Defining a circuit#

For simplicity, we’ll use a single-qubit circuit with ten Pauli $$X$$ gates that compiles to the identity, defined below.

import pennylane as qml

def circuit():
for _ in range(10):
qml.PauliX(wires=0)
return qml.expval(qml.PauliZ(0))

In this example, we will use the probability of the ground state as our observable to mitigate, the expectation value of which should evaluate to one in the noiseless setting.

High-level usage#

As of version 0.19 of PennyLane, and 0.11 of Mitiq, PennyLane comes with out of the box support for error mitigation. This makes it very easy to use zero-noise extrapolation when working with PennyLane, regardless of where the circuit is being executed.

We define the executor function in the following code block. As we are using IBMQ backends, we first load our account.

Note: Using an IBM quantum computer requires a valid IBMQ account. See https://quantum-computing.ibm.com/ for instructions to create an account, save credentials, and get access to online quantum computers.

First, we get our devices set up depending on whether we would like to use real hardware, or a simulator.

import qiskit

USE_REAL_HARDWARE = False

if qiskit.IBMQ.stored_account() and USE_REAL_HARDWARE:
dev = qml.device(
"qiskit.ibmq",
wires=1,
backend="ibmq_qasm_simulator",
provider=provider
)
else:
noise_strength = 0.05
dev_noise_free = qml.device("default.mixed", wires=1)
dev = qml.transforms.insert(
qml.AmplitudeDamping,
noise_strength
)(dev_noise_free)

With dev set to the desired device, we can now use PennyLane’s mitigate_with_zne function, in conjuction with a noise scaling method, and inference technique from Mitiq.

from mitiq.zne.scaling import fold_global
from mitiq.zne.inference import RichardsonFactory

scale_factors = [1, 2, 3]
noise_scale_method = fold_global

device_circuit = qml.QNode(circuit, dev)
error_mitigated_device_circuit = qml.transforms.mitigate_with_zne(
device_circuit,
scale_factors,
noise_scale_method,
RichardsonFactory.extrapolate,
)

We can now test the mitigated version of the circuit against the unmitigated to ensure it is working as expected.

unmitigated = device_circuit()
mitigated = error_mitigated_device_circuit()
print(f"Unmitigated result {unmitigated:.3f}")
print(f"Mitigated result   {mitigated:.3f}")

Unmitigated result 0.609
Mitigated result   0.937

As the ideal, desired result is 1.000, the mitigated result performs much better than unmitigated.

Options#

In this section we will discuss the many different options for noise scaling and extrapolation that can be passed into PennyLane’s mitigate_with_zne function.

The following code block shows an example of using linear extrapolation with five different (noise) scale factors.

from mitiq.zne.inference import LinearFactory

scale_factors = [1.0, 1.5, 2.0, 2.5, 3.0]
noise_scale_method = fold_global
mitigated = qml.transforms.mitigate_with_zne(
device_circuit,
scale_factors,
noise_scale_method,
LinearFactory.extrapolate,
)()

print(f"Mitigated result {mitigated:.3f}")

Mitigated result 0.787

To specify a different noise scaling method, we can pass a different function for the argument scale_noise. This function should input a circuit and scale factor and return a circuit. The following code block shows an example of scaling noise by local folding instead of global folding.

from mitiq.zne.scaling import fold_gates_at_random

scale_factors = [1.0, 1.5, 2.0, 2.5, 3.0]
noise_scale_method = fold_gates_at_random
mitigated = qml.transforms.mitigate_with_zne(
device_circuit,
scale_factors,
noise_scale_method,
LinearFactory.extrapolate,
)()

print(f"Mitigated result {mitigated:.3f}")

Mitigated result 0.782

Further options are described and elaborated in our article on additional options in ZNE.

Decorator usage#

Finally, it is perhaps more common to define a circuit using a decorator when you know in advance you would like an error mitigated value. For this our circuit will be defined as above, but we will use decorators to indicate which device we would like to run it on, and that we would like to error-mitigate it.

from mitiq.zne.scaling import fold_gates_from_left

@qml.transforms.mitigate_with_zne([1, 2, 3], fold_gates_from_left, RichardsonFactory.extrapolate)
@qml.qnode(dev)
def circuit():
for _ in range(10):
qml.PauliX(wires=0)
return qml.expval(qml.PauliZ(0))

print(f"Zero-noise extrapolated value: {circuit():.3f}")

Zero-noise extrapolated value: 0.937

Finally, more information about using PennyLane together with Mitiq can be found in PennyLane’s tutorial on error mitigation.


text:
# Rule of Simplification/Also known as

This extra level of wordage has not been adopted by $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is argued that it may cause clarity to suffer.


text:
# Shortest code to produce infinite output

Write the shortest code you can that produces an infinite output.

That's all. You code will only be disqualified if it stops producing output at some point. As always in code golf, the shortest code wins.

Here's a list of answers that I think are really clever, so they can get credit:

## Leaderboard

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Leaderboard

	Author	Language	Size

Winners by Language

Language	User	Score

{{PLACE}}	{{NAME}}	{{LANGUAGE}}	{{SIZE}}	Link

{{LANGUAGE}}	{{NAME}}	{{SIZE}}	Link

• All answers disqualified because at some point the Earth will be swallowed by the sun, and at some point the universe will die :P – Doorknob Nov 9 '13 at 20:00
• Does "infinite until your computer crashes" count? <_< – Izkata Nov 10 '13 at 1:39
• If I write mine in Piet, can I count the pixels of the text the other programs used? I believe the smallest possible repeating Piet program would be 6 pixels. That beats Befunge if "off" pixels still count. – DampeS8N Nov 12 '13 at 20:27
• @Izkata So any answer that crashes your computer is also allowed :D – ɐɔıʇǝɥʇuʎs Jul 11 '14 at 20:11
• @Doorknob So really, the challenge is to produce infinite output in a finite amount of time. Sounds easy enough. – Sanchises Apr 10 '15 at 21:15

# Braingolf, 6 bytes

[1+!_]

Try it online!

# Decimal, 12 bytes

11D91D30191D

Try it online!

Explanation:

11D   ; push whatever's in RAM to the stack as an INT
91D   ; declare jump 1
301   ; print
91D   ; goto jump 1

Another version that does not invoke undefined behavior:

82D   ; builtin - push random INT to stack
91D   ; declare jump 1
301   ; print
91D   ; goto jump 1

## Check, 3 bytes (non-competing)

#<#

Outputs infinite newlines.

Check is my new esolang. It uses a combination of 2D and 1D semantics.

### Explanation

This uses a bit of a hack in the language - namely, that < means "print a newline" in 1D mode, and "move left" in 2D mode.

The IP first runs into a #, which turns it into 2D mode. However, it runs into <, which immediately points it back. It runs into the first # again and switches back to 1D mode. The IP then hits <, which outputs a newline, and then hits the second #, which switches it back to 2D mode again. The IP wraps around to the first #, where it switches back to 1D mode. It hits < again, printing another newline, and the process repeats.

• ...how is that 7 bytes? – MD XF Jun 6 '17 at 4:11
• @MDXF I guess I awarded myself extra bytes for not being able to count. – Esolanging Fruit Jun 6 '17 at 5:00

# Unreadable, 16 bytes

'"""""'"""'"'"""

Try it online!

• Aren't Unreadable programs supposed to use the > '"""""'"""'"'""" markdown, to live up to its name? – caird coinheringaahing Dec 10 '17 at 14:52

# MOO, 39 38 bytes

while(!suspend(player:tell()))endwhile

Huge abuse of side effects; player:tell() by default does not return a value, which means it implicitly returns zero, and the resulting suspend(0) is necessary to avoid running out of ticks.

# Cubically, 3 bytes

%0)

This should look like this:

(%0)
(    open loop that can always be jumped to
%0   print 0th face sum as integer (0)
) jump back to loop regardless of faces

However, ) will just jump back to the start of the file if no jump point is provided.

# Dyalog APL, 5 bytes

⍞←⍣⊢0

⍞← character output

(f⍣g)Y power operator repeatedly applies left operand to the argument until (f Y) g Y returns true. In this case f is the assignment to output and g is ⊢ which returns the right argument, which is always 0 as the assignment doesn't modify it.

Note that the right operand could also be any of ⊣≠≢

I hesitate to add a link to try it online :)

• Nothing to worry about, TIO limits you to 128 kb and 1 minute: Try it online! – Adám Aug 22 '17 at 15:47

# JavaScript, 20 bytes

for(;;)console.log()

Sends infinite undefined output.

# Pyth - 2 bytes

#0

Exlanation:

#0
#  Loop forever
0 Print 0

# Dreaderef, 98 7 bytes

5?1 1-1

Try it online!

## Explanation:

In a more readable form (which can still be fed into the preprocessor), this program is:

numo 0
deref 1 -1

The numo (numeric output) statement does just about what you'd expect: it outputs 0. I could have used chro, but that makes it output unprintables and doesn't affect the byte count.

To understand the second line, you have to know a little bit about Dreaderef:

• Programs are loaded into the data space at the beginning of execution.
• The pointer to the current instruction is located at position -1.

What deref 1 -1 does is:

• Dereferences 1 (takes the value of the cell at index 1). This always returns 0.
• Stores the result in cell -1, which is effectively a GOTO to instruction 0.

Dreaderef doesn't have a "load constant" instruction, so the best way to load a constant into a cell is to use some sort of arithmetic instruction or copy it from some other cell with deref, which is what I do here.

# Forth, 24 bytes

Defines then calls a word that loops forever, pushing and printing <0> each iteration.

: f begin .s 0 until ; f

Try it online

With a do-loop (25 bytes):

: f 0 0 do 0 .s +loop ; f

Prints <1> 0  forever.

• I know this is a bit late, but technically the do loop isn't infinite, as eventually the counter will wrap around and reach 0 again (albeit it will possibly take years depending on speed and implementation). Use : f begin .s 0 until ; instead (plus it saves a byte) – reffu Feb 12 '18 at 14:15
• @reffu That's actually a byte longer. You have to call the function. – mbomb007 Feb 12 '18 at 16:33
• good point, I didn't realize this one had slightly different rules than the typical codegolf challenge. – reffu Feb 12 '18 at 17:07

# TIS-n11, 12 bytes

@0
MOV 0 ANY

Try it online!

Just shoves a bunch of 0\n to the output. On TIO, reached the 128KiB cutoff quite quickly. Without the -n flag, prints null bytes instead.

If you are familiar with TIS-100, you will be used to a 3x4 array of computational nodes, however, this solutions only uses one such node, nominally giving the zeroes downward as output.

The only instruction that send data out of a computational node is MOV, and that requires a source and a destination. the smallest possible source is a single digit number, and the smallest possible destination is UP. However, to produce actual output, the data needs to travel downwards, restricting the choice to either DOWN or ANY.

# Gol><>, 1 byte

D

Try it online!

D is a command designed for debugging, which prints the entire stack (from bottom to top) between brackets. D on empty stack generates []\n to stdout, and the IP loops through the row endlessly.

# Alternative solution, 1 byte

Credits to Jo King

N

Try it online!

Unlike ><>, Gol><> has infinite zeroes at the bottom of the stack, which allows to print the top without pushing anything. Any of Nno will do; N prints 0\n, n prints 0, and o prints null bytes.

Alternatively, there are many 2-byte solutions:

• One of 0-9a-f, l, or m followed by one of NnoD; Try it online!
• Any of 0-9a-f pushes a hexadecimal digit to the stack, l pushes the current stack size, and m pushes -1.
• N prints the top as number with newline, n is the same without newline, and o prints as a char.
• S"; Try it online!
• Initially S" is interpreted as one command which prints the encountered chars up to ending ". The next loop of S" is interpreted as a literal S and then closing ", which prints a single S to stdout.
• " followed by any of NnoD; Try it online!
• Use " as both opening and closing quote, then print the char.

# Ahead, 1 byte

O

Prints 0 forever. O pops the stack and prints that value as a number written out (as opposed to a Unicode character). Like Befunge, the empty stack pops 0, so this will print 0.

Since this is a 1x1 board, the head has nowhere to go and will stay on this cell forever, outputting forever.

## Prolog (14 bytes)

a:-write(0),a.

Simple tail recursive predicate in swi-prolog. Called like so a.

• Welcome to PPCG! – Laikoni Jun 6 '18 at 14:16
• @Laikoni it's nice to be here :) thank you. – Khepu Jun 6 '18 at 16:59

# Flobnar, 4 bytes

0,_@

Try it online!

Outputs 0x00 bytes indefinitely.

This works because _ is said to get the conditional from the cell on the other side of it, and then use its value to choose between the east and west cells. It is expected to be evaluated from a vertical direction, but it doesn't have to be.

@ indicates the starting point for the program and evaluates to the return value of the cell to its west (_). _ evaluates what's on the other side (0,) to print 0. , always has a return value of 0, however, and this causes the horizontal if to choose the branch to its east. This is @, which evaluates the cell to its west, continuing the cycle.

Flobnar doesn't have decimal output by default, but the linked implementation has a -d flag which enables . for decimal output. This allows a slightly prettier version that outputs printable characters instead.

0._@

Try it online!

# √ å ı ¥ ® Ï Ø ¿ , 3 bytes

(o)

Will output:

===== OUTPUT =====

0 0 0 0 0 etc.

==================

followed by execution information.

TIO Compiler. Enter the code as the first input and None as the second.

• It seems that the language was implemented recently. Should this be marked as non-competing? – fergusq Mar 13 '17 at 18:47
• Oh yeah forgot to do that. Update added – caird coinheringaahing Mar 13 '17 at 18:48

# Taxi, 377 369 bytes

-8 bytes by getting rid of the quotes.

Go to Post Office:w 1 l 1 r 1 l.
[B]
! is waiting at Writer's Depot.
! is waiting at Writer's Depot.
! is waiting at Writer's Depot.
Go to Zoom Zoom:s 1 l 1 l 4 l 2 r.
Go to Writer's Depot:w.
Pickup a passenger going to Post Office.
Pickup a passenger going to Post Office.
Pickup a passenger going to Post Office.
Go to Post Office:n 3 r 2 r 3 r 1 l.
Switch to plan B.

The repetition of the passengers is needed for the driver to get enough fare to get enough gas to continue forever.

Try it online!

# Perl 6, 8 bytes

Thanks to nwellnhof for this solution.

.say xx*

Try it online!

Prints (Any) infinitely.

# Perl 6, 10 bytes

1...!*.say

Try it online!

Defines a list that increments until the result of printing the current number is false (which never happens).

# Aheui (esotope), 6 bytes

박망

Try it online!

or simillar codes with same length like following;

반망
받망
...
박맣
...

will produce infinite output.

# Clean, 7 bytes

f=[0:f]

Try it online!

Gives [0,0,0,0,...

# Gol><>, 1 byte

n

There we go, I feel like this cheats a little bit, but it works! Now to golf this more... ;)

Try it online!

# Muriel, 39 bytes

A:"\";.A;@\"A:\\\"\"+|A+A";@"A:\""+|A+A

Try it online!

This is near identical to the Muriel quine, except I've changed the print command in the last instruction to an eval command, and the addition of .A; in order to print something (in this case, the shortest thing was a part of the source code ("+|A+A";@"A:\""+|A+A) itself).

# Voovoos, 3 bytes

x.<

Pretty much just endlessly bounces between the . and <
. pops the top of the stack and prints it, but if you pop an empty stack in voovoos you get 0, so it just endlessly prints 0.

# TI-BASIC, 5 bytes

Disp 1:prgmA

This solution requires that the program's name is prgmA.

Displays 1 on a new line forever, or at least until the calculator runs out of free RAM.

Here's an alternative 7 byte solution that is guaranteed to never terminate:

While 1:Disp 1:End

It has the same output as the above solution.

Notes:

• Each successive prgmA call uses up 16 bytes of free RAM.

• If there aren't 16 free bytes, then the program crashes with an ERR:MEMORY error.

# YABALL, 3 bytes

!.?

Try it online!

Explanation

!                  Go off of "reverse mode" (a mode where the IP goes left instead of right) if in reverse mode, otherwise no-op
.                  Print current cell, which is NUL
?                  Go in "reverse mode"

Basically an infinite loop printing NULs.

The commands are executed like so: !.?.!.?.!.? ...

## MINOL, 5 bytes

GOTO0

Any expression in MINOL that isn't entered as part of a program (that is, isn't assigned a line number by the user) is considered line 0. While seemingly not a part of the official spec (which, as far as I can tell, is a series of letters sent to and subsequently reformatted and published by 'Dr. Dobb's Journal of Computer Calisthenics and Orthodontia' in 1976), lines executed this way always seem to spit out a newline. This obviously wouldn't be the case in a full program, which would take 11 bytes (inc. the trailing CR):

1 PR:GOTO1

# Turing Machine But Way Worse, 27 13 bytes

0 0 0 1 0 1 0

Try it online!

Outputs infinite NUL characters.

# Triangular, 4 bytes

,%0/

Prints 0 repeatedly.

Try it online!

Ungolfed:

  ,
% 0
/

# Underload, 8 bytes

(:S:^):^

Endlessly outputs :S:^

Try it Online!


text:
# Computing train value of planetary gear

1. Dec 10, 2012

### lucfuture

2. Relevant equations
e=(product of number of driving teeth)/(product of number of driven teeth)

e=(n$_{out}$-n$_{arm}$)/(n$_{in}$-n$_{arm}$)

3. The attempt at a solution
e=(N$_{p1}$/N$_{s1}$)*(N$_{s2}$/N$_{p2}$)
e=1.25926

e should be 0.7941 however

From there out it seems pretty trivial. I just need to get the right e value.

2. Dec 10, 2012

### lucfuture

Problem statement

File size:
42.8 KB
Views:
79


text:
# Introduction to ZK

For anyone who wants to learn how zero knowledge proofs work
A zero knowledge proof (ZKP or just ZK) is a proof of computation satisfying:
• Succinctness: the size of the proof is constant (or in some cases logarithmic) in the size of the computation
• Zero knowledge: certain inputs or intermediate variables of the computation can be hidden from the verifier of the proof
Contrary to the name, the succinctness property is more central to current applications of ZK, whereas the zero knowledge aspect is often not used and can be turned off. Succinctness allows us to compress expensive computations into ZKPs that are computationally cheap to verify:
The general dynamic of a zero knowledge proof is that a Prover generates a ZKP of an expensive computation and sends the (constant sized) ZKP to a Verifier. Because the ZKP is constant sized, the Verifier can run a cheap constant time verification algorithm on the ZKP: if it passes, then the Verifier is assured that the Prover executed the claimed computation truthfully. Most importantly, the Verifier has this assurance without trusting the Prover. The precise assurance is of the form
under certain cryptographic assumptions, ___ is true with probability
$2^{-100}$
.

## How does it all work?

There are now developer toolkits with varying levels of abstraction (Rust/Go/Javascript API libraries, domain-specific languages) to translate arbitrary NP computations to ZK circuits. Given computation-specific inputs, a ZK circuit generates a ZK proof to be submitted to the Verifier.
This step of translation from familiar computation to ZK circuit is tricky: at their core, ZK "circuits" are closer in spirit to circuits in hardware chips than normal computer programs (in CS lingo: ZK circuits are not specified in a Turing complete language). There are still many improvements to be made in ZK circuit design. This is where we need brilliant minds (that's you!) to join us and innovate.

### Circuit Design: Arithmetizations

In a ZK circuit, a computation is represented as a collection of vectors together with imposed constraints (aka relations) between certain entries in the vectors. For example, computing 1 + 1 = 2 could be saying we have a vector (a, b, c) and constraints
a == 1, b == 1, a + b == c
One way to think about constraints is that in a trustless environment, all entries of the vector can be adversarially selected, and your only safeguards are the constraints you impose on these numbers. Continuing the example, if we didn't have the last constraint a + b == c, then I could supply c = 0 and all our constraints would still be satisfied!
There are different ways to translate a standard computation into such a collection of vectors and constraints - these are called arithmetizations. Nowadays there are developer "front-ends" with different levels of abstraction and customizability for translating a ZK circuit into an arithmetization.
The choice of an arithmetization and a front-end for writing in that arithmetization is the closest thing to choosing a programming language in ZK.

### Prover-Verifier Dynamic

Once we have an arithmetization (vectors + constraints), the prover-verifier dynamic goes as follows:
1. 1.
The Prover sends the Verifier some commitment to the vectors and further commitments (details omitted) to the constraints.
2. 2.
The Verifier sends the Prover some random numbers
3. 3.
Prover uses the previous commitment, along with some cryptography, to give a proof that the supplied vectors actually satisfies the claimed constraints (aka, the computation did what it claimed to do).
The above procedure is interactive, since the Verifier needs to provide randomness, but we remark that there is a general way to make it non-interactive: see Fiat-Shamir Heuristic. In the non-interactive protocol, the prover does all of their steps first and sends the result to the verifier. The Fiat-Shamir Heuristic allows the verifier to then verify the proof with the same assurance of correctness as if the whole process had been interactive.

### Polynomial Commitments

We skipped over some cryptographic details in our overview of the prover-verifier dynamic. Namely, what are commitments? One can think of a commitment as a more expressive hash: it pins down the vector so you can't change it later, but still allows you to perform some operations on the hashes which tell you useful information.
The question of how to commit to vectors is an area of active research. First, most vector commitments translate a vector into a polynomial (by Lagrange interpolation) and work with the polynomial. Then, broadly speaking, they currently fall into two categories:
• Use elliptic curve cryptography (not quantum-secure, additional assumptions for security, slower runtime)
• Hash the polynomials and do some sampling (proof sizes are larger, additional assumptions for security)
For an overview of polynomial commitments from the API perspective, see this video.
The choice of which polynomial commitment scheme to use is extremely important for the performance and security of the entire ZKP process. The speed of proof generation (step 3 in the prover-verifier dynamic) and cost of proof verification hinge upon this choice.
In the real world, after you have designed your [hardware] circuit its performance is governed by the laws of physics. In ZK, circuit performance is governed by the laws of math, and polynomial commitment schemes specify which laws apply.

### Summary

To choose a ZK proving stack and start building:
1. 1.
First, choose which arithmetization to use, along with a developer front-end for writing circuits in that arithmetization.
2. 2.
Choose what polynomial commitment scheme to prover/verifier will use. Often this choice is baked into the choice of arithmetization.
3. 3.
Find an existing library that generates proofs from a given arithmetization. (Not recommended: rolling your own.)

#### Axiom's Stack

1. 1.
We use the PLONKish arithmetization with the Halo2 frontend.
2. 2.
We use the KZG polynomial commitment scheme.
3. 3.
We use Privacy Scaling Explorations' fork of the Halo2 backend, which supports KZG commitments.

### How does it all work on Ethereum?

We described the prover-verifier dynamic in a general off-chain setting above. How does it translate to using ZK SNARKs on-chain?
Everything is largely the same. The main difference is that the Verifier is now a smart contract. This means that a smart contract runs the algorithm to verify a ZK SNARK supplied by the Prover. This enables powerful modularity and composability: the smart contract can programmatically perform further actions depending on whether the SNARK is verified correctly or not. For more details about how to produce SNARK verifier smart contracts, see this page.

## Challenges

Why is ZK not used more prevalently if it can compress any computation? One reason is that only recently has the dev tooling and cryptographic systems become expressive and stable enough for people to actual build on top of them.
There are also some intrinsic challenges related to the nature of ZK circuits:

While proof size and proof verification time are constant, the runtime to generate a proof is far from it. Right now, the estimated overhead of generating a proof for a particular computation is around 100-1000x. This is an active engineering problem with many facets for improvement:
• Improving circuit design - this involves finding the optimal way to translate a computation into an arithmetization.
• General performance engineering - some of the open source code used for proof generation was developed for other purposes, and serious performance optimization has not been applied yet.
• Choice of proving system: the combination of arithmetization and polynomial commitment scheme forms a proving system. New research in this area can lead to step change improvements in performance.
• Hardware: many core algorithms (Fast Fourier Transform, Elliptic Curve Multiscalar Multiplication) involved in the polynomial commitments can be parallelized or otherwise optimized using GPUs, FPGAs, ASICs.

### VM / Turing completeness

As mentioned above, a ZK circuit in its purest form is not Turing complete (you cannot have recursive functions or infinite loops). It also does not behave in the way we are used to a general purpose VM (e.g., LLVM) behaving. For example, the notion of an "if" statement is very different: assuming boolean a, to replicate
if a:
f(b)
else:
g(b)
in a circuit, we need to compute both f(b) and g(b) and then return a * f(b) + (1 - a) * g(b) (assuming that a is either 0 or 1).
There are ways to build general or customized VMs from ZK circuits using the principles of recursion and aggregation of ZK circuits. For example, to create a ZKP of f(g(x)), you would create ZKP A for y == g(x) and then a ZKP B that verifies the proof A: y == g(x) and further computes f(y). This is a more advanced topic which we will write about in more depth later, but for now we direct the interested reader to this blog post.

### Numerical architecture

A fundamental difference between traditional compute and all ZK circuits is the numerical system the compute is applied on top of. In traditional architecture, we work in the world of bits: numbers are uint32, uint64, int32, int64, etc. Meanwhile, due to the cryptographic underpinnings behind ZK, all ZK circuits involve modular arithmetic, i.e., numbers are element in a finite field. This usually means there is a special prime number p, and then all operations are performed modulo p.
This difference means that:
• Operations that are cheap for bits (bit shifts, AND, XOR, OR) are expensive to implement in ZK.
• Since ZK circuits still need to be implemented in traditional architectures, the implementation of things like finite field arithmetic adds another layer of overhead to all computations.
There are continual developments to overcome this challenge, such as ZK friendly hashes or using "lookup tables" for ZK-unfriendly operations, but for now it is still a source of difficulties for performance and development.

## Closing Remarks

The potential of things ZK can build is tremendous and only increasing exponentially. We believe that a key ingredient in successfully building new ZK applications is to have a good understanding of the current features and limitations of ZKPs.


text: The rest of the Group 1 carbonates don't decompose at Bunsen temperatures, although at higher temperatures they will. It consists of one carbon atom surrounded by three oxygen atoms, in a trigonal planar arrangement, with D3h molecular symmetry. Whereas the polarizing power decreases down the group with increase in the … Mafic rocks. Learn vocabulary, terms, and more with flashcards, games, and other study tools. (g) the preparation of crystals of soluble salts, such as copper(II) sulfate, from insoluble bases and carbonates (i) the test used to identify SO₄²⁻ ions; A/AS level. 1.6 The Periodic Table (j) trends in solubility of Group 2 hydroxides and sulfates Method. The Group 1 hydrogencarbonates are stable enough to exist as solids, although they do decompose easily on heating. Unit 1: THE LANGUAGE OF CHEMISTRY, STRUCTURE OF MATTER AND SIMPLE REACTIONS. Quantity. Mineral - Mineral - Carbonates: The carbonate minerals contain the anionic complex (CO3)2–, which is triangular in its coordination—i.e., with a carbon atom at the centre and an oxygen atom at each of the corners of an equilateral triangle. Most other carbonates are insoluble. This, in turn, increases Earth temperature. Generally only the carbonates of the Group 1 elements and the ammonium ion are soluble in water; most other carbonates are insoluble. High surface soil pH, ranging from 7.4 to 8.4, will have an effect on … All other group 1 carbonates are stable in Bunsen flame. Hence group-2 carbonates are more covalent than the carbonates of group-1. Group 1: Alkali Metals and Carbonates (X 2 CO 3) is the reaction between Li, Na, K, Rb and Cs with CO 3. Start studying The Effects of Heat on the Carbonates and Nitrates in Group 1. Carbonates are widely used in industry, such as in iron smelting, as a raw material for Portland cement and lime manufacture, in the composition of ceramic glazes, and more. * The polarizing power of the group-1 metal ions (M +) is less than the polarizing power of group-2 metal ions (M 2+). Unit 1: CHEMICAL SUBSTANCES, REACTIONS and ESSENTIAL RESOURCES. Copyright 2021 WILX. Aragonite CaCO 3; Cerussite PbCO 3; Strontianite SrCO 3; Witherite BaCO 3; Rutherfordine UO 2 CO 3; Natrite Na 2 CO 3; Anhydrous carbonates with compound formulas. Add to basket. Middle Devonian Transverse Group in Charlevoix and Emmet counties, Michigan – Page 1 of 4 Figure 1. Carbonates. All the carbonates in this Group undergo thermal decomposition to give the metal oxide and carbon dioxide gas. Until recently carbonate deposits have not been found on Mars via remote sensing or in situ missions, even though Martian meteorites contain small amounts. Group 1 nitrates except lithium decompose to form nitrites and oxygen gas. Important organocarbonates include dimethyl carbonate, the cyclic compounds ethylene carbonate and propylene carbonate, and the phosgene replacement, triphosgene. Carbonate minerals are extremely varied and ubiquitous in chemically precipitated sedimentary rock. Insoluble ionic compounds include: All carbonates apart from group-1 carbonates and ammonium carbonates. Thermal stability of alkali metal hydrides and carbonates (1 answer) Closed 1 year ago. Lattice Energy. All except Lithium are soluble in water and stable to heat. Officials from the ICHD are asking community members to keep an eye on their website https://ioniacounty.org/health/health-department/ for details. It is the conjugate base of the hydrogen carbonate (bicarbonate) ion, HCO−3, which is the conjugate base of H2CO3, carbonic acid. Historically it was extracted from the ashes of plants growing in sodium-rich soils. Kemper, F., Molster, F.J., Jager, C. and Waters, L.B.F.M. In Group 1, lithium carbonate behaves in the same way, producing lithium oxide and carbon dioxide: (5) L i 2 C O 3 ( s) → L i 2 O ( s) + C O 2. Calcium ion also has a smaller radius and so a higher charge density giving it a greater polarising power allowing it to distort the bonding These compounds are also known as organocarbonates or carbonate esters, and have the general formula ROCOOR′, or RR′CO3. Volume increases from Li to Cs 4. Source(s): JPH Chemistry. Rubidium was discovered (1861) spectroscopically by German scientists Robert Bunsen and Gustav Kirchhoff and named Group 1 metals most clearly show the effect of increasing size and mass on the decent of a group. Insoluble ionic compounds include: All carbonates apart from group-1 carbonates and ammonium carbonates. Also, for the same period the group 1 carbonate is relatively more stable. Eine Demonstration des Hochdruck-Injektionsverfahren zeigen wir Ihnen in diesem Video. Carbonate is the most abundant foreign ion in biological apatites, often referred to as “carbonated apatites,” where it mostly replaces PO43− (Elliott, 1994; LeGeros, 1965). 8 i. understand experimental procedures to show: patterns in thermal decomposition of Group 1 and 2 nitrates and carbonates; Wales. An unusual trend in the group 1 elements is the smooth decrease in the melting and boiling points from Li to Cs. Group 2 Heating carbonates. To test what you've learned from this lesson- Answer to question 23 the calcium ion has a larger charge (2+) thatn the potassium ion (1+). Among them, microsomal PGES-1 (mPGES-1) is an inducible form, and we showed earlier that its expression is upregulated during microglial activation. A carbonate salt forms when a positively charged ion, M+, M2+, or M3+, associates with the negatively charged oxygen atoms of the ion by forming electrostatic attractions with them, forming an ionic compound: Most carbonate salts are insoluble in water at standard temperature and pressure, with solubility constants of less than 1×10−8. Start a free trial of Quizlet Plus by Thanksgiving | Lock in 50% off all year Try it free. LA-ICP-MS. µXRF. The carbonates of the alkali metals are water-soluble; all others are insoluble. Ionia County officials: Not enough vaccine yet to vaccinate group 1B population, They say they currently have only a small amount of vaccine that is already spoken for in planned clinics, Firefighters battle blaze at America’s Best, Michigan State Capitol Commission to meet regarding open carry ban. Other minerals have been proposed which would fit the observations. 5mm. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The first major bump in the road for Michigan came Thursday, when officials from the Ionia County Health Department (ICHD) announced that the county does not have enough vaccine to cover all of this large group at this time. Solid Compounds. LOCATION The Middle Devonian Traverse Group is a sequence of carbonates and shales randomly exposed as … Groundwater may have existed at Gusev[6] and Meridiani Planum. The resource is set out as teachers' notes followed by the students' page which presents the task to be investigated. All of these decrease down the group. All the carbonates in this group undergo thermal decomposition to the metal oxide and carbon dioxide gas. IF-G-NP. Although the carbonate salts of most metals are insoluble in water, the same is not true of the bicarbonate salts. It has a molecular mass of 60.01 g/mol and carries a total formal charge of −2. They say they currently have only a small amount of vaccine that is already spoken for in planned clinics, and that they do not expect more vaccine until close to the end of January. LIBS. IONIA, Mich. (WILX) - Wednesday, Governor Whitmer announced that Michigan will be moving to Phase 1B of COVID-19 vaccination starting next week. Lithium carbonate, and other carbonates of group 1, do not decarboxylate readily. Wir sind für Sie da! It works as a buffer in the blood as follows: when pH is low, the concentration of hydrogen ions is too high, so one exhales CO2. Browse. Thermal decomposition is the term given to splitting up a compound by heating it. This will cause the equation to shift left,[further explanation needed] essentially decreasing the concentration of H+ ions, causing a more basic pH. The stability of group 1 and 2 carbonates increases down the group therefore you can expect that the lower the metal in the periodic table the more stable is its corresponding carbonate. Appearance: white powder, fragrance-free; Sodium Carbonate (Na 2 CO 3) Inorganic carbonates comprise many minerals (see carbonate mineral) and are the principal constituents of limestones and dolomites; they also comprise the hard parts of many marine invertebrates. [citation needed]. It is a major factor in climate change and the long-term carbon cycle, due to the large number of marine organisms (especially coral) which are formed of calcium carbonate. 20mm. The lattice energies. Major … All hydroxides, apart from group-1 … Phase 1B … What's chemistry without combining a bunch of stuff together, right? The carbonates of the alkali metals are water-soluble; all others are insoluble. The decomposition temperatures again increase as you go down the Group. These compounds are also known as organocarbonates or carbonate esters, and have the general formula ROCOOR′, or RR′CO3. As a result, Cs (melting point = 28.5°C) is one of only three metals (the others are Ga and Hg) that are liquids at body temperature (37°C). The metals of group 1, magnesium, calcium, strontium, and barium also form hydrogen carbonates—compounds that contain the hydrogen carbonate anion, $\text{HCO}_3^{\;\;-}$, also known as the bicarbonate anion. Select method. Both carbonates and nitrates of Group 2 elements become more thermally stable down the group. 1 Structures Expand this section. Only available in 5mm diameter. Explore the properties of group 1 and 2 by examining whether precipitates form when you add drops of sulphate or carbonate solution to group 1 or 2 metal ions Downloads The periodic table - solubility of sulphates and carbonates of groups 1 and 2 - teacher Atomic and ionic radii, volume : Atomic and ionic radii increases from Li to Fr due to presence of extra shell of electrons. So, solubility should decrease from Li to Cs. Size. Chemical Characteristics: Low solubility in water. Trace element concentrations in (bio) carbonates are used as proxies in paleo-climate research (foraminifera, corals, pteropods, mussel shells, red algae, stalactites, etc. Their solubility increases down the group since their lattice energy decreases more rapidly than their hydration energy in the group. These anionic groups are strongly bonded individual units and do not share oxygen atoms with one another. IONIA, Mich. (WILX) - Wednesday, Governor Whitmer announced that Michigan will be moving to Phase 1B of COVID-19 vaccination starting next week. group 1 carbonates are thermally very stable but group 2 carbonates are thermally unstable, the unstability increasing with cationic size. Recharacterized after milling. Anonymous. (2001) The mineral composition and spatial distribution of the dust ejecta of NGC 6302. International Union of Pure and Applied Chemistry, "Pyroclastic Activity at Home Plate in Gusev Crater, Mars", "Overview of the Opportunity Mars Exploration Rover Mission to Meridiani Planum: Eagle Crater to Purgatory Ripple", Carbonate/bicarbonate/carbonic acid equilibrium in water: pH of solutions, buffer capacity, titration and species distribution vs. pH computed with a free spreadsheet, https://en.wikipedia.org/w/index.php?title=Carbonate&oldid=991310907, Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference, Short description is different from Wikidata, Pages using collapsible list with both background and text-align in titlestyle, Articles containing unverified chemical infoboxes, Wikipedia articles needing clarification from January 2018, Articles with unsourced statements from February 2019, Creative Commons Attribution-ShareAlike License, This page was last edited on 29 November 2020, at 11:41. The name may also refer to a carbonate ester, an organic compound containing the carbonate group C(=O)(O–)2. GCSE. Not getting into the energetics one can safely state that the reason for these trend is the polarising power of the metal cation. The rest of the Group 1 carbonates do not decompose at laboratory temperatures, although at higher temperatures this becomes possible. Other groups in 1B will be contacted directly by a representative of the County, or can call the Health Department at (616) 527-5341. The decomposition temperatures again increase as you go down the Group. Alkali metal bicarbonates on heating decompose to give respective carbonates 2MHCO 3 M 2 CO 3 + CO 2 + H 2 O All carbonates and bicarbonates are water soluble. The ease of thermal decomposition on carbonates and nitrates (see table) the strength of covalent bonds in M 2. When the partial pressure of CO2 is reduced, for example when a can of soda is opened, the equilibrium for each of the forms of carbonate (carbonate, bicarbonate, carbon dioxide, and carbonic acid) shifts until the concentration of CO2 in the solution is equal to the solubility of CO2 at that temperature and pressure. The first resource is a differentiated worksheet with the questions designed around the style of AQA, Edexcel and OCR exam papers and test students on every aspect of the topic including the reactions, observations, trends, theory of charge density/polarisation and finishes with a few questions … In geology and mineralogy, the term "carbonate" can refer both to carbonate minerals and carbonate rock (which is made of chiefly carbonate minerals), and both are dominated by the carbonate ion, CO2−3. s- block elements www.spiroacademy.com About 30,000 tons were produced in 1989. Organic carbonates are esters; that is, compounds in which the hydrogen atoms of carbonic acid have been replaced by carbon-containing combining groups such as ethyl, C 2 H 5 . Carbonates of metal: Thermal stabilityThe carbonates of alkali metals except lithium carbonate are stable to heat. [7], Except where otherwise noted, data are given for materials in their. Submit your order. 3 Chemical and Physical Properties Expand this section. Carbonates are readily decomposed by acids. WJEC Chemistry. While important, this phase is also made more difficult by having a larger pool of eligible recipients. The tonnage of CO2−3 is on a geological scale and may all be redissolved into the sea and released to the atmosphere, increasing CO2 levels even more. Thus sodium carbonate is basic, sodium bicarbonate is weakly basic, while carbon dioxide itself is a weak acid. I want to help you achieve the grades you (and I) know you are capable of; these grades are the stepping stone to your future. Contents. Electronic configuration : ns1 2. Carbonates. (From Grant and Hackh's Chemical Dictionary, 5th ed) MeSH. The most common are calcite or calcium carbonate, CaCO3, the chief constituent of limestone (as well as the main component of mollusc shells and coral skeletons); dolomite, a calcium-magnesium carbonate CaMg(CO3)2; and siderite, or iron(II) carbonate, FeCO3, an important iron ore. Whereas, the carbonates of transition elements may be colored. Contents. When group 2 carbonates thermally decompose they produce the metal oxide and carbon dioxide For example: MgCO3 ----> MgO + CO2 If "X" represents any one of the elements: XCO3 ----> XO + CO2 As you go down the group more heat energy is required to thermally decompose the carbonate As you go down the group the carbonates become more thermally stable. The latticeenergies. Help, due tomorrow! Search. CVS Health (1) Fischer Search Group (1) Prime Time Contracting, Inc (1) Ascension (1) Experience Level. Grades 11 & 12(Indian) 1 0. Phase 1B is a particularly important benchmark in the process of vaccinations as it includes individuals over 65. SOLUBILITY OF COMPOUNDS (GROUP 1) Solubility of a compound mainly depends on two factors . Eight resources on the thermal decomposition of the group 1 and 2 nitrates and carbonates. Sodium carbonate, Na 2 CO 3, (also known as washing soda, soda ash and soda crystals) is the inorganic compound with the formula Na 2 CO 3 and its various hydrates. BHVO-2-NP. Increased solubility of carbonate through increased temperatures results in lower production of marine calcite and increased concentration of atmospheric carbon dioxide. Ends in 01d 12h 05m 13s. solubility stability alkali-metals. This means that the enthalpy change from the carbonate to the oxide becomes more negative so more heat is needed to decompose it. Carbonates of group-1 and group-2 elements are colorless. Carbonated water is formed by dissolving CO2 in water under pressure. Minerals. Unstable to heat. How many milliliters of.125 M sodium Carbonate … The solubility of Group 1 compounds The carbonates The least soluble Group 1 … The effective hydrated ionic radii. Save your resume. Multiple large announcements are planned for when the ICHD is ready to start vaccinating the over-65 group. Density : Densities are quite low and increases from Li to Cs. Eight resources on the thermal decomposition of the group 1 and 2 nitrates and carbonates. To test what you've learned from this lesson- Answer to question 23 the calcium ion has a larger charge (2+) thatn the potassium ion (1+). FK-N-NP. Note: the Group 1 metal carbonates are hydroscopicthey absorb water from the air. Subgroup 1K – Carbonates Soil criteria in this subgroup are the same as Group 1 except in the upper 12 inches of the soil profile free carbonates range between 5 and 15 percent calcium carbonate equivalent. By the same principle, when the pH is too high, the kidneys excrete bicarbonate (HCO−3) into urine as urea via the urea cycle (or Krebs–Henseleit ornithine cycle). Lithium Carbonate (Li 2 CO 3) Uses: drug development. By removing the bicarbonate, more H+ is generated from carbonic acid (H2CO3), which comes from CO2(g) produced by cellular respiration. Start studying The Effects of Heat on the Carbonates and Nitrates in Group 1. Mid Level (12) Senior Level (5) Entry Level (2) Forgot to save your resume? In organic chemistry a carbonate can also refer to a functional group within a larger molecule that contains a carbon atom bound to three oxygen atoms, one of which is double bonded. Lattice Energy. The effect of heat on the Group 2 carbonates. From Li to Cs, due to larger ion size, hydration enthalpy decreases. Carbonates of group-1 and group-2 elements are colorless. Group 1 carbonates don't decompose at Bunsen temperatures, although at higher temperatures they will. WJEC Chemistry. Thermal stability increases down the group because the size of the cation (positive ion) increases, so the lattice energy of the carbonate decreases, but the lattice energy of the oxide decreases faster. Systematic additive IUPAC name for carbonate anion is trioxidocarbonate(2−). Rubidium is the second most reactive metal and is very soft, with a silvery-white lustre. Learn vocabulary, terms, and more with flashcards, games, and other study tools. By contrast, the least soluble Group 1 carbonate is lithium carbonate. 4 Spectral Information Expand this section. The metals of group 1, magnesium, calcium, strontium, and barium also form hydrogen carbonates —compounds that contain the hydrogen carbonate anion, HCO − 3 HCO 3 −, also known as the bicarbonate anion. All the carbonates in this Group undergo thermal decomposition to give the metal oxide and carbon dioxide gas. In more acid conditions, aqueous carbon dioxide, CO2(aq), is the main form, which, with water, H2O, is in equilibrium with carbonic acid – the equilibrium lies strongly towards carbon dioxide. Learn the chemistry of limestone. 1 GROUP -1 ( ALKALI METALS) 1. All of these carbonates are white solids, and the oxides that are produced are also white solids. 13mm. 32mm. In organic chemistry a carbonate can also refer to a functional group within a larger molecule that contains a carbon atom bound to three oxygen atoms, one of which is double bonded. All the carbonates in this Group undergo thermal decomposition to give the metal oxide and carbon dioxide gas. The stability of group 1 and 2 carbonates increases down the group therefore you can expect that the lower the metal in the periodic table the more stable is its corresponding carbonate. Production. Use for to create your resume on Indeed and apply to jobs quicker. In the case of metal ions with insoluble carbonates, such as CaCO3, formation of insoluble compounds results. The carbonate ion is the simplest oxocarbon anion. This structure is incompatible with the observed symmetry of the ion, which implies that the three bonds are equally long and that the three oxygen atoms are equivalent. Solubility. The thermal stability of the hydrogencarbonates. 1 Structures Expand this section. Sublimation and melting point of covalent bonds in M 2 1 year ago important reversible REACTIONS control the above balance. ) MeSH carbonated water is formed by dissolving CO2 group 1 carbonates water and stable to heat the lab interconversion. Deposits and lithium salts in underground brine pools most reactive metal and is very soft, with D3h symmetry... Announcements are planned for when the ICHD is ready to start vaccinating the over-65 group compound by it! Compounds are also white solids this group undergo thermal decomposition to give the metal cation of... 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text:
# Serega and Fun

Serega loves fun. However, everyone has fun in the unique manner. Serega has fun by solving query problems. One day Fedor came up with such a problem.

You are given an array a consisting of n positive integers and queries to it. The queries can be of two types:

1. Make a unit cyclic shift to the right on the segment from l to r (both borders inclusive). That is rearrange elements of the array in the following manner:a[l], a[l + 1], …, a[r - 1], a[r] → a[r], a[l], a[l + 1], …, a[r - 1].
2. Count how many numbers equal to k are on the segment from l to r (both borders inclusive).

Fedor hurried to see Serega enjoy the problem and Serega solved it really quickly. Let’s see, can you solve it?

Input

The first line contains integer n (1 ≤ n ≤ 105) — the number of elements of the array. The second line contains n integers a[1], a[2], …, a[n] (1 ≤ a[i] ≤ n).

The third line contains a single integer q (1 ≤ q ≤ 105) — the number of queries. The next q lines contain the queries.

As you need to respond to the queries online, the queries will be encoded. A query of the first type will be given in format: 1 l i r i. A query of the second type will be given in format: 2 l i r i k i. All the number in input are integer. They satisfy the constraints: 1 ≤ l i, r i, k i ≤ n.

To decode the queries from the data given in input, you need to perform the following transformations:li = ((l i + lastans - 1) modn) + 1; ri = ((r i + lastans - 1) modn) + 1; ki = ((k i + lastans - 1) modn) + 1.

Where lastans is the last reply to the query of the 2-nd type (initially, lastans = 0). If after transformation l i is greater than r i, you must swap these values.

Output

For each query of the 2-nd type print the answer on a single line.

Examples

input

76 6 2 7 4 2 571 3 62 2 4 22 2 4 72 2 2 51 2 61 1 42 1 7 3

output

2100

input

88 4 2 2 7 7 8 881 8 82 8 1 71 8 11 7 32 8 8 31 1 41 2 71 4 5

output

20

Solution:

#include <cstring>
#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <memory.h>
#include <cassert>

using namespace std;

const int N = 200010;
const int MAGIC = 474;

int len;
int a[N];

int cnt, from[N], to[N];
vector <int> where[N];

void init() {
cnt = 1;
from[0] = 1;
to[0] = len;
for (int i = 1; i <= len; i++) {
where[i].clear();
}
for (int i = 1; i <= len; i++) {
where[a[i]].push_back(i);
}
}

int get_count(int k, int from, int to) {
int sz = where[k].size();
if (sz == 0) {
return 0;
}
int low = 0, high = sz;
while (low < high) {
int mid = (low + high) >> 1;
if (where[k][mid] < from) {
low = mid + 1;
} else {
high = mid;
}
}
int X = low;
low = -1, high = sz - 1;
while (low < high) {
int mid = (low + high + 1) >> 1;
if (where[k][mid] > to) {
high = mid - 1;
} else {
low = mid;
}
}
int Y = low;
return (Y - X + 1);
}

int new_from[N], new_to[N];

int bounds[4], bound_id;

void cut(int b0, int b1, int b2) {
bounds[0] = b0;
bounds[1] = b1;
bounds[2] = b2;
bounds[3] = 123456789;
bound_id = 0;
int new_cnt = 0;
int sum = 0;
for (int i = 0; i < cnt; i++) {
int new_sum = sum + (to[i] - from[i] + 1);
if (new_sum <= bounds[bound_id]) {
if (new_sum == bounds[bound_id]) {
bound_id++;
}
new_from[new_cnt] = from[i];
new_to[new_cnt] = to[i];
new_cnt++;
sum = new_sum;
continue;
}
int get = bounds[bound_id] - sum;
new_from[new_cnt] = from[i];
new_to[new_cnt] = from[i] + get - 1;
new_cnt++;
from[i] += get;
sum += get;
bound_id++;
i--;
}
cnt = new_cnt;
for (int i = 0; i < cnt; i++) {
from[i] = new_from[i];
to[i] = new_to[i];
}
}

int new_a[N];

int main() {
scanf("%d", &len);
for (int i = 1; i <= len; i++) {
scanf("%d", a + i);
}
init();
int tt;
scanf("%d", &tt);
int lastans = 0;
for (int qq = 1; qq <= tt; qq++) {
int com;
scanf("%d", &com);
if (com == 2) {
int ll, rr, kk;
scanf("%d %d %d", &ll, &rr, &kk);
ll = ((ll + lastans - 1) % len) + 1;
rr = ((rr + lastans - 1) % len) + 1;
kk = ((kk + lastans - 1) % len) + 1;
if (ll > rr) {
swap(ll, rr);
}
int ans = 0;
int sum = 0;
for (int i = 0; i < cnt; i++) {
int x = from[i], y = to[i];
int begin = sum + 1, end = sum + (y - x + 1);
if (ll > begin) {
x += ll - begin;
}
if (rr < end) {
y -= end - rr;
}
if (x <= y) {
ans += get_count(kk, x, y);
}
sum = end;
}
lastans = ans;
printf("%d\n", ans);
} else {
int ll, rr;
scanf("%d %d", &ll, &rr);
ll = ((ll + lastans - 1) % len) + 1;
rr = ((rr + lastans - 1) % len) + 1;
if (ll > rr) {
swap(ll, rr);
}
if (ll == rr) {
continue;
}
cut(ll - 1, rr - 1, rr);
int first = -1;
int sum = 0;
for (int i = 0; i < cnt; i++) {
if (sum == ll - 1) {
first = i;
}
sum += to[i] - from[i] + 1;
if (sum == rr) {
int p = from[i];
for (int j = i; j > first; j--) {
from[j] = from[j - 1];
to[j] = to[j - 1];
}
from[first] = to[first] = p;
break;
}
}
}
if (cnt > MAGIC) {
int pos = 0;
for (int i = 0; i < cnt; i++) {
for (int j = from[i]; j <= to[i]; j++) {
new_a[++pos] = a[j];
}
}
for (int i = 1; i <= len; i++) {
a[i] = new_a[i];
}
init();
}
}
return 0;
}


text:
# Find the riemann sum for f(x)=x+x^2?

Jul 14, 2018

${\int}_{a}^{b} f \left(x\right) \text{d} x = {\lim}_{n \to \infty} \frac{1}{n} ^ 2 {\sum}_{i = a n}^{b n} \left(i + {i}^{2} / n\right)$

#### Explanation:

A Riemann sum is a certain type of approximations for an integral. The easiest and simplest method to use is a rectangle sum; where we sum the area of $n$ rectangles with equal width to approximate the area under a curve; as $n$ approaches infinity, the sum approaches the true value of the net area.

Here's a Riemann sum with $4$ rectangles only:

This is just an example. Instead, consider $n$ rectangles.

Let $f$ be a function. Say, we wish to find the area under $f$ from $a$ to $b$. For the moment, let's only do the case $a = 0$ and $b = 1$. We will generalise later.

Now; imagine ${x}_{0} , {x}_{1} , {x}_{2} , \ldots {x}_{n}$ to be some "marks" on the x-axis such that the width of the i-th rectangle is ${x}_{i} - {x}_{i - 1}$ and ${x}_{0} = 0$.

Let ${\Delta}_{i} = {x}_{i} - {x}_{i - 1}$ be the width of the i-th rectangle. As we want ${\Delta}_{i}$ to be constant, we can more easily call it $\textcolor{red}{\Delta}$.

Another property about our x-axis marks is that the lenght/non-width side of the i-th rectangle is going to be $f \left({x}_{i}\right)$. Hence, the its area is going to be

$\text{Area"_i = "width" xx "lenght} = \Delta f \left({x}_{i}\right)$

Since we want the area until $b = 1$, we must have ${x}_{n} = 1$.

The areas of the rectangles approximate the areas under the graph of $f$. If there are $n$ rectangles, then:

${\int}_{0}^{1} f \left(x\right) \text{d} x \approx {\sum}_{i = 1}^{n} \left({x}_{i} - {x}_{i - 1}\right) f \left({x}_{i}\right) = {\sum}_{i = 1}^{n} \Delta f \left({x}_{i}\right)$

One way to make ${\Delta}_{i}$ constant is by defining ${x}_{i} = \frac{i}{n}$. Then,

${\Delta}_{i} = \frac{i}{n} - \frac{i - 1}{n} = \frac{1}{n}$, $\forall i \in \mathbb{N}$.

By this definition, ${x}_{0} = 0$ and ${x}_{n} = 1$, as desired.

${\int}_{0}^{1} f \left(x\right) \text{d} x \approx {\sum}_{i = 1}^{n} \frac{1}{n} f \left(\frac{i}{n}\right)$

The more rectangles we have, the better the sum approximates the integral. Thus, we can say that

${\int}_{0}^{1} f \left(x\right) \text{d} x = {\lim}_{n \to \infty} {\sum}_{i = 0}^{n} \frac{1}{n} f \left(\frac{i}{n}\right)$

Generalisation: Since we want the area from $a$ to $b$, the very first rectangle must have lenght $f \left(a\right)$ and the last one $f \left(b\right)$; to do this, we can change the lower and upper bounds of the sum operator:

$\textcolor{b l u e}{{\int}_{a}^{b} f \left(x\right) \text{d} x = {\lim}_{n \to \infty} {\sum}_{i = a n}^{b n} \frac{1}{n} f \left(\frac{i}{n}\right)}$

This is one of the many Riemann Sum formulae for a function $f$.

In our case, $f \left(x\right) = x + {x}^{2}$:

$\therefore {\int}_{a}^{b} f \left(x\right) \text{d} x \approx {\sum}_{i = a n}^{b n} \frac{1}{n} \left(\frac{i}{n} + {i}^{2} / {n}^{2}\right)$

$\therefore {\int}_{a}^{b} f \left(x\right) \text{d} x \approx \frac{1}{n} ^ 2 {\sum}_{i = a n}^{b n} \left(i + {i}^{2} / n\right)$

if there are $n$ rectangles, as stated already.