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Which of the following can be inferred from the data above?

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Which of the following can be inferred from the data above? [#permalink]

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Which of the following statements can be inferred from the data above?

I. The Klein family's annual income more than doubled from 1985 to 1995.
II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than $40,000. A. I only B. II only C. I and III only D. II and III only E. I, II, and III Hi This is an GMAT Prep Question.Please let me know the correct answer and why it correct? Moreover, why We cannot infer II from the data above? Originally posted by vikrantgulia on 31 Dec 2013, 02:01. Last edited by Bunuel on 14 Jul 2014, 03:16, edited 1 time in total. Edited the question. Senior Manager Status: Final Lap Joined: 25 Oct 2012 Posts: 270 Concentration: General Management, Entrepreneurship GPA: 3.54 WE: Project Management (Retail Banking) Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 31 Dec 2013, 04:36 vikrantgulia wrote: Hi This is an GMAT Prep Question.Please let me know the correct answer and why it correct? Moreover, why We cannot infer II from the data above? Hi there, The answer IMO is C : Statements I and III must be true according to the figure above I- The klein family's annual income more than doubled from 1985 to 1995 : True, In 1985 it was$25K while in 1995 it was more than $60K II- The klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995 That is not true because the annual income increased by approximately$15K from 1985 to 1990 while it increased by approximately $20K from 1990 to 1995 Notice that these values are obtained by substracting the value of the annual income of the latter year from that of the former III- The klein family's average (arithmetic mean) annual income for the period shown was greater than$ 40K.
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .
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Re: Which of the following can be inferred from the data above? [#permalink]

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13 Jul 2014, 17:03
For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under $40,000. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8077 Location: Pune, India Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 14 Jul 2014, 02:32 7 This post received KUDOS Expert's post 1 This post was BOOKMARKED dolphinitida wrote: For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under$40,000.

What you are talking about is the median. Year 6 will give you the median income. The mean needn't lie at the middle value.
Say, for a set of numbers 4, 5, 5, 6, 7, 20, 22, mean will not be 6 (the middle value). Median will be 6 but mean will be much higher.

Another way to analyze statement III is this:

First 3 points are at most 15000 less than 40,000.
Next 4 points are almost 40,000 each.
The next point is at least 15,000 more than 40,000
The next 3 points are at least 20,000 more than 40,000.

So overall, the distance of the points higher than 40,000 is more than the distance of the points lower than 40,000. This is just your deviation concept applied visually.

For more on deviation, check: http://www.veritasprep.com/blog/2012/05 ... eviations/
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews VP Joined: 09 Jun 2010 Posts: 1196 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 02 May 2015, 07:45 this question require us to approximate. not easy. but if you practice often, it would be easy _________________ visit my facebook to help me. on facebook, my name is: thang thang thang Intern Joined: 13 Jul 2015 Posts: 37 Location: Singapore GMAT 1: 730 Q50 V39 WE: Operations (Investment Banking) Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 01 Oct 2015, 00:37 vikrantgulia wrote: Attachment: GMAT.png Which of the following statements can be inferred from the data above? I. The Klein family's annual income more than doubled from 1985 to 1995. II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than$40,000.

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

Hi
This is an GMAT Prep Question.Please let me know the correct answer and why it correct?
Moreover, why We cannot infer II from the data above?

Bunuel, could you comment if this is an actual GMATPrep Quant question or an IR question?
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Re: Which of the following can be inferred from the data above? [#permalink]

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01 Oct 2015, 16:36
Hi SamuelWitwicky,

Most Test Takers see 1 question in the Quant section that involves a complex-looking chart, graph or table; the prompt is usually more complex than an 'average' question and takes the better part of 3 minutes to solve. Thus, this type of question COULD show up in your Quant section (but you won't see many of them).

As complex as the prompt looks, very little actual math is required to solve it. Some general estimation and pattern-matching will get you the correct answer.

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Intern Joined: 13 Jul 2015 Posts: 37 Location: Singapore GMAT 1: 730 Q50 V39 WE: Operations (Investment Banking) Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 01 Oct 2015, 19:04 EMPOWERgmatRichC wrote: Hi SamuelWitwicky, Most Test Takers see 1 question in the Quant section that involves a complex-looking chart, graph or table; the prompt is usually more complex than an 'average' question and takes the better part of 3 minutes to solve. Thus, this type of question COULD show up in your Quant section (but you won't see many of them). As complex as the prompt looks, very little actual math is required to solve it. Some general estimation and pattern-matching will get you the correct answer. GMAT assassins aren't born, they're made, Rich Hey thanks for the reply. ok if that is so, i'd like to understand the wording of the question so that I don't make the same mistake again. The question stem states, which of the following can be inferred from the graph right? I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8077 Location: Pune, India Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 01 Oct 2015, 19:32 2 This post received KUDOS Expert's post SamuelWitwicky wrote: I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. There is a difference: Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle) This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Which of the following can be inferred from the data above? [#permalink]

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01 Oct 2015, 19:50
VeritasPrepKarishma wrote:
SamuelWitwicky wrote:
I feel statement II can be inferred to be wrong from the graph.

And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No.

There is a difference:

Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only.

On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995?
When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle)

This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it.

ok got it! thanks a lot Karishma
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Re: Which of the following can be inferred from the data above? [#permalink]

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26 Dec 2015, 00:16
Rock750 wrote:
vikrantgulia wrote:
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .

How did you take last three numbers as 60. they look more close to 65 to me. Deciding the xact values make me waste a lot of time on these questions. Any help will be much appreciated. Thanks
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Re: Which of the following can be inferred from the data above? [#permalink]

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27 Dec 2015, 20:43
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Expert's post
NoHalfMeasures wrote:
Rock750 wrote:
vikrantgulia wrote:
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .

How did you take last three numbers as 60. they look more close to 65 to me. Deciding the xact values make me waste a lot of time on these questions. Any help will be much appreciated. Thanks

Use the deviation method I have discussed above.

If you do want to take values, take approximate values:
25, 26, 26, 35, 36, 38, 38, 56, 63, 63, 63

Mind you, the average you will get will not be very close to 40 so a few points here and there on your approximation will make no difference.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 29 Nov 2015 Posts: 13 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 29 Dec 2015, 11:31 VeritasPrepKarishma wrote: SamuelWitwicky wrote: I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. There is a difference: Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle) This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it. So just to be sure, in PS and DS, when the question asks 'can you infer...' it means can you get an exact value of this (true/false, yes/no, etc)? But in IR, when the question asks 'can you infer...' it means is there sufficient information provided in the paragraphs and tables for you to ARRIVE at an answer or DEDUCE an answer? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8077 Location: Pune, India Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 29 Dec 2015, 21:48 2 This post received KUDOS Expert's post sarathvr wrote: VeritasPrepKarishma wrote: SamuelWitwicky wrote: I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. There is a difference: Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle) This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it. So just to be sure, in PS and DS, when the question asks 'can you infer...' it means can you get an exact value of this (true/false, yes/no, etc)? But in IR, when the question asks 'can you infer...' it means is there sufficient information provided in the paragraphs and tables for you to ARRIVE at an answer or DEDUCE an answer? No, the difference does not lie in whether the question is asked in PS/DS or IR. The difference lies in the way the question is asked. Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. This is a statement. To infer it, it needs to be true. If this is not true, you say that you cannot infer this from the graph. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Which of the following can be inferred from the data above? [#permalink]

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21 Nov 2016, 08:23
Responding to a pm:
Quote:
For this particular question (URL link above), I chose Option E as I can infer all the statements, two of which(I and III) are right and one (II) is false.
Kindly explain where am I going wrong?

You cannot infer statement II. Here is why:

Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true ONLY.

On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995?
When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle)

This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 30 Aug 2016 Posts: 5 Location: Italy Schools: HEC Dec"18 (I) GMAT 1: 700 Q47 V41 GPA: 3.83 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 03 Jan 2017, 11:18 Just a quick note Once you realized that I. must be true and II. must not, then you can ignore statement III. and directly mark C: C is the only answer in which I. is true and II. is not! Intern Joined: 05 Oct 2016 Posts: 10 Location: India GMAT 1: 600 Q50 V30 GMAT 2: 600 Q50 V30 GMAT 3: 600 Q50 V30 GPA: 3.4 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 16 Feb 2017, 08:29 VeritasPrepKarishma wrote: dolphinitida wrote: For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under$40,000.

What you are talking about is the median. Year 6 will give you the median income. The mean needn't lie at the middle value.
Say, for a set of numbers 4, 5, 5, 6, 7, 20, 22, mean will not be 6 (the middle value). Median will be 6 but mean will be much higher.

Another way to analyze statement III is this:

First 3 points are at most 15000 less than 40,000.
Next 4 points are almost 40,000 each.
The next point is at least 15,000 more than 40,000
The next 3 points are at least 20,000 more than 40,000.

So overall, the distance of the points higher than 40,000 is more than the distance of the points lower than 40,000. This is just your deviation concept applied visually.

For more on deviation, check: http://www.veritasprep.com/blog/2012/05 ... eviations/

this saves a lot of time. i kept addiing all the values and calculating average, eating lot of time
thanks for the post
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Re: Which of the following can be inferred from the data above? [#permalink]

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07 Mar 2018, 06:02
I did a calculation mistake while evaluating statement 3 - marked answer A as option. Learning: I should not do calculation in mind, rather I should actually write calculations in a piece of paper.
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Re: Which of the following can be inferred from the data above? [#permalink]

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07 Mar 2018, 06:46
11 data points : Income in 1000
85 -> 25
86 -> 26
87 -> 26
88 -> 35
89 -> 40
90 -> 44
91 -> 44
92 -> 58
93 -> 64
94 -> 64
95 -> 64

I. The Klein family's annual income more than doubled from 1985 to 1995.
Income in 95 = 64 which is more than twice of 25 ( income from 85 ) - True : B, D are out

II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.

90 - 85 = 44 - 25 = 19
95 - 90 = 64 - 44 = 20

so 19 > 20 - No So E is out

III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than \$40,000.

to find average - Let's divide data points by 11 directly rather than summing and then dividing by 11
$$\frac{25}{11}$$ + $$\frac{26}{11}$$ + $$\frac{26}{11}$$ + $$\frac{35}{11}$$ + $$\frac{40}{11}$$ +
$$\frac{44}{11}$$ + $$\frac{44}{11}$$ + $$\frac{58}{11}$$ + $$\frac{64}{11}$$ + $$\frac{64}{11}$$ +
$$\frac{64}{11}$$

= 2 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5 + 5 + 5 [ just taking integer values ]
= 8 + 10 + 9 + 15
= 18 + 24
= 42 > 40 hence answer = C

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
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Re: Which of the following can be inferred from the data above?   [#permalink] 07 Mar 2018, 06:46
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# Demonstration of the trapped-ion quantum CCD computer architecture

## Abstract

The trapped-ion quantum charge-coupled device (QCCD) proposal1,2 lays out a blueprint for a universal quantum computer that uses mobile ions as qubits. Analogous to a charge-coupled device (CCD) camera, which stores and processes imaging information as movable electrical charges in coupled pixels, a QCCD computer stores quantum information in the internal state of electrically charged ions that are transported between different processing zones using dynamic electric fields. The promise of the QCCD architecture is to maintain the low error rates demonstrated in small trapped-ion experiments3,4,5 by limiting the quantum interactions to multiple small ion crystals, then physically splitting and rearranging the constituent ions of these crystals into new crystals, where further interactions occur. This approach leverages transport timescales that are fast relative to the coherence times of the qubits, the insensitivity of the qubit states of the ion to the electric fields used for transport, and the low crosstalk afforded by spatially separated crystals. However, engineering a machine capable of executing these operations across multiple interaction zones with low error introduces many difficulties, which have slowed progress in scaling this architecture to larger qubit numbers. Here we use a cryogenic surface trap to integrate all necessary elements of the QCCD architecture—a scalable trap design, parallel interaction zones and fast ion transport—into a programmable trapped-ion quantum computer that has a system performance consistent with the low error rates achieved in the individual ion crystals. We apply this approach to realize a teleported CNOT gate using mid-circuit measurement6, negligible crosstalk error and a quantum volume7 of 26 = 64. These results demonstrate that the QCCD architecture provides a viable path towards high-performance quantum computers.

## Access options

from\$8.99

All prices are NET prices.

## Data availability

The data presented in this manuscript are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was made possible by a large group of people, and we thank the entire Honeywell Quantum Solutions team for their many contributions.

## Author information

Authors

### Contributions

J.M.P., J.M.D., C.F., J.P.G., S.A.M., M.S.A. and B.N. all contributed to the experimental design, construction and data collection. C.H.B. and K.M. contributed to the theory, quantum circuit design and data analysis. All authors contributed to this manuscript.

### Corresponding author

Correspondence to D. Hayes.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Extended data

is available for this paper at https://doi.org/10.1038/s41586-021-03318-4.

Peer review information Nature thanks Winfried Hensinger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 System details.

a, Close-up schematic of the trap region used in this work. Dimensions are in micrometres. We refer to the orange gating region on the left as zone 1 and the blue gating region as zone 2. b, Transport primitive library and associated heating estimates. For the estimates, we fit four ion-crystal spin-flip data to a model that assumes that all modes are at the same temperature. The fitted temperature increases are converted to units of quanta/mode (we note that the inequality holds for all modes). The times shown do not include interpolation between different operations or small delays in the electronics, which increase the time for every operation by ~10%. The interzone shift is a linear shift between the two gate zones, whereas an intrazone shift moves ions within a single gate zone by 110 μm for single-qubit addressing. c, Times for qubit operations, transport and cooling. Circuits can be run using two different measurement protocols. For circuits in which all measurements are made at the end, we use the high-fidelity measurement setting. Circuits containing mid-circuit measurements use shorter-duration measurements to minimize the crosstalk error on idle qubits. The shorter detection time measurement error is ~7 × 10−3, about twice as large as those reported in Table 1. Mid-circuit measurements induce an error of ~1% on neighbouring idle qubits, as measured by a Ramsey experiment. There are three different cooling stages used during transport (stages 1 and 2) and before gates (stage 3) and are either implemented through Doppler or sideband (SB) cooling. d, Construction of a phase-insensitive TQ gate. The Mølmer–Sørensen interaction generates the unitary $${U}_{{\rm{MS}}}=\exp [-{\rm{i}}\frac{{\rm{\pi }}}{4}{(X\sin \varphi +Y\cos \varphi )}^{\otimes 2}]$$ (orange), whose basis is determined by the optical phase ϕ. SQ operations driven by the same laser beams generate the unitary $${U}_{{\rm{SQ}}}=\exp [-{\rm{i}}\frac{{\rm{\pi }}}{4}(X\cos \,\varphi +Y\sin \,\varphi )]$$ (blue) and are applied globally to both qubits. The resulting composite gate is, up to a global phase, given by $${U}_{zz}=\exp (-{\rm{i}}\frac{{\rm{\pi }}}{4}Z\otimes Z)$$ (green).

### Extended Data Fig. 2 Estimated gate errors and randomized benchmarking results.

a, SQ RB for all four qubits. b, TQ RB results for zone 1. c, TQ RB results for zone 2. d, An estimated error budget for a single TQ gate operation. e, Correlation parameters for simultaneous TQ RB. Each square represents a value of δi,j with the numbers being scaled by 10−4. All are within one standard deviation of zero. f, Addressability error estimates for simultaneous RB. The quantities $${\alpha }_{z}^{{\rm{both}}}$$ and γz refer to simultaneous RB decay rates and their deviation from individual RB decay rates, as described in the text.

### Extended Data Fig. 3 Teleported CNOT gate experiment results.

a, Circuit implementing a teleported CNOT gate, with q0 and q3 the control and target qubit, respectively. b, Bar plots showing the distribution of measurement outcomes when qubits q0 and q3 are prepared and measured in the {|0, |1} and {|+, |−} bases. c, The teleported CNOT gate time budget.

### Extended Data Fig. 4 QV N = 4 results.

a, Example of the transport operations needed for part of an N = 4 QV circuit. Trap regions are labelled as load zone (L); auxiliary zone 1 (A1); gate zone 1, left (G1L); gate zone 1, centre (G1C); gate zone 1, right (G1R); auxiliary zone 2 (A2); gate zone 2, left (G2L); gate zone 2, centre (G2C); and gate zone 2, right (G2R). b, An N = 4 QV circuit time budget.

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Pino, J.M., Dreiling, J.M., Figgatt, C. et al. Demonstration of the trapped-ion quantum CCD computer architecture. Nature 592, 209–213 (2021). https://doi.org/10.1038/s41586-021-03318-4

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## Olivier Bournez ; Amaury Pouly - A Universal Ordinary Differential Equation

lmcs:4437 - Logical Methods in Computer Science, February 28, 2020, Volume 16, Issue 1 - https://doi.org/10.23638/LMCS-16(1:28)2020
A Universal Ordinary Differential Equation

Authors: Olivier Bournez ; Amaury Pouly

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\varphi$ on the reals, and for any positive continuous function $\epsilon(t)$, it has a $\mathcal{C}^\infty$ solution with $| y(t) - \varphi(t) | < \epsilon(t)$ for all $t$. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE \emph{never} has a unique solution, even with a finite number of conditions of the form $y^{(k_i)}(a_i)=b_i$. The question whether one can require the solution that approximates $\varphi$ to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a \textbf{fixed} polynomial ODE such that for any $\varphi$ and $\epsilon(t)$ there exists some initial condition that yields a solution that is $\epsilon$-close to $\varphi$ at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.

Volume: Volume 16, Issue 1
Published on: February 28, 2020
Accepted on: February 28, 2020
Submitted on: April 11, 2018
Keywords: Mathematics - Classical Analysis and ODEs,Computer Science - Computational Complexity,Computer Science - Logic in Computer Science,Electrical Engineering and Systems Science - Systems and Control,Mathematics - Dynamical Systems


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# Questions about equivalance relations and equivalance classes. . .

• Aug 31st 2011, 01:54 PM
Dakaa
Questions about equivalance relations and equivalance classes. . .
http://i54.tinypic.com/1z2q1dl.png

for the first question, can i assume the non-empty set A has more than one element, and say a = something and b = something to give example.
• Aug 31st 2011, 07:56 PM
FernandoRevilla
Re: Questions about equivalance relations and equivalance classes. . .
Quote:

Originally Posted by Dakaa
http://i54.tinypic.com/1z2q1dl.png

for the first question, can i assume the non-empty set A has more than one element, and say a = something and b = something to give example.

No, you can't. Choose for example $A=\{0\}$ . A little help: Reflexive For all $x\in\mathbb{Z}$ we have $x-x=0$ . But $0\in A$ because being $A\neq \emptyset$ , choosing $a\in A$ we have $a-a=0\in A$ . Conclusion: $xRx$ .

Try the rest.
• Sep 4th 2011, 07:13 PM
Dakaa
Re: Questions about equivalance relations and equivalance classes. . .
Quote:

Originally Posted by FernandoRevilla
No, you can't. Choose for example $A=\{0\}$ . A little help: Reflexive For all $x\in\mathbb{Z}$ we have $x-x=0$ . But $0\in A$ because being $A\neq \emptyset$ , choosing $a\in A$ we have $a-a=0\in A$ . Conclusion: $xRx$ .

Try the rest.

Yup, did the rest, now moving on to question 2.