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Convert Acre To Square Feet Calculator
The acre is the most commonly used unit for measuring vast land areas. Acre is used in imperial and US customary systems. As stated in the International yard and pound agreement of 1959, an acre is
equivalent to 4,046.8564224 square meters.
If we go back to the medieval era, there is an interesting story about an acre. The land area ploughed by the farmer with the help of an ox in a day was known as an acre.
The acre is denoted by ac, Which means 1 acre can also be written as 1ac. One acre is equal to 43560 square ft.
Easy and free online unit converter calculator to calculate the units of length, area, height, volume etc.
Click here to Cubic Meter To Square Feet Converter Calculators quickly and easily.
The square foot is generally used to define the carpet, built-up and super area of the property. Square feet are used in imperial and US customary systems. One square foot can also be defined as an
area equivalent to the square with sides that are 1 foot.
If you want to covert the land area into square feet, you have to multiply the acres by 43560. We can covert the acre into square ft. with the following formula:
Square feet = acres X 43,560
For example, If you want to convert the 10-acre land area into square feet multiply 10-acre land with 43560.
10 acre X 43560 =435600 square feet.
Check out to know more about MM To Square Feet Converter & how to convert Hectare To Acre Unit Converter? | {"url":"https://www.velocityhousing.in/calculator/acre-to-square-feet-unit-converter-unit-conversion-1-acre-to-square-feet","timestamp":"2024-11-13T18:24:38Z","content_type":"text/html","content_length":"44787","record_id":"<urn:uuid:ac9f6c8d-eab3-462a-b3c3-1640dcae72f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00349.warc.gz"} |
A zero padding apparatus and method for fixed length signaling information are disclosed. A zero padding apparatus according to an embodiment of the present invention includes a processor configured
to generate a LDPC information bit string by deciding a number of groups whose all bits are to be filled with 0 using a difference between a length of the LDPC information bit string and a length of
a BCH-encoded bit string, selecting the groups using a shortening pattern order to fill all the bits of the groups with 0, and filling at least a part of remaining groups, which are not filled with
0, with the BCH-encoded bit string; and memory configured to provide the LDPC information bit string to an LDPC encoder.
Latest Electronics and Telecommunications Research Institute Patents:
This application is a Continuation Application of U.S. Pat. Application No. 17/891,189, filed on Aug. 19, 2022, which is a Continuation Application of U.S. Pat. Application No. 17/202,475, filed on
Mar. 16, 2021, now U.S. Pat. No. 11,463,475, issued on Oct. 4, 2022, which is a Continuation Application of U.S. Pat. Application No. 16/390,316, filed on Apr. 22, 2019, now U.S. Pat. No. 10,992,319,
issued on Apr. 27, 2021, which is a Continuation Application of U.S. Pat. Application No. 15/553,458 filed on Aug. 24, 2017, now U.S. Pat. No. 10,320,418, which claims the benefit under 35 USC 119(a)
of PCT Application No. PCT/KR2016/001756, filed on Feb. 23, 2016, which claims the benefit of Korean Patent Application No. 10-2015-0028060, filed Feb. 27, 2015, Korean Patent Application No.
10-2015-0031947, filed Mar. 6, 2015, and Korean Patent Application No. 10-2016-0020636 filed Feb. 22, 2016 in the Korean Intellectual Property Office, the entire disclosure of which is incorporated
herein by reference for all purposes.
The present invention relates to channel encoding and modulation techniques for the transmission of signaling information, and more particularly to encoding and decoding apparatuses for effectively
transmitting signaling information in a next generation digital broadcasting system.
Bit-Interleaved Coded Modulation (BICM) is bandwidth-efficient transmission technology, and is implemented in such a manner that an error-correction coder, a bit-by-bit interleaver and a high-order
modulator are combined with one another.
BICM can provide excellent performance using a simple structure because it uses a low-density parity check (LDPC) coder or a Turbo coder as the error-correction coder. Furthermore, BICM can provide
high-level flexibility because it can select modulation order and the length and code rate of an error correction code in various forms. Due to these advantages, BICM has been used in broadcasting
standards, such as DVB-T2 and DVB-NGH, and has a strong possibility of being used in other next-generation broadcasting systems.
Such BICM may be used not only for the transmission of data but also for the transmission of signaling information. In particular, channel encoding and modulation techniques for the transmission of
signaling information need to be more robust than channel encoding and modulation techniques for the transmission of data.
Therefore, in particular, there is a pressing need for new channel encoding and modulation techniques for the transmission of signaling information.
DISCLOSURE Technical Problem
An object of the present invention is to provide channel encoding and modulation techniques that are appropriate for the transmission of signaling information via a broadcast system channel.
Another object of the present invention is to provide a new zero padding technique that is optimized for the transmission of signaling information.
Technical Solution
In order to accomplish the above objects, the present invention provides a zero padding apparatus, including: a processor configured to generate a LDPC information bit string by deciding a number of
groups whose all bits are to be filled with 0 using a difference between a length of the LDPC information bit string and a length of a BCH-encoded bit string, selecting the groups using a shortening
pattern order to fill all the bits of the groups with 0, and filling at least a part of remaining groups, which are not filled with 0, with the BCH-encoded bit string; and memory configured to
provide the LDPC information bit string to an LDPC encoder.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the processor may decide a group which is partially to be filled with 0 using the shortening pattern order, and fill a number of bits corresponding to a number decided by using the
difference between the length of the LDPC information bit string and the length of the BCH-encoded bit string with 0 from the start of the group.
In this case, the shortening pattern order may be defined for 9 groups.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
In this case, the LDPC codeword may be used for parity permutation which is performed by group-wise interleaving corresponding to an order of group-wise interleaving and a part of parity bits of the
LDPC codeword is punctured after the parity permutation.
Furthermore, according to an embodiment of the present invention, there is provided a zero padding method, including: deciding a number of groups whose all bits are to be filled with 0 using a
difference between a length of an LDPC information bit string and a length of a BCH-encoded bit string; selecting the groups using a shortening pattern order; filling all the bits of the groups with
0; and generating the LDPC information bit string by filling at least a part of remaining groups which are not filled with 0, with the BCH-encoded bit string.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information.
In this case, the zero padding method may further comprise providing the LDPC information bit string to an LDPC encoder.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the zero padding method may further comprise deciding a group which is partially to be filled with 0 using the shortening pattern order; and filling a number of bits corresponding to a
number decided by using the difference between the length of the LDPC information bit string and the length of the BCH-encoded bit string with 0 from the start of the group.
In this case, the shortening pattern order may be defined for 9 groups.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
In this case, the LDPC codeword may be used for parity permutation which is performed by group-wise interleaving corresponding to an order of group-wise interleaving and a part of parity bits of the
LDPC codeword is punctured after the parity permutation.
Furthermore, according to an embodiment of the present invention, there is provided an inverse zero padding apparatus, including: memory configured to receive an LDPC information bit string from an
LDPC decoder; and a processor configured to select groups whose all bits are filled with 0 using a shortening pattern order and generate a BCH-encoded bit string from the LDPC information bit string
using remaining groups exclusive of the groups.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the shortening pattern order may be defined for 9 groups.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
Advantageous Effects
According to the present invention, the channel encoding and modulation techniques that are appropriate for the transmission of signaling information via a broadcast system channel are provided.
Furthermore, in the present invention, shortening and puncturing are optimized according to the amount of signaling information in the construction of BICM for the transmission of signaling
information, thereby being able to efficiently transmit/receive the signaling information.
FIG. 1 is a block diagram showing a signaling information encoding/decoding system according to an embodiment of the present invention;
FIG. 2 is an operation flowchart showing a signaling information encoding method according to an embodiment of the present invention;
FIG. 3 is an operation flowchart showing a signaling information decoding method according to an embodiment of the present invention;
FIG. 4 is a diagram showing a broadcast signal frame according to an embodiment of the present invention;
FIG. 5 is a diagram showing the structure of a parity check matrix (PCM) corresponding to an LDPC code according to an embodiment of the present invention;
FIG. 6 is a diagram showing an example of the operation of the zero padding unit shown in FIG. 1;
FIG. 7 is a diagram showing an example of the operation of the parity permutation unit shown in FIG. 1;
FIG. 8 is a diagram showing an example of the operation of the zero removing unit shown in FIG. 1;
FIG. 9 is a block diagram showing a zero padding apparatus according to an embodiment of the present invention; and
FIG. 10 is an operation flowchart showing a zero padding method according to an embodiment of the present invention.
The present invention will be described in detail below with reference to the accompanying drawings. Repeated descriptions and descriptions of well-known functions and configurations that have been
deemed to make the gist of the present invention unnecessarily obscure will be omitted below. The embodiments of the present invention are intended to fully describe the present invention to persons
having ordinary knowledge in the art to which the present invention pertains. Accordingly, the shapes, sizes, etc. of components in the drawings may be exaggerated to make the description obvious.
Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
FIG. 1 is a block diagram showing a signaling information encoding/decoding system according to an embodiment of the present invention.
Referring to FIG. 1, the signaling information encoding/decoding system includes a signaling information encoding apparatus 100, and a signaling information decoding apparatus 300.
The signaling information encoding apparatus 100 and the signaling information decoding apparatus 300 perform communication through the medium of a wireless channel 200.
The signaling information encoding apparatus 100 channel-encodes and modulates signaling information, such as L1-Basic, L1-Detail or the like.
The signaling information encoding apparatus 100 includes a segmentation unit 110, a scrambling unit 120, a BCH encoder 130, a zero padding unit 140, an LDPC encoder 150, a parity permutation unit
160, a parity puncturing unit 170, a zero removing unit 180, a bit interleaving unit 190, and a constellation mapping unit 195.
The signaling information encoding apparatus 100 shown in FIG. 1 may be viewed as corresponding to a Bit-Interleaved Coded Modulation (BICM) apparatus. In this case, the error correction encoder of
the BICM apparatus may be viewed as corresponding to the segmentation unit 110, the scrambling unit 120, the BCH encoder 130, the zero padding unit 140, the LDPC encoder 150, the parity permutation
unit 160, the parity puncturing unit 170, and the zero removing unit 180 that are shown in FIG. 1.
When the length of the signaling information is longer than a preset length, the segmentation unit 110 segments the signaling information into a plurality of groups in order to segment the signaling
information into a plurality of LDPC codewords and then transmit the LDPC codewords. That is, when the signaling information cannot be contained in a single LDPC codeword, the segmentation unit may
determine the number of codewords in which the signaling information is to be contained, and then may segment the signaling information according to the determined number of codewords.
For example, when the length of the signaling information is fixed like L1-Basic, the signaling information encoding apparatus 100 may not include the segmentation unit 110.
For example, when the length of the signaling information is variable like L1-Detail, the signaling information encoding apparatus 100 may include the segmentation unit 110.
The scrambling unit 120 performs scrambling in order to protect the signaling information. In this case, the scrambling may be performed using various methods that are known in the present technical
The BCH encoder 130 performs BCH encoding using a BCH parity whose parity length N[bch_Parity] is 168 bits.
In this case, the BCH encoding may be the same as BCH encoding for LDPC code in which the length of data BICM is 16200.
In this case, a BCH polynomial used for the BCH encoding may be expressed in Table 1 below, and the BCH encoding expressed in Table 1 may have 12-bit error correction capability:
Code Length N[ldpc] =16200 g[1](x) 1+x+x^3+x^5+x^14 g[2](X) 1+x^6+x^8+x^11+x^14 g[3](X) 1+x+x^2+x^6+x^9+x^10+x^14 g[4](X) 1+x^4+x^7+x^8+x^10+x^12+x^14 g[5](X) 1+x^2+x^4+x^6+x^8+x^9+x^11+x^13+x^14 g
[6](X) 1+x^3+x^7+x^8+x^9+x^13+x^14 g[7](x) 1+x^2+x^5+x^6+x^7+x^10+x^11+x^13+x^14 g[8](X) 1+x^5+x^8+x^9+x^10+x^11+x^14 g[9](X) 1+x+x^2+x^3+x^9+x^10+x^14 g[10](X) 1+x^3+x^6+x^9+x^11+x^12+x^14 g[11](X)
1+x^4+x^11+x^12+x^14 g[12](X) 1+x+x^2+x^3+x^5+x^6+x^7+x^8+x^10+x^13+x^14
After the BCH encoding has been performed, the zero padding unit 140 performs zero padding or shortening.
In this case, the zero padding means that part of a bit string is filled with bit “0.”
As a result of the BCH encoding, the length of the bit string may be expressed by N[bch] = K[sig] + N[bch_Parity]. In this case, K[sig] may be the number of information bits of the BCH encoding. For
example, when K[sig] is fixed to 200 bits like L1-Signaling information, N[bch] may be 368 bits.
When the LDPC encoder 150 uses an LDPC code whose code rate is 3/15 and whose length is 16200, the information length K[ldpc] of the LDPC code is 3240 bits. In this case, since information that is to
be actually transmitted is N[bch] bits and the length of the information part of the LDPC code is K[ldpc] bits, zero padding, i.e., the process of filling bits corresponding to K[ldpc] - N[bch] with
bit “0,” is performed. K[ldpc] - N[bch] may be 2872 in case of L1-Basic information.
In this case, the order of the zero padding plays an important role in determining the performance of the encoder, and the order of the zero padding may be expressed as shortening pattern order.
In this case, the bits padded with zeros are used only for LDPC encoding, and are not actually transmitted.
The LDPC information bits composed of K[ldpc] bits is segmented into N[info_group] groups, as shown in Equation 1 below. For example, when K[ldpc] is 3240, N[info_group] is 9, and thus the LDPC
information bits may be grouped into 9 groups.
where Z[j] is a group composed of 360 bits.
The part of K[ldpc] bits that is zero-padded is determined according to the following procedure.
(Step 1) First, the number of groups in which all the bits thereof will be padded with “0” is calculated using Equation 2 below:
For example, when K[ldpc] is 3240 and N[bch] is 368, N[pad] may be 7. The fact that N[pad] is 7 indicates that the number of groups in which all the bits thereof will be padded with “0” is 7.
(Step 2) When N[pad] is not 0, zero padding is performed on N[pad] groups in the order of Z[πs(0)],Z[πs(1)],...,Z[πs(Npad-1)] according to the shortening pattern order π[s](j) of Table 2 below. In
this case, π[s](j) may refer to the shortening pattern order of a j -th bit group.
When N[pad] is 0, the above procedure is omitted.
MOD E N[group] π[s](j) (0 ≤ j < N[group]) π[s](0) π[s](1) π[s](2) π[s](3) π[s](4) π[s](5) π[s](6) π[s](7) π[s](8) L1-Basic 9 4 1 5 2 8 6 0 7 3
The shortening pattern order of Table 2 above indicates that zero padding targets are selected in the order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd
group indexed as 2, a 9th group indexed as 8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3. That is, when only 7 groups are selected
as zero padding targets in the example of Table 2 above, a total of 7 groups, i.e., the 5th group indexed as 4, the 2nd group indexed as 1, the 6th group indexed as 5, the 3rd group indexed as 2, the
9th group indexed as 8, the 7th group indexed as 6, and the 1st group indexed as 0, are selected as the zero padding targets.
In particular, the shortening pattern order of Table 2 above may be optimized for fixed length signaling information.
When the number of groups in which all the bits thereof will be padded with “0” and the corresponding groups are determined, all the bits of the determined groups are filled with “0.”
(Step 3) Additionally, for a group corresponding to Z[πs] (N[pad]), bits corresponding to (K[ldpc] - N[bch] - 360 × N[pad]) from the start of the group are additionally zero-padded. In this case, the
fact that zero padding is performed from the start of the corresponding group may indicate that zero padding is performed from a bit corresponding to a smaller index.
(Step 4) After the zero padding has been all completed, an LDPC information bit string is generated by sequentially mapping BCH-encoded N[bch] bits to a remaining part that has not been zero-padded.
The LDPC encoder 150 performs LDPC encoding using K[ldpc] and which has been zero-padded and to which signaling information has been mapped.
In this case, the LDPC encoder 150 may correspond to an LDPC codeword whose code rate is 3/15 and whose length is 16200. The LDPC codeword is a systematic code, and the LDPC encoder 150 generates an
output vector, such as that of Equation 3 below:
For example, when K[ldpc] is 3240, parity bits may be 12960 bits.
The parity permutation unit 160 performs group-wise parity interleaving on a parity part, not an information part, as a preliminary task for parity puncturing.
In this case, the parity permutation unit 160 may perform parity interleaving using Equation 4 below:
where Y is a j-th group-wise interleaved bit group, and π(j) is the order of group-wise interleaving, which may be defined in Table 3 below:
Mode N[group] Order of Group-wise interleaving π(j) (9 ≤ j < 45) 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 L1-Basic 45 20 23 25 32 38
That is, the parity permutation unit 160 outputs 3240 bits (9 bit groups) corresponding to information bits among the 16200 bits (45 bit groups) of the LDPC codeword without change, groups 12960
parity bits into 36 bit groups each including 360 bits, and interleave the 36 bit groups in the order of group-wise interleaving corresponding to Table 3 above.
The order of group-wise interleaving of Table 3 indicates that a 21th group indexed as 20 is located at a 10th group location indexed as 9, a 24rd group indexed as 23 is located at a 11st group
location indexed as 10, a 26th group indexed as 25 is located at a 12nd group location indexed as 11,..., and a 18th bit group indexed as 17 is located at a 45th group location indexed as 44.
In this case, the bit group (the bit group indexed as 20) at a front location may correspond to most important parity bits, and the bit group (the bit group indexed as 17) at a rear location may
correspond to least important parity bits.
In particular, the order of group-wise interleaving of Table 3 may be optimized for fixed length signaling information.
After the parity interleaving (parity permutation) has been completed, the parity puncturing unit 170 may puncture the partial parities of the LDPC codeword. The punctured bits are not transmitted.
In this case, after the parity interleaving has been completed, parity repetition in which part of the parity-interleaved LDPC parity bits is repeated may be performed before parity puncturing is
The parity puncturing unit 170 calculates a final puncturing size, and punctures bits corresponding to the calculated final puncturing size. The final puncturing size corresponding to the number of
bits to be punctured may be calculated according to the length N[bch] of the BCH-encoded bit string as follows:
(Step 1) A temporary puncturing size N[punc_temp] is calculated using Equation 5 below:
where K[ldpc] is the length of the LDPC information bit string, N[bch] is the length of the BCH-encoded bit string, A is a first integer, and B is a second integer.
In this case, the difference K[ldpc] - N[bch] between the length of the LDPC information bit string and the length of the BCH-encoded bit string may correspond to a zero padding length or a
shortening length.
The parameters for puncturing required for the calculation of Equation 5 may be defined as in Table 4 below:
N[bch] K[ldpc] A B n N[ldpc_parity] η[Mod] L1-Basic 368 3240 0 9360 0 12960 2
where N[ldpc_parity] is the number of parity bits of the LDPC codeword, and η[MOD] is a modulation order. In this case, the modulation order may be 2, which is indicative of QPSK.
In particular, the parameters for puncturing of Table 4 may be optimized for fixed length signaling information.
(Step 2) The temporary number of transmission bits N[FFC_temp] is calculated using the calculated temporary puncturing size N[punc_temp] and N[ldpc_parity] of Table 4, as shown in Equation 6 below:
(Step 3) The number of transmission bits N[FFC] is calculated using the temporary number of transmission bits N[FFC_temp], as shown in Equation 7 below:
The number of transmission bits N[FFC] is the sum of the length of the information part and the length of the parity part after the completion of the puncturing.
(Step 4) A final puncturing size N[punc] is calculated using the calculated number of transmission bits N[FFC] as shown in Equation 8 below:
where the final puncturing size N[punc] is the size of parities that need to be punctured.
That is, the parity puncturing unit 170 may puncture the last N [punc] bits of the whole LDPC codeword on which the parity permutation and the repetition have been performed.
The zero removing unit 180 removes zero-padded bits from the information part of the LDPC codeword.
The bit interleaving unit 190 performs bit interleaving on the zero-removed LDPC codeword. In this case, the bit interleaving may be performed using a method in which the direction in which the LDPC
codeword is recorded in memory of a preset size and the direction in which the LDPC codeword is read therefrom are made different.
The constellation mapping unit 195 performs symbol mapping. For example, the constellation mapping unit 195 may be implemented using a QPSK method.
The signaling information decoding apparatus 300 demodulates and channel-decodes signaling information, such as L1-Basic, L1-Detail, or the like.
The signaling information decoding apparatus 300 includes a constellation de-mapping unit 395, a bit de-interleaving unit 390, an inverse zero removing unit 380, an inverse parity puncturing unit 370
, an inverse parity permutation unit 360, an LDPC decoder 360, an inverse zero padding unit 340, a BCH decoder 330, an inverse scrambling unit 320, and an inverse segmentation unit 310.
The signaling information decoding apparatus 300 shown in FIG. 1 may be viewed as corresponding to a Bit-Interleaved Coded Modulation (BICM) decoding apparatus. In this case, the error correction
decoder of the BICM decoding apparatus may be viewed as corresponding to the inverse zero removing unit 380, the inverse parity puncturing unit 370, the inverse parity permutation unit 360, the LDPC
decoder 360, the inverse zero padding unit 340, the BCH decoder 330, the inverse scrambling unit 320 and the inverse segmentation unit 310 that are shown in FIG. 1.
The inverse segmentation unit 310 performs the inverse operation of the segmentation unit 110.
The inverse scrambling unit 320 performs the inverse operation of the scrambling unit 120.
The BCH decoder 330 performs the inverse operation of the BCH encoder 130.
The inverse zero padding unit 340 performs the inverse operation of the zero padding unit 140.
In particular, the inverse zero padding unit 340 may receive an LDPC information bit string from the LDPC decoder 350, may select groups whose all bits are filled with 0 using shortening pattern
order, and may generate a BCH-encoded bit string from the LDPC information bit string using groups exclusive of the former groups.
The LDPC decoder 350 performs the inverse operation of the LDPC encoder 150.
The inverse parity permutation unit 360 performs the inverse operation of the parity permutation unit 160.
In particular, the inverse parity permutation unit 360 may segment the parity bits of the LDPC codeword into a plurality of groups, and may group-wise de-interleave the groups using the order of
group-wise interleaving, thereby generating an LDPC codeword that is to be LDPC-decoded.
The inverse parity puncturing unit 370 performs the inverse operation of the parity puncturing unit 170.
In this case, the inverse parity puncturing unit 370 may calculate a temporary puncturing size using a first integer, multiplied by the difference between the length of the LDPC information bit
string and the length of the BCH-encoded bit string, and a second integer different from the first integer, may calculate the temporary number of transmission bits using the difference between the
sum of the length of the BCH-encoded bit string and 12960 and the temporary puncturing size, may calculate the number of transmission bits using the temporary number of transmission bits and
modulation order, may calculate a final puncturing size using the temporary number of transmission bits, the number of transmission bits and the temporary number of transmission bits, and may
generate an LDPC codeword to be provided to the inverse parity permutation unit 360 by taking into account the final puncturing size.
The inverse zero removing unit 380 performs the inverse operation of the zero removing unit 180.
The bit de-interleaving unit 390 performs the inverse operation of the bit interleaving unit 190.
The constellation de-mapping unit 395 performs the inverse operation of the constellation mapping unit 195.
FIG. 2 is an operation flowchart showing a signaling information encoding method according to an embodiment of the present invention.
Referring to FIG. 2, the signaling information encoding method according to the embodiment of the present invention includes segmenting signaling information into a plurality of groups first at step
At step S210, when the length of the signaling information is longer than a preset length, the signaling information is segmented into a plurality of groups in order to segment the signaling
information into a plurality of LDPC codewords and then transmit the LDPC codewords. That is, when the signaling information cannot be contained in a single LDPC codeword, the number of codewords in
which the signaling information is to be contained may be determined and then the signaling information may be segmented according to the determined number of codewords at step S210.
For example, when the length of the signaling information is variable like L1-Detail, the signaling information encoding method may include step S210.
For example, when the length of the signaling information is fixed like L1-Basic, the signaling information encoding method may not include step S210.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing scrambling in order to protect the signaling information at step S220.
In this case, the scrambling may be performed using various methods that are known in the present technical field.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing BCH encoding using a BCH parity whose parity length N[bch_Parity] is
168 bits at step S230.
Step S230 may be performed by the BCH encoder 130 shown in FIG. 1.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing zero padding or shortening after the BCH encoding has been performed at
step S240.
In this case, the zero padding may be performed by the zero padding unit 140 shown in FIG. 1.
Since information that is to be actually transmitted is N[bch] bits and the length of the information part of the LDPC code is K[ldpc] bits, zero padding, i.e., the process of filling bits
corresponding to K[ldpc] - N[bch] with bit “0,” is performed at step S240.
The zero padding of step S240 may be performed according to the shortening pattern order of Table 2.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing LDPC encoding using K[ldpc] and which has been zero-padded and to which
signaling information has been mapped at step S250.
In this case, step S250 may be performed by an LDPC encoder corresponding to an LDPC codeword whose code rate is 3/15 and whose length is 16200.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing group-wise parity interleaving on a parity part, not an information
part, as a preliminary task for parity puncturing at step S260.
In this case, at step S260, the group-wise parity interleaving may be performed according to the order of group-wise interleaving of Equation 4 and Table 3.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes puncturing the partial parities of the LDPC codeword after the parity interleaving
(parity permutation) has been completed at step S270.
At step S270, the punctured bits are not transmitted.
In this case, after the parity interleaving has been completed, parity repetition in which part of the parity-interleaved LDPC parity bits is repeated may be performed before parity puncturing is
The parity puncturing of step S270 may be performed by the parity puncturing unit 170 shown in FIG. 1.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing zero removing, i.e., the process of removing the zero-padded bits from
the information part of the LDPC codeword, at step S280.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing bit interleaving on the zero-removed LDPC codeword at step S290. In
this case, step S290 may be performed using a method in which the direction in which the LDPC codeword is recorded in memory of a preset size and the direction in which the LDPC codeword is read
therefrom are made different.
Furthermore, the signaling information encoding method according to the embodiment of the present invention includes performing symbol mapping at step S295.
FIG. 3 is an operation flowchart showing a signaling information decoding method according to an embodiment of the present invention.
Referring to FIG. 3, the signaling information decoding method according to the embodiment of the present invention includes performing constellation de-mapping on a signal received via an antenna at
step S310.
In this case, step S310 may correspond to the inverse operation of step S295 shown in FIG. 2, and may be performed by the constellation de-mapping unit 395 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing bit de-interleaving at step S320.
In this case, step S320 may correspond to the inverse operation of step S290 shown in FIG. 2, and may be performed by the bit de-interleaving unit 390 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse zero removing at step S330.
In this case, step S330 may correspond to the inverse operation of step S280 shown in FIG. 2, and may be performed by the inverse zero removing unit 380 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse parity puncturing at step S340.
In this case, step S340 may correspond to the inverse operation of step S270 shown in FIG. 2, and may be performed by the inverse parity puncturing unit 370 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse parity permutation at step S350.
In this case, step S350 may correspond to the inverse operation of step S260 shown in FIG. 2, and may be performed by the inverse parity permutation unit 360 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing LDPC decoding at step S360.
In this case, step S360 may correspond to the inverse operation of step S250 shown in FIG. 2, and may be performed by the LDPC decoder 350 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse zero padding at step S370.
In this case, step S370 may correspond to the inverse operation of step S240 shown in FIG. 2, and may be performed by the inverse zero padding unit 340 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing BCH decoding at step S380.
In this case, step S380 may correspond to the inverse operation of step S230 shown in FIG. 2, and may be performed by the BCH decoder 330 shown FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse scrambling at step S390.
In this case, step S390 may correspond to the inverse operation of step S220 shown in FIG. 2, and may be performed by the inverse scrambling unit 320 shown in FIG. 1.
Furthermore, the signaling information decoding method according to the embodiment of the present invention includes performing inverse segmentation at step S395.
In this case, step S395 may correspond to the inverse operation of step S210 shown in FIG. 2, and may be performed by the inverse segmentation unit 310 shown in FIG. 1.
FIG. 4 is a diagram showing a broadcast signal frame according to an embodiment of the present invention.
Referring to FIG. 4, the broadcast signal frame according to the embodiment of the present invention may include a bootstrap 421, a preamble 423, and data symbols 425.
The preamble 423 includes signaling information.
In an example shown in FIG. 4, the preamble 423 may include L1-Basic information 431 and L1-Detail information 433.
In this case, the L1-Basic information 431 may be fixed-length signaling information.
For example, the L1-Basic information 431 may correspond to fixed length 200 bits.
In this case, the L1-Detail information 433 may be variable length signaling information.
For example, the L1-Detail information 433 may correspond to 200 to 2352 bits.
An LDPC (low-density parity check) code is known as a code very close to the Shannon limit for an additive white Gaussian noise (AWGN) channel, and has the advantages of asymptotically excellent
performance and parallelizable decoding compared to a turbo code.
Generally, an LDPC code is defined by a low-density parity check matrix (PCM) that is randomly generated. However, a randomly generated LDPC code requires a large amount of memory to store a PCM, and
requires a lot of time to access memory. In order to overcome these problems, a quasi-cyclic LDPC (QC-LDPC) code has been proposed. A QC-LDPC code that is composed of a zero matrix or a circulant
permutation matrix (CPM) is defined by a PCM that is expressed by the following Equation 9:
In this equation, J is a CPM having a size of L×L, and is given as Equation 10 below. In the following description, L may be 360.
Furthermore, J^i is obtained by shifting an L×L identity matrix 7 (J^0) to the right i (0 ≤ i < L) times, and J^∞ is an L×L zero matrix. Accordingly, in the case of a QC-LDPC code, it is sufficient
if only index exponent i is stored in order to store J^i, and thus the amount of memory required to store a PCM is considerably reduced.
FIG. 5 is a diagram showing the structure of a PCM corresponding to an LDPC code according to an embodiment of the present invention.
Referring to FIG. 5, the sizes of matrices A and C are g × K and (N-K-g) × (K+g), respectively, and are composed of an L×L zero matrix and a CPM, respectively. Furthermore, matrix Z is a zero matrix
having a size of g × (N-K-g), matrix D is an identity matrix having a size of (N-K-g) × (N-K-g), and matrix B is a dual diagonal matrix having a size of g × g. In this case, the matrix B may be a
matrix in which all elements except elements along a diagonal line and neighboring elements below the diagonal line are 0, and may be defined as Equation 11 below:
where I[L×L] is an identity matrix having a size of L×L.
That is, the matrix B may be a bit-wise dual diagonal matrix, or may be a block-wise dual diagonal matrix having identity matrices as its blocks, as indicated by Equation 11 above. The bit-wise dual
diagonal matrix is disclosed in detail in Korean Patent Application Publication No. 2007-0058438, etc.
In particular, it will be apparent to those skilled in the art that when the matrix B is a bit-wise dual diagonal matrix, it is possible to perform conversion into a Quasi-cyclic form by applying row
or column permutation to a PCM including the matrix B and having a structure shown in FIG. 5.
In this case, N is the length of a codeword, and K is the length of information.
The present invention proposes a newly designed QC-LDPC code whose code rate is 3/15 and whose codeword length is 16200, as shown in Table 5 below. That is, the present invention proposes an LDPC
code that is designed to receive information having a length of 3240 and generate an LDPC codeword having a length of 16200.
Table 5 shows the sizes of the matrices A, B, C, D and Z of the QC-LDPC code according to the present invention:
Code rate Length Sizes A B C D Z 3/15 16200 1080 × 3240 1080 × 1080 11880 × 4320 11880 × 11880 1080 × 11880
The newly designed LDPC code may be represented in the form of a sequence, an equivalent relationship is established between the sequence and the matrix (parity bit check matrix), and the sequence
may be represented as shown the following table:
1st row: 8 372 841 4522 5253 7430 8542 9822 10550 11896 11988 2nd row: 80 255 667 1511 3549 5239 5422 5497 7157 7854 11267 3rd row: 257 406 792 2916 3072 3214 3638 4090 8175 8892 9003 4th row: 80 150
346 1883 6838 7818 9482 10366 10514 11468 12341 5th row: 32 100 978 3493 6751 7787 8496 10170 10318 10451 12561 6th row: 504 803 856 2048 6775 7631 8110 8221 8371 9443 10990 7th row: 152 283 696 1164
4514 4649 7260 7370 11925 11986 12092 8th row: 127 1034 1044 1842 3184 3397 5931 7577 11898 12339 12689 9th row: 107 513 979 3934 4374 4658 7286 7809 8830 10804 10893 10th row: 2045 2499 7197 8887
9420 9922 10132 10540 10816 11876 11st row: 2932 6241 7136 7835 8541 9403 9817 11679 12377 12810 12nd row: 2211 2288 3937 4310 5952 6597 9692 10445 11064 11272
An LDPC code that is represented in the form of a sequence is being widely used in the DVB standard.
According to an embodiment of the present invention, an LDPC code presented in the form of a sequence is encoded, as follows. It is assumed that there is an information block S =(s[0],s[1],...,s
[K-1]) having an information size K . The LDPC encoder generates a codeword Λ=(λ[0],λ[1],λ[2],...,λ[N-1]) having a size of N=K+M[1]+M[2] using the information block S having a size K. In this case, M
[1] = g, and M[2]=N-K-g. Furthermore, M[1] is the size of a parity corresponding to the dual diagonal matrix B, and M[2] is the size of a parity corresponding to the identity matrix D. The encoding
process is performed as follows:
• Initialization:
• $(12)$
• First information bit λ[0] is accumulated at parity bit addresses specified in the 1st row of the sequence of the above table. For example, in an LDPC code whose length is 16200 and whose code
rate is 3/15, an accumulation process is as follows:
• $p 8 = p 8 ⊕ λ 0 p 372 = p 372 ⊕ λ 0 p 841 = p 841 ⊕ λ 0 p 4522 = p 4522 ⊕ λ 0 p 5253 = p 5253 ⊕ λ 0$
• $p 7430 = p 7430 ⊕ λ 0 p 8542 = p 8542 ⊕ λ 0 p 9822 = p 9822 ⊕ λ 0 p 10550 = p 10550 ⊕ λ 0$
• $p 11896 = p 11896 ⊕ λ 0 p 11988 = p 11988 ⊕ λ 0$
• where the addition ⊕ occurs in GF(2).
• The subsequent L-1 information bits, i.e., λ[m], m = 1,2,...,L - 1, are accumulated at parity bit addresses that are calculated by the following Equation 13:
• $(13)$
• where x denotes the addresses of parity bits corresponding to the first information bit λ[0], i.e., the addresses of the parity bits specified in the first row of the sequence of Table, Q[1]=M[1]
/L, Q[2]=M[2]/L, and L=360. Furthermore, Q[1] and Q[2] are defined in the following Table 2. For example, for an LDPC code whose length is 16200 and whose code rate is 3/15, M[1] = 1080, Q[1]=3,
M[2]=11880, Q[2] =33 and L=360, and the following operations are performed on the second bit λ[1], using Equation 13 above:
• $p 11 = p 11 ⊕ λ 1 p 375 = p 375 ⊕ λ 1 p 4555 = p 4555 ⊕ λ 1 p 5286 = p 5286 ⊕ λ 1$
• $p 7463 = p 7463 ⊕ λ 1 p 8575 = p 8575 ⊕ λ 1 p 9855 = p 9855 ⊕ λ 0 p 10583 = p 10583 ⊕ λ 1$
• $p 11929 = p 11929 ⊕ λ 1 p 12021 = p 12021 ⊕ λ 1$
Table 6 shows the sizes of M[1], Q[1], M[2] and Q[2] of the designed QC-LDPC code:
Code rate Length Sizes M[1] M[2] Q[1] Q[2] 3/15 16200 1080 11880 3 33
- The addresses of parity bit accumulators for new 360 information bits ranging from λ[L] to λ[2L-1] are calculated and accumulated from Equation 13 using the second row of the sequence.
- In a similar manner, for all groups composed of new L information bits, the addresses of parity bit accumulators are calculated and accumulated from Equation 13 using new rows of the sequence.
- After all the information bits ranging from λ[0] to λ[K-1] have been exhausted, the operations of Equation 14 below are sequentially performed from i=1:
• Thereafter, when a parity interleaving operation, such as that of Equation 15 below, is performed, parity bits corresponding to the dual diagonal matrix B are generated:
• $(15)$
When the parity bits corresponding to the dual diagonal matrix B have been generated using K information bits λ[0], λ[1],...,λ[K-1], parity bits corresponding to the identity matrix D are generated
using the M[1] generated parity bits λ[K],λ[K+1],...,λ[K+M1-1].
• For all groups composed of L information bits ranging from λ[K] to λ[K+M1-1], the addresses of parity bit accumulators are calculated using the new rows (starting with a row immediately
subsequent to the last row used when the parity bits corresponding to the dual diagonal matrix B have been generated) of the sequence and Equation 13, and related operations are performed.
• When a parity interleaving operation, such as that of Equation 16 below, is performed after all the bits ranging from λ[K] to λ[K+M1-1] have been exhausted, a parity corresponding to the identity
matrix D is generated:
• $(16)$
FIG. 6 is a diagram showing an example of the operation of the zero padding unit shown in FIG. 1.
Referring to FIG. 6, a zero padding operation in the case where the shortening pattern order is [4 1 5 2 8 6 0 7 3] can be seen.
In the example shown in FIG. 6, the length of the LDPC information bit string is 3240, and thus LDPC information bits include 9 groups each composed of 360 bits.
First, when the number of groups for which all the bits thereof are filled with 0 is determined using Equation 2, (3240-368/360) = 7.9, and thus 7 groups are determined to be the groups for which all
the bits thereof are filled with 0.
Furthermore, since the shortening pattern order is [4 1 5 2 8 6 0 7 3], a total of 7 groups, i.e., a 5th group 610 indexed as 4, a 2nd group 620 indexed as 1, a 6th group 630 indexed as 5, a 3rd
group 640 indexed as 2, a 9th group 650 indexed as 8, a 7th group 660 indexed as 6 and a 1st group 670 indexed as 0, are selected, and all the bits of the groups are filled with 0.
Furthermore, since an 8th group 680 indexed as 7 is next to the 1st group 670 indexed as 0, 352 (= 3240 - 368 - (360 × 7)) bits from the beginning of the 8th group 680 indexed as 7 are filled with 0.
After the zero padding has been completed, the BCH-encoded bit string of N[bch] (=368) bits is sequentially mapped to a total of 368 bits, i.e., the 360 bits of the 4th group 690 indexed as 3 and the
remaining 8 bits of the 8th group 680 indexed as 7.
FIG. 7 is a diagram showing an example of the operation of the parity permutation unit shown in FIG. 1.
Referring to FIG. 7, a parity permutation operation in the case where the order of group-wise interleaving corresponding to the sequence [20 23 25 32 38 41 18 9 10 11 31 24 14 15 26 40 33 19 28 34 16
39 27 30 21 44 43 35 42 36 12 13 29 22 37 17] can be seen.
K[ldpc] (=3240) information bits are not interleaved, and 36 groups each composed of 360 bits (a total of 12960 bits) become an interleaving target.
Since the order of group-wise interleaving corresponds to the sequence [20 23 25 32 38 41 18 9 10 11 31 24 14 15 26 40 33 19 28 34 16 39 27 30 21 44 43 35 42 36 12 13 29 22 37 17], the parity
permutation unit locates a 21st group indexed as 20 at a 10th group location 710 indexed as 9, a 24th group indexed as 23 at a 11th group location 720 indexed as 10, ..., a 38th group indexed as 37
at a 44th group location 730 indexed as 43, and a 18th bit group indexed as 17 at a 45th group location 740 indexed as 44.
FIG. 8 is a diagram showing an example of the operation of the zero removing unit shown in FIG. 1.
Referring to FIG. 8, it can be seen that the zero removing unit generates signaling information for transmission by removing zero-padded parts from the information part of an LDPC codeword.
FIG. 9 is a block diagram showing a zero padding apparatus according to an embodiment of the present invention.
Referring to FIG. 9, the zero padding apparatus according to the embodiment of the present invention includes a processor 920, and memory 910.
The processor 920 generates a LDPC information bit string by deciding a number of groups whose all bits are to be filled with 0 using a difference between a length of the LDPC information bit string
and a length of a BCH-encoded bit string (K[ldpc] - N[bch]), selecting the groups using a shortening pattern order to fill all the bits of the groups with 0, and filling at least a part of remaining
groups, which are not filled with 0, with the BCH-encoded bit string.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information. In this case, the fixed length signaling information may be L1-Basic information.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the processor 920 may decide a group which is partially to be filled with 0 using the shortening pattern order, and fill a number of bits corresponding to a number decided by using the
difference between the length of the LDPC information bit string and the length of the BCH-encoded bit string (K[ldpc]-N[bch]) with 0 from the start of the group.
In this case, the shortening pattern order may be defined for 9 groups as shown in the Table 2.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
In this case, the LDPC codeword may be used for parity permutation which is performed by group-wise interleaving corresponding to an order of group-wise interleaving shown in the Table 3 and a part
of parity bits of the LDPC codeword is punctured after the parity permutation.
The memory 910 provides the LDPC information bit string to an LDPC encoder.
The zero padding apparatus shown in FIG. 9 may correspond to the zero padding unit 140 shown in FIG. 1.
Furthermore, the structure shown in FIG. 9 may correspond to an inverse zero padding apparatus. In this case, the inverse zero padding apparatus may correspond to the inverse zero padding unit 340
shown in FIG. 1.
When the structure shown in FIG. 9 corresponds to the inverse zero padding apparatus, the memory 910 receives the LDPC information bit string from the LDPC decoder.
The processor 920 selects groups whose all bits are filled with 0 using a shortening pattern order and generates the BCH-encoded bit string from the LDPC information bit string using remaining groups
exclusive of the groups.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information. In this case, the fixed length signaling information may be L1-Basic information.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the shortening pattern order may be defined for 9 groups as shown in the Table 2.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
FIG. 10 is an operation flowchart showing a zero padding method according to an embodiment of the present invention.
Referring to FIG. 10, the zero padding method according to the embodiment of the present invention includes deciding the number of groups whose all bits are to be filled with 0 using a difference
between a length of an LDPC information bit string and a length of a BCH-encoded bit string at step S1010.
In this case, the BCH-encoded bit string may correspond to fixed length signaling information. In this case, the fixed length signaling information may be L1-Basic information.
In this case, each of the groups may include 360 bits, the LDPC information bit string may include 3240 bits, and the LDPC encoder may correspond to an LDPC codeword whose length is 16200 and whose
code rate is 3/15.
In this case, the LDPC codeword may be used for parity permutation which is performed by group-wise interleaving corresponding to an order of group-wise interleaving and a part of parity bits of the
LDPC codeword is punctured after the parity permutation.
Furthermore, the zero padding method according to the embodiment of the present invention includes selecting the groups using a shortening pattern order at step S1020.
In this case, the shortening pattern order may be defined for 9 groups as shown in the Table 2.
In this case, the shortening pattern order may correspond to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group indexed as 2, a 9th group indexed as
8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
Furthermore, the zero padding method according to the embodiment of the present invention includes filling all the bits of the selected groups with 0 at step S1030.
Furthermore, the zero padding method according to the embodiment of the present invention includes generating the LDPC information bit string by filling at least a part of remaining groups which are
not filled with 0, with the BCH-encoded bit string at step S1040.
Although it is not shown in FIG. 10, the zero padding method according to the embodiment of the present invention may further include providing the LDPC information bit string to an LDPC encoder.
Moreover, the zero padding method according to the embodiment of the present invention may further include deciding a group which is partially to be filled with 0 using the shortening pattern order;
and filling the number of bits corresponding to the number decided by using the difference between the length of the LDPC information bit string and the length of the BCH-encoded bit string with 0
from the start of the group.
As described above, the zero padding apparatus, the zero padding method and the inverse zero padding apparatus according to the present invention are not limited to the configurations and methods of
the above-described embodiments, but some or all of the embodiments may be selectively combined such that the embodiments can be modified in various manners.
1. A zero padding apparatus, comprising:
a processor configured to generate a low-density parity check (LDPC) information bit string by deciding a number of groups whose all bits are to be filled with 0 using a difference between a
length of the LDPC information bit string and a length of a BCH-encoded bit string, selecting the groups using a shortening pattern order to fill all the bits of the groups with 0, and filling at
least a part of remaining groups, which are not filled with 0, with at least a part of the BCH-encoded bit string; and
memory configured to provide the LDPC information bit string to an LDPC encoder,
wherein each of the groups includes 360 bits, the LDPC information bit string includes 3240 bits, and the LDPC encoder corresponds to an LDPC codeword whose length is 16200 bits and whose code
rate is 3/15, and
wherein the LDPC codeword is generated by performing accumulation using a sequence corresponding to a parity check matrix (PCM).
2. The zero padding apparatus of claim 1, wherein the shortening pattern order corresponds to an order of a 5th group indexed as 4, a 2th group indexed as 1, a 6th group indexed as 5, a 3rd group
indexed as 2, a 9th group indexed as 8, a 7th group indexed as 6, a 1st group indexed as 0, a 8th group indexed as 7, and a fourth group indexed as 3.
3. The zero padding apparatus of claim 2, wherein the LDPC codeword is used for parity permutation which is performed by group-wise interleaving corresponding to an order of group-wise interleaving
and a part of parity bits of the LDPC codeword is punctured after the parity permutation.
4. The zero padding apparatus of claim 2, wherein the BCH-encoded bit string corresponds to fixed length signaling information.
5. The zero padding apparatus of claim 2, wherein the processor decides a group which is partially to be filled with 0 using the shortening pattern order, and fills a number of bits corresponding to
a number decided by using the difference between the length of the LDPC information bit string and the length of the BCH-encoded bit string with 0 from the start of the group.
International Classification: H03M 13/11 (20060101); H03M 13/29 (20060101); H03M 13/25 (20060101); H03M 13/27 (20060101); H03M 13/00 (20060101); | {"url":"https://patents.justia.com/patent/20230344449","timestamp":"2024-11-08T14:58:59Z","content_type":"text/html","content_length":"166166","record_id":"<urn:uuid:6a5b8eba-50d0-4d7e-8d3b-5297d6373bde>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00225.warc.gz"} |
A generic model of thinning and stand density effects on forest
Issue Ann. For. Sci.
Volume 66, Number 8, December 2009
Article Number 815
Number of page(s) 11
DOI https://doi.org/10.1051/forest/2009073
Published online 25 November 2009
Ann. For. Sci. 66 (2009) 815
Original article
A generic model of thinning and stand density effects on forest growth, mortality and net increment
Un modèle générique des effets de l’éclaircie et de la densité des peuplements sur la croissance des forêts, la mortalité et l’accroissement net
^1 IIASA International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria
^2 Institute of Silviculture, Department of Forest- and Soil Sciences, University of Natural Resources and Applied Life Sciences (BOKU) Vienna, Peter Jordan Straße 82, 1190 Wien, Austria
^* Corresponding author: franklin@iiasa.ac.at
Received: 30 October 2008
Revised: 24 February 2009
Accepted: 24 March 2009
• For assessing forest thinning effects at large (i.e. continental) scale, data scarcity and technical limitations prevent the application of localized or individual-based thinning models.
• Here we present a simple general framework to analyze and predict the effects of thinning on growth and mortality, including the following stand density development. The effects are modeled in
relative terms so that the model can be parameterized based on any thinning experiment that includes an un-thinned control, regardless of site conditions and stand age.
• The model was tested against observed thinning effects on growth and mortality from five temperate and boreal species (all species pooled r^2 = 0.51). It predicted a maximum increase in net stem
biomass increment of 16% and a reduction in density-related mortality of 75% compared to un-thinned conditions at stand densities of around 70% of the maximum (increment optimal density).
• A sensitivity analysis revealed overlapping ranges of near optimal density (net increment within 95% of optimal) among all tested species, suggesting that one thinning scenario can be used for many
species. The simple and general formulation of thinning effects based on only five parameters allows easy integration with a wide range of generic forest growth models.
• Pour évaluer les effets de l’éclaircie en forêt à une large échelle (c’est-à-dire continentale), la rareté des données et des limitations techniques empêchent l’application de modèles d’éclaircie
localisés ou individuels.
• Ici, nous présentons un simple cadre général pour analyser et prédire les effets de l’éclaircie sur la croissance et la mortalité, y compris le développement suivant de la densité du peuplement.
Les effets sont modélisés en termes relatifs, de sorte que le modèle peut être paramétré sur la base de n’importe quelle expérience d’éclaircie qui inclut un témoin non éclairci, indépendamment des
conditions du site et de l’âge du peuplement.
• Le modèle a été testé contre les effets de l’éclaircie observés sur la croissance et la mortalité de cinq espèces tempérées et boréales (r^2 = 0,51) pour toutes les espèces mises en pool). Il a
prédit une augmentation maximale de l’accroissement net de la biomasse des troncs de 16 % et une réduction de la mortalité liée à la densité de 75 % par rapport aux conditions de non éclaircie de
densité de peuplement de l’ordre de 70 % du maximum (densité de l’accroissement optimal).
• Une analyse de sensibilité a révélé des écarts de chevauchements près de la densité optimale (accroissement net dans 95 % de l’optimal) entre toutes les espèces testées, suggérant que un scénario
d’éclaircie peut être utilisé pour de nombreuses espèces. La simple et générale formulation des effets de l’éclaircie basée sur seulement cinq paramètres permet une intégration facile avec une large
gamme de modèles génériques de croissance des forêts.
Key words: dead wood / optimal density / forest model / closure / forestry
Mots clés : bois mort / densité optimale / modèles de forêts / fermeture / foresterie
© INRA, EDP Sciences, 2009
Thinning is an important means to pursue silvicultural objectives (e.g. selection of desired tree species, promotion of stability and stem quality) that generates income opportunities during the long
rotation periods in temperate and boreal forest ecosystems. As a centrepiece in stand level forest management, thinning has received considerable attention in forest research. A number of empirical
studies and thinning trials were initiated to quantify the effect of different thinning intensities, intervals and structures (i.e., thinning from above, thinning from below) mainly on stemwood
growth (e.g. Assmann, 1961; Pretzsch, 2005). Besides a focus on an improved physiological understanding of thinning effects, various approaches of representing such effects in the framework of forest
models have been presented (Söderbergh and Ledermann, 2003), ranging from empirical approaches to physiology-based models.
The former are strongly linked to the advent of empirical, individual tree growth models in the second half of the 20th century. Such concepts, starting with Newnham (1964), relate individual tree
growth to a tree’s environment (i.e. competition) and are thus able to simulate the liberating effect of thinning on the remaining individuals. A number of competition indices have been developed for
this purpose, resulting in distance-dependent and distance-independent tree growth models (e.g. Hynynen et al., 2005; Monserud and Sterba, 1999; Siitonen et al., 1999; Sterba and Monserud, 1997). For
parameterization of these empirical models for local to national applications, data sets ranging from research trials to national forest inventories have been used (Hasenauer, 2006). Such empirical
models are widely used in forest management planning at the operational scale today (e.g. Crookston and Dixon, 2005).
In contrast, physiology-based approaches have been mainly used to investigate how the processes affecting growth and allocation change as a result of thinning. For example, Zeide (2001) presented a
process-oriented framework to thinning effects related to closure and environmental variables. Recently, Petritsch et al. (2007) introduced prescribed time-lags in a biogeochemical process model to
achieve a better representation of growth and allocation regimes after thinning. Although offering a generalized framework, process-based models are still limited with regard to management support
mostly due to their data and parameterization requirements. Moreover, process based models often suffer from a lack of precision in longer term predictions for scenario analyses, due to compounding
of errors (Mason and Dzierzon, 2006). In contrast, empirical models have been found to achieve high precision within their domain of development and parameterization, yet are of limited generality.
In large scale scenario models for policy support the challenge is thus to provide a thinning framework that is (i) general enough to be applicable at continental scale and (ii) confined in
complexity to correspond to the limited structural information available in models operating at that scale and in order to facilitate parameterization. Consequently, although thinning is a powerful
intervention towards various strategic management goals in forestry, large scale modeling approaches have been strongly limited in addressing thinning effects. Large scale carbon budgeting tools
have, for instance, widely neglected the effects of thinnings (e.g. Kurz and Apps, 1999). In EFISCEN, the most widely applied continental scale scenario model in Europe, the representation of
thinning is limited by the model structure of age-volume matrices (Schelhaas et al., 2007). For example, the thinning effect on growth is essentially independent of development stage in EFISCEN and
changed mortality patterns as an effect of thinning are simplified to a complete cancellation of mortality in thinned stands. The problem that the modelled thinning effect is not realistically
dependent on the development stage applies also to the approach taken by Böttcher et al. (2008) although in their model, mortality in response to thinning is reduced in a more realistic gradual
fashion. More realistic stand and individual-based thinning models relying on empirical findings of growth and yield studies, such as the MELA system (Siitonen et al., 1999), Motti (Hynynen et al.,
2005), and Prognaus (Sterba and Monserud, 1997), have to date not been applied and validated at continental scale to our knowledge.
Considering the limitations in the current state of the art continental scale scenario models, our overall objective was to develop a general and simple thinning framework applicable at large scales,
based on a quantitative density – growth – mortality relationship. Notwithstanding the findings on the importance of the local context of thinning effects, our aim is to simplify and generalize
available knowledge for use in policy support frameworks. The purpose is to derive a model framework predicting stand responses to thinning, or a sequence of thinnings, in terms of density, growth
and mortality. We develop models for thinning effects on growth and mortality separately which are then combined to predict the total effect in terms of net stand growth and mortality. The effects
are modelled as functions of density relative to maximum stand density (closure). Omitting absolute density numbers, the model can be generalized to any forest stand conditions regardless of absolute
density and biomass values. With respect to stem biomass productivity, this framework should also allow an assessment of potential “optimal” closure and associated thinning scenarios and their
consequences for mortality (dead wood production). The underlying principles, based on stand level forest functioning, are evaluated for five boreal-temperate tree species.
2. MATERIALS AND METHODS
2.1. Theory and models
The model is based on the relationship between the self-thinning limit, i.e. the maximal number of trees that can coexist on a fixed area (N[max]; cf. symbols in Tab. I) and the mean tree size (b).
The effects of thinning, an imposed reduction in density, are modeled based on how much the resulting density deviates from the maximum density (self-thinning limit) and how this affects growth and
mortality dynamically as the stand re-closes. As we are using relative measures of growth, mortality and density, the model can be applied to both biomass and volume data. The general results and
conclusions are not dependent on the combination of the thinning framework with a particular growth model. However, for some numerical illustrations, a growth model (Franklin et al., 2009) is used in
this paper. The model is evaluated for even-aged stands.
2.2. Stand density and self thinning
To model the number of trees and the effects of the changes in forest stand density during the development of a stand, a self-thinning equation is used. For stands growing at maximum density (closed
stands) $See PDF$(1)where N[max] is number of trees and b is the biomass of the average tree. For a fixed area, the number of trees (N) is decreasing as b and the total stand biomass (B) are
increasing. Although individual stands do generally not grow continuously along this line due to discontinuous mortality events, for our purposes, a self-thinning line (Eq. (1)) is a sufficient
approximation of stand development of a closed stand. Although the existence of a universal α (slope of the self-thinning line) has been suggested based on empirical evidence (Reineke, 1933), theory
for local competition and probability theory (Dewar and Porte, 2008), empirical studies have shown substantial variability in α, e.g. among species (Weller, 1987). In our analysis we allow α to vary
among species.
Net B increment of a closed stand can be expressed as $See PDF$(2)where the first and second terms represent self-thinning mortality and gross increment, respectively. To represent the relative size
difference between trees that die in self-thinning and the mean tree, b in the self-thinning term in Equation (2) is multiplied by a factor q= biomass of a dying tree/mean tree biomass. By inserting
equation (1) in the expression above, it can be shown that mortality in a closed stand (dB[d] / dt[c]; Eq. (3)) is proportional to the increment of live biomass (dB / dt[c]) $See PDF$(3)It is natural
that q is less than 1, since smaller trees are suppressed by larger, e.g. in terms of light absorption, and therefore are more likely to die in the self-thinning process. In reality q may change with
age as the stand develops from a left skewed size distribution towards a normal distribution for old stands (Coomes and Allen, 2007). However, for simplicity and as we mainly focus on managed forests
within a relatively narrow age-span (i.e., managed ecosystems and not old growth forests), we assume that q is constant over the time periods of our thinning response observations.
Figure 1
Closure (c) after a thinning. The figure shows thinning removal of N[0] − N[t] trees, which are smaller than the average tree (thinning from below). This leads to an increase in mean tree size (b)
from b[0] before thinning to b[t] after the thinning. c after the thinning is given by N after thinning (N[t]) divided by the N[max] (solid line) corresponding to b[t] (N[max t]). The dashed line
shows the development after thinning.
2.3. Thinning effects
Thinning causes reduction of density and associated reduction in resource use and competition, which increase the growth of the remaining trees and reduces their mortality rate. On the stand scale
this effect can be divided into two effects. First, the total stand production (NPP) is reduced (although very slightly for light thinning) because of the reduced resource capture (Zeide, 2004).
Second, the self-thinning (density dependent) mortality is reduced, which is linked to the improved growth of the remaining trees.
To model these effects in a way that is independent of site productivity we use the concept relative density or closure (c; Garcia, 1990), i.e. the number of trees relative to the maximum number of
trees (N[max], given by the self-thinning limit) for a fixed individual tree size (b).
If the mean size of trees removed are the same as that of the remaining trees, c after thinning from a closed stand (c = 1) is simply equal to removed volume/total volume or removed number of trees/
total number of trees. As described in Figure 1, if the removed trees are of a different mean size than the remaining trees, c after thinning (c[t]) is calculated based on the new b and N after the
thinning, b[t] and N[t], respectively, using equation (1), which yields .
Figure 2
The response of growth rate (u[g]) and mortality (u[m]) to stand closure (c). Dotted straight lines show the responses immediately after thinning before the trees have utilized the new growing
space available. The solid lines show the acclimated responses and the dashed lines show the effect of a thinning to c = 0.5 followed by acclimation.
After thinning there is a reduction in stand growth rate through reduced resource absorption, where stem-wood growth relative to that of a closed stand is related to closure (c) according to u[g]
(Eq. (4)). However, the growth reduction is not proportional to the reduction in c because, although they are fewer, each of the remaining trees receives more light and other resources. For example,
if only a small fraction of the trees are removed in a dense canopy with some depth, the total light absorption declines very little due to the overlapping of tree crowns in the path of incident
light (for non zenith angles). However, if the density is very low (no overlapping of tree crowns in the path of light) a reduction in density will lead to a proportional reduction in light
absorption. Thus, the slope of the change in relative stand radiation absorption in response to reduced c (du[g] / dc) declines with density as du[g] / dc → 0 when c → 1, and du[g] / dc > 1 when c →
0, which is a pattern described by equation (4). We expect similar relationships in case resources other than light are more limiting, such as water and nutrients, although this is less clear (see
discussion). $See PDF$(4)γ[g] is determined from experimental data collected from the literature (see below).
Some studies have shown that the growth of individual trees does not increase immediately after a thinning and that the trees must physiologically acclimate before they can fully utilize the
increased growing space (e.g. Pukkala et al., 2002). This acclimation can for example be related to redistribution of nitrogen and adjustment of photosynthetic capacity (Medhurst and Beadle, 2005).
However, a time-lag in growth response after thinning is not always observed (Norgrove and Hauser, 2002) and its duration may be small compared to the total duration of the thinning effect on stand
growth (Pukkala et al., 2002). Thus, for simplicity we neglect this effect and exclude from our analysis growth measurements representing very short periods directly after thinning where there was
evidence that the thinning effect on growth of individual trees ((dB / dt + dB / dt[d]) / c) has not yet peaked.
In addition to the growth rate, the self-thinning rate is affected by closure (c). The density dependent self-thinning mortality (dd-mortality) is strongly negatively correlated with growth rate
within a stand (Wyckoff and Clark, 2002) implying that increased growth rate of individual trees diminishes dd-mortality. If there is a minimum growth rate required to survive (cf. Keane et al., 2001
), mortality could theoretically be completely cancelled after a thinning since the growth rate of all trees will be raised above this minimum growth limit. However, due to temporal variability in
growth rates there will be some risk of mortality even if the mean value of growth rate is above the limit for survival. Depending on growth rate variability there will therefore be a more or less
strong reduction in dd-mortality rate in response to thinning. Striving for simplicity we assume that Equation (5) captures the overall response of dd-mortality to c. $See PDF$(5)A similar equation
for adding density dependent self-thinning to a growth model has successfully been used to predict forest growth in agreement with yield tables (Tang et al., 1994).
Mortality other than dd-mortality is included in the growth model as density independent mortality (di-mortality), see below.
2.4. Data preparation and parameterization of thinning effects
Data was collected from studies that included both thinned and un-thinned control stands, and where the dominant form of mortality was dd-mortality, as opposed to random, disturbance or age related
mortality. Datasets and data points (Tab. II, Tab. A available at www.afs-journal.org) from plots or years that had been subject to significant mortality other than dd-mortality, such as insect
attack and storm damage, were excluded. In addition to that indicated in the original publications, stands subject to non dd-mortality were identified by an increase in mortality per tree after
thinning (dd-mortality per tree decreases after thinning).
Table II
Thinning experiments – control plot data.
To determine c of the thinned stands, we used a self-thinning equation (Eq. (1)) parameterized for the corresponding control stands to calculate the maximum number of trees (N[max]) at the given
individual tree size (b) of the thinned stands. N of the thinned stands was then divided by N[max] to obtain c of the thinned stand. This means that the control and thinned stands are compared at the
same b. By division with the values for control plots over the same observation period, relative thinning effects on gross growth and mortality were calculated as functions of c and then used to fit
the parameters of Equations (4) and (5). In using this method it is implicitly assumed that growth and mortality rate do not change significantly during the period of observation, which is true for
the approximately linearly growing stands used in this study or if the observation period is short.
Figure 3
Development of tree numbers (N) and closure (c) after thinning of a closed stand (B = B[0], N = N[0], c = 1) at two levels of biomass removal, 25% (N[1], c[1] dashed lines) and 50% (N[2], c[2]
dotted lines). Parameter values: k = 3000, γ[m] = 4, α = − 1.5.
The relative size of self-thinned trees to the mean tree size (q, Eq. (3)) was estimated for all control plot observations.
2.5. Growth and mortality dynamics of thinned stands
For net increment (dB / dt) the total effect of a thinning depends on the relative strength of the effects on stand growth and mortality. For a moderate thinning, the avoided self-thinning usually
more than compensates for the reduction in total stand production so that net stem increment is enhanced. After a thinning, the c of the thinned stand increases with time and may eventually approach
a fully closed stand (c = 1; Fig. 3).
In a closed stand, net increment (dB / dt[c]) and mortality (dB[d] / dt[c]) are linked according to equation (3). In a thinned (open) stand, total stem wood production is reduced according to
equation (4) and self thinning (second term in Eq. (6)) is reduced according to equation (5) as functions of closure (c) $See PDF$(6)As this study focuses on the analysis of managed stands where
density independent mortality is low, density independent mortality is ignored in equation (6). However for other applications and for our illustrated scenario below, density independent mortality is
added to the framework. By definition density independent mortality (dB[di] / dt, Eq. (7)) does not change as a function of c. For spruce stands mortality has been shown to be higher for very small
and very big trees (Monserud and Sterba, 1999), i.e. slow growing stages (in terms of absolute growth rate) of stand development, and is therefore modeled to be a function of the net increment (dB /
dt) relative to its maximum over the stand development (dB / dt[max]) $See PDF$(7)Because of its independence of c, density independent mortality in this study only plays a role in the illustration
of thinning scenarios below, where m was assumed to be 1% per year.
Using equations (1)–(6), the stand behavior in terms of biomass (B) and mortality (B[d]) can be derived as functions of closure (c) and time for thinned stands.
The relationship between c, N and B can also be described independently of time and dB / dt using equations (1), (3), (5) and (6). To illustrate their behavior, we derive analytical expressions for
the development of c and N after a thinning removal of a fraction f[t] of the standing biomass just before thinning (B[0]), as functions of B development after the thinning: $See PDF$(8) $See PDF$(9)
In case the mean size of the trees removed in a thinning is not equal to the mean tree size of the stand, B[0] in Equations (8) and (9) must be recalculated to match b after thinning (b[t], Fig. 1).
The behavior of N and c after a thinning (Eqs. (8) and (9)) is illustrated in Figure 3. The development of c and N after a thinning of a closed stand depend on the fraction biomass removed in the
thinning (f[t]), the maximum N as a function of B (the self thinning equation, Eq. (1)) and the density response of self thinning mortality (γ[m], Eq. (5)).
Because the Equations (4)–(9) above are independent of the functional form of the basic increment function (dB / dt[c]), the framework can be used to describe the thinning response in combination
with any basic growth function. In summary, the parameters that are needed to add a thinning response to any growth model of standing biomass (B) are the parameters of the self thinning equation k,
and α, the growth and mortality response parameters γ[g] and γ[m], and the closure before thinning, c[0]. To obtain a time dependent response (as opposed to only B dependent) it is also necessary to
know the relative size of self-thinning compared to surviving trees (q), see below.
2.6. Optimal closure and thinning
To determine the optimal c in terms of net stem biomass increment we consider net increment dB / dt (Eq. (6)) as c is reduced. Net increment is decreased due to reduced gross growth at the same time
as it is increased due to reduced self thinning mortality (Fig. 2). The maximum net increment is obtained where the sum of the two effects on dB / dt (Eq. (6)) is maximized. Using Equation (3),
Equation (6) can be reformulated as $See PDF$(10)dB / dt is maximized by setting Equation (10) = 0, inserting the functions for u[m] and u[g] (Eqs. (4) and (5)) and solving for c, $See PDF$(11)To
obtain the optimal c, Equation (11) must be solved numerically.
A sensitivity analysis of optimal c was conducted with regard to the density effects on mortality (γ[m]) and growth (γ[g]) and relative size of self-thinned trees (q). Net increment gain in a thinned
stand was analyzed in comparison to un-thinned (c = 1) stands.
The equations above illustrate the responses of a forest stand to a specified thinning. To describe the development of a forest stand managed by successive thinnings and calculate an optimal thinning
level in terms of maximized biomass increment, it is also necessary to describe how and when thinnings are initiated.
In order to illustrate an example of an optimal thinning scenario based on our model and the optimal c, a thinning scenario was designed to keep c within a near-optimal range. We base our thinning
scenario on the assumption that the purpose of the thinning is to enhance biomass productivity. In addition, although we do not explicitly consider any economical factors, the thinning scenario
should not be unrealistic in relation to practical applications that include management costs and benefits. The number of thinnings should not be too high, since very frequent thinnings would be
cost-intensive. Furthermore we presume that no thinning occurs before the stand has reached a certain biomass and tree size because very early thinnings would yield relatively little harvest of thin
stems (low value) at the same time as costs are high due to the large number of stems that need to be cut. The point of first thinning also depends on the planting density (for planted stands) as the
stand must have a sufficiently high closure (c) before thinning is beneficial (see below). We set the timing of the first thinning equal to the point of peak growth, which in forest stand simulations
with realistic planting densities (much lower than maximum density (N[max]), i.e. c ≪ 1) nearly coincided with c reaching the computed limit for optimal thinning (c ≈ 0.85).
3. RESULTS
3.1. Density effects on growth and mortality
Table III
Thinning effects on gross increment and mortality .
Figure 4
Relative gross stem-wood increment (a) and mortality (b) as functions of closure (c). Solid lined are best fits for all species pooled (see Tab. II). One point for Scots pine was outside of the
plotted range (relative mortality = 2.14, c = 0.95).
Table III and Figure 4 show the fitted parameters and the data for relative growth and mortality as functions of c for different species and for all species pooled. There are statistically
significant differences in γ[g] (p = 0.0004) and γ[m] (p = 0.044) among species although the number of data points is small and confidence intervals are large for most species. Residual analysis
showed that the c effect on gross increment is reasonably well described by equation (4) with a r^2 = 0.41 for all species pooled, and with higher r^2 for the species specific fits (Tab. III). For
mortality, the agreement between model and data is better, r^2 = 0.60 for all species pooled, but lower for the pine species (Tab. III). A residual analysis revealed a tendency for overestimation of
mortality at high c and underestimation at low c for the pine species.
3.2. Optimal closure
Figure 5
Closure (c) that maximizes net increment (gross stem growth – mortality) and relative net increment (RNI, net increment – net increment of closed stands, at the same mean tree biomass) for
different species. Lines show the range where RNI > 95% of optimum. Parameter values in Tables I–III.
Optimization of current net increment (Eq. (6)) with respect to c resulted in optimal c between 0.53 and 0.88 and increases in relative net increment compared to un-thinned conditions (RNI) between
3.5% and 17% (Fig. 5). However, the ranges of c where RNI is within 95% of optimal overlap for all species for c between 0.71 and 0.81. For all species pooled, the optimal c was 0.68 which led to RNI
= 16% and a 75% reduction in mortality (dead wood production) compared to un-thinned conditions. The value of RNI was more variable among species than the optimal c and the 95% of optimum intervals
did not overlap except within the groups broad leafed species (gold birch and monarch birch) and needle leafed species (lodgepole pine, Scots pine and hemlock), which had a lower RNI than the
Figure 6
Sensitivity of optimal closure (c) and optimal relative net increment (RNI, net increment – net increment of closed stands, at the same mean tree biomass) for intervals, starting to the left in the
figure, for q (=1 to 0.08, dashed line), γ[g] (= 4 to 1.5, solid line) and γ[m] (=1.8 to 8.6 dotted line). The parameter intervals correspond to the range of estimated values (Tab. II). Baseline
parameter values: q = 0.2, γ[g] = 3.5, γ[m] = 4, α = − 1.75, corresponding to the values for all species combined, expect for the slightly lower value of α.
The sensitivity of optimal c and RNI to the parameters q, γ[g] and γ[m] was estimated by varying these parameters through the range of parameter values observed in our data set (Tab. III). Baseline
parameter values were taken from the results for all species pooled except for α. For α a slightly higher value of –1.75 was chosen because this value is more in line with self-thinning studies that
are based on much larger data sets than ours, e.g. (Weller, 1987). The results (Fig. 6) show that optimal c is relatively insensitive to variation in all parameters except that it increases if γ[g]
is significantly reduced. RNI on the other hand is sensitive to the relative size of self-thinned trees to the mean tree size (q), whereas it is much less sensitive to γ[g] and γ[m].
3.3. A thinning scenario application
We illustrated the application of optimal closure in constructing a thinning scenario based on the thinning effects framework and the numerical results for optimal closure.
To maximize current net increment in a thinning scenario, c should be kept as near the optimum as possible. In theory this could be achieved by frequent thinnings that never allow the stand to
deviate from the optimal c. In practice, thinnings are associated with costs and a very frequent thinning regime is not realistic. However, the peak of net increment as a function of c is not
particularly steep (Fig. 5), which means that some variation around the optimal density, for example between 0.5 and 0.9, does not severely reduce net increment (≈5% reduction compared to optimal c,
for the pooled species parameter values; Fig. 5). At the same time, the frequency of thinnings can be significantly reduced compared to a narrower interval of allowed c values.
Based on the above results, a thinning scenario is outlined as follows. After peak growth rate has been reached, the first thinning is performed according to a specified level (f[t], fraction stem
biomass removed). After the thinning the relative density of the stand (c) will increase and eventually approach a closed stand (Fig. 3). The next thinning is triggered when the stand has reached a
specified c = c[t]. Thereafter, new thinnings are triggered each time c reaches c[t]. Thus, a thinning scenario is specified by the B at initial thinning (at maximum growth rate), the fraction stem
biomass removed (f[t]) and the value of c that triggers a new thinning (c[t]). Figure 7 illustrates a thinning scenario compared to an un-thinned stand. In practice, it is however necessary to
consider the development stage and the timing of final harvest before a thinning is done.
Figure 7
Examples of the development of un-thinned (solid line) and a thinned stand (dashed line) simulated with a forest growth model for un-thinned stands (Franklin et al., 2009) combined with the
thinning scenario presented here, including the effect of competition un-related mortality (Eq. (7)). Thinnings that reduce closure (c) to c = 0.6 are triggered when c > 0.85. The stand is Scots
pine planted at 3000 trees per ha. Net increment (C) is gross biomass growth – mortality.
4. DISCUSSION
4.1. General assumptions
Our aim was to develop a model of thinning effects general enough to be applied to any even-aged forest stand and yet simple enough to be used without re-formulation for all forests in temperate or
boreal regions. For this reason the model was based on the concept of closure (c), i.e. the stand density relative to maximum stand density and relative rates of growth and mortality, i.e. growth and
mortality relative to a stand at maximum density (c = 1). Although a conceptually very similar approach has been presented before (Pretzsch, 2005), time based approaches are far more common.
Commonly, effects of different densities after thinning are compared at the same points in time, which means that the responses will be strongly time dependent due to the closing of the stand over
time and the differing growth rates at different densities. In contrast, our use of c, the density compared to maximum density at the same mean tree biomass (b), provides an invariable basis for
evaluating density effects measured over different time spans. This model implies that site conditions affect only the rate of change but not the path of relative density and biomass development of a
stand, which has ample empirical support (Long et al., 2004).
In using c as the single control of density effects, independent of age or tree size, we implicitly assume that competition among trees and stand size structure qualitatively does not change
significantly during stand development. Stability of size structure and competitive interactions in self-thinning stands of plants has been shown to result from asymmetric competition (larger trees
affecting smaller trees much more than the other way around) (Hara, 1993). For older stands the role of competition may, however, change, leading to changes in size distributions (Coomes and Allen,
2007). Although this study does not include old-growth stands, equation (7) shows how the effect of reduced competition and relatively increased density independent mortality in older stands can be
included in our model. A change in size distribution may also affect the relative size difference between self-thinned and surviving trees (q), which would violate our assumption of constant q. For
the two studies with more than one record of q there was a clear increase over time for Scots pine (average +64% over 10 y) but not for lodgepole pine (+3.8% over 10 y). The increase in q for Scots
pine coincides with a reduction in increment of 32%, which did not occur in the Lodgepole pine stand, and indicates that net growth rate of the Scots pine stand has passed its peak and is declining.
Such aging of stands is associated with changes in size distribution and an increase in density independent mortality, as discussed above. Thus, density independent mortality of larger trees may
explain the apparent increase in q for Scots pine.
Because competition related mortality risk is strongly negatively correlated with individual tree growth rate within a stand (Wyckoff and Clark, 2002) one may expect that it is possible to make our
mortality effect (u[m]) a function of the effect on growth (u[g]) and thereby simplify the model. However, we did not find any correlation between individual growth (u[g] / c) and mortality risk (u[m
] / c) in response to stand closure among species. This result indicates that mortality risk is affected by other factors in addition to growth rate. For example, because mortality in contrast to
growth is a threshold event for the individual, it could respond stronger to environmental variability than growth. Even if the mean resource availability and climate conditions over time are
sufficient for survival, temporal fluctuations could cause mortality. Furthermore, it has been shown that while growth is closely linked to stand density, an additional density measure, gap fraction,
affects mortality (Zeide, 2005). Clearly, our separate effects of density (c) on mortality (u[m]) and growth (u[g]) cannot be uniquely combined to a common density effect, at least not without
introducing a more complex density measure, which, however, would not be compatible with our aim of parsimony in order to facilitate wide applicability of the model.
It is implicit in our model that gross increment is maximized at c = 1, which has been showed in previous studies of optimal density (Zeide, 2004). However, Zeide (2004) did not explicitly include
mortality and did not find a simple relationship for optimal net increment due to interacting effects of stand density and b. In contrast, by our comparison of c values at a common b we avoid this
interaction of effects and obtain an expression for optimal net increment as a trade-off between explicit mortality and growth effects.
4.2. Optimal closure
The estimated ranges of near optimum (> 95% of maximum) c in Figure 5 show that these ranges overlap for all species between c = 0.71 to c = 0.81, indicating that a c around 0.75 may, in our
framework, be generally applicable for optimal thinning (to maximize net increment) for most species. The range of near optimal c for all species pooled c = 0.5 to 0.9 is in general agreement with
the results of thinning trials of (Assmann, 1961), reporting relative stand densities (basal area) between 0.6 and 0.9 as threshold for achieving optimum stem productivity. The sensitivity analysis
indicated that optimal c is quite robust with respect to variation in parameter values although it is slightly increased with decreasing closure effect on growth (γ[g]), which acts to reduce the
positive response of growth of remaining trees after a thinning (Fig. 6).
In comparison to optimal c, the maximum relative gain in net increment (RNI) is more variable among species, although the range of 4 to 16% is rather limited (Fig. 5). RNI is also more sensitive than
optimal c to parameter changes, particularly to the size ratio of self-thinning trees to the mean tree, q (Fig.6). The reason for the model sensitivity to q is that this parameter determines the
mortality rate (dead wood production) relative to net increment and thereby controls how much mortality can be avoided (and added to net increment) by thinning. Our data showed a variation in q
between 0.06 and 1.27, where the latter value (Monarch birch) was considered unrealistic and was therefore set to 1. Such a high estimate of q may indicate that the mortality was not due to
competition and self-thinning and therefore not representative, as discussed above. If Monarch birch is excluded, our range of q is between 0.06 and 0.321. For additional assessment of q variability,
a simulation experiment was conducted with the hybrid patch model PICUS v1.4 (Seidl et al., 2005) that models mortality at the individual tree level. Starting from generic homogeneous initial stand
conditions 100 y simulations over an array of species and environmental conditions (cf. Seidl et al., 2009) resulted in values of q between 0.5 and 0.25.
Comparing species specific results, model results for Scots pine, both in terms of optimal c (= 0.9) and the low increase in net increment in thinned compared to closed stands (RNI = 4%) is in
agreement with results by Mäkinen and Isomäki (2004). For lodgepole pine our result for the gross and net increment as a function of c are in reasonable agreement with the findings of Cochran and
Dahms (2000). Our model study suggests that the potential increase in RNI may be higher for the deciduous broad-leafed species than for the needle leafed species (Fig. 5), which was previously
observed when comparing beech (RNI ≈ 20%) and spruce (RNI ≈ 10%) (Pretzsch, 2005). Although there were significant differences in the closure effects on growth and mortality, γ[g] and γ[m], among
species, these differences could be misleading because they could also be related to site differences. There are also differences in the self-thinning parameter α and the relative size of
self-thinned trees (q) among the species but more data is clearly needed to obtain confident conclusions about species or site effects. However, the requirement that data must include both increment
and mortality in terms of both numbers and biomass (or volume) substantially limits the availability of suitable data.
An potential factor of uncertainty in the estimation of stemwood productivity gain of thinning is related to the potential delay in growth response after a thinning due to a phase of physiological
acclimation as discussed above (Materials and methods –thinning effects). Such an effect would reduce the potential gain of a thinning but would not affect the optimal c. The delay could
significantly affect the optimal frequency of thinnings if the phase of acclimation covers a significant proportion of the growing time, which, however, was not found in the experiments analyzed in
our study.
4.3. Limitations and potential extensions
An approach based on self-thinning as a function of mean tree size does not adequately represent strongly age- heterogeneous stands and is not applicable to management systems such as continuous
cover forestry or variable retention systems, which are increasingly discussed as alternatives to even-aged management strategies (cf. Seidl et al., 2008).
Due to limited data availability, our model does not include effects of site conditions on the thinning response. This is a clear limitation of the model as several studies point at considerably
varying responses with local site conditions (e.g., Pretzsch, 2005). Underlying these empirical observations are, however, complex interactions of environmental and physiological factors, such as
soil fertility, soil depth, water availability, climate and disturbance. These interactions are rarely represented at the data resolution available at continental scales and they are quantitatively
not well understood in terms of general relationships valid across species and environmental conditions, which prevent their inclusion in our model. Interactions among resource availability, climate
and density relationships remain an interesting topic for further research and potential development of more detailed models.
Considering that a main aspect of thinning, besides focusing growth on the remaining individuals, is to improve stand stability and quality, neglecting these aspects in taking a mean tree approach
might represent another limitation of the presented model. While generic effects on the stand collective can be mimicked by means of the relative size of removed trees (e.g., thinning from above,
thinning from below – see Fig. 1) the approach lacks the flexibility of individual-tree models with regard to spatially heterogeneous and selective approaches (cf. Hasenauer, 2006) Furthermore,
although we consider thinning-related biomass (or volumetric) changes of the mean tree in our model, allocational shifts in stem allometry between e.g., height and diameter increment are not
explicitly included. A simple way to obtain the effects on diameter growth would be to assume that mean tree height is not affected by thinning (e.g. Cochran and Dahms, 2000) and use allometric
functions to derive the effects on diameter directly from biomass results.
Overall, considering that model complexity is strongly determined by the intended domain of application we believe that the presented approach is a reasonable compromise between ecological realism
and general, large-scale applicability.
4.4. Conclusions for model applicability
For prognostic modeling our use of relative, site- and productivity-independent changes in gross growth (u[g]) and mortality (u[m]) allows integration with any existing forest increment model for
un-thinned stands. In combination with equations for tree numbers (Eq. (8)) and closure (Eq. (9)), stand development in terms of biomass, dead wood and associated tree numbers can be obtained for an
arbitrary thinning scenario. The simplicity of the approach makes it potentially useful for large scale, continental and global modeling of net increment and mortality where site information is
limited or non-existing, or where computation time (e.g. in dynamic optimization frameworks) or ease of mathematical integration is an issue. For example, the approach could add realism to large
scale policy related analyses (e.g. Böttcher et al., 2008) or for analyses using global dynamic vegetation models (e.g. Zaehle et al., 2006).
The presented framework is an attempt to transfer the detailed accumulated knowledge on thinning responses represented in detailed growth and yield models (e.g., overviews in Hasenauer, 2006;
Söderbergh and Ledermann, 2003) and thinning experiments to model frameworks at larger spatial scales in a general and physiologically meaningful set of equations. This approach necessarily
sacrifices complexity and details compared to comprehensive individual-based models (Crookston and Dixon, 2005; Hyytiäinen et al., 2004; Sterba and Monserud, 1997). However, the selected mean tree
approach is a structural advancement of the state of the art compared to structurally simple scenario tools applied at continental scale (e.g. Schelhaas et al., 2007). In conclusion, introducing a
generic thinning framework as presented in this study in large scale scenario analyses of forest resource development could significantly increase their realism with regard to the silvicultural
decision space in forest management.
Online material
We thank Hannes Böttcher at IIASA for helpful comments on the manuscript. We are furthermore grateful to two anonymous reviewers for helping to improve an earlier version of the manuscript. This
research received funding from the European Community’s 7th Framework Programme (FP7) under the grant 212535, Climate Change - Terrestrial Adaptation and Mitigation in Europe (CC-TAME),
www.cctame.eu, and from the 6th Framework Programme (FP6) under the grant SSPI551 CT-2003/503614 (INSEA).
All Tables
Table II
Thinning experiments – control plot data.
Table III
Thinning effects on gross increment and mortality .
All Figures
Figure 1
Closure (c) after a thinning. The figure shows thinning removal of N[0] − N[t] trees, which are smaller than the average tree (thinning from below). This leads to an increase in mean tree size (b)
from b[0] before thinning to b[t] after the thinning. c after the thinning is given by N after thinning (N[t]) divided by the N[max] (solid line) corresponding to b[t] (N[max t]). The dashed line
shows the development after thinning.
In the text
Figure 2
The response of growth rate (u[g]) and mortality (u[m]) to stand closure (c). Dotted straight lines show the responses immediately after thinning before the trees have utilized the new growing
space available. The solid lines show the acclimated responses and the dashed lines show the effect of a thinning to c = 0.5 followed by acclimation.
In the text
Figure 3
Development of tree numbers (N) and closure (c) after thinning of a closed stand (B = B[0], N = N[0], c = 1) at two levels of biomass removal, 25% (N[1], c[1] dashed lines) and 50% (N[2], c[2]
dotted lines). Parameter values: k = 3000, γ[m] = 4, α = − 1.5.
In the text
Figure 4
Relative gross stem-wood increment (a) and mortality (b) as functions of closure (c). Solid lined are best fits for all species pooled (see Tab. II). One point for Scots pine was outside of the
plotted range (relative mortality = 2.14, c = 0.95).
In the text
Figure 5
Closure (c) that maximizes net increment (gross stem growth – mortality) and relative net increment (RNI, net increment – net increment of closed stands, at the same mean tree biomass) for
different species. Lines show the range where RNI > 95% of optimum. Parameter values in Tables I–III.
In the text
Figure 6
Sensitivity of optimal closure (c) and optimal relative net increment (RNI, net increment – net increment of closed stands, at the same mean tree biomass) for intervals, starting to the left in the
figure, for q (=1 to 0.08, dashed line), γ[g] (= 4 to 1.5, solid line) and γ[m] (=1.8 to 8.6 dotted line). The parameter intervals correspond to the range of estimated values (Tab. II). Baseline
parameter values: q = 0.2, γ[g] = 3.5, γ[m] = 4, α = − 1.75, corresponding to the values for all species combined, expect for the slightly lower value of α.
In the text
Figure 7
Examples of the development of un-thinned (solid line) and a thinned stand (dashed line) simulated with a forest growth model for un-thinned stands (Franklin et al., 2009) combined with the
thinning scenario presented here, including the effect of competition un-related mortality (Eq. (7)). Thinnings that reduce closure (c) to c = 0.6 are triggered when c > 0.85. The stand is Scots
pine planted at 3000 trees per ha. Net increment (C) is gross biomass growth – mortality.
In the text | {"url":"https://www.afs-journal.org/articles/forest/full_html/2009/08/f09069/f09069.html","timestamp":"2024-11-05T23:00:27Z","content_type":"text/html","content_length":"183831","record_id":"<urn:uuid:3f250d41-2172-48d9-b497-cbe057a4b5d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00361.warc.gz"} |
e-day - A mathematical holiday celebrated on February 7th
MaRDI is a Germany-wide initiative to address challenges in mathematical research. It is aimed mainly at mathematicians to change how we do mathematics using FAIR (Findable, Accessible,
Interoperable, and Reusable) research data. This way, it fits very well with our own open-source philosophy.
IMAGINARY actively participates in this initiative by assisting in communicating and...Lire la suite | {"url":"https://www.imaginary.org/fr/node/1379","timestamp":"2024-11-07T01:13:45Z","content_type":"text/html","content_length":"33325","record_id":"<urn:uuid:63531baf-cedc-4a31-8009-2c8f827fd8f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00217.warc.gz"} |
DSolve returns true
7200 Views
3 Replies
1 Total Likes
I keep checking videos and online tutorials via youtube and Mathematica website however, I still can't get a simple DSolve equation to work. Why is this giving me true? I am trying to Solve it so I
can plot solutions and slope fields. Equation is Dy/Dx = x-y
In: Clear[x, y]
In: DSolve[y'[x] == x - y[x], y[x], x]
Out: DSolve[True, y[x], x]
3 Replies
Values of parameters (and functions and anything else you define in a Mathematica session) are independent of the window that you are in (i.e, they're defined in a Kernel session, not a window
generally--there's an exception to this that is subtle).
If you want to clear out all the definitions and value assignments you have made in a Mathematica session then execute this
and if you want to remove them completely from the Mathematica session then execute
Using Mathematica 7, I got a solution
(* Commands 1*)
Clear[x, y];
DSolve[ y'[x] == x - y[x], y[x], x]
{{y[x] -> -1 + x + E^-x C[1]}}
The only time I was able to reproduce the "DSolve[True...." message was when I had the sequence of commands as follows,
Clear[x, y];
y'[x] = x - y[x];
DSolve[y'[x] == x - y[x], y[x], x]
What's happening is that y'[x] is known to be x- y[x], and the DSolve is doing a compare of y-y[x] == y-y[x], returning True. Then it can't solve True for y[x] in x, resulting in the question being
Would you try exiting the kernel, restart fresh, and re-typing (* commands 1*) only.
Restarted and it works! Is there a way to clear data after each problem or should I do each problem in a new window after restarting?
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/332674","timestamp":"2024-11-07T15:53:44Z","content_type":"text/html","content_length":"104646","record_id":"<urn:uuid:7fb5f5aa-b98f-4eee-95d5-beac6d7e56c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00026.warc.gz"} |
Mastering Formulas In Excel: Which Of These Formulas Uses Relative Cel
Mastering formulas in Excel is a crucial skill for anyone working with data. One of the key elements of formulas in Excel is understanding relative cell references. These references are essential for
creating dynamic and flexible formulas that can be easily applied to different sets of data. In this blog post, we will explore the importance of understanding relative cell references in formulas
and highlight which of these formulas utilize them.
Key Takeaways
• Understanding relative cell references is crucial for mastering formulas in Excel.
• Relative cell references are essential for creating dynamic and flexible formulas.
• Mastering relative cell references can streamline formula creation and make formulas easily applicable to different data sets.
• It is important to differentiate between relative and absolute cell references and know when to use each type in formulas.
• Seeking additional resources and practicing using relative cell references can further proficiency in Excel formula creation.
Understanding Relative Cell References
Excel formulas often use cell references to perform calculations or functions. Understanding relative cell references is crucial for mastering formulas in Excel.
A. Define relative cell references in Excel formulas
Relative cell references in Excel formulas are used to indicate a cell's position in relation to the cell containing the formula. When the formula is copied to another cell, the references are
adjusted relative to the new location.
B. Provide examples of formulas that use relative cell references
Examples of formulas that use relative cell references include:
• SUM: =SUM(A1:A5) - This formula sums the values in cells A1 to A5. When copied to another cell, the references adjust accordingly.
• AVERAGE: =AVERAGE(B1:B10) - This formula calculates the average of values in cells B1 to B10.
• IF: =IF(C1>10, "Yes", "No") - This formula returns "Yes" if the value in cell C1 is greater than 10, otherwise it returns "No".
C. Explain the significance of relative cell references in complex formulas
Relative cell references are crucial in complex formulas as they allow for the formula to be easily replicated across multiple cells without having to manually adjust the references. This saves time
and reduces the risk of errors when working with large datasets or complex calculations.
Formulas that Use Relative Cell References
When working with Excel formulas, it's essential to understand how relative cell references can be utilized to your advantage. Here are some common Excel formulas that make use of relative cell
List common Excel formulas that utilize relative cell references
• =SUM
• =AVERAGE
• =MIN
• =MAX
• =COUNT
Provide a brief description of each formula and its purpose
The =SUM formula is used to add up a range of cells. It is a versatile formula that can be applied to various types of data, such as numbers or dates.
The =AVERAGE formula calculates the average value of a range of cells. This is useful for analyzing data sets and identifying trends.
The =MIN formula returns the smallest value in a given range of cells. It can be helpful in identifying the lowest point in a dataset.
The =MAX formula does the opposite of the =MIN formula, returning the largest value in a range of cells.
The =COUNT formula counts the number of cells in a range that contain numbers. This can be useful for tracking data completeness or identifying outliers.
Discuss the benefits of using relative cell references in these formulas
Relative cell references are advantageous because they allow formulas to be copied and pasted to different cells, and the references will adjust automatically. This makes it easy to apply the same
formula to multiple sets of data without having to manually update the cell references. It also simplifies the process of creating dynamic formulas that can adapt to changes in the dataset.
Formulas that Use Absolute Cell References
In Excel, cell references are crucial in creating formulas that automatically update when new data is added or removed. Understanding the difference between relative and absolute cell references is
essential in mastering formulas in Excel.
Differentiate between relative and absolute cell references
Relative cell references change when a formula is copied to another cell. For example, if a formula refers to cell A1 and is copied to B1, the formula will automatically update to refer to B1 instead
of A1.
Absolute cell references, on the other hand, remain constant when the formula is copied to another cell. They are denoted by the use of a dollar sign before the column and row reference, such as
Identify Excel formulas that use absolute cell references
Some of the Excel formulas that use absolute cell references include:
• VLOOKUP: When using VLOOKUP, absolute cell references are crucial to ensure that the lookup range remains constant.
• HLOOKUP: Similar to VLOOKUP, HLOOKUP also requires absolute cell references to maintain the consistency of the lookup range.
• IF: Absolute cell references are commonly used in nested IF formulas to ensure that certain conditions remain constant.
Explain the importance of understanding when to use absolute cell references
Understanding when to use absolute cell references is crucial for ensuring the accuracy and reliability of Excel formulas. By using absolute cell references, you can prevent errors and ensure that
your formulas consistently refer to the correct cells, even when copied to different locations.
Additionally, mastering the use of absolute cell references can significantly improve the efficiency of your Excel workflow, as it allows for the seamless replication and updating of formulas without
the fear of unintended changes.
Best Practices for Mastering Formulas with Relative Cell References
Mastering relative cell references is essential for effective use of Excel formulas. Understanding how to use these references can make your formulas more dynamic and versatile. Here are some best
practices to keep in mind when working with relative cell references:
Offer tips for effectively using relative cell references in Excel formulas
• Understand the difference between relative, absolute, and mixed cell references: Relative cell references adjust when the formula is copied to a new location, while absolute references remain
constant. Mixed references are partially absolute and partially relative.
• Use relative references to create dynamic formulas: By using relative cell references, you can create formulas that automatically adjust to new data, making your spreadsheet more flexible and
• Utilize the fill handle: The fill handle in Excel allows you to quickly copy and fill a series of cells with a formula, with the references automatically adjusting based on their relative
Discuss the potential pitfalls of not understanding relative cell references
• Error-prone formulas: If you don't understand how relative references work, you may end up with formulas that produce incorrect results when copied to different locations.
• Increased workload: Without using relative references, you may find yourself manually adjusting formulas for each new set of data, which can be time-consuming and prone to errors.
• Limited flexibility: Absolute cell references can limit the flexibility of your spreadsheet, making it more difficult to adapt to changes in your data.
Provide examples of how mastering relative cell references can streamline formula creation
• Summing a range of cells: By using relative cell references, you can easily sum a range of cells and then copy the formula to other locations without having to manually adjust the references.
• Calculating percentage changes: Relative references allow you to calculate percentage changes based on the relative position of the cells, making it easy to apply the formula to different data
• Creating dynamic charts: Using relative references in formulas for chart data ranges can make your charts automatically update as new data is added, saving you time and effort.
Resources for Learning More About Relative Cell References
When it comes to mastering Excel formulas, understanding relative cell references is crucial. Here are some resources that can help you dive deeper into this important concept:
A. Online tutorials or courses for mastering Excel formulas
• Microsoft Excel Official Training: The official training provided by Microsoft for Excel offers comprehensive tutorials on relative cell references and other formula-related topics.
• LinkedIn Learning: This platform offers a wide range of courses on Excel formulas, including detailed explanations of relative cell references.
B. Books or websites that provide in-depth explanations of relative cell references
• Excel Bible by John Walkenbach: This book is a comprehensive guide to all things Excel, including detailed explanations of relative cell references and how to use them effectively in formulas.
• Exceljet.net: This website provides in-depth explanations of various Excel features, including detailed articles on relative cell references.
C. Personal recommendations for furthering Excel formula knowledge
• Practice, Practice, Practice: The best way to master relative cell references and Excel formulas in general is to practice regularly. Create sample spreadsheets and experiment with different
formulas to solidify your understanding.
• Join Excel Communities: Online forums and communities dedicated to Excel are great places to learn from others, ask questions, and share your knowledge. Sites like Stack Overflow and Excel forums
can be incredibly helpful.
In conclusion, mastering formulas with relative cell references is crucial for anyone looking to excel in Excel. Understanding how to use relative cell references can greatly enhance the
functionality and flexibility of your formulas, allowing you to easily apply them to different sets of data. It's important for readers to practice using relative cell references in their Excel
formulas in order to become more proficient in creating complex calculations and analyses. By doing so, they can unlock the full potential of Excel as a powerful tool for data manipulation and
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How can you tell if a cipher is secure?
How can you tell if a cipher is secure? posted December 2014
How can you tell if a cipher is secure?
I was asked that question during an interview a while ago. Back then it troubled me because it seemed so basic and yet and I had no idea how to answer it. I became vivid and didn't know what to say
and later I didn't get the job. I thought it would be interesting to write down how I would answer this question now.
We're in the dark
If we have no idea how the algorithm works and can only do some playing with it (or worse, we can only listen to people using it) then we can safely call it a Black box. This is pretty hard to attack
and also pretty hard to talk about security here because it might be not secure at all (like Enigma during the war) but we would still have a hard time telling.
We're almost in the dark
If I talk about Blackboxes then I should talk about Whiteboxes. If you have no idea what your cipher is doing, but you can analyze it, then you call that a Whitebox. For example your phone, your
credit card, and so on, they all use cryptographic tools that you can take a look at since it is in your own hands. Cryptography was not thought for that kind of model. It was primarily born to hide
private conversations from other curious parties and ciphers were supposed to be used by good people only. In our modern world, clever people found a new use for cryptography in our everyday-devices.
And it quickly became problematic since a lot of new attacks rose up.
It doesn't mean that attacking a whitebox is easy though. People use reverse engineering, heuristics, side-channel attacks (fault injection, power analysis, sound analysis...) and a lot of other
techniques to break them. There are a lot of researches, especially in the domain of smartcards.
We're in the clear
In cryptography, Kerckhoffs's principle was stated by Auguste Kerckhoffs in the 19th century: A cryptosystem should be secure even if everything about the system, except the key, is public
OK. Now let's talk about a cipher we can see and touch and play with anyway we want.
We will not talk about the implementation of the cipher because it will have the same problems the whitebox have: it will be subject to new attacks depending where and how you implement it. It is a
whole new level of research and precautions you have to take. Think about side-channel attacks, bugs (buffer overflow), human errors, padding attacks (if we chose a bad mode of operation for
example), etc...
So we'll place ourselves in a more restrictive model.
There is also a lot of different kind of ciphers (and it would be an history class to write about it) so I'll just talk about the most used type of ciphers nowadays: Stream ciphers and Block ciphers
(both symmetric ciphers) and Asymmetric ciphers
Although let's just say that it was proven by Shannon in the 40s that the One Time Pad is the only cipher having perfect secrecy.
What kind of secrecy?
There are multiple kinds of secrecy in the literature. Perfect Secrecy that we just talked about. The kind of secrecy that doesn't leak any information about the plaintext and the key (although it
might leak the maximum length of it). And if you read more about it you will see that it is not practical thus almost never used.
So we defined a weaker type of secrecy: Semantic secrecy.
There are multiple definitions but they are equivalent to IND-CPA or IND-CCA or IND-CCA2 depending on what you want from your cipher.
• IND here means Indistinguishable
• CPA here means "under Chosen Plaintext Attack".
• CCA means "under Chosen Ciphertext Attack" and it is a stronger definition of secrecy.
• CCA2 means "under Adaptive Chosen Ciphertext Attack"
Note that Non-Malleability has nothing to do with secrecy. For example the One Time Pad is perfectly secure, yet it is Malleable. Although you can prove that a cipher is non-malleable under chosen
ciphertext attack and it would be the same thing as ind-cca. Also, some kind of ciphers are malleable on purpose, see Homomorphic encryption.
There are also talks about Provable secrecy where we reduce the whole cipher/cryptosystem to the solidity of a problem difficult to compute. It's done more for Asymmetric encryption that generally
relies on math more than symmetric encryption. For example RSA is provably secure because its entire security relies on the Factoring Problem, which states that it is increasingly very difficult to
compute p and q in n = p * q n a large number and p, q primes.
A good cipher
So if we want to prove that our cipher is secure. We would have to prove that an Adversary would have no advantage in a guessing game where he would send us two plaintexts and we would send him back
one of the encrypted plaintext (chosen at random) expecting him to guess which one it is (see image above).
This is difficult to prove. For asymmetric encryption we'd rather reduce that to other assumptions. For symmetric encryption we would have to make sure its encrypted ciphertexts look random every
time. Like an ideal cipher.
for Stream Ciphers the randomness of the ciphertexts depends a lot on the randomness of the Pseudo Random Number Generator you are building it with (PRNG).
for Block Ciphers there are two things that are important: Confusion and Diffusion. Confusion states that a small change in the key has to change the whole ciphertext, Diffusion states that a small
change in the plaintext has to change the whole ciphertext. In AES for example, Confusion is done by using a Key Derivation Function to create several subkeys and XOR them to the internal state on
multiple rounds. Diffusion is done during the different rounds with a mix of Permutations and Substitution (that's why we call AES a substitution-permutation network).
A cryptanalyst is the black beast of cryptographers. His job is to attack our lovely ciphers. But this is useful in knowing how secure is a cipher. AES has been considered a solid cipher because it
is build on solid principles like the avalanche principle (confusion and diffusion) but not only because of that. Because it has resisted to known attacks since it has been born. A good cipher should
resist multiple years of attacks. A good cipher should withstand the efforts of cryptanalyst in time.
What does a cryptanalyst do to break a cipher?
The first good answer is bruteforce or exhaustive search. If we can simply bruteforce a cipher then it is obviously not good. When you do this you have to consider the birthday paradox and
time-memory trade off attacks. tl;dr: it doesn't take as long as you think to bruteforce something because of probabilities.
And here is why AES is considered a solid cipher again. There are no known attacks better than bruteforcing.
For cryptographers, a cryptographic "break" is anything faster than a brute force
It is known (thanks to Shannon in 1949) that in a known plaintext attack we can build an equation linking the plaintext with the ciphertext and where the unknowns are the bits of the key. Here the
algebraic/linear attack would be to solve the system we now have. If it is more complicated than that, we often talk about second order attack, third order attack, etc...
In a Differential cryptanalysis what we do is we try to notice a correlation in the differences between the internal state of several messages getting encrypted. Often to find a subkey and then
recover the key Total Break.
So the answer could be summed up like this: After researching how to make a good cipher (reducing it to a known hard math problem if it's an asymmetric cipher; making sure of it having nice criterion
(confusion & diffusion) in the case of a block cipher; making sure of the non-correlation of its PNRG and the randomness of its PNRG if it's a stream cipher), you would try to break it with known
attacks. Then you would enlarge the model you are working with to see if your cipher is still as secure, and if not how you would modify it or use it to make it secure. For example with textbook RSA
you can forge your own packets from it (if you have c = m^e you can create 2^e * c which would successfully decrypt as 2*m) and its deterministic properties (that makes it not semantically secure)
leak information ( (c/n) = (m^e/p) (m^e/q) = (m/p)(m/q) = (m/n) with (a/b) being the symbol of Legendre/Jacobi here). Without a padding system like OAEP RSA leaks the Jacobi Symbol of m.
I'm still a student. Those are information I've gathered and this is my understanding of it.
Totally anon
Really great post! I learned something from this! I would like to add something though, the heuristic of 'making a good cipher by reducing it to a known hard math problem if it's an asymmetric
cipher' won't always work, the Merkle–Hellman knapsack cryptosystem is an example of this.
link: https://en.wikipedia.org/wiki/Merkle%E2%80%93Hellman_knapsack_cryptosystem
Jim Lebeau
Thanks for posting this.
Discrete*, not discreet (in this case) :)
woops, thanks anonymous!
How can you tell if a cipher is secure?
For a cipher with all known proven facts about it, assuming there is a short-cut in the search space, you need to:
- determine how long the secret must be maintained to protect its information
- over-estimate the adversaries computing capabilities (including growth over the secrets lifetime)
- "accurately" estimate any potential short-cuts (50% reduction in search space, etc)
- use truly random processes for key-generation
- derive a sufficiently long key-length that satisfies the constraints above
Faith based reasoning:
- Trust that P != NP
- Ignore Quantum Computing
Unless using a one-time-pass with a truly random key then, can you feel 100% secure? I don't think so, but you should still breathe easy.
I never studied this stuff, but this just seems like sound reasoning.
I made a cipher and would like some tips on making it better or if it's crackable. I'll give an example of it below then I will link the key and rules I used for making it.
Woops wrong link
Here you go.
A few years ago I came up with this cipher, which I called DN-6. Polialphabetical with double encryption. I found it the best way to do it is by hand. I know the task is/can be tedious, but well
worth it if it can secure the info. The polialphabetical pad can only be used once and once alone.
How do you know a cipher is secure? You don't know. You can only prove that it isn't secure by breaking it using either a known or previously unknown attack.
I created an encryption program, mainly for the fun of it. You can see it at the homepage. It's called Alkemi.
Anyway, I encrypted some text and placed the ciphertext for downloading. It's called Challenge.txt. I know about Kerchhoff's principle. But I'm curious: If all you have is the ciphertext and some
idea of how the encryption works (from reading the website), can it still be broken?
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Putting min() and max() Into Action – Real Python
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Weierstrass elliptic function: Transformations
Transformations and argument simplifications
Argument involving basic arithmetic operations
Addition formulas
Translation by half-periods
Half-angle formulas
Multiple arguments
Argument involving numeric multiples of variable
Double angle formulas
Argument involving symbolic multiples of variable
Determinants involving derivatives
Related transformations
Halving half-period
Third of half-period
General fractions of half-periods
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Marcus–Hush–Chidsey kinetics at electrode–electrolyte interfaces
Electrochemical kinetics at electrode–electrolyte interfaces limit the performance of devices including fuel cells and batteries. While the importance of moving beyond Butler–Volmer kinetics and
incorporating the effect of electronic density of states of the electrode has been recognized, a unified framework that incorporates these aspects directly into electrochemical performance models is
still lacking. In this work, we explicitly account for the density functional theory-calculated density of states numerically in calculating electrochemical reaction rates for a variety of
electrode–electrolyte interfaces. We first show the utility of this for two cases related to Li metal electrodeposition and stripping on a Li surface and a Cu surface (anode-free configuration). The
deviation in reaction rates is minor for cases with flat densities of states such as Li, but is significant for Cu due to nondispersive d-bands creating large variation. Finally, we consider a
semiconducting case of a solid-electrolyte interphase consisting of LiF and Li[2]CO[3] and note the importance of the Fermi level at the interface pinned by the redox reaction occurring there. We
identify the asymmetry in reaction rates as a function of discharge/charge naturally within this approach.
Reactions at electrode–electrolyte interfaces control the limits of operation of various electrochemical devices. In the case of batteries, interfacial kinetics control the rate of reactions at the
anode and cathode, which ultimately leads to the limits on fast discharge and charge capability of modern Li-ion batteries.^1,2 Fast charging is an important requirement for electric vehicles,^3
while fast discharge is necessary for electric vertical take-off and landing aircraft.^4 In particular, there is significant interest in modifying the reaction rates of electrodeposition and
stripping of Li metal electrodes by modifying the nature of the so-called solid electrolyte interphase (SEI).^5
The typical kinetic rate law used for Li metal electrodes is Butler–Volmer.^6 Recently, Boyle et al.^7 highlighted the importance of moving beyond Butler–Volmer kinetics and incorporating the effects
of reorganization within the Marcus–Hush–Chidsey (MHC) formalism^8,9 within the low overpotential limit. In an accompanying paper for this special issue, we extend that analysis and show the
importance of accounting for the full MHC kinetic rate behavior.^10 However, most treatments assume that the electrode density of states (DOS) is constant and independent of the energy states/
overpotential. In the conventional treatment, this approach has no explanatory power for any asymmetry observed in the Tafel response. Asymmetric Marcus Theory^11,12 has sought to address this issue
by arguing that force constants (that is, the curvature of Gibbs free energy vs reaction coordinate) differ between reactants and products. However, as we show, in many cases, the impact of the DOS’s
energy dependence on predicted reaction rates can also be large.
In this work, we incorporate the effect of the DOS at the electrode using density functional theory (DFT) calculations directly and numerically show its importance in a variety of cases. The approach
can be thought of as a generalization of Gerischer’s model^13 expressed in the “language” of MHC. We begin by discussing the effect of the DOS for Li electrodeposition and stripping for Li metal
electrodes. Given the relatively flat nature of the DOS of Li metal, it affects the kinetic rates to a lesser extent. Next, we discuss the case of metal electrodeposition on Cu. Given the DOS
associated with the more localized d-band states, there is significant deviation between the kinetic behavior when accounting for its variation compared to assuming a constant DOS. This highlights
that the electronic state of the metal is an important factor in determining the high-rate performance of anode-free cells.^14 Finally, we discuss the case of an electrochemical redox reaction with
the density of states determined by SEI components, for example, LiF and Li[2]CO[3].
We also provide both our full analysis code open-source on GitHub as well as a user-friendly interface for visualization of the effect of different model parameters on the results presented herein
that readers can view online.
A. Theoretical approach
The insight of Chidsey^9 that it is critical to consider the occupation of electronic states when assessing rate constants within a Marcus-type theory was important. Building on this insight, the
rate expressions developed in MHC theory (we here adapt notation from Ref. 15) are
$kred∝∫−∞∞|V(ε )|2exp−(λ−eη+ε)24λkBT fFD(ε )dε,$
$kox∝∫−∞∞|V(ε )|2exp−(λ+eη−ε)24λkBT1−fFD(ε )dε,$
for the reductive and oxidative directions, respectively, where V($ε$) denotes a coupling constant between the (presumed localized) states on the electrolyte molecule and the (presumed
plane-wave-like) states in the solid electrode. This coupling constant includes the density of states as well as an overlap integral term. λ is the reorganization energy, e is the electronic charge,
η is the applied overpotential, k[B] is Boltzmann’s constant, and T is the absolute temperature. All energies are measured relative to the Fermi level E[f]. f[FD], denoting the Fermi-Dirac
distribution for the occupation of state by electrons, is given by
and its complement, (1 − f[FD]), is the distribution for holes.
In prior analyses, either explicitly or implicitly, the |V($ε$)|^2 term is presumed to have weak energy dependence and brought outside the integral. This allows the simplification presented in Ref.
16, taking advantage of the fact that the integral is over the whole real line, and the Fermi–Dirac distributions for electrons and holes can be equivalently thought of as complements of each other,
or equal but with the energy coordinate running in the opposite direction,
$kox/red∝∫−∞∞exp−(ε−λ±eη)24λkBTdε1+expε kBT.$
However, this treatment neglects the electrode’s DOS, which generally does have strong variations with energy, especially over scales larger than a few k[B]T. The most extreme example of this is in a
semiconductor, where the value is identically zero at and near the Fermi level. This, as we will show, can lead to dramatically different rate constant predictions. However, even in the case of a
transition metal (which often has large spikes in the DOS due to highly localized d-bands), this assumption can break down.
In our analysis, we consider adapted versions of Eqs. (1) and (2), where we explicitly account for the electrode DOS $D(ε )$,
$kred∝∫D(ε )exp−(λ−eη+ε)24λkBTexpε kBT1+expε kBTdε,$
$kox∝∫D(ε )exp−(λ+eη−ε)24λkBTdε1+expε kBT.$
Because DOSs are not defined over an infinite range of energies, we must truncate the integrals in our calculations. In practice, this makes no discernible difference in the predicted rate constants
because the Gaussian terms fall off dramatically within a relatively small energy range from E[f].
We also note that the dependence of the density of states (and to a lesser extent, the coupling constant that is also part of the |V($ε$)|^2 term in Eqs. (1) and (2)) has been described before,^17
but is generally either assumed to be weak, or is modeled with simple functional forms that presume, e.g., spherical Fermi surfaces or other approximations that may or may not be valid across the
wide diversity of relevant surfaces. To our knowledge, this is the first analysis that explicitly accounts for the DFT-calculated DOS numerically in directly calculating rates for electrochemical
reactions at electrode–electrolyte interfaces.
B. Computational approach
In this work, we use the Julia programming language to calculate the integrals described above using the adaptive Gauss–Kronrod quadrature^18 as implemented in the QuadGK.jl package.
Density functional theory calculations for Li and Cu were conducted using the GPAW package^19 through the Atomic Simulation Environment package.^20 Ion–electron interactions were treated using the
projector augmented wave approach.^21 For all calculations, a grid spacing of 0.16 Å and a 4 × 4 × 1 Monkhorst–Pack k-mesh were used.^22 To improve self-consistent field convergence, Fermi smearing
was applied to electron occupation with a width of 0.05 eV. All relaxations and analysis, unless otherwise specified, were conducted using the Bayesian error estimation functional with van der Waals
correction (BEEF-vdW) exchange-correlation functional. DOSs were calculated with a 100 meV Gaussian smearing width.
Calculations for LiF/Li[2]CO[3] were performed in Quantum Espresso according to methods described in Ref. 23.
A. Li metal anode
The first case is the redox processes occurring in a Li metal electrode. The overall redox reaction is given by
where the forward reaction denotes electrodeposition (charging), and the backward reaction denotes stripping (discharging). The reaction rates are dependent on the organic electrolyte used,^7 and the
analysis is presented in the case of an EC:DEC electrolyte. Figure 1 shows experimental data and best-fit results for each model (results modeled with Eq. (4) are indicated in figure legends by “MHC”
and from Eqs. (5) and (6) by “MHC + DOS”). Note that over the experimental overpotential range of ±0.2 V, the two models fit equally well. This is due to the fact that the DOS is relatively flat over
this range. Over the full range of the plot, one can start to see variations between the models, but only modest ones.
This case illustrates why the importance of this more generalized analysis could have been missed previously—at relatively modest overpotentials, many DOSs are “close enough” to flat that predictions
of the traditional MHC model and the one we present here differ little. We will now turn to cases where the two models show larger discrepancies.
B. Cu current collector
We next examine the case of depositing Li on an anode-free configuration on top of a Cu current collector.^14 This is given by
where * represents a site on a Cu surface. In this case, the electronic states of the Cu surface become important. The DOS (a) and predicted rate constants (b) for the Cu(111) surface are shown in
Fig. 2. We chose Cu(111) as it has the most desirable nucleation characteristics among the low Miller index surfaces,^14 in addition to the lowest surface energy.^24 Note that due to the high density
of states in the Cu d-bands starting ∼1.5 eV below the equilibrium E[f], a jump of roughly an order of magnitude in the rate constant is observed at this overpotential. An overpotential of this
magnitude is entirely feasible, particularly in the context of the fast charge/discharge rates currently targeted for many applications.
It is worth noting that the scale of variation in DOS (i.e., an order of magnitude) is approximately equal to the change in the predicted rate constant. Indeed, one should expect this provided λ is
much smaller than the overpotential window of interest, since λ essentially parametrizes a sliding Gaussian filter through which the DOS impacts k via the denominator of the first exponential terms
in Eqs. (5) and (6).
C. Effect of solid-electrolyte-interphase (SEI)
We will now consider the case where the Li electrode is covered with an inorganic solid electrolyte interphase due to the spontaneous reaction with the organic electrolyte typically used in Li-metal
batteries. While the exact nature of the SEI remains an active area of research, it is widely acknowledged that some of the inorganic phases present include Li[2]CO[3], LiF, Li[2]O, etc.^25 We
consider a model SEI consisting of an interface between LiF and Li[2]CO[3], building on the work of Pan et al.^26 Here, we consider a redox reaction, e.g., Li deposition, occurring at the SEI
component. Most SEI components are semiconducting or insulating by design and, hence, the kinetics for this semiconductor–electrolyte interface are quite different.
Figure 3 shows the results for this system. Because the DOS is identically zero throughout the ∼4 eV bandgap and the equilibrium E[f] position of the pristine interface is at midgap, there is a
discrepancy of as much as eight orders of magnitude between the predictions of the two models, even at low overpotentials, indicating the criticality of considering the energy dependency of the DOS
in this context. This further substantiates the remark above regarding the scale of variation of the DOS—because it goes to zero (over a window of many λ) in the bandgap, this scale tends to
infinity, and the scale of the discrepancy in k likewise diverges.
D. Effects of model parameters and assumptions
Several parameters in the form of energy scales control the behavior of these models. Some of their behavior has already been alluded to above, and we examine these effects more closely here. First
is the position of the equilibrium Fermi level E[f]. Thus far, we have assumed it to be at the position predicted by DFT for a pristine surface. However, many system variables can influence this
energy. Bulk or surface defects can shift E[f]. In an electrochemical system, the redox reaction sets the Fermi level at the electrode–electrolyte interface and is an independent variable.^27
Figure 4(a) shows the effect of shifting E[f] on the SEI case. In particular, we presume that an alternative redox couple (or equivalently, a particular voltage on the Li/Li^+ scale) is chosen such
that the equilibrium E[f] is near the valence band maximum of the SEI. In this case, the behavior at low overpotentials exhibits a marked asymmetry: at negative η (pushing further into the valence
bands), k rises sharply, while at positive η (pushing back into the bandgap), it drops exponentially until reaching midgap, whereupon it rises again as the conduction band minimum begins to have an
impact. Such an asymmetry depending on the redox potential has been recognized^28 and observed^29 for Li–O[2] batteries where discharge product, Li[2]O[2], is a wide-band gap insulator similar to the
LiF/Li[2]CO[3] interface considered here.
Figure 4(b) shows the effect of varying the reorganization energy λ. As mentioned previously, the mathematical effect of λ is primarily to act as a Gaussian smoothing filter on the density of states.
However, an increased λ also increases the offset between the filters in the positive and negative directions, with the overall result being a slowly varying and slightly outward-shifted k vs η
To better build the intuition for the impact of these parameters, we have built an interactive visualization, which can be viewed at https://nbviewer.jupyter.org/github/aced-differentiate/MHC_DOS/
blob/master/dataviz.ipynb. It features the DOSs for all materials considered in this work and allows the viewer to modify E[f], λ, and the temperature and see in real time how rate constant
predictions change.
It is worth noting that we have assumed that the electronic states of the electrode do not change along the reaction coordinate. This is exact for outer-sphere electron transfer reactions and is only
approximate for inner-sphere electron transfer.^30 For inner-sphere electron transfer, incorporating the electronic density of states along the reaction coordinate may become important. For the
electrodeposition cases considered in this work, the approximation made is expected to be valid for the case of depositing Li on Li, while for the case of depositing Li on Cu may require re-visiting.
In this work, we explore the role of the electronic density of states at the electrode using density functional theory calculations and incorporate that numerically into a generalized
Marcus–Hush–Chidsey kinetics formalism. We discuss three cases, electrodeposition on (i) Li surface and (ii) Cu surface, and (iii) redox reactions at an SEI component chosen here to be the LiF/Li[2]
CO[3] interface. In the case of the Li surface, due to a relatively flat density of states, we find minor deviations from the standard MHC rate law. In the case of a Cu surface, mimicking an
anode-free Li metal battery, the d-bands of Cu can significantly modify the rate constants, especially at high rates of charge and discharge. Finally, we show the drastic changes in reaction rates
for a semiconducting interface and note the importance of the location of the Fermi level, which is pinned by the redox reactions occurring at that interface. We show strong asymmetric behavior in
the SEI case, similar to that noted for Li[2]O[2] in Li–O[2] batteries.
We believe this approach has broad promise for improved understanding of electrokinetics at solid–liquid interfaces such as those investigated here and could also be extended in the future to
solid–solid interfaces such as those discussed in Refs. 31 and 32 by the inclusion of computed DOSs for the materials on both sides of the interface.
We have also made all analysis codes available open-source so that the community can take advantage of these models.
R.K. and V.V. gratefully acknowledge support from the Advanced Research Projects Agency Energy (ARPA-E) under Grant No. DE-AR0001211. This research was also supported by the Carnegie Mellon
University Manufacturing Futures Initiative, which was made possible by the Richard King Mellon Foundation. We thank Yumin Zhang, Shang Zhu, and Zeeshan Ahmad for sharing structures used in this
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15 Famous Mathematicians and Their Contributions
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15 Famous Mathematicians and Their Contributions
Table of Contents
1. Introduction
2. Famous Mathematicians list and their contributions
3. Conclusion
Being a mathematician is hard. Only a few people have mastered this subject and achieved fame. Of those, there have been some famous Indian mathematicians. In this article, we will discuss some of
the famous mathematicians and their contributions to Mathematics.
Riddles about famous Mathematicians
Here is a PDF that consists of riddles about famous mathematicians. Click on the download button to solve them.
📥 Riddles about famous Mathematicians Download
15 Famous Mathematicians and Their Contributions
1. Euclid
Euclid was one among the famous mathematicians, and he was known as the ‘Father of Geometry.’ His famous Geometry contribution is referred to as the Euclidean geometry, which is there in the Geometry
chapter of class IX. He spent all his life working for mathematics and set a revolutionary contribution to Geometry.
2. Pythagoras
‘Pythagoras theorem’ is very popular and an important mathematical formula to solve mathematical problems. Since childhood, all of us have struggled to solve mathematical problems applying the
Pythagoras theorem. Pythagoras discovered this prominent theorem and he became the father of ‘Pythagoras theorem.’ Pythagoras was from Greece, but he fled off to India in the latter part of his life.
3. Archimedes
Archimedes was a Greek Scientist, a great mathematician, and a Physician. He also worked his entire life in search of discovering mathematical formulas that are related to Physics. The best
contribution of Archimedes in mathematics is known as the invention of compound pulleys, antiquity, and screw pump. Students of class X have to study these chapters of compound pulleys and antiquity.
4. Thales of Miletus
Thales of Miletus was one of the most famous mathematicians from Greece. He was very skilled in Geometry and used Geometry as a means to calculate the heights of pyramids and measure the distance of
a ship from the shore. As he was also a philosopher, in the ‘Thales’ theorem’ he tries to apply Geometry by using deductive reasoning and derives the conclusion of four corollaries. You can learn
more about Thales of Miletus here.
5. Aristotle
Aristotle was a great scholar and he had vast knowledge in various areas, including Physics, mathematics, geology, metaphysics, medicine, biology, and psychology. He was a student of Plato, and both
of them together discovered many philosophical theories and contributed to mathematics and Platonism. He combines mathematics and philosophy and in his treaties, and uses mathematical science in
three principal ways.
6. Diophantus
His algebraic equations are quite easy and unique, so he was popularly known as the ‘Father of Algebra.’ He wrote a series of books on Algebra. He later gained popularity for his book Arithmetica,
where a brief description with examples was given on the best solution for all the algebraic equations and the theory related to the number.
7. Eratosthenes
Eratosthenes was a world-famous mathematician known for his unbelievable and exact calculation. He was the only mathematician who put efforts to calculate the earth’s circumference and calculated the
Earth’s axis tilt. Both his calculations are exact, and so he became famous worldwide.
8. Hipparchus
Like Geometry, trigonometry chapters are also important for class IX and X students. The founder of trigonometry was an intelligent mathematician and mythologist Hipparchus. He discovered the first
trigonometric table in mathematics. He was the first person to develop a well-grounded process by which people can predict solar eclipses.
9. Hero of Alexandria
Heron’s formula in mathematics is applied by students from class VI onwards. Yes, he was the one who discovered the square root of numbers. In today’s mathematics, his formula is known as Heron’s
formula. So, he gained popularity and became known as the ‘Hero of Alexandria.’
10. Ptolemy
Ptolemy was a mathematician; he was also a geographer, musician, writer, and astronomer. His contributions to mathematics were incredible. He wrote about mathematics, and among them, his best treaty
was called Almagest. He also believed that in the Universe, the position of the Earth was in the center.
11. Xenocrates
Xenocrates was a famous mathematician from Greek. He had written a series of books on mathematics. He emphasizes the theory of numbers in mathematics, and all his written books were based on the
theory of numbers, and geometry. He could easily calculate the syllables from an alphabet.
12. Anaxagoras
He was a great mathematician and an astronomer. Because of his outstanding knowledge of mathematics and cosmology, he discovered the exact clarification of eclipses and stated that the Sun is larger
than Peloponnese.
13. Hypatia
She was a famous mathematician and a philosopher. She was the first woman to give importance to mathematics. She was a genius, and for many young women, she became an inspiration and encouraged them
to pursue their dreams. In Alexandria's history, she was the last famous mathematician.
14. Antiphon
Antiphon discovered the value of Pi. This renowned mathematician was the first one who calculated the upper bound and lowers bound values of Pi by inscribing and circumscribing around a circle, the
polygon, and processed finally to calculate the areas of the polygon. His idea of calculating a polygon area became very famous, and it changed mathematics for the world.
15. Diocles
Diocles was a profound geometer and mathematician. He was renowned for his discovery in the subdivision of geometry. In the books of Mathematics, the ‘Geometry curve’ is known by his name as the
‘Cissoid of Diocles.’ To find out a solution to doubling the cube, the method of Cissoid of Diocles was used.
We have discussed above the list of famous mathematicians and their contributions to mathematics. However, they all are from Greek. There are famous Indian mathematicians also like Srinivasa
Ramanujan, Aryabhata, Shakuntala Devi, and many more.
If you wish to know more about famous Indian mathematicians and their mathematics contributions, then comment in this section. For any question related to this article about ‘famous mathematicians
and their contributions, ’ feel free to ask.
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A Step-by-Step Approach to Using SAS for Factor Analysis and Structural Equation Modeling, Second Edition, 2nd Edition
Book description
This easy-to-understand guide makes SEM accessible to all users. This second edition contains new material on sample-size estimation for path analysis and structural equation modeling. In a single
user-friendly volume, students and researchers will find all the information they need in order to master SAS basics before moving on to factor analysis, path analysis, and other advanced statistical
Product information
• Title: A Step-by-Step Approach to Using SAS for Factor Analysis and Structural Equation Modeling, Second Edition, 2nd Edition
• Author(s):
• Release date: March 2013
• Publisher(s): SAS Institute
• ISBN: 9781629592442 | {"url":"https://www.oreilly.com/library/view/a-step-by-step-approach/9781599942308/","timestamp":"2024-11-08T05:42:24Z","content_type":"text/html","content_length":"119188","record_id":"<urn:uuid:d0641d66-eb25-4a8d-b551-8cc8c87fc642>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00317.warc.gz"} |
Lab 4: Forms
University of Washington, CSE 190 M
Lab 4: Forms
Except where otherwise noted, the contents of this document are Copyright 2010 Marty Stepp and Jessica Miller.
original lab idea and code by Kevin Wallace; revised by Brian Le and Marty Stepp
Basic lab instructions
• You may want to bring your textbook to labs to look up syntax and examples.
• We encourage you to talk to your classmates; it's okay to share code and ideas during lab.
• You don't need to finish all of the exercises. Do as much as you can in the allotted time.
• Before you leave, check in with a TA to get credit.
• Resources: (make sure it's installed!); ; textbook Chapter 6
Today's lab exercises
This week, we'll write code to allow a student to submit a credit card number in exchange for a good grade in this class. (Just a joke!)
The HTML page buyagrade.html will have a form that submits to sucker.php. Then sucker.php will output a web page to confirms that the student's money was taken.
Download the HTML file below (right-click, Save Target As...) to get started:
Exercise : Creating a Form (~20 min)
(See example screenshot on next slide.)
Let's create an HTML form that POSTs its submitted data to a PHP program on a server.
• Download buyagrade.html. Upload it to Webster and point your web browser to it on Webster.
• Modify the HTML file by turning part of its contents into an HTML form.
□ Give name attributes to the form controls so they will be sent as query parameters; the names are up to you.
□ Some form controls (such as radio buttons) need value attributes.
□ Fill the Section drop-down list with choices MA through MH.
• Test your form parameters by temporarily setting its action attribute to:
□ http://webster.cs.washington.edu/params.php
Exercise , output
Your page should look like this. You can also try our (don't peek at the source!).
Exercise : Displaying Form Data (~15 min)
(See example screenshot on next slide.)
Now write the PHP page to handle the submitted form data. Start from the skeleton above.
• Tell your buyagrade.html to POST its data to sucker.php.
• Modify sucker.php to display the name, credit card number and type (Visa or MasterCard).
□ For now, the page doesn't need to save this information on the server.
□ Embed variables' values in HTML using PHP expression blocks such as <?= $foo ?>.
Exercise , output
Your page should look like this. You can also try our (don't peek at the source!).
Exercise : Save Form Data (~15 min)
(See next two slides for example screenshot and file permission issues.)
Modify your sucker.php page to save the submitted data to a file suckers.txt.
Exercise , file permissions
If you see an error about "Permission denied", right-click the suckers.txt file in SSH and choose Properties. In the box that appears, check the permission boxes for "Write" permission. Do the same
for your lab4 folder itself; and also give the lab4 folder the "Execute" permission.
Exercise , output
Your page should look like this. You can also try our (don't peek at the source!).
Exercise : Basic Data Validation (~10 min)
(See example screenshot on next slide.)
Update sucker.php to verify that all parameters exist and that the user did not leave any blank.
• You can check whether a particular parameter has been passed using the PHP function isset.
• You can check whether an element of $_REQUEST is blank by comparing it to the empty string, "".
□ You can also use the element itself as a Boolean test to see if it is "truthy" or "falsy".
• If the user did not fill in every field correctly, show an error message like on the next slide.
□ Give a "Try again" link that directs back to the initial buyagrade.html page.
Exercise , output
Your error message should look like this. You can also try our (don't peek at the source!).
Exercise : Credit Card Check (~10 min)
(See example screenshot on next slide.)
Update your sucker.php file to further check the validity of the credit card number:
• Check to make sure the credit card number is composed of exactly 16 characters.
• Check to make sure that a Visa card number starts with a 4 and a MasterCard number starts with a 5.
• If the credit card doesn't pass the above checks, show an error like on the next slide.
Exercise , output
Your error message should look like this. You can also try our (don't peek at the source!).
Exercise : (h4x0rz only): Luhn Algorithm
Update your sucker.php file to check credit card numbers with the classic .
• The above Luhn link describes the algorithm and contains a poorly written implementation.
□ Or try to code it yourself. The next slides describe the algorithm's implementation.
• If the credit card doesn't pass the above checks, show the same error as from the previous exercise.
Exercise , algorithm description
from Building Java Programs:
A valid credit card number passes a digit-sum test known as the Luhn checksum algorithm. Luhn's algorithm states that if you sum the digits of the number in a certain way, the total sum must be a
multiple of 10 for a valid number. Systems that accept credit cards perform a Luhn test before contacting the credit card company for final verification.
The algorithm for summing the digits is the following: Consider each digit of the credit card to have a zero-based index: the first is at index 0, and the last is at index 15. Start from the
rightmost digit (index 15) and process each digit one at a time. For digits at odd-numbered indexes (15, 13, etc.), simply add that digit to the cumulative sum. For digits at even-numbered
indexes (14, 12, etc), double the digit's value, then if that doubled value is less than 10, add it to the sum. If the doubled number is 10 or greater, add each of its digits separately into the
• The next slide contains pseudo-code for the algorithm.
Exercise , algorithm pseudocode
from Building Java Programs:
sum = 0.
for (each digit of CC number, starting from the last index):
if index is odd:
add digit to sum.
double the digit's value.
if doubled value < 10:
add doubled value to sum.
split doubled value into its two digits.
add first digit to sum.
add second digit to sum.
• 4111111111111111 and 4408041254369873 are example numbers that pass the Luhn algorithm.
If you finish them all...
If you finish all the exercises, you can add any other content or code you like to your page.
If the lab is over or almost over, check with a TA and you may be able to be dismissed.
Great work! | {"url":"https://courses.cs.washington.edu/courses/cse190m/10su/labs/lab4-buyagrade.shtml","timestamp":"2024-11-09T00:33:22Z","content_type":"application/xhtml+xml","content_length":"16641","record_id":"<urn:uuid:79ba92d4-6163-468b-ac7f-a10a0d6aec3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00668.warc.gz"} |
Conditional Decoupling of Quantum Information
Insights from quantum information theory show that correlation measures based on quantum entropy are fundamental tools that reveal the entanglement structure of multipartite states. In that spirit,
Groisman, Popescu, and Winter [Phys. Rev. A 72, 032317 (2005), 10.1103/PhysRevA.72.032317] showed that the quantum mutual information I (A ;B ) quantifies the minimal rate of noise needed to erase
the correlations in a bipartite state of quantum systems A B . Here, we investigate correlations in tripartite systems A B E . In particular, we are interested in the minimal rate of noise needed to
apply to the systems A E in order to erase the correlations between A and B given the information in system E , in such a way that there is only negligible disturbance on the marginal B E . We
present two such models of conditional decoupling, called deconstruction and conditional erasure cost of tripartite states A B E . Our main result is that both are equal to the conditional quantum
mutual information I (A ;B |E )—establishing it as an operational measure for tripartite quantum correlations.
Physical Review Letters
Pub Date:
July 2018
□ Quantum Physics;
□ Condensed Matter - Statistical Mechanics;
□ Computer Science - Information Theory;
□ High Energy Physics - Theory
6 pages, 1 figure, see companion paper at arXiv:1609.06994 | {"url":"https://ui.adsabs.harvard.edu/abs/2018PhRvL.121d0504B/abstract","timestamp":"2024-11-06T19:29:03Z","content_type":"text/html","content_length":"40677","record_id":"<urn:uuid:f05cdc3e-e556-41f5-b50a-163e6886df14>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00263.warc.gz"} |
Can you fix the K-means train results?
kamegai_satoshi Partner, Dataiku DSS Core Designer, Dataiku DSS Core Concepts, Registered Posts: 1 Partner
We fixed the seed in K-means, but the clusters we got for each train were different. Specifically, the Variables importance and silhouette were different.
Are there any other settings needed besides seed to fix the results? We think that changing results every time under the same conditions is a big problem for business use.
How to reproduce it is as follows.
• Create a project by importing Sample project(name=Predicting Churn).
• Changing the setting of Visual Analysis(name=Clustering customers into segments).
□ Algorithms > KMeans > Seed = 1000
• Run the TRAIN of "Clustering customers into segments".
CoreyS Dataiker Alumni, Dataiku DSS Core Designer, Dataiku DSS Core Concepts, Registered Posts: 1,150 ✭✭✭✭✭✭✭✭✭
Hi, @kamegai_satoshi
! Can you provide any further details on the thread to assist users in helping you find a solution (insert examples like DSS version etc.) Also, can you let us know if you’ve tried any fixes
already?This should lead to a quicker response from the community.
EmmaH Dataiker Posts: 4 Dataiker
Hi @kamegai_satoshi
Dataiku uses the Scikit-learn KMeans clustering algorithm in the Visual ML, so the seed that you're defining is the random_state parameter. This parameter uses the same randomly-initialized point
as a centroid each time you rerun the algorithm.
From the KMeans doc:
If the algorithm stops before fully converging (because of ``tol`` or
``max_iter``), ``labels_`` and ``cluster_centers_`` will not be consistent,
i.e. the ``cluster_centers_`` will not be the means of the points in each
cluster. Also, the estimator will reassign ``labels_`` after the last
iteration to make ``labels_`` consistent with ``predict`` on the training
Essentially, the algorithm hits the max number of iterations before it can assign a consistent label.
A solution for this in Dataiku could be to add a custom python model with an increased number of iterations:
from sklearn.cluster import KMeans
clf = KMeans(n_clusters=5, n_init=1, random_state = 1000) | {"url":"https://community.dataiku.com/discussion/comment/14969#Comment_14969","timestamp":"2024-11-15T00:47:24Z","content_type":"text/html","content_length":"407492","record_id":"<urn:uuid:2af7052e-a92b-4e85-8260-1933d9770709>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00244.warc.gz"} |
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CPC '21 Contest 1 P1 - AQT and Fractions
Remember to use this editorial only when stuck, and not to copy-paste code from it. Please be respectful to the problem author and editorialist.
Submitting an official solution before solving the problem yourself is a bannable offence.
Subtask 1
The number of possible inputs is very small in this subtask. Here are some solutions:
• The only time when the digits infinitely repeat is when and , so we can just check for this specific case and use something like the to_string(int) method in C++ to count the number of digits
past the decimal.
• Hardcode every possible input.
Subtask 2
is always a multiple of , so the decimal will never repeat infinitely, so we can convert the fraction to a string and just check the number of digits. Note that we cannot simply just count the number
of digits in , as the fraction may not be completely reduced. Make sure to be careful about precision (i.e. using arbitrary precision integers).
Time Complexity: per test case.
Subtask 3
This subtask exists to reward solutions that would get full marks but fail due to precision or overflow errors.
Subtask 4
One possible solution is to observe that a finite decimal value for can only get so long with the input constraints, so we can perform long division to find the value of and assume that the answer is
infinite if our decimal gets too long.
Time Complexity: per test case.
Alternatively, start by assuming that is fully reduced (if it is not the case, reduce the fraction first). Next, notice the following:
• The only way for the decimal to be finite is if for some where .
• Again let where and . The number of digits to the right of the decimal is always . We can show that this is true with the following:
□ Let and . We can rewrite as and notice that either or (or both).
□ Every power of (the portion) adds additional digit by 'shifting' the current decimal to the right.
□ Every additional power of or (either or respectively) will always increase the number of digits by as the fraction is already reduced.
□ Thus, if the number of digits is , otherwise the number of digits is , which is equivalent to .
Overall, this solution leads to a simpler implementation.
Time Complexity: per test case.
• Could someone explain why dividing by 2 or 5 increases the digits after the decimal point by 1? (given that the original number without the decimal point has no factor of 2 or 5)
If there were just one non-zero digit after the decimal point, I could see why this would be true, since I could literally just test out the possible digits.
But I can't seem to convince myself of this when multiple non-zero digits are involved; for example, I know that if we have 0.01 divided by 2 the 1 gets "halved" and we get 0.005; but where x is
a placeholder for some nonzero digit, if we divide x.x1 by 2, the 1 getting "halved" contributes a 0.005, but the other nonzero digits also get "halved," so what's to keep those contributions
from adding enough to the thousandths place that the last 5 ticks up to a bigger (more to the left) place? [I hope that made sense lol] | {"url":"https://dmoj.ca/problem/cpc21c1p1/editorial","timestamp":"2024-11-05T12:19:15Z","content_type":"text/html","content_length":"29928","record_id":"<urn:uuid:53fba62a-5f66-4406-987a-42d0ee0575de>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00181.warc.gz"} |
Cycle fun
I've forgotten where I saw this---perhaps it was a Project Euler problem. In any case, you can see the pattern. It's a nice (fairly simple) exercise in Python to construct a list of lists and the
functions to modify it. The great thing is that once you have the inner 3 x 3 circle correct, the rest is automatic. The code for this is in cycle.py.
The real reason I'm posting about it is that it makes a fun plotting project for matplotlib, especially in combination with the prime number generator that we worked up the other day (in primes.py).
The graphic at the top of the post was obtained by modifying the array to hold n % 10 for each value, and using the matplotlib color map called "winter." I think for that one I had n = 50, so that's
a 101 x 101 grid.
The prime example :) has n = 40 and uses the 'autumn' colormap. That's the one that script.py is set up to generate at the moment. Is there a pattern or is it just random?
[UPDATE: I did a bit of testing. In tracing the results for the prime plot using a small n, I discovered that matplotlib has shown the matrix in a different way than I printed it at the top of the
post. Their "heatmap" goes vertically for the first step and cycles clockwise; it is both a mirror image and rotated 180 degrees from the way I did it. Sorry. But, now that I'm done, I think I can
just leave it that way... ]
Zipped project files on Dropbox (here). | {"url":"https://telliott99.blogspot.com/2010/12/cycle-fun.html","timestamp":"2024-11-12T22:36:19Z","content_type":"application/xhtml+xml","content_length":"80724","record_id":"<urn:uuid:2191af28-be73-4d9b-86e4-077e55c5229c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00434.warc.gz"} |
6.5: Friction (Part 2)
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Friction and the Inclined Plane
One situation where friction plays an obvious role is that of an object on a slope. It might be a crate being pushed up a ramp to a loading dock or a skateboarder coasting down a mountain, but the
basic physics is the same. We usually generalize the sloping surface and call it an inclined plane but then pretend that the surface is flat. Let’s look at an example of analyzing motion on an
inclined plane with friction.
A skier with a mass of 62 kg is sliding down a snowy slope at a constant velocity. Find the coefficient of kinetic friction for the skier if friction is known to be 45.0 N.
The magnitude of kinetic friction is given as 45.0 N. Kinetic friction is related to the normal force N by f[k] = \(\mu_{k}\)N; thus, we can find the coefficient of kinetic friction if we can find
the normal force on the skier. The normal force is always perpendicular to the surface, and since there is no motion perpendicular to the surface, the normal force should equal the component of the
skier’s weight perpendicular to the slope. (See Figure \(\PageIndex{1}\), which repeats a figure from the chapter on Newton’s laws of motion.)
Figure \(\PageIndex{1}\): The motion of the skier and friction are parallel to the slope, so it is most convenient to project all forces onto a coordinate system where one axis is parallel to the
slope and the other is perpendicular (axes shown to left of skier). The normal force \(\vec{N}\) is perpendicular to the slope, and friction \(\vec{f}\) is parallel to the slope, but the skier’s
weight \(\vec{w}\) has components along both axes, namely \(\vec{w}_{y}\) and \(\vec{w}_{x}\). The normal force \(\vec{N}\) is equal in magnitude to \(\vec{w}_{y}\), so there is no motion
perpendicular to the slope. However, \(\vec{f}\) is less than \(\vec{w}_{x}\) in magnitude, so there is acceleration down the slope (along the x-axis).
We have
\[N = w_{y} = w \cos 25^{o} = mg \cos 25^{o} \ldotp\]
Substituting this into our expression for kinetic friction, we obtain
\[f_{k} = \mu_{k} mg \cos 25^{o},\]
which can now be solved for the coefficient of kinetic friction \(\mu_{k}\).
Solving for \(\mu_{k}\) gives
\[\mu_{k} = \frac{f_{k}}{N} = \frac{f_{k}}{w \cos 25^{o}} = \frac{f_{k}}{mg \cos 25^{o}} \ldotp\]
Substituting known values on the right-hand side of the equation,
\[\mu_{k} = \frac{45.0\; N}{(62\; kg)(9.80\; m/s^{2})(0.906)} = 0.082 \ldotp\]
This result is a little smaller than the coefficient listed in Table 6.1 for waxed wood on snow, but it is still reasonable since values of the coefficients of friction can vary greatly. In
situations like this, where an object of mass m slides down a slope that makes an angle \(\theta\) with the horizontal, friction is given by f[k] = \(\mu_{k}\) mg cos \(\theta\). All objects slide
down a slope with constant acceleration under these circumstances.
We have discussed that when an object rests on a horizontal surface, the normal force supporting it is equal in magnitude to its weight. Furthermore, simple friction is always proportional to the
normal force. When an object is not on a horizontal surface, as with the inclined plane, we must find the force acting on the object that is directed perpendicular to the surface; it is a component
of the weight.
We now derive a useful relationship for calculating coefficient of friction on an inclined plane. Notice that the result applies only for situations in which the object slides at constant speed down
the ramp.
An object slides down an inclined plane at a constant velocity if the net force on the object is zero. We can use this fact to measure the coefficient of kinetic friction between two objects. As
shown in Example \(\PageIndex{1}\), the kinetic friction on a slope is f[k] = \(\mu_{k}\) mg cos \(\theta\). The component of the weight down the slope is equal to mg sin \(\theta\) (see the
free-body diagram in Figure \(\PageIndex{1}\)). These forces act in opposite directions, so when they have equal magnitude, the acceleration is zero. Writing these out,
\[\mu_{k} mg \cos \theta = mg \sin \theta \ldotp\]
Solving for \(\mu_{k}\), we find that
\[\mu_{k} = \frac{mg \sin \theta}{mg \cos \theta} = \tan \theta \ldotp\]
Put a coin on a book and tilt it until the coin slides at a constant velocity down the book. You might need to tap the book lightly to get the coin to move. Measure the angle of tilt relative to the
horizontal and find \(\mu_{k}\). Note that the coin does not start to slide at all until an angle greater than \(\theta\) is attained, since the coefficient of static friction is larger than the
coefficient of kinetic friction. Think about how this may affect the value for \(\mu_{k}\) and its uncertainty.
Atomic-Scale Explanations of Friction
The simpler aspects of friction dealt with so far are its macroscopic (large-scale) characteristics. Great strides have been made in the atomic-scale explanation of friction during the past several
decades. Researchers are finding that the atomic nature of friction seems to have several fundamental characteristics. These characteristics not only explain some of the simpler aspects of
friction—they also hold the potential for the development of nearly friction-free environments that could save hundreds of billions of dollars in energy which is currently being converted
(unnecessarily) into heat.
Figure \(\PageIndex{2}\) illustrates one macroscopic characteristic of friction that is explained by microscopic (small-scale) research. We have noted that friction is proportional to the normal
force, but not to the amount of area in contact, a somewhat counterintuitive notion. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area because only
high spots touch. When a greater normal force is exerted, the actual contact area increases, and we find that the friction is proportional to this area.
Figure \(\PageIndex{2}\): Two rough surfaces in contact have a much smaller area of actual contact than their total area. When the normal force is larger as a result of a larger applied force, the
area of actual contact increases, as does friction.
However, the atomic-scale view promises to explain far more than the simpler features of friction. The mechanism for how heat is generated is now being determined. In other words, why do surfaces get
warmer when rubbed? Essentially, atoms are linked with one another to form lattices. When surfaces rub, the surface atoms adhere and cause atomic lattices to vibrate—essentially creating sound waves
that penetrate the material. The sound waves diminish with distance, and their energy is converted into heat. Chemical reactions that are related to frictional wear can also occur between atoms and
molecules on the surfaces. Figure \(\PageIndex{3}\) shows how the tip of a probe drawn across another material is deformed by atomic-scale friction. The force needed to drag the tip can be measured
and is found to be related to shear stress, which is discussed in Static Equilibrium and Elasticity. The variation in shear stress is remarkable (more than a factor of 1012 ) and difficult to predict
theoretically, but shear stress is yielding a fundamental understanding of a large-scale phenomenon known since ancient times—friction.
Figure \(\PageIndex{3}\): The tip of a probe is deformed sideways by frictional force as the probe is dragged across a surface. Measurements of how the force varies for different materials are
yielding fundamental insights into the atomic nature of friction.
Describe a model for friction on a molecular level. Describe matter in terms of molecular motion. The description should include diagrams to support the description; how the temperature affects the
image; what are the differences and similarities between solid, liquid, and gas particle motion; and how the size and speed of gas molecules relate to everyday objects.
The two blocks of Figure \(\PageIndex{4}\) are attached to each other by a massless string that is wrapped around a frictionless pulley. When the bottom 4.00-kg block is pulled to the left by the
constant force \(\vec{P}\), the top 2.00-kg block slides across it to the right. Find the magnitude of the force necessary to move the blocks at constant speed. Assume that the coefficient of kinetic
friction between all surfaces is 0.400.
Figure \(\PageIndex{4}\): (a) Each block moves at constant velocity. (b) Free-body diagrams for the blocks.
We analyze the motions of the two blocks separately. The top block is subjected to a contact force exerted by the bottom block. The components of this force are the normal force N[1] and the
frictional force −0.400 N[1]. Other forces on the top block are the tension T in the string and the weight of the top block itself, 19.6 N. The bottom block is subjected to contact forces due to the
top block and due to the floor. The first contact force has components −N[1] and 0.400 N[1], which are simply reaction forces to the contact forces that the bottom block exerts on the top block. The
components of the contact force of the floor are N[2] and 0.400 N[2]. Other forces on this block are −P, the tension T, and the weight –39.2 N. Solution Since the top block is moving horizontally to
the right at constant velocity, its acceleration is zero in both the horizontal and the vertical directions. From Newton’s second law,
\[\sum F_{x} = m_{2} a_{x}\] \[\sum F_{y} = m_{1} a_{y}\]
\[T - 0.400\; N_{1} = 0\] \[N_{1} - 19.6\; N = 0 \ldotp\]
Solving for the two unknowns, we obtain N[1] = 19.6 N and T = 0.40 N[1] = 7.84 N. The bottom block is also not accelerating, so the application of Newton’s second law to this block gives
\[\sum F_{x} = m_{2} a_{x}\] \[\sum F_{y} = m_{1} a_{y}\]
\[T - P + 0.400\; N_{1} + 0.400\; N_{2} = 0\] \[N_{2} - 39.2\; N - N_{1} = 0 \ldotp\]
The values of N[1] and T were found with the first set of equations. When these values are substituted into the second set of equations, we can determine N[2] and P. They are
\[N_{2} = 58.8\; N\; and\; P = 39.2\; N \ldotp\]
Understanding what direction in which to draw the friction force is often troublesome. Notice that each friction force labeled in Figure \(\PageIndex{4}\) acts in the direction opposite the motion of
its corresponding block.
A 50.0-kg crate rests on the bed of a truck as shown in Figure \(\PageIndex{5}\). The coefficients of friction between the surfaces are \(\mu_{k}\) = 0.300 and \(\mu_{s}\) = 0.400. Find the
frictional force on the crate when the truck is accelerating forward relative to the ground at (a) 2.00 m/s^2, and (b) 5.00 m/s^2.
Figure \(\PageIndex{5}\): (a) A crate rests on the bed of the truck that is accelerating forward. (b) The free-body diagram of the crate.
The forces on the crate are its weight and the normal and frictional forces due to contact with the truck bed. We start by assuming that the crate is not slipping. In this case, the static frictional
force fs acts on the crate. Furthermore, the accelerations of the crate and the truck are equal.
1. Application of Newton’s second law to the crate, using the reference frame attached to the ground, yields
\[\begin{split} \sum F_{x} & = ma_{x} \\ f_{s} & = (50.0\; kg)(2.00\; m/s^{2}) \\ & = 1.00 \[\begin{split} \sum F_{y} & = ma_{y} \\ N - 4.90 \times 10^{2}\; N & = (50.0\; kg)(0) \\ N & = 4.90 \
\times 10^{2}\; N \end{split}\] times 10^{2}\; N \ldotp \end{split}\]
We can now check the validity of our no-slip assumption. The maximum value of the force of static friction is $$\mu_{s} N = (0.400)(4.90 \times 10^{2}\; N) = 196\; N,$$whereas the actual force of
static friction that acts when the truck accelerates forward at 2.00 m/s^2 is only 1.00 x 10^2 N. Thus, the assumption of no slipping is valid.
2. If the crate is to move with the truck when it accelerates at 5.0 m/s^2, the force of static friction must be $$f_{s} = ma_{x} = (50.0\; kg)(5.00\; m/s^{2}) = 250\; N \ldotp$$Since this exceeds
the maximum of 196 N, the crate must slip. The frictional force is therefore kinetic and is $$f_{k} = \mu_{k} N = (0.300)(4.90 \times 10^{2}\; N) = 147\; N \ldotp$$The horizontal acceleration of
the crate relative to the ground is now found from $$\begin{split} \sum F_{x} & = ma_{x} \\ 147\; N & = (50.0\; kg)a_{x}, \\ so\; a_{x} & = 2.94\; m/s^{2} \ldotp \end{split}\]
Relative to the ground, the truck is accelerating forward at 5.0 m/s^2 and the crate is accelerating forward at 2.94 m/s^2. Hence the crate is sliding backward relative to the bed of the truck with
an acceleration 2.94 m/s^2 − 5.00 m/s^2 = −2.06 m/s^2.
Earlier, we analyzed the situation of a downhill skier moving at constant velocity to determine the coefficient of kinetic friction. Now let’s do a similar analysis to determine acceleration. The
snowboarder of Figure \(\PageIndex{6}\) glides down a slope that is inclined at \(\theta\) = 13° to the horizontal. The coefficient of kinetic friction between the board and the snow is \(\mu_{k}\) =
0.20. What is the acceleration of the snowboarder?
Figure \(\PageIndex{6}\): (a) A snowboarder glides down a slope inclined at 13° to the horizontal. (b) The free-body diagram of the snowboarder.
The forces acting on the snowboarder are her weight and the contact force of the slope, which has a component normal to the incline and a component along the incline (force of kinetic friction).
Because she moves along the slope, the most convenient reference frame for analyzing her motion is one with the x-axis along and the y-axis perpendicular to the incline. In this frame, both the
normal and the frictional forces lie along coordinate axes, the components of the weight are mg sin θ along the slope and mg cos \(\theta\) at right angles into the slope , and the only acceleration
is along the x-axis (a[y ]= 0).
We can now apply Newton’s second law to the snowboarder:
\[\begin{split} \sum F_{x} & = ma_{x} \\ mg \sin \theta - \mu_{k} N & = ma_{x} \end{split}\] \[\begin{split} \sum F_{y} & = ma_{y} \\ N - mg \cos \theta &= m(0) \ldotp \end{split}\]
From the second equation, N = mg cos \(\theta\). Upon substituting this into the first equation, we find
\[\begin{split} a_{x} & = g(\sin \theta - \mu_{k} \cos \theta) \\ & = g(\sin 13^{o} - 0.520 \cos 13^{o}) = 0.29\; m/s^{2} \ldotp \end{split}\]
Notice from this equation that if \(\theta\) is small enough or \(\mu_{k}\) is large enough, a[x] is negative, that is, the snowboarder slows down.
The snowboarder is now moving down a hill with incline 10.0°. What is the skier’s acceleration? | {"url":"https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/06%3A_Applications_of_Newton's_Laws/6.05%3A_Friction_(Part_2)","timestamp":"2024-11-09T10:53:49Z","content_type":"text/html","content_length":"158524","record_id":"<urn:uuid:da389e16-662f-4b5f-a9af-ade445b96d8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00556.warc.gz"} |
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How to arbitrage in stock index futures?
Generally speaking, the greater the volatility, the greater the return. Is there any situation where the volatility is small and the return is large? Yes, that is hedging arbitrage
When we establish two investment targets and their fluctuation directions are opposite, then as a whole, whether the market rises or falls, after the profits and losses are offset, the fluctuations
are limited.
Many people understand that hedging is used to control risks, because after hedging, you can lose less or no money even if the market falls.But at the same time, hedging also limits profits because
when the market goes up, you don't make money.
Some people will definitely ask, why do hedging? It’s better to buy financial management from a bank.The reason is that after hedging, apart from offsetting the profits and losses caused by market
fluctuations, what else can you gain? After the implementation of some hedging strategies, you can obtain a fixed-income product, such as the hedging of short stock index futures orders and spot
prices; after the implementation of other hedging strategies, you can obtain cheap short options or long options, such as grade A and Convertible bonds for hedging
The following Arbitrage Master will describe in detail the hedging strategy of stock index futures and discount grade A
Overview of stock index futures
The so-called stock index futures are standardized futures contracts with a certain stock index as the underlying asset. The price quoted by the buyer and seller is the stock index price level after
a certain period of time. When the contract expires Later, stock index futures were delivered through cash settlement of the spread.
Compared with stock trading, there are many obvious differences between stock index futures trading:
(1) Stock index futures contracts have expiration dates and cannot be held indefinitelyNormally you can hold stocks after buying them, but stock index futures contracts have a definite expiration
date.Therefore, when trading stock index futures, you must pay attention to the contract expiration date to decide whether to close the position in advance or wait for the contract to expire for cash
(2) Stock index futures adopt margin tradingWhen trading stock index futures, investors do not need to pay the full amount of the contract value, but only need to pay a certain proportion of funds as
a performance guarantee. Currently, stock trading in my country requires the payment of the full amount of the stock value.Since stock index futures are margin transactions, the loss may even exceed
the investment principal, which is also different from stock trading.
(3) In terms of trading direction, stock index futures trading can be short-sellingStock index futures trading is a two-way transaction. You can buy first and then sell, or you can sell first and
then buy. However, the stock market in some countries does not have a short-selling mechanism. Stocks can only be bought first and then sold. Short selling is not allowed. At this time, stock trading
is One-way transaction
(4) In terms of settlement method, stock index futures trading adopts a same-day debt-free settlement system.In stock index futures trading, the exchange must settle the trading margin on the same
day. If the account margin balance is insufficient, it must be replenished within the specified time, otherwise the position may be forcibly closed.Stock transactions are full-price transactions,
which do not require additional funds from investors, and the book profits and losses are not settled after buying the stock and before selling it.
Stock index futures hedging method
The stock spot market and the stock index futures market are closely connectedAccording to the system design of stock index futures, the futures price will be equal to the price of the underlying
index in the spot market on the expiration date of the contract.However, in actual market conditions, the futures index price is often affected by a variety of factors and deviates from its
reasonable theoretical price. The price gap between the futures index and the spot index often appears to be too large or too small, resulting in a gap between the futures market and the spot market.
We call this kind of simultaneous trading across the futures market and the current market "futures and current arbitrage".
When the spot index is undervalued and the futures contract of a certain delivery month is overvalued, investors can sell the futures contract and buy spot at the same time to establish an arbitrage
position; when the gap between spot and futures prices tends to Normally, the futures contract is closed and the spot is sold at the same time to obtain arbitrage profits. This strategy is called
"forward arbitrage."In forward arbitrage, there are five options for buying spot:
First, buy ETF funds
For example, we want to arbitrage the price difference between the CSI 300 spot and stock index futures, but buying CSI 300 stocks is too troublesome. At this time, we can choose the CSI 300 ETF fund
as an alternative to the spotWhen we find that the deviation between the stock index futures and the spot exceeds the expected value, we buy the CSI 300 ETF fund and short the stock index futures
position. When the deviation between the two is reasonable, we sell the CSI 300 ETF and close the short position in the stock index futures.If the deviations never converge, then the deviations
between the two will inevitably converge until the settlement date.
The second is to buy spot stocks
There are liquidity restrictions in the CSI 300 ETF fund. If a large amount of funds is used for forward futures arbitrage, the brokerage can provide professional trading software and buy 300 CSI 300
constituent stocks at the same time to replace the CSI 300 ETF fund. , the closing strategy is the same as that of ETF funds
The third is to buy closed-end funds
There is a discount in closed-end funds, so using closed-end funds as spot money can simultaneously earn the difference between the discount of closed-end funds and the current spot price.Since there
are no closed-end funds that completely track the CSI 300 Index, this type of futures arbitrage cannot achieve complete hedging of positions. There may be deviations between the CSI 300 Index and the
closed-end funds.Therefore, in this type of arbitrage, the profit is not fixed, and there is even the possibility of loss.
The fourth is to buy graded funds
If there is an overall discount or overall premium in the tiered funds, you can use the tiered funds as spot goods, while doing discount arbitrage or premium arbitrage, while using stock index
futures short orders to protect
For hedging of stock index futures, the key is the premium and discount of the corresponding point of the futures.
Stock index futures premium structure
As shown in Table 8-1, the price of IF stock index futures contract is higher than the price of CSI 300 spot, which is called "futures index premium"
If the price of IF stock index futures contract is lower than the spot price of CSI 300, it is called "futures index discount", as shown in Table 8-2
Stock index futures discount structure
The reasonable state of stock index futures should be premium (excluding dividends), because investors can save money and interest by using CSI 300 futures to replace the CSI 300 spot. , so people
who invest in futures are willing to pay a certain premium to buyIn addition, stocks always rise in the long term. Considering the cost of capital and the long-term return rate of the stock market, I
think an annual premium of about 6% is appropriate.
Annualized premium and discount = premium and discount ratio × 12 months/(contract expiration date - current date)
But if the premium is very high, for example, higher than 20% (annualized), then hedging It makes sense, because the long-term return rate of the stock market will not be so high, and hedging is
equivalent to receiving a fixed annual return of 20%, which exceeds the upward slope of the index.
Stock Index Futures Hedging Case
Since December 2014, the CSI 300 Index has set off a major rise, and the stock index futures have an annualized rate of more than 30% relative to the CSI 300 spot.In terms of spot selection, I chose
tiered funds at that time.
The reason for choosing tiered funds is that tiers A and B are in a state of super discount at this time. Hedging with short orders in stock index futures can not only earn premium income, but also
earn discount arbitrage income from tiered funds.
The final result was very good. Not only did I earn premium income from futures, but I also earned arbitrage income from spot prices.
Thoughts on the high premium of stock index futures
When I chose to hedge against short orders in stock index futures, there were many objections. The most typical voice was that "stock index futures are the weather vane of spot prices." They believed
that futures prices were high. Water, the spot price will definitely rise sharply, and if you hedge, you will lose the opportunity to make money.Does the premium of stock index futures really mean
that the spot price will definitely rise in the future? At the end of 2014, I wrote an article and shared it with you
Stock index futures should have a reasonable premium relative to spot prices. How high should this premium be? Roughly equivalent to the risk-free rate
Assuming that an investor wants to invest in the CSI 300 Index, he can buy the CSI 300 ETF fund with 1 million yuan, or he can buy 1 futures index with 150,000 yuan, and then use the remaining
850,000 yuan to buy bonds, etc. low risk productsIf the stock index futures and the CSI 300 spot are equal, then rational investors will choose to buy futures + bonds instead of the CSI 300 index
Expressed as an equation:
300ETF=0.15×long futures index +0.85×bond
If the risk-free interest rate is 6%, then the futures index has a reasonable premium. According to the current Shanghai and Shenzhen stock exchange rates, The 300 index is around 3400 points, and
the reasonable premium for the month is around 15 points.
In the past two weeks, the premium of the futures index has increased to 100 points, which is converted into a risk-free interest rate of about 30% to 40%. This will inevitably attract a large number
of arbitrage orders. The arbitrage model is: buy N× 1 million 300 ETFs, and sell N hands of futures short positions to hedgeCurrently, you can choose IF1501 short order. This Friday, the price
premium will still be 120 points, which is equivalent to a risk-free return of 3% a month.
Now comes the problem. A large number of arbitrage orders to buy 300 ETF, or directly buy CSI 300 constituent stocks will cause the CSI 300 index to rise sharply, futures bulls will make profits, and
more arbitrage orders will enter. , the CSI 300 Index continues to rise
This is a positive feedback process, that is, the more arbitrage orders there are, the faster the CSI 300 Index will rise.Therefore, in the past two weeks, the CSI 300 Index has been stronger than
any other index, and the 28-20 split is very obvious.
If you think about it carefully, in fact, those of us who do futures arbitrage are lending money to those who are long on the CSI 300 Index at a monthly interest rate of 3%, so in essence we still
rely on leverage to rise.
Think one more step. In any stock market, borrowing money to trade stocks with a monthly interest rate of 3% will eventually lead to a dead end.Then when the premium of stock index futures converges
or even turns into a discount, the arbitrage order will inevitably close the position, that is, sell 300 ETFs or 300 heavyweight stocks, and close the short order.So at this time, the CSI 300 Index
will be under heavy selling pressure. If the selling pressure is heavy enough and triggers the selling of margin trading, what will be the result?
The final result was as expected. The CSI 300 Index suffered two sharp dives. After the current index's rise converged, it resumed its upward trend.
Reprinted in "The Road to Low-Risk Investment (2nd Edition)" | {"url":"https://yourmoneydouble.com/blogs/reduce-investment-risk/how-to-arbitrage-in-stock-index-futures","timestamp":"2024-11-02T18:30:41Z","content_type":"text/html","content_length":"234795","record_id":"<urn:uuid:f3462815-3cd1-4d12-a0dc-462aea33bdfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00034.warc.gz"} |
What Does Arc Me-an in X Y?
Posted by | Mar 16, 2020 | Invitation to Bid | 0 |
In the event you need to understand what does arc me an in mathematics
It may also be utilised to specify the exact distance between 2 things. The unit’s definition has been in existence for quite a while also it is used in technical areas in addition to in math
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Units of measurement writing research papers are just that – there to give you something to work with. This would not be possible without measuring something and putting the measurement in terms of
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If you didn’t know that you can take all those units and also make them into a one-way partnership, then what is the idea? They are just numbers. The units are already there and also we just have to
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The units of measurement http://bestresearchpaper.com proved formulated for those who were using stone tools. There was A rock tool a tool that weighed lots. All these units were applied to quantify
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You might observe the similarities between this particular unit and the notion of radians, if you wanted to learn very well what does arc me-an in math afterward. The arc is your shortest distance
between 2 points onto a world or a ellipse. The number of amounts that these units could be quantified in is actually a different story nonetheless.
Units are not the same as units of mass. In geometry, the components of measure are generally called inches, feet, miles, etc.,. In certain instances, the units are separate inside the same unit.
If you would like to learn what exactly does arc me an in mathematics afterward you will require to find the answer to this question,”Exactly what does arc me-an in math?” https://
digital.library.unt.edu/ark:/67531/metadc1036213/m2/1/high_res_d/4594140.pdf Once you find out the size of the arc, then you will possess the answer.
Units are available in lots of distinctive things and it’s all in accordance with that which exactly they symbolize. You should always don’t forget that every unit is different from the others and
you can’t utilize you to compare exactly the other. So as to find the most suitable solution, Every unit must be contrasted to the other components.
Units come in all different sizes. This is why some people confuse one measurement unit for another. It is all relative and each specific measurement has a certain size that has to be compared to the
other units.
You can find units that are not founded on size whatsoever. As an example, we will get all different units of this circumference of a circle. Each circle has got its measurements that are exact,
nonetheless it’s important to bear in mind that the shapes of the ring each possess their particular dimensions.
These are the questions that you should ask yourself whenever you’re currently working on this machine. It’s crucial you can figure out the replies for your math 22, to know the components. The
components are also the real key to solving a number of one’s problems. Those that are complicated such as for example differential equations are not solved minus the employment of their units.
In which you ought to figure out just how exactly does arc means in mathematics, the very next time you are focusing over a challenge, you will be able to quickly and easily discover the solution. As
you have the units on your grasp. Therefore you may readily figure the clear answer out. | {"url":"https://ppmu.bohol.gov.ph/what-does-arc-me-an-in-x-y-2","timestamp":"2024-11-06T12:14:30Z","content_type":"text/html","content_length":"117528","record_id":"<urn:uuid:4b6c2d5b-4f0c-41da-8d56-cb8fe740ba0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00733.warc.gz"} |
Test your understanding of probability and the 0-1 probability scale with this self marking quiz.
This is level 1: calculating the probability of a single event occuring. Type in your answers as fractions in their lowest terms. You can earn a trophy if you get at least 7 questions correct.
There are 7 balls on a snooker table. Three of the balls are red, one is blue, one pink, one yellow and one white.
If one ball is chosen at random, what is the probability it is yellow?
If one ball is chosen at random, what is the probability it is red?
If one ball is chosen at random, what is the probability it is not white?
What is the probability that the number of candles on the cake is less than five?
What is the probability that the card is a Queen?
There are nine counters in a bag numbered from one to nine. If a counter is drawn at random from the bag what is the probability that the number on the counter is not a multiple of three?
United and City are playing each other in the final. It has been calculated that the probability of United winning is three fifths. What is the probability of United not winning?
© Transum Mathematics 1997-2024
Scan the QR code below to visit the online version of this activity.
Description of Levels
Level 1 - Calculating the probability of a single event occuring
Level 2 - Calculating the probability of a single event occuring from a two-way table
Level 3 - Calculating the probability of a single event occuring then deciding the expected number of times it will occur in a given number of trials
Level 4 - Finding the probability of a combined event occuring by drawing a table to represent the possibility space
Level 5 - Calculating probability of an event occuring by listing all the possible outcomes
Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent.
Curriculum Reference
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent.
If you are not familiar with standard playing cards here is a guide.
There are 52 cards in a pack (usually called a deck). The cards are grouped into four suits of 13 cards each. these suits are called clubs, diamonds, hearts and spades. Click on a tab below to see
the cards in each of these suits.
Hearts ♥ Diamonds ♦ Clubs ♣ Spades ♠
Hearts ♥ Ace of Hearts
Two of Hearts Three of Hearts Four of Hearts Five of Hearts
Six of Hearts Seven of Hearts Eight of Hearts Nine of Hearts
Ten of Hearts Jack of Hearts Queen of Hearts King of Hearts | {"url":"https://www.transum.org/Software/Online_Exercise/Probability/","timestamp":"2024-11-10T11:53:10Z","content_type":"text/html","content_length":"60249","record_id":"<urn:uuid:2abd06d1-7e86-42fd-9e83-45067e6dda2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00865.warc.gz"} |
Graph theory
Graph theory lecture + practice, 2024/2025 autumn
Tuesday 13:00-16:00, Grünwald Géza room
Practice: There will be 2 tests: on 22nd October and on 10th December. In the first week of the exam period, one of the midterms can be retaken on request by the student. Additionally, bonus points
can be earned during the semester.
Lecture: During the exam period, written exams will be announced to test the students' knowledge. Additionally, bonus points can be earned during the semester.
Grades (for both courses):
0% – 50%: fail (1)
51% – 62%: pass (2)
63% – 75%: satisfactory (3)
76% – 87%: good (4)
88% – 100%: excellent (5).
1. Basics (presentation). Multigraphs and (simple) graphs. Loops and parallel edges. Degrees. Handshake lemma. Complete graph. Complement of a graph. Isomorphic graphs.
2. Graph realization problem (presentation). Degree sequences. Realization by multigraphs. Realization by loopless multigraphs (without a proof). Realization by simple graphs: Havel—Hakimi-lemma (and
3. Connectivity and trees (presentation). Vertex removal, edge removal. Sub(multi)graphs, spanning and induced subgraphs. Walks, trails, paths, cycles. Connected and disconnected graphs. Connected
components. Trees. Leaves, number of edges in a tree. Rooted trees. Spanning trees.
4. Graph algorithms. Breadth-first transversal, depth-first transversal. Kruskal's algorithm for finding minimum cost spanning tree. Dijkstra's algorithm for finding shortest path in weighted graphs.
5. Network flows. Network, feasible flows. Value of a flow. Cuts. Capacity of a cut, determining flow value using a cut. Maximum flow, minimum cut theorem. Augmenting paths. Finding a feasible flow
with greater flow value using an augmenting path. Ford—Fulkerson-algorithm.
6. Applications of network flows. Project selection problem.
7. Matchings (presentation). Matchings, and the parameter ν(G). Matchings in bipartite graphs: Kőnig sets and the marriage theorem, Hungarian method + worked-out example (in Hungarian).
8. Vertex coloring. Proper vertex coloring, chromatic number. Cliques and the parameter ω(G). Lower bound for χ(G) in terms of ω(G). Upper bound for χ(G) in terms of maximum degree, greedy coloring
9. Eulerian trail, Hamiltonian path, Hamiltonian cycle. Eulerian trail. Euler's theorem. Hamiltonian path and cycle. Dirac's theorem (without a proof). A method for proving the non-existence of
Hamiltonian path / cycle.
The proofs are needed in the same detail (no proof / partial proof / sketch of the proof / full proof etc.) as it was presented in the lecture.
Sample 1st test
Sample exam #1
Sample exam #2
1. Basics. Realization of degree sequences.
2. Connectivity. Trees
3. Basic graph algorithms
4. Network flows
5. Applications of network flows
6. Matchings in bipartite graphs
Components: #1 (a graph with 4 components), #2 (a graph with 3 components), #3 (a graph with 3 components).
Tree: #1, #2. Spanning tree: #1, #2.
Tree traversals: breadth-first search, depth-first search.
Matchings: #1 (non-perfect), #2 (perfect), #3 (matching in a bipartite graph), #4 (matching in a bipartite graph that covers A), #5 (perfect matching in a bipartite graph). Kőnig set: #1, #2.
Proper (vertex) coloring: #1, #2. Four color map problem / Four color theorem #1, #2.
Edge coloring: #1, #2.
Eulerian trails: #1 (closed Eulerian trail), #2 (closed Eulerian trail), #3 (non-closed Eulerian trail), #4 (non-closed Eulerian trail).
Hamiltonian path: #1, #2. Hamiltonian cycle: #1, #2, #3.
These pictures were mostly collected from the web (see the URLs for the source).
Béla Csaba, Péter Hajnal, Gábor V. Nagy: Graph theory for MSc students in computer science (2019) [Note, textbook]
Douglas B. West: Introduction to Graph Theory (Prentice Hall)
Reinhard Diestel: Graph Theory (Springer-Verlag, free online version)
Benny Sudakov: Graph Theory (electronic lecture notes) | {"url":"http://www.math.u-szeged.hu/~ngaba/graph/","timestamp":"2024-11-14T13:42:31Z","content_type":"text/html","content_length":"10633","record_id":"<urn:uuid:60f11a4a-de4c-4e06-9665-87ef707d7d7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00297.warc.gz"} |
Symmetric designs, difference sets, and autocorrelations of finite binary sequences
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Symmetric designs with parameters [...] are very regular structures useful in the design of experiments, and [...]-difference sets are a common means of constructing them, as well as being
interesting subsets of groups in their own right. We investigate [...]-symmetric designs and [...]-difference sets, especially those satisfying v = 4n + 1, where n = k [...]. These must have
parameters of the form [...], [...], [...] for [...]. Such difference sets exist for t = 1, 2. Generalizing work of M. Kervaire and S. Eliahou, we conjecture that abelian difference sets with these
parameters do not exist for [...], and we prove this for large families of values of t (or n). In particular, we eliminate all values of t except for a set of density 0. The theory of biquadratic
reciprocity is especially useful here, to determine whether or not certain primes are biquadratic, and hence semi-primitive, modulo the factors of v.
Cyclic difference sets also correspond to [...]-sequences of length v with constant periodic autocorrelation. Such sequences are of interest in communications theory, especially if they also have
small aperiodic correlations. We find and prove bounds on the aperiodic correlations of a binary sequence that has constant periodic correlations. As an example, such sequences corresponding to
cyclic difference sets satisfying v = 4n have aperiodic autocorrelations bounded absolutely by (3/2)n, with a similar bound in the case of cyclic Hadamard difference sets (those satisfying v = 4n -
Finally, we present an alternative construction of a class of symmetric designs due to A.E. Brouwer which includes the v = 4n + 1 designs as a special case.
Item Type: Thesis (Dissertation (Ph.D.))
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Mathematics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Wilson, Richard M.
Thesis Committee: • Wilson, Richard M. (chair)
• Wales, David B.
• Luxemburg, W. A. J.
• Doran, William
Defense Date: 23 May 1995
Record Number: CaltechETD:etd-09052007-084407
Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09052007-084407
DOI: 10.7907/m1he-d221
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 3338
Collection: CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On: 11 Sep 2007
Last Modified: 16 Apr 2021 23:27
Thesis Files
PDF (Broughton_wj_1995.pdf) - Final Version
See Usage Policy.
Repository Staff Only: item control page | {"url":"https://thesis.library.caltech.edu/3338/","timestamp":"2024-11-08T04:40:20Z","content_type":"application/xhtml+xml","content_length":"27474","record_id":"<urn:uuid:c20757a5-e1bb-498f-92a3-1e48fd87c2b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00214.warc.gz"} |
Decision Trees: Theory and Implementation
Decision Trees: Decision Trees are the Supervised Machine learning algorithm that can be used for Classification and Regression problems. A classification problem identifies the set of categories or
groups to which an observation belongs. A Decision Tree uses a tree or graph-like model of decision. Each internal node represents a "test" on attributes, and each branch represents the outcome of
the test. Each leaf node represents a class label (decision taken after computing all features).
Common Examples:
Let's draw a Decision Tree for calculating the XNOR table. The table is as follows:
INPUT-1 INPUT-2 OUTPUT
X1 X2 Y
Here we have three features, INPUT-1, INPUT-2, and OUTPUT.
Let's make the feature INPUT-1 the root node and build a Decision Tree.
Source: alliance.seas.uppnn.edu
Here we have split our features at each level depending upon the requirement, and at last, we have final decisions. In actual practice, the data is not as simple as this. The complexity of trees
would increase as the data tend towards more practical real-world examples. Let's have a look at a similar kind of example:
Suppose we have to predict whether a student has been invited for an interview or not. We have three features, college type, is-intern, and has-project, and based on these, we have to make the
Decision Tree.
TIER-1/TIER-2 TIER-3 IS-INTERN HAS-PROJECT GOT A CALL
YES YES YES YES
YES YES NO YES
YES NO YES YES
YES NO NO NO
YES YES NO NO
YES NO YES NO
YES YES YES YES
In the Machine Learning algorithm, we try to find a nearly accurate result and not the exact one. The task would be very complex if we train the algorithm to get the precise result every time.
Similarly, the best version of the Decision Tree is very expensive and computationally impossible to build. So we try to develop a perfect Decision Tree that will give decent accuracy and be much
easier to make. According to the above Decision Tree, if a student is from tier-3 college and has a fantastic internship(say Google) but doesn't have any project, they will not get an interview call,
which is practically not feasible. Mistakes will come in the way when we have exceptions that don't match the data upon which the Decision Tree is built.
Mathematics behind Decision Tree:
Following are the steps that we need to follow to build a very good Decision Tree:-
1) We need to select the target attribute/feature from the given attributes. In the above example, the target attribute is “GOT A CALL,” which has two classes of values, either yes or no.
2) Calculate the information gain of the selected target attribute
Information Gain = -PP+N log2(PP+N) - NP+N log2(NP+N)
Where P = number of the elements in class 1 of the target attribute.
N = number of the elements in class 2 of the target attribute.
In our case, the target attribute has two classes, “YES” and “NO”. So, P=5, and N=5.
3) Now for the remaining features/attributes.
1. We calculate the entropy for all the Categorical variables.
2. We take average information entropy for the current attribute.
The entropy of a given attribute is given by:
Entropy = i=1v Information gain of that i th categorical variable x Probability of i th categorical variable
Here 'V is the number of categorical variables in that attribute.
For example, if the attribute is "IS-INTERN," then it has two categorical variables, "YES" and "NO."
I.G of “YES” = -47log2(47) say E1
I.G of “NO” = -37log2(37) say E2
The entropy of the attribute “IS-INTERN” = E1+E2
4) Wel will now find the Gain of each attribute (except the target attribute).
Gain of Individual attribute = Information Gain of the target attribute - Entropy of that attribute.
5) The attribute having maximum Gain value will be our Root node. We will divide the root node into subsets that contain possible values for the best feature.
6) Recursively make new decision trees using the subsets of the dataset created in step 5, till the point where we don't have any features to split upon. | {"url":"https://www.naukri.com/code360/library/decision-trees-theory-and-implementation","timestamp":"2024-11-08T10:56:05Z","content_type":"text/html","content_length":"413387","record_id":"<urn:uuid:b4716aa6-1db4-4ab3-8200-8962e4388df5>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00375.warc.gz"} |
Consider a set S of integers from 1 to 250. A denotes the set of integers divisible by 3,8 denotes the set of integersdivisible by
Consider a set S of integers from 1 to 250. A denotes the set of integers divisible by 3,8 denotes the set of integersdivisible by 5 and C denotes the set of integers divisible by 7 then
1 thought on “Consider a set S of integers from 1 to 250. A denotes the set of integers divisible by 3,8 denotes the set of integersdivisible by”
1. Answer:
sorry I don’t know this answer
Leave a Comment | {"url":"https://wiki-helper.com/consider-a-set-s-of-integers-from-1-to-250-a-denotes-the-set-of-integers-divisible-by-3-8-denote-39735540-6/","timestamp":"2024-11-12T20:48:19Z","content_type":"text/html","content_length":"126240","record_id":"<urn:uuid:87df7eb7-750d-4571-9aa7-1b64826438bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00596.warc.gz"} |
Verifiable Distributed Aggregation Functions
Internet-Draft VDAF November 2023
Barnes, et al. Expires 23 May 2024 [Page]
Intended Status:
Verifiable Distributed Aggregation Functions
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multi-party protocols for computing aggregate statistics over user measurements. These protocols are designed
to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers
to detect if a malicious or misconfigured client submitted an measurement that would result in an invalid aggregate result.¶
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current
Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as
reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 23 May 2024.¶
Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License
text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
[TO BE REMOVED BY RFC EDITOR: The source for this draft and and the reference implementation can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf.]¶
The ubiquity of the Internet makes it an ideal platform for measurement of large-scale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial
overlap, however, between information that is valuable to measure and information that users consider private.¶
For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to
guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition.¶
In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given
disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy.¶
Most prior approaches to this problem fall under the rubric of "differential privacy (DP)" [Dwo06]. Roughly speaking, a data aggregation system that is differentially private ensures that the degree
to which any individual measurement influences the value of the aggregate result can be precisely controlled. For example, in systems like RAPPOR [EPK14], each user samples noise from a well-known
distribution and adds it to their measurement before submitting to the aggregation server. The aggregation server then adds up the noisy measurements, and because it knows the distribution from
whence the noise was sampled, it can estimate the true sum with reasonable precision.¶
Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically,
depending on the "amount" of noise a client adds to its measurement, it may be possible for a curious aggregator to make a reasonable guess of the measurement's true value. Indeed, the more noise the
clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility.¶
The ideal goal for a privacy-preserving measurement system is that of secure multi-party computation (MPC): No participant in the protocol should learn anything about an individual measurement beyond
what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal.¶
VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of non-colluding aggregation servers. As long as a subset of the servers executes the protocol
honestly, VDAFs guarantee that no measurement is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed measurements
that would otherwise garble the result of the computation can be detected and removed from the set of measurements. We refer to this property as "robustness".¶
In addition to these MPC-style security goals of privacy and robustness, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the
aggregate result itself does not leak too much information about any one measurement.¶
• TODO(issue #94) Provide guidance for local and central DP and point to it here.¶
The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees.
Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems
supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research
Group (ISRG), Google, Apple, and others [ENPA].¶
The VDAF abstraction laid out in Section 5 represents a class of multi-party protocols for privacy-preserving measurement proposed in the literature. These protocols vary in their operational and
security requirements, sometimes in subtle but consequential ways. This document therefore has two important goals:¶
1. Providing higher-level protocols like [DAP] with a simple, uniform interface for accessing privacy-preserving measurement schemes, documenting relevant operational and security requirements, and
specifying constraints for safe usage:¶
1. General patterns of communications among the various actors involved in the system (clients, aggregation servers, and the collector of the aggregate result);¶
2. Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and¶
3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
2. Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions.¶
This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶
• The aforementioned Prio system [CGB17] allows for the privacy-preserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶
1. Each client shards its measurement into a sequence of additive shares and distributes the shares among the aggregation servers.¶
2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate.¶
3. Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result.¶
The difficult part of this system is ensuring that the servers hold shares of a valid, aggregatable value, e.g., the measurement is an integer in a specific range. Thus Prio specifies a
multi-party protocol for accomplishing this task.¶
In Section 7 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant
performance gains.¶
• More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar for solving the t-heavy-hitters problem in a privacy-preserving manner. Here each client holds a bit-string of length n,
and the goal of the aggregation servers is to compute the set of strings that occur at least t times. The core primitive used in their protocol is a specialized Distributed Point Function (DPF) [
GI14] that allows the servers to "query" their DPF shares on any bit-string of length shorter than or equal to n. As a result of this query, each of the servers has an additive share of a bit
indicating whether the string is a prefix of the client's string. The protocol also specifies a multi-party computation for verifying that at most one string among a set of candidates is a prefix
of the client's string.¶
In Section 8 we describe a VDAF called Poplar1 that implements this functionality.¶
Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the client-generated measurements are prepared for aggregation. Because these constructions are
based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this
process may require multiple round trips over the network.¶
There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also
defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.¶
• OPEN ISSUE Decide if we should give one or two example DAFs. There are natural variants of Prio3 and Poplar1 that might be worth describing.¶
The remainder of this document is organized as follows: Section 3 gives a brief overview of DAFs and VDAFs; Section 4 defines the syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6
defines various functionalities that are common to our constructions; Section 7 describes the Prio3 construction; Section 8 describes the Poplar1 construction; and Section 9 enumerates the security
considerations for VDAFs.¶
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described
in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is
usually handled by raising an exception.¶
A variable with type Bytes is a byte string. This document defines several byte-string constants. When comprised of printable ASCII characters, they are written as Python 3 byte-string literals
(e.g., b'some constant string').¶
A global constant VERSION of type Unsigned is defined, which algorithms are free to use as desired. Its value SHALL be 8.¶
This document describes algorithms for multi-party computations in which the parties typically communicate over a network. Wherever a quantity is defined that must be be transmitted from one party to
another, this document prescribes a particular encoding of that quantity as a byte string.¶
• OPEN ISSUE It might be better to not be prescriptive about how quantities are encoded on the wire. See issue #58.¶
Some common functionalities:¶
+---->| Aggregator 0 |----+
| +--------------+ |
| ^ |
| | |
| V |
| +--------------+ |
| +-->| Aggregator 1 |--+ |
| | +--------------+ | |
+--------+-+ | ^ | +->+-----------+
| Client |---+ | +--->| Collector |--> Aggregate
+--------+-+ +->+-----------+
| ... |
| |
| | |
| V |
| +----------------+ |
+--->| Aggregator N-1 |---+
Input shares Aggregate shares
Figure 1: Overall data flow of a (V)DAF
In a DAF- or VDAF-based private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶
• To submit an individual measurement, the Client shards the measurement into "input shares" and sends one input share to each Aggregator. We sometimes refer to this sequence of input shares
collectively as the Client's "report".¶
• The Aggregators refine their input shares into "output shares".¶
□ Output shares are in one-to-one correspondence with the input shares.¶
□ Just as each Aggregator receives one input share of each measurement, if this process succeeds, then each aggregator holds one output share.¶
□ In VDAFs, Aggregators will need to exchange information among themselves as part of the validation process.¶
• Each Aggregator combines the output shares in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result.¶
• The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch.¶
Aggregators are a new class of actor relative to traditional measurement systems where Clients submit measurements to a single server. They are critical for both the privacy properties of the system
and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by non-collusion among Aggregators, and Aggregators are the entities that
perform validation of Client measurements. Thus Clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to
correctly run the protocol.¶
Within the bounds of the non-collusion requirements of a given (V)DAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an
Aggregator, effectively using the other Aggregator(s) to augment a basic client-server protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to the higher-level protocol making use of the (V)DAF to arrange for the required information to be
delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting
measurements may be anonymous.¶
By way of a gentle introduction to VDAFs, this section describes a simpler class of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs do not provide verifiability of the
computation. Clients must therefore be trusted to compute their input shares correctly. Because of this fact, the use of a DAF is NOT RECOMMENDED for most applications. See Section 9 for additional
A DAF scheme is used to compute a particular "aggregation function" over a set of measurements generated by Clients. Depending on the aggregation function, the Collector might select an "aggregation
parameter" and disseminates it to the Aggregators. The semantics of this parameter is specific to the aggregation function, but in general it is used to represent the set of "queries" that can be
made on the measurement set. For example, the aggregation parameter is used to represent the candidate prefixes in Poplar1 Section 8.¶
Execution of a DAF has four distinct stages:¶
Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶
A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF
defines the associated constants and types enumerated in the following table.¶
Table 1: Constants and types defined by each concrete DAF.
Parameter Description
ID Algorithm identifier for this DAF. A 32-bit, unsigned integer.
SHARES Number of input shares into which each measurement is sharded.
NONCE_SIZE Size of the nonce passed by the application.
RAND_SIZE Size of the random byte string passed to sharding algorithm.
Measurement Type of each measurement.
PublicShare Type of each public share.
InputShare Type of each input share.
AggParam Type of aggregation parameter.
OutShare Type of each output share.
AggShare Type of the aggregate share.
AggResult Type of the aggregate result.
These types define the inputs and outputs of DAF methods at various stages of the computation. Some of these values need to be written to the network in order to carry out the computation. In
particular, it is RECOMMENDED that concrete instantiations of the Daf interface specify a method of encoding the PublicShare, InputShare, and AggShare.¶
Each DAF is identified by a unique, 32-bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 17.¶
In order to protect the privacy of its measurements, a DAF Client shards its measurements into a sequence of input shares. The shard method is used for this purpose.¶
• Daf.shard(measurement: Measurement, nonce: bytes[Daf.NONCE_SIZE], rand: bytes[Daf.RAND_SIZE]) -> tuple[PublicShare, list[InputShare]] is the randomized sharding algorithm run by each Client. The
input rand consists of the random bytes consumed by the algorithm. This value MUST be generated using a cryptographically secure pseudorandom number generator (CSPRNG). It consumes the
measurement and produces a "public share", distributed to each of the Aggregators, and a corresponding sequence of input shares, one for each Aggregator. The length of the output vector MUST be
Once an Aggregator has received the public share and one of the input shares, the next step is to prepare the input share for aggregation. This is accomplished using the following algorithm:¶
• Daf.prep(agg_id: Unsigned, agg_param: AggParam, nonce: bytes[NONCE_SIZE], public_share: PublicShare, input_share: InputShare) -> OutShare is the deterministic preparation algorithm. It takes as
input the public share and one of the input shares generated by a Client, the Aggregator's unique identifier, the aggregation parameter selected by the Collector, and a nonce and returns an
output share.¶
The protocol in which the DAF is used MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of input_share in the sequence of input shares
output by the Client.¶
Concrete DAFs implementations MAY impose certain restrictions for input shares and aggregation parameters. Protocols using a DAF MUST ensure that for each input share and aggregation parameter
agg_param, Daf.prep is only called if Daf.is_valid(agg_param, previous_agg_params) returns True, where previous_agg_params contains all aggregation parameters that have previously been used with the
same input share.¶
DAFs MUST implement the following function:¶
• Daf.is_valid(agg_param: AggParam, previous_agg_params: set[AggParam]) -> Bool: Checks if the agg_param is compatible with all elements of previous_agg_params.¶
Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result:¶
• Daf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) -> AggShare is the deterministic aggregation algorithm. It is run by each Aggregator a set of recovered output shares.¶
Aggregator 0 Aggregator 1 Aggregator SHARES-1
============ ============ ===================
out_share_0_0 out_share_1_0 out_share_[SHARES-1]_0
out_share_0_1 out_share_1_1 out_share_[SHARES-1]_1
out_share_0_2 out_share_1_2 out_share_[SHARES-1]_2
... ... ...
out_share_0_B out_share_1_B out_share_[SHARES-1]_B
| | |
V V V
+-----------+ +-----------+ +-----------+
| aggregate | | aggregate | ... | aggregate |
+-----------+ +-----------+ +-----------+
| | |
V V V
agg_share_0 agg_share_1 agg_share_[SHARES-1]
Figure 3: Aggregation of output shares. `B` indicates the number of measurements in the batch.
For simplicity, we have written this algorithm in a "one-shot" form, where all output shares for a batch are provided at the same time. Many DAFs may also support a "streaming" form, where shares are
processed one at a time.¶
Implementation note: For most natural DAFs (and VDAFs) it is not necessary for an Aggregator to store all output shares individually before aggregating. Typically it is possible to merge output
shares into aggregate shares as they arrive, merge these into other aggregate shares, and so on. In particular, this is the case when the output shares are vectors over some finite field and
aggregating them involves merely adding up the vectors element-wise. Such is the case for Prio3 Section 7 and Poplar1 Section 8.¶
After the Aggregators have aggregated a sufficient number of output shares, each sends its aggregate share to the Collector, who runs the following algorithm to recover the following output:¶
• Daf.unshard(agg_param: AggParam, agg_shares: list[AggShare], num_measurements: Unsigned) -> AggResult is run by the Collector in order to compute the aggregate result from the Aggregators'
shares. The length of agg_shares MUST be SHARES. num_measurements is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic.¶
Aggregator 0 Aggregator 1 Aggregator SHARES-1
============ ============ ===================
agg_share_0 agg_share_1 agg_share_[SHARES-1]
| | |
V V V
| unshard |
Figure 4: Computation of the final aggregate result from aggregate shares.
• QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [DAP
].) Or is this out-of-scope of this document? See https://github.com/ietf-wg-ppm/ppm-specification/issues/19.¶
Securely executing a DAF involves emulating the following procedure.¶
def run_daf(Daf,
agg_param: Daf.AggParam,
measurements: list[Daf.Measurement],
nonces: list[bytes[Daf.NONCE_SIZE]]):
out_shares = [[] for j in range(Daf.SHARES)]
for (measurement, nonce) in zip(measurements, nonces):
# Each Client shards its measurement into input shares and
# distributes them among the Aggregators.
rand = gen_rand(Daf.RAND_SIZE)
(public_share, input_shares) = \
Daf.shard(measurement, nonce, rand)
# Each Aggregator prepares its input share for aggregation.
for j in range(Daf.SHARES):
Daf.prep(j, agg_param, nonce,
public_share, input_shares[j]))
# Each Aggregator aggregates its output shares into an aggregate
# share and sends it to the Collector.
agg_shares = []
for j in range(Daf.SHARES):
agg_share_j = Daf.aggregate(agg_param,
# Collector unshards the aggregate result.
num_measurements = len(measurements)
agg_result = Daf.unshard(agg_param, agg_shares,
return agg_result
Figure 5: Execution of a DAF.
The inputs to this procedure are the same as the aggregation function computed by the DAF: An aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a
"benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure point-to-point channels. In reality, these channels need to be instantiated
by some "wrapper protocol", such as [DAP], that realizes these channels using suitable cryptographic mechanisms. Moreover, some fraction of the Aggregators (or Clients) may be malicious and diverge
from their prescribed behaviors. Section 9 describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.¶
Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Client-generated measurements. Evaluation of a VDAF involves the same
four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the
computation's correctness. Accommodating this interaction will require syntactic changes.¶
Overall execution of a VDAF comprises the following stages:¶
In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: Verification of the validity of the recovered output shares. This process ensures that aggregating the output
shares will not lead to a garbled aggregate result.¶
The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF.¶
Table 2: Constants and types defined by each concrete VDAF.
Parameter Description
ID Algorithm identifier for this VDAF.
VERIFY_KEY_SIZE Size (in bytes) of the verification key (Section 5.2).
RAND_SIZE Size of the random byte string passed to sharding algorithm.
NONCE_SIZE Size (in bytes) of the nonce.
ROUNDS Number of rounds of communication during the Preparation stage (Section 5.2).
SHARES Number of input shares into which each measurement is sharded (Section 5.1).
Measurement Type of each measurement.
PublicShare Type of each public share.
InputShare Type of each input share.
AggParam Type of aggregation parameter.
OutShare Type of each output share.
AggShare Type of the aggregate share.
AggResult Type of the aggregate result.
PrepState Aggregator's state during preparation.
PrepShare Type of each prep share.
PrepMessage Type of each prep message.
Some of these values need to be written to the network in order to carry out the computation. In particular, it is RECOMMENDED that concrete instantiations of the Vdaf interface specify a method of
encoding the PublicShare, InputShare, AggShare, PrepShare, and PrepMessage.¶
Each VDAF is identified by a unique, 32-bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 17. The following method is defined for every VDAF:¶
def domain_separation_tag(Vdaf, usage: Unsigned) -> Bytes:
Format domain separation tag for this VDAF with the given usage.
return format_dst(0, Vdaf.ID, usage)
It is used to construct a domain separation tag for an instance of Xof used by the VDAF. (See Section 6.2.)¶
Sharding transforms a measurement into input shares as it does in DAFs (cf. Section 4.1); in addition, it takes a nonce as input and produces a public share:¶
• Vdaf.shard(measurement: Measurement, nonce: bytes[Vdaf.NONCE_SIZE], rand: bytes[Vdaf.RAND_SIZE]) -> tuple[PublicShare, list[InputShare]] is the randomized sharding algorithm run by each Client.
Input rand consists of the random bytes consumed by the algorithm. It consumes the measurement and the nonce and produces a public share, distributed to each of Aggregators, and the corresponding
sequence of input shares, one for each Aggregator. Depending on the VDAF, the input shares may encode additional information used to verify the recovered output shares (e.g., the "proof shares"
in Prio3 Section 7). The length of the output vector MUST be SHARES.¶
In order to ensure privacy of the measurement, the Client MUST generate the random bytes and nonce using a CSPRNG. (See Section 9 for details.)¶
To recover and verify output shares, the Aggregators interact with one another over ROUNDS rounds. Prior to each round, each Aggregator constructs an outbound message. Next, the sequence of outbound
messages is combined into a single message, called a "preparation message", or "prep message" for short. (Each of the outbound messages are called "preparation-message shares", or "prep shares" for
short.) Finally, the preparation message is distributed to the Aggregators to begin the next round.¶
An Aggregator begins the first round with its input share and it begins each subsequent round with the previous prep message. Its output in the last round is its output share and its output in each
of the preceding rounds is a prep share.¶
This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing the
measurement shares for aggregation.¶
Aggregator 0 Aggregator 1 Aggregator SHARES-1
============ ============ ===================
input_share_0 input_share_1 input_share_[SHARES-1]
| | ... |
V V V
+-----------+ +-----------+ +-----------+
| prep_init | | prep_init | | prep_init |
+-----------+ +------------+ +-----------+
| | ... |
V V V
+----------------------------------------------+ \
| prep_shares_to_prep | |
+----------------------------------------------+ |
| | ... | |
V V V | x ROUNDS
+-----------+ +-----------+ +-----------+ |
| prep_next | | prep_next | | prep_next | |
+-----------+ +-----------+ +-----------+ |
| | ... | |
V V V /
... ... ...
| | ... |
V V V
out_share_0 out_share_1 out_share_[SHARES-1]
Figure 6: VDAF preparation process on the input shares for a single measurement. At the end of the computation, each Aggregator holds an output share or an error.
To facilitate the preparation process, a concrete VDAF implements the following methods:¶
• Vdaf.prep_init(verify_key: bytes[Vdaf.VERIFY_KEY_SIZE], agg_id: Unsigned, agg_param: AggParam, nonce: bytes[Vdaf.NONCE_SIZE], public_share: PublicShare, input_share: InputShare) -> tuple
[PrepState, PrepShare] is the deterministic preparation-state initialization algorithm run by each Aggregator to begin processing its input share into an output share. Its inputs are the shared
verification key (verify_key), the Aggregator's unique identifier (agg_id), the aggregation parameter (agg_param), the nonce provided by the environment (nonce, see Figure 7), the public share
(public_share), and one of the input shares generated by the Client (input_share). Its output is the Aggregator's initial preparation state and initial prep share.¶
It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key prior to generating and processing reports. (See Section 9 for details.)¶
Protocols using the VDAF MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of input_share in the sequence of input shares output by
the Client.¶
Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1.¶
• Vdaf.prep_next(prep_state: PrepState, prep_msg: PrepMessage) -> Union[tuple[PrepState, PrepShare], OutShare] is the deterministic preparation-state update algorithm run by each Aggregator. It
updates the Aggregator's preparation state (prep_state) and returns either its next preparation state and its message share for the current round or, if this is the last round, its output share.
An exception is raised if a valid output share could not be recovered. The input of this algorithm is the inbound preparation message.¶
• Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: list[PrepShare]) -> PrepMessage is the deterministic preparation-message pre-processing algorithm. It combines the prep shares generated
by the Aggregators in the previous round into the prep message consumed by each in the next round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS states. The Aggregator enters the first state on using the initialization algorithm, and the update algorithm advances the
Aggregator to the next state. Thus, in addition to defining the number of rounds (ROUNDS), a VDAF instance defines the state of the Aggregator after each round.¶
• TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class.¶
The preparation-state update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that
an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators
recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like
Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
Note that it is possible for a VDAF to specify ROUNDS == 0, in which case each Aggregator runs the preparation-state update algorithm once and immediately recovers its output share without
interacting with the other Aggregators. However, most, if not all, constructions will require some amount of interaction in order to ensure validity of the output shares (while also maintaining
• OPEN ISSUE accommodating 0-round VDAFs may require syntax changes if, for example, public keys are required. On the other hand, we could consider defining this class of schemes as a different
primitive. See issue#77.¶
Similar to DAFs (see Section 4.3), VDAFs MAY impose restrictions for input shares and aggregation parameters. Protocols using a VDAF MUST ensure that for each input share and aggregation parameter
agg_param, the preparation phase (including Vdaf.prep_init, Vdaf.prep_next, and Vdaf.prep_shares_to_prep; see Section 5.2) is only called if Vdaf.is_valid(agg_param, previous_agg_params) returns
True, where previous_agg_params contains all aggregation parameters that have previously been used with the same input share.¶
VDAFs MUST implement the following function:¶
• Vdaf.is_valid(agg_param: AggParam, previous_agg_params: set[AggParam]) -> Bool: Checks if the agg_param is compatible with all elements of previous_agg_params.¶
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):¶
• Vdaf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) -> AggShare is the deterministic aggregation algorithm. It is run by each Aggregator over the output shares it has computed for a
batch of measurements.¶
The data flow for this stage is illustrated in Figure 3. Here again, we have the aggregation algorithm in a "one-shot" form, where all shares for a batch are provided at the same time. VDAFs
typically also support a "streaming" form, where shares are processed one at a time.¶
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):¶
• Vdaf.unshard(agg_param: AggParam, agg_shares: list[AggShare], num_measurements: Unsigned) -> AggResult is run by the Collector in order to compute the aggregate result from the Aggregators'
shares. The length of agg_shares MUST be SHARES. num_measurements is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic.¶
The data flow for this stage is illustrated in Figure 4.¶
Secure execution of a VDAF involves simulating the following procedure.¶
def run_vdaf(Vdaf,
verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
agg_param: Vdaf.AggParam,
nonces: list[bytes[Vdaf.NONCE_SIZE]],
measurements: list[Vdaf.Measurement]):
out_shares = []
for (nonce, measurement) in zip(nonces, measurements):
# Each Client shards its measurement into input shares.
rand = gen_rand(Vdaf.RAND_SIZE)
(public_share, input_shares) = \
Vdaf.shard(measurement, nonce, rand)
# Each Aggregator initializes its preparation state.
prep_states = []
outbound = []
for j in range(Vdaf.SHARES):
(state, share) = Vdaf.prep_init(verify_key, j,
# Aggregators recover their output shares.
for i in range(Vdaf.ROUNDS-1):
prep_msg = Vdaf.prep_shares_to_prep(agg_param,
outbound = []
for j in range(Vdaf.SHARES):
out = Vdaf.prep_next(prep_states[j], prep_msg)
(prep_states[j], out) = out
# The final outputs of the prepare phase are the output shares.
prep_msg = Vdaf.prep_shares_to_prep(agg_param,
outbound = []
for j in range(Vdaf.SHARES):
out_share = Vdaf.prep_next(prep_states[j], prep_msg)
# Each Aggregator aggregates its output shares into an
# aggregate share. In a distributed VDAF computation, the
# aggregate shares are sent over the network.
agg_shares = []
for j in range(Vdaf.SHARES):
out_shares_j = [out[j] for out in out_shares]
agg_share_j = Vdaf.aggregate(agg_param, out_shares_j)
# Collector unshards the aggregate.
num_measurements = len(measurements)
agg_result = Vdaf.unshard(agg_param, agg_shares,
return agg_result
Figure 7: Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter, a list of measurements, and a nonce for each measurement. This document does not specify how the nonces are chosen, but security requires
that the nonces be unique. See Section 9 for details. As explained in Section 4.6, the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol
In each round of preparation, each Aggregator writes a prep share to some broadcast channel, which is then processed into the prep message using the public prep_shares_to_prep() algorithm and
broadcast to the Aggregators to start the next round. In this section we describe some approaches to realizing this broadcast channel functionality in protocols that use VDAFs.¶
The state machine of each Aggregator is shown in Figure 8.¶
State transitions are made when the state is acted upon by the host's local inputs and/or messages sent by the peers. The initial state is Start. The terminal states are Rejected, which indicates
that the report cannot be processed any further, and Finished(out_share), which indicates that the Aggregator has recovered an output share out_share.¶
class State:
class Start(State):
class Continued(State):
def __init__(self, prep_state):
self.prep_state = prep_state
class Finished(State):
def __init__(self, output_share):
self.output_share = output_share
class Rejected(State):
def __init__(self):
Note that there is no representation of the Start state as it is never instantiated in the ping-pong topology.¶
For convenience, the methods described in this section are defined in terms of opaque byte strings. A compatible Vdaf MUST specify methods for encoding public shares, input shares, prep shares, prep
messages, and aggregation parameters. Minimally:¶
Implementations of Prio3 and Poplar1 MUST use the encoding scheme specified in Section 7.2.7 and Section 8.2.6 respectively.¶
For VDAFs with precisely two Aggregators (i.e., Vdaf.SHARES == 2), the following "ping pong" communication pattern can be used. It is compatible with any request/response transport protocol, such as
Let us call the initiating party the "Leader" and the responding party the "Helper". The high-level idea is that the Leader and Helper will take turns running the computation locally until input from
their peer is required:¶
• For a 1-round VDAF (e.g., Prio3 (Section 7)), the Leader sends its prep share to the Helper, who computes the prep message locally, computes its output share, then sends the prep message to the
Leader. Preparation requires just one round trip between the Leader and the Helper.¶
• For a 2-round VDAF (e.g., Poplar1 (Section 8)), the Leader sends its first-round prep share to the Helper, who replies with the first-round prep message and its second-round prep share. In the
next request, the Leader computes its second-round prep share locally, computes its output share, and sends the second-round prep message to the Helper. Finally, the Helper computes its own
output share.¶
• In general, each request includes the Leader's prep share for the previous round and/or the prep message for the current round; correspondingly, each response consists of the prep message for the
current round and the Helper's prep share for the next round.¶
The Aggregators proceed in this ping-ponging fashion until a step of the computation fails (indicating the report is invalid and should be rejected) or preparation is completed. All told there there
are ceil((Vdaf.ROUNDS+1)/2) requests sent.¶
Each message in the ping-pong protocol is structured as follows (expressed in TLS syntax as defined in Section 3 of [RFC8446]):¶
enum {
} MessageType;
struct {
MessageType type;
select (Message.type) {
case initialize:
opaque prep_share<0..2^32-1>;
case continue:
opaque prep_msg<0..2^32-1>;
opaque prep_share<0..2^32-1>;
case finish:
opaque prep_msg<0..2^32-1>;
} Message;
These messages are used to transition between the states described in Section 5.7. They are encoded and decoded to or from byte buffers as described Section 3 of [RFC8446]) using the following
• encode_ping_pong_message(message: Message) -> bytes encodes a Message into an opaque byte buffer.¶
• decode_pong_pong_message(encoded: bytes) -> Message decodes an opaque byte buffer into a Message, raising an error if the bytes are not a valid encoding.¶
The Leader's initial transition is computed with the following procedure:¶
def ping_pong_leader_init(
vdaf_verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
agg_param: bytes,
nonce: bytes[Vdaf.NONCE_SIZE],
public_share: bytes,
input_share: bytes,
) -> tuple[State, bytes]:
(prep_state, prep_share) = Vdaf.prep_init(
Vdaf.decode_input_share(0, input_share),
outbound = Message.initialize(
return (Continued(prep_state), encode_ping_pong_message(outbound))
return (Rejected(), None)
The output is the State to which the Leader has transitioned and an encoded Message. If the Leader's state is Rejected, then processing halts. Otherwise, if the state is Continued, then processing
The Leader sends the outbound message to the Helper. The Helper's initial transition is computed using the following procedure:¶
def ping_pong_helper_init(
vdaf_verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
agg_param: bytes,
nonce: bytes[Vdaf.NONCE_SIZE],
public_share: bytes,
input_share: bytes,
inbound_encoded: bytes,
) -> tuple[State, bytes]:
(prep_state, prep_share) = Vdaf.prep_init(
Vdaf.decode_input_share(1, input_share),
inbound = decode_ping_pong_message(inbound_encoded)
if inbound.type != 0: # initialize
return (Rejected(), None)
prep_shares = [
Vdaf.decode_prep_share(prep_state, inbound.prep_share),
return Vdaf.ping_pong_transition(
return (Rejected(), None)
Procedure ping_pong_transition() takes in the prep shares, combines them into the prep message, and computes the next prep state of the caller:¶
def ping_pong_transition(
agg_param: Vdaf.AggParam,
prep_shares: list[Vdaf.PrepShare],
prep_state: Vdaf.PrepState,
) -> (State, bytes):
prep_msg = Vdaf.prep_shares_to_prep(agg_param,
out = Vdaf.prep_next(prep_state, prep_msg)
if type(out) == Vdaf.OutShare:
outbound = Message.finish(Vdaf.encode_prep_msg(prep_msg))
return (Finished(out), encode_ping_pong_message(outbound))
(prep_state, prep_share) = out
outbound = Message.continue(
return (Continued(prep_state), encode_ping_pong_message(outbound))
The output is the State to which the Helper has transitioned and an encoded Message. If the Helper's state is Finished or Rejected, then processing halts. Otherwise, if the state is Continued, then
processing continues.¶
Next, the Helper sends the outbound message to the Leader. The Leader computes its next state transition using the function ping_pong_leader_continued:¶
def ping_pong_leader_continued(
agg_param: bytes,
state: State,
inbound_encoded: bytes,
) -> (State, Optional[bytes]):
return Vdaf.ping_pong_continued(
def ping_pong_continued(
is_leader: bool,
agg_param: bytes,
state: State,
inbound_encoded: bytes,
) -> (State, Optional[bytes]):
inbound = decode_ping_pong_message(inbound_encoded)
if inbound.type == 0: # initialize
return (Rejected(), None)
if !isinstance(state, Continued):
return (Rejected(), None)
prep_msg = Vdaf.decode_prep_msg(state.prep_state, inbound.prep_msg)
out = Vdaf.prep_next(state.prep_state, prep_msg)
if type(out) == tuple[Vdaf.PrepState, Vdaf.PrepShare] \
and inbound.type == 1:
# continue
(prep_state, prep_share) = out
prep_shares = [
Vdaf.decode_prep_share(prep_state, inbound.prep_share),
if is_leader:
return Vdaf.ping_pong_transition(
elif type(out) == Vdaf.OutShare and inbound.type == 2:
# finish
return (Finished(out), None)
return (Rejected(), None)
return (Rejected(), None)
If the Leader's state is Finished or Rejected, then processing halts. Otherwise, the Leader sends the outbound message to the Helper. The Helper computes its next state transition using the function
def ping_pong_helper_continued(
agg_param: bytes,
state: State,
inbound_encoded: bytes,
) -> (State, Optional[bytes]):
return Vdaf.ping_pong_continued(
They continue in this way until processing halts. Note that, depending on the number of rounds of preparation that are required, there may be one more message to send before the peer can also finish
processing (i.e., outbound != None).¶
The ping-pong topology of the previous section is only suitable for VDAFs involving exactly two Aggregators. If more Aggregators are required, the star topology described in this section can be used
• TODO Describe the Leader-emulated broadcast channel architecture that was originally envisioned for DAP. (As of DAP-05 we are going with the ping pong architecture described in the previous
This section describes the primitives that are common to the VDAFs specified in this document.¶
Both Prio3 and Poplar1 use finite fields of prime order. Finite field elements are represented by a class Field with the following associated parameters:¶
A concrete Field also implements the following class methods:¶
A field element is an instance of a concrete Field. The concrete class defines the usual arithmetic operations on field elements. In addition, it defines the following instance method for converting
a field element to an unsigned integer:¶
Likewise, each concrete Field implements a constructor for converting an unsigned integer into a field element:¶
Each concrete Field has two derived class methods, one for encoding a vector of field elements as a byte string and another for decoding a vector of field elements.¶
def encode_vec(Field, data: Vec[Field]) -> Bytes:
encoded = Bytes()
for x in data:
encoded += to_le_bytes(x.as_unsigned(), Field.ENCODED_SIZE)
return encoded
def decode_vec(Field, encoded: Bytes) -> Vec[Field]:
L = Field.ENCODED_SIZE
if len(encoded) % L != 0:
raise ERR_DECODE
vec = []
for i in range(0, len(encoded), L):
encoded_x = encoded[i:i+L]
x = from_le_bytes(encoded_x)
if x >= Field.MODULUS:
raise ERR_DECODE # Integer is larger than modulus
return vec
Figure 9: Derived class methods for finite fields.
Finally, Field implements the following methods for representing a value as a sequence of field elements, each of which represents a bit of the input.¶
def encode_into_bit_vector(Field,
val: Unsigned,
bits: Unsigned) -> Vec[Field]:
Encode the bit representation of `val` with at most `bits` number
of bits, as a vector of field elements.
if val >= 2 ** bits:
# Sanity check we are able to represent `val` with `bits`
# number of bits.
raise ValueError("Number of bits is not enough to represent "
"the input integer.")
encoded = []
for l in range(bits):
encoded.append(Field((val >> l) & 1))
return encoded
def decode_from_bit_vector(Field, vec: Vec[Field]) -> Field:
Decode the field element from the bit representation, expressed
as a vector of field elements `vec`.
bits = len(vec)
if Field.MODULUS >> bits == 0:
raise ValueError("Number of bits is too large to be "
"represented by field modulus.")
decoded = Field(0)
for (l, bit) in enumerate(vec):
decoded += Field(1 << l) * bit
return decoded
Figure 10: Derived class methods to encode integers into bit vector representation.
The following auxiliary functions on vectors of field elements are used in the remainder of this document. Note that an exception is raised by each function if the operands are not the same length.¶
def vec_sub(left: Vec[Field], right: Vec[Field]):
Subtract the right operand from the left and return the result.
return list(map(lambda x: x[0] - x[1], zip(left, right)))
def vec_add(left: Vec[Field], right: Vec[Field]):
"""Add the right operand to the left and return the result."""
return list(map(lambda x: x[0] + x[1], zip(left, right)))
Figure 11: Common functions for finite fields.
Some VDAFs require fields that are suitable for efficient computation of the discrete Fourier transform, as this allows for fast polynomial interpolation. (One example is Prio3 (Section 7) when
instantiated with the generic FLP of Section 7.3.3.) Specifically, a field is said to be "FFT-friendly" if, in addition to satisfying the interface described in Section 6.1, it implements the
following method:¶
• Field.gen() -> Field returns the generator of a large subgroup of the multiplicative group. To be FFT-friendly, the order of this subgroup MUST be a power of 2. In addition, the size of the
subgroup dictates how large interpolated polynomials can be. It is RECOMMENDED that a generator is chosen with order at least 2^20.¶
FFT-friendly fields also define the following parameter:¶
• GEN_ORDER: Unsigned is the order of a multiplicative subgroup generated by Field.gen().¶
The tables below define finite fields used in the remainder of this document.¶
Table 3: Parameters for the finite fields used in this document.
Parameter Field64 Field128 Field255
MODULUS 2^32 * 4294967295 + 1 2^66 * 4611686018427387897 + 1 2^255 - 19
ENCODED_SIZE 8 16 32
Generator 7^4294967295 7^4611686018427387897 n/a
GEN_ORDER 2^32 2^66 n/a
VDAFs in this specification use extendable output functions (XOFs) to extract short, fixed-length strings we call "seeds" from long input strings and expand seeds into long output strings. We specify
a single interface that is suitable for both purposes.¶
XOFs are defined by a class Xof with the following associated parameter and methods:¶
• SEED_SIZE: Unsigned is the size (in bytes) of a seed.¶
• Xof(seed: Bytes[Xof.SEED_SIZE], dst: Bytes, binder: Bytes) constructs an instance of Xof from the given seed, domain separation tag, and binder string. (See below for definitions of these.) The
seed MUST be of length SEED_SIZE and MUST be generated securely (i.e., it is either the output of gen_rand or a previous invocation of the XOF).¶
• xof.next(length: Unsigned) returns the next length bytes of output of xof.¶
Each Xof has two derived methods. The first is used to derive a fresh seed from an existing one. The second is used to compute a sequence of field elements.¶
def derive_seed(Xof,
seed: Bytes[Xof.SEED_SIZE],
dst: Bytes,
binder: Bytes):
"""Derive a new seed."""
xof = Xof(seed, dst, binder)
return xof.next(Xof.SEED_SIZE)
def next_vec(self, Field, length: Unsigned):
"""Output the next `length` elements of `Field`."""
m = next_power_of_2(Field.MODULUS) - 1
vec = []
while len(vec) < length:
x = from_le_bytes(self.next(Field.ENCODED_SIZE))
x &= m
if x < Field.MODULUS:
return vec
def expand_into_vec(Xof,
seed: Bytes[Xof.SEED_SIZE],
dst: Bytes,
binder: Bytes,
length: Unsigned):
Expand the input `seed` into vector of `length` field elements.
xof = Xof(seed, dst, binder)
return xof.next_vec(Field, length)
Figure 12: Derived methods for XOFs.
This section describes XofTurboShake128, an XOF based on the TurboSHAKE128 [TurboSHAKE]. This XOF is RECOMMENDED for all use cases within VDAFs. The length of the domain separation string dst passed
to XofTurboShake128 MUST NOT exceed 255 bytes.¶
class XofTurboShake128(Xof):
"""XOF wrapper for TurboSHAKE128."""
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed, dst, binder):
self.l = 0
self.m = to_le_bytes(len(dst), 1) + dst + seed + binder
def next(self, length: Unsigned) -> Bytes:
self.l += length
# Function `TurboSHAKE128(M, D, L)` is as defined in
# Section 2.2 of [TurboSHAKE].
# Implementation note: Rather than re-generate the output
# stream each time `next()` is invoked, most implementations
# of TurboSHAKE128 will expose an "absorb-then-squeeze" API that
# allows stateful handling of the stream.
stream = TurboSHAKE128(self.m, 1, self.l)
return stream[-length:]
Figure 13: Definition of XOF XofTurboShake128.
While XofTurboShake128 as described above can be securely used in all cases where a XOF is needed in the VDAFs described in this document, there are some cases where a more efficient instantiation
based on fixed-key AES is possible. For now, this is limited to the XOF used inside the Idpf Section 8.1 implementation in Poplar1 Section 8.3. It is NOT RECOMMENDED to use this XOF anywhere else.
The length of the domain separation string dst passed to XofFixedKeyAes128 MUST NOT exceed 255 bytes. See Security Considerations Section 9 for a more detailed discussion.¶
class XofFixedKeyAes128(Xof):
XOF based on a circular collision-resistant hash function from
fixed-key AES.
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed, dst, binder):
self.length_consumed = 0
# Use TurboSHAKE128 to derive a key from the binder string and
# domain separation tag. Note that the AES key does not need
# to be kept secret from any party. However, when used with
# IdpfPoplar, we require the binder to be a random nonce.
# Implementation note: This step can be cached across XOF
# evaluations with many different seeds.
dst_length = to_le_bytes(len(dst), 1)
self.fixed_key = TurboSHAKE128(dst_length + dst + binder, 2, 16)
self.seed = seed
def next(self, length: Unsigned) -> Bytes:
offset = self.length_consumed % 16
new_length = self.length_consumed + length
block_range = range(
int(self.length_consumed / 16),
int(new_length / 16) + 1)
self.length_consumed = new_length
hashed_blocks = [
self.hash_block(xor(self.seed, to_le_bytes(i, 16))) \
for i in block_range
return concat(hashed_blocks)[offset:offset+length]
def hash_block(self, block):
The multi-instance tweakable circular correlation-robust hash
function of [GKWWY20] (Section 4.2). The tweak here is the key
that stays constant for all XOF evaluations of the same Client,
but differs between Clients.
Function `AES128(key, block)` is the AES-128 blockcipher.
lo, hi = block[:8], block[8:]
sigma_block = concat([hi, xor(hi, lo)])
return xor(AES128(self.fixed_key, sigma_block), sigma_block)
XOFs are used to map a seed to a finite domain, e.g., a fresh seed or a vector of field elements. To ensure domain separation, the derivation is needs to be bound to some distinguished domain
separation tag. The domain separation tag encodes the following values:¶
The following algorithm is used in the remainder of this document in order to format the domain separation tag:¶
def format_dst(algo_class: Unsigned,
algo: Unsigned,
usage: Unsigned) -> Bytes:
"""Format XOF domain separation tag for use within a (V)DAF."""
return concat([
to_be_bytes(VERSION, 1),
to_be_bytes(algo_class, 1),
to_be_bytes(algo, 4),
to_be_bytes(usage, 2),
It is also sometimes necessary to bind the output to some ephemeral value that multiple parties need to agree on. We call this input the "binder string".¶
This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of
quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶
• Each measurement is encoded as a vector over some finite field.¶
• Measurement validity is determined by an arithmetic circuit evaluated over the encoded measurement. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The
circuit's output is a single field element: if zero, then the measurement is said to be "valid"; otherwise, if the output is non-zero, then the measurement is said to be "invalid".¶
• The aggregate result is obtained by summing up the encoded measurement vectors and computing some function of the sum.¶
At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each
Aggregator. Next, in the preparation phase, the Aggregators carry out a multi-party computation to determine if their shares correspond to a valid measurement (as determined by the arithmetic
circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its shares locally. Finally, the Collector sums up the aggregate shares and computes
the aggregate result.¶
This VDAF does not have an aggregation parameter. Instead, the output share is derived from the measurement share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful
use of the aggregation parameter.¶
As the name implies, Prio3 is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from
techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of
generality as the original construction.¶
The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating measurements. Prio3 can be
thought of as a transformation of a particular class of FLPs into a VDAF.¶
The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete
FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in
Appendix "Test Vectors".¶
Conceptually, an FLP is a two-party protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is
distributed among the Aggregators. The Client generates a "proof" of its measurement's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some computation
on its measurement share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid
measurement. (See Section 7.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms of an abstract base class Flp that specifies the set of methods and parameters a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 4.¶
Table 4: Constants and types defined by a concrete FLP.
Parameter Description
PROVE_RAND_LEN Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof
QUERY_RAND_LEN Length of the query randomness, the number of random field elements consumed by the verifier
JOINT_RAND_LEN Length of the joint randomness, the number of random field elements consumed by both the prover and verifier
MEAS_LEN Length of the encoded measurement (Section 7.1.1)
OUTPUT_LEN Length of the aggregatable output (Section 7.1.1)
PROOF_LEN Length of the proof
VERIFIER_LEN Length of the verifier message generated by querying the measurement and proof
Measurement Type of the measurement
AggResult Type of the aggregate result
Field As defined in (Section 6.1)
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1):¶
• Flp.prove(meas: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field]) -> Vec[Field] is the deterministic proof-generation algorithm run by the prover. Its inputs are the encoded
measurement, the "prover randomness" prove_rand, and the "joint randomness" joint_rand. The prover randomness is used only by the prover, but the joint randomness is shared by both the prover and
• Flp.query(meas: Vec[Field], proof: Vec[Field], query_rand: Vec[Field], joint_rand: Vec[Field], num_shares: Unsigned) -> Vec[Field] is the query-generation algorithm run by the verifier. This is
used to "query" the measurement and proof. The result of the query (i.e., the output of this function) is called the "verifier message". In addition to the measurement and proof, this algorithm
takes as input the query randomness query_rand and the joint randomness joint_rand. The former is used only by the verifier. num_shares specifies how many shares were generated.¶
• Flp.decide(verifier: Vec[Field]) -> Bool is the deterministic decision algorithm run by the verifier. It takes as input the verifier message and outputs a boolean indicating if the measurement
from which it was generated is valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the measurement and
proof, the query-generation algorithm outputs a share of the verifier message. Furthermore, the privacy property of the FLP system ensures that the verifier message reveals nothing about the
measurement other than whether it is valid. Therefore, to decide if a measurement is valid, the Aggregators will run the query-generation algorithm locally, exchange verifier shares, combine them to
recover the verifier message, and run the decision algorithm.¶
The query-generation algorithm includes a parameter num_shares that specifies the number of shares that were generated. If these data are not secret shared, then num_shares == 1. This parameter is
useful for certain FLP constructions. For example, the FLP in Section 7.3 is defined in terms of an arithmetic circuit; when the circuit contains constants, it is sometimes necessary to normalize
those constants to ensure that the circuit's output, when run on a valid measurement, is the same regardless of the number of shares.¶
An FLP is executed by the prover and verifier as follows:¶
def run_flp(flp, meas: Vec[Flp.Field], num_shares: Unsigned):
joint_rand = flp.Field.rand_vec(flp.JOINT_RAND_LEN)
prove_rand = flp.Field.rand_vec(flp.PROVE_RAND_LEN)
query_rand = flp.Field.rand_vec(flp.QUERY_RAND_LEN)
# Prover generates the proof.
proof = flp.prove(meas, prove_rand, joint_rand)
# Shard the measurement and the proof.
meas_shares = additive_secret_share(meas, num_shares, flp.Field)
proof_shares = additive_secret_share(proof, num_shares, flp.Field)
# Verifier queries the meas shares and proof shares.
verifier_shares = [
for meas_share, proof_share in zip(meas_shares, proof_shares)
# Combine the verifier shares into the verifier.
verifier = flp.Field.zeros(len(verifier_shares[0]))
for verifier_share in verifier_shares:
verifier = vec_add(verifier, verifier_share)
# Verifier decides if the measurement is valid.
return flp.decide(verifier)
Figure 14: Execution of an FLP.
The proof system is constructed so that, if meas is valid, then run_flp(Flp, meas, 1) always returns True. On the other hand, if meas is invalid, then as long as joint_rand and query_rand are
generated uniform randomly, the output is False with overwhelming probability.¶
We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5-round, public-coin, interactive
oracle proof system in their paper.¶
The type of measurement being aggregated is defined by the FLP. Hence, the FLP also specifies a method of encoding raw measurements as a vector of field elements:¶
• Flp.encode(measurement: Measurement) -> Vec[Field] encodes a raw measurement as a vector of field elements. The return value MUST be of length MEAS_LEN.¶
For some FLPs, the encoded measurement also includes redundant field elements that are useful for checking the proof, but which are not needed after the proof has been checked. An example is the
"integer sum" data type from [CGB17] in which an integer in range [0, 2^k) is encoded as a vector of k field elements, each representing a bit of the integer (this type is also defined in Section
7.4.2). After consuming this vector, all that is needed is the integer it represents. Thus the FLP defines an algorithm for truncating the encoded measurement to the length of the aggregated output:¶
• Flp.truncate(meas: Vec[Field]) -> Vec[Field] maps an encoded measurement (e.g., the bit-encoding of the measurement) to an aggregatable output (e.g., the singleton vector containing the
measurement). The length of the input MUST be MEAS_LEN and the length of the output MUST be OUTPUT_LEN.¶
Once the aggregate shares have been computed and combined together, their sum can be converted into the aggregate result. This could be a projection from the FLP's field to the integers, or it could
include additional post-processing.¶
• Flp.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult maps a sum of aggregate shares to an aggregate result. The length of the input MUST be OUTPUT_LEN. num_measurements is the
number of measurements that contributed to the aggregated output.¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affine-aggregatable encodings (AFEs)" from [CGB17].¶
To improve soundness, the prover can construct multiple unique proofs for its measurement such that the verifier will only accept the measurement once all proofs have been verified. Notably, several
proofs using a smaller field can offer the same level of soundness as a single proof using a large field.¶
To generate these proofs for a specific measurement, the prover calls Flp.prove multiple times, each time using an independently generated prover and joint randomness string. The verifier checks each
proof independently, each time with an independently generated query randomness string. It accepts the measurement only if all the decision algorithm accepts on each proof.¶
See Section 9.5 below for discussions on choosing the right number of proofs.¶
This section specifies Prio3, an implementation of the Vdaf interface (Section 5). It has two generic parameters: an Flp (Section 7.1) and a Xof (Section 6.2). It also has an associated constant,
PROOFS, with a value within the range of [1, 256), denoting the number of FLPs generated by the Client (Section 7.1.2). The value of PROOFS is 1 unless explicitly specified.¶
The associated constants and types required by the Vdaf interface are defined in Table 5. The methods required for sharding, preparation, aggregation, and unsharding are described in the remaining
subsections. These methods refer to constants enumerated in Table 6.¶
Table 5: VDAF parameters for Prio3.
Parameter Value
VERIFY_KEY_SIZE Xof.SEED_SIZE
RAND_SIZE Xof.SEED_SIZE * (1 + 2 * (SHARES - 1)) if Flp.JOINT_RAND_LEN == 0 else Xof.SEED_SIZE * (1 + 2 * (SHARES - 1) + SHARES)
NONCE_SIZE 16
ROUNDS 1
SHARES in [2, 256)
Measurement Flp.Measurement
AggParam None
PublicShare Optional[list[bytes]]
InputShare Union[tuple[list[Flp.Field], list[Flp.Field], Optional[bytes]], tuple[bytes, bytes, Optional[bytes]]]
OutShare list[Flp.Field]
AggShare list[Flp.Field]
AggResult Flp.AggResult
PrepState tuple[list[Flp.Field], Optional[Bytes]]
PrepShare tuple[list[Flp.Field], Optional[Bytes]]
PrepMessage Optional[bytes]
Table 6: Constants used by Prio3.
Variable Value
USAGE_MEAS_SHARE: Unsigned 1
USAGE_PROOF_SHARE: Unsigned 2
USAGE_JOINT_RANDOMNESS: Unsigned 3
USAGE_PROVE_RANDOMNESS: Unsigned 4
USAGE_QUERY_RANDOMNESS: Unsigned 5
USAGE_JOINT_RAND_SEED: Unsigned 6
USAGE_JOINT_RAND_PART: Unsigned 7
This section describes the process of recovering output shares from the input shares. The high-level idea is that each Aggregator first queries its measurement and share of proof(s) locally, then
exchanges its share of verifier(s) with the other Aggregators. The shares of verifier(s) are then combined into the verifier message(s) used to decide whether to accept.¶
In addition, for FLPs that require joint randomness, the Aggregators must ensure that they have all used the same joint randomness for the query-generation algorithm. To do so, they collectively
re-derive the joint randomness from their measurement shares just as the Client did during sharding.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the joint randomness parts. This leaves open the possibility that the Client cheated by, say,
forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint
randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their shares of verifier(s).¶
The definitions of constants and a few auxiliary functions are defined in Section 7.2.6.¶
def prep_init(Prio3, verify_key, agg_id, _agg_param,
nonce, public_share, input_share):
k_joint_rand_parts = public_share
(meas_share, proofs_share, k_blind) = \
Prio3.expand_input_share(agg_id, input_share)
out_share = Prio3.Flp.truncate(meas_share)
# Compute the joint randomness.
joint_rand = []
k_corrected_joint_rand, k_joint_rand_part = None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_part = Prio3.joint_rand_part(
agg_id, k_blind, meas_share, nonce)
k_joint_rand_parts[agg_id] = k_joint_rand_part
k_corrected_joint_rand = Prio3.joint_rand_seed(
joint_rands = Prio3.joint_rands(k_corrected_joint_rand)
# Query the measurement and proof share.
query_rands = Prio3.query_rands(verify_key, nonce)
verifiers_share = []
for _ in range(Prio3.PROOFS):
proof_share, proofs_share = front(
Prio3.Flp.PROOF_LEN, proofs_share)
query_rand, query_rands = front(
Prio3.Flp.QUERY_RAND_LEN, query_rands)
if Prio3.Flp.JOINT_RAND_LEN > 0:
joint_rand, joint_rands = front(
Prio3.Flp.JOINT_RAND_LEN, joint_rands)
verifiers_share += Prio3.Flp.query(meas_share,
prep_state = (out_share, k_corrected_joint_rand)
prep_share = (verifiers_share, k_joint_rand_part)
return (prep_state, prep_share)
def prep_next(Prio3, prep, prep_msg):
k_joint_rand = prep_msg
(out_share, k_corrected_joint_rand) = prep
# If joint randomness was used, check that the value computed by the
# Aggregators matches the value indicated by the Client.
if k_joint_rand != k_corrected_joint_rand:
raise ERR_VERIFY # joint randomness check failed
return out_share
def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
# Unshard the verifier shares into the verifier message.
verifiers = Prio3.Flp.Field.zeros(
Prio3.Flp.VERIFIER_LEN * Prio3.PROOFS)
k_joint_rand_parts = []
for (verifiers_share, k_joint_rand_part) in prep_shares:
verifiers = vec_add(verifiers, verifiers_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
# Verify that each proof is well-formed and the input is valid
for _ in range(Prio3.PROOFS):
verifier, verifiers = front(Prio3.Flp.VERIFIER_LEN, verifiers)
if not Prio3.Flp.decide(verifier):
raise ERR_VERIFY # proof verifier check failed
# Combine the joint randomness parts computed by the
# Aggregators into the true joint randomness seed. This is
# used in the last step.
k_joint_rand = None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand = Prio3.joint_rand_seed(k_joint_rand_parts)
return k_joint_rand
Figure 18: Preparation state for Prio3.
Every input share MUST only be used once, regardless of the aggregation parameters used.¶
Aggregating a set of output shares is simply a matter of adding up the vectors element-wise.¶
def aggregate(Prio3, _agg_param, out_shares):
agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return agg_share
Figure 20: Aggregation algorithm for Prio3.
To unshard a set of aggregate shares, the Collector first adds up the vectors element-wise. It then converts each element of the vector into an integer.¶
def unshard(Prio3, _agg_param,
agg_shares, num_measurements):
agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for agg_share in agg_shares:
agg = vec_add(agg, agg_share)
return Prio3.Flp.decode(agg, num_measurements)
Figure 21: Computation of the aggregate result for Prio3.
This section defines a number of auxiliary functions referenced by the main algorithms for Prio3 in the preceding sections.¶
The following methods are called by the sharding and preparation algorithms.¶
def helper_meas_share(Prio3, agg_id, k_share):
return Prio3.Xof.expand_into_vec(
def helper_proofs_share(Prio3, agg_id, k_share):
return Prio3.Xof.expand_into_vec(
byte(Prio3.PROOFS) + byte(agg_id),
Prio3.Flp.PROOF_LEN * Prio3.PROOFS,
def expand_input_share(Prio3, agg_id, input_share):
(meas_share, proofs_share, k_blind) = input_share
if agg_id > 0:
meas_share = Prio3.helper_meas_share(agg_id, meas_share)
proofs_share = Prio3.helper_proofs_share(agg_id, proofs_share)
return (meas_share, proofs_share, k_blind)
def prove_rands(Prio3, k_prove):
return Prio3.Xof.expand_into_vec(
Prio3.Flp.PROVE_RAND_LEN * Prio3.PROOFS,
def query_rands(Prio3, verify_key, nonce):
return Prio3.Xof.expand_into_vec(
byte(Prio3.PROOFS) + nonce,
Prio3.Flp.QUERY_RAND_LEN * Prio3.PROOFS,
def joint_rand_part(Prio3, agg_id, k_blind, meas_share, nonce):
return Prio3.Xof.derive_seed(
byte(agg_id) + nonce + Prio3.Flp.Field.encode_vec(meas_share),
def joint_rand_seed(Prio3, k_joint_rand_parts):
"""Derive the joint randomness seed from its parts."""
return Prio3.Xof.derive_seed(
def joint_rands(Prio3, k_joint_rand_seed):
"""Derive the joint randomness from its seed."""
return Prio3.Xof.expand_into_vec(
Prio3.Flp.JOINT_RAND_LEN * Prio3.PROOFS,
This section defines serialization formats for messages exchanged over the network while executing Prio3. It is RECOMMENDED that implementations provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the remainder we use S as an alias for Prio3.Xof.SEED_SIZE and F as an alias for Prio3.Field.ENCODED_SIZE. XOF seeds are
represented as follows:¶
Field elements are encoded in little-endian byte order (as defined in Section 6.1) and represented as follows:¶
When joint randomness is not used, the prep message is the empty string. Otherwise the prep message consists of the joint randomness seed computed by the Aggregators:¶
struct {
Prio3Seed k_joint_rand;
} Prio3PrepMessageWithJointRand;
Aggregate shares are structured as follows:¶
struct {
Prio3Field agg_share[F * Prio3.Flp.OUTPUT_LEN];
} Prio3AggShare;
This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and the extensions to their proof system made
here. The construction is specified in Section 7.3.3.¶
• OPEN ISSUE We're not yet sure if specifying this general-purpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two
reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a
way that improves performance, but breaks compatibility with the general-purpose FLP.¶
In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the general-purpose construction. The reference implementation can be found at
• OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the encoded measurement: If the circuit output is zero, then the measurement is deemed valid;
otherwise, if the circuit output is non-zero, then the measurement is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the
measurement. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the measurement.¶
Suppose for a moment that the validity circuit C is affine, meaning its only operations are addition and multiplication-by-constant. In particular, suppose the circuit does not contain a
multiplication gate whose operands are both non-constant. Then to decide if a measurement x is valid, each Aggregator could evaluate C on its share of x locally, broadcast the output share to its
peers, then combine the output shares locally to recover C(x). This is true because for any SHARES-way secret sharing of x it holds that¶
C(x_shares[0] + ... + x_shares[SHARES-1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES-1])
(Note that, for this equality to hold, it may be necessary to scale any constants in the circuit by SHARES.) However this is not the case if C is not-affine (i.e., it contains at least one
multiplication gate whose operands are non-constant). In the proof system of [BBCGGI19], the proof is designed to allow the (distributed) verifier to compute the non-affine operations using only
linear operations on (its share of) the measurement and proof.¶
To make this work, the proof system is restricted to validity circuits that exhibit a special structure. Specifically, an arithmetic circuit with "G-gates" (see [BBCGGI19], Definition 5.2) is
composed of affine gates and any number of instances of a distinguished gate G, which may be non-affine. We will refer to this class of circuits as 'gadget circuits' and to G as the "gadget".¶
As an illustrative example, consider a validity circuit C that recognizes the set L = set([0], [1]). That is, C takes as input a length-1 vector x and returns 0 if x[0] is in [0,2) and outputs
something else otherwise. This circuit can be expressed as the following degree-2 polynomial:¶
C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]
This polynomial recognizes L because x[0]^2 = x[0] is only true if x[0] == 0 or x[0] == 1. Notice that the polynomial involves a non-affine operation, x[0]^2. In order to apply [BBCGGI19], Theorem
4.3, the circuit needs to be rewritten in terms of a gadget that subsumes this non-affine operation. For example, the gadget might be multiplication:¶
Mul(left, right) = left * right
The validity circuit can then be rewritten in terms of Mul like so:¶
C(x[0]) = Mul(x[0], x[0]) - x[0]
The proof system of [BBCGGI19] allows the verifier to evaluate each instance of the gadget (i.e., Mul(x[0], x[0]) in our example) using a linear function of the measurement and proof. The proof is
constructed roughly as follows. Let C be the validity circuit and suppose the gadget is arity-L (i.e., it has L input wires.). Let wire[j-1,k-1] denote the value of the jth wire of the kth call to
the gadget during the evaluation of C(x). Suppose there are M such calls and fix distinct field elements alpha[0], ..., alpha[M-1]. (We will require these points to have a special property, as we'll
discuss in Section 7.3.1.1; but for the moment it is only important that they are distinct.)¶
The prover constructs from wire and alpha a polynomial that, when evaluated at alpha[k-1], produces the output of the kth call to the gadget. Let us call this the "gadget polynomial". Polynomial
evaluation is linear, which means that, in the distributed setting, the Client can disseminate additive shares of the gadget polynomial that the Aggregators then use to compute additive shares of
each gadget output, allowing each Aggregator to compute its share of C(x) locally.¶
There is one more wrinkle, however: It is still possible for a malicious prover to produce a gadget polynomial that would result in C(x) being computed incorrectly, potentially resulting in an
invalid measurement being accepted. To prevent this, the verifier performs a probabilistic test to check that the gadget polynomial is well-formed. This test, and the procedure for constructing the
gadget polynomial, are described in detail in Section 7.3.3.¶
The FLP described in the next section extends the proof system of [BBCGGI19], Section 4.2 in three ways.¶
First, the validity circuit in our construction includes an additional, random input (this is the "joint randomness" derived from the measurement shares in Prio3; see Section 7.2). This allows for
circuit optimizations that trade a small soundness error for a shorter proof. For example, consider a circuit that recognizes the set of length-N vectors for which each element is either one or zero.
A deterministic circuit could be constructed for this language, but it would involve a large number of multiplications that would result in a large proof. (See the discussion in [BBCGGI19], Section
5.2 for details). A much shorter proof can be constructed for the following randomized circuit:¶
C(meas, r) = r * Range2(meas[0]) + ... + r^N * Range2(meas[N-1])
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here meas is the length-N input and r is a random field element. The gadget circuit Range2 is the "range-check" polynomial described
above, i.e., Range2(x) = x^2 - x. The idea is that, if meas is valid (i.e., each meas[j] is in [0,2)), then the circuit will evaluate to 0 regardless of the value of r; but if meas[j] is not in [0,2)
for some j, the output will be non-zero with high probability.¶
The second extension implemented by our FLP allows the validity circuit to contain multiple gadget types. (This generalization was suggested in [BBCGGI19], Remark 4.5.) This provides additional
flexibility for designing circuits by allowing multiple, non-affine sub-components. For example, the following circuit is allowed:¶
C(meas, r) = r * Range2(meas[0]) + ... + r^L * Range2(meas[L-1]) + \
r^(L+1) * Range3(meas[L]) + ... + r^N * Range3(meas[N-1])
where Range3(x) = x^3 - 3x^2 + 2x. This circuit checks that the first L inputs are in range [0,2) and the last N-L inputs are in range [0,3). Of course, the same circuit can be expressed using a
sub-component that the gadgets have in common, namely Mul, but the resulting proof would be longer.¶
Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the fixed points alpha[0], ..., alpha[M-1], other than to require that the points are distinct. In this document, the fixed
points are chosen so that the gadget polynomial can be constructed efficiently using the Cooley-Tukey FFT ("Fast Fourier Transform") algorithm. Note that this requires the field to be "FFT-friendly"
as defined in Section 6.1.2.¶
The FLP described in Section 7.3.3 is defined in terms of a validity circuit Valid that implements the interface described here.¶
A concrete Valid defines the following parameters:¶
Table 7: Validity circuit parameters.
Parameter Description
GADGETS A list of gadgets
GADGET_CALLS Number of times each gadget is called
MEAS_LEN Length of the measurement
OUTPUT_LEN Length of the aggregatable output
JOINT_RAND_LEN Length of the random input
Measurement The type of measurement
AggResult Type of the aggregate result
Field An FFT-friendly finite field as defined in Section 6.1.2
Each gadget G in GADGETS defines a constant DEGREE that specifies the circuit's "arithmetic degree". This is defined to be the degree of the polynomial that computes it. For example, the Mul circuit
in Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has degree 2. Hence, the arithmetic degree of this gadget is 2.¶
Each gadget also defines a parameter ARITY that specifies the circuit's arity (i.e., the number of input wires).¶
Gadgets provide a method to evaluate their circuit on a list of inputs, eval(). The inputs can either belong to the validity circuit's field, or the polynomial ring over that field.¶
A concrete Valid provides the following methods for encoding a measurement as an input vector, truncating an input vector to the length of an aggregatable output, and converting an aggregated output
to an aggregate result:¶
• Valid.encode(measurement: Measurement) -> Vec[Field] returns a vector of length MEAS_LEN representing a measurement.¶
• Valid.truncate(meas: Vec[Field]) -> Vec[Field] returns a vector of length OUTPUT_LEN representing an aggregatable output.¶
• Valid.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult returns an aggregate result.¶
Finally, the following methods are derived for each concrete Valid:¶
def prove_rand_len(self):
"""Length of the prover randomness."""
return sum(g.ARITY for g in Valid.GADGETS)
def query_rand_len(self):
"""Length of the query randomness."""
return len(Valid.GADGETS)
def proof_len(self):
"""Length of the proof."""
length = 0
for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS):
P = next_power_of_2(1 + g_calls)
length += g.ARITY + g.DEGREE * (P - 1) + 1
return length
def verifier_len(self):
"""Length of the verifier message."""
length = 1
for g in self.GADGETS:
length += g.ARITY + 1
return length
Figure 22: Derived methods for validity circuits.
This section specifies FlpGeneric, an implementation of the Flp interface (Section 7.1). It has as a generic parameter a validity circuit Valid implementing the interface defined in Section 7.3.2.¶
• NOTE A reference implementation can be found in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/flp_generic.py.¶
The FLP parameters for FlpGeneric are defined in Table 8. The required methods for generating the proof, generating the verifier, and deciding validity are specified in the remaining subsections.¶
In the remainder, we let [n] denote the set {1, ..., n} for positive integer n. We also define the following constants:¶
• Let H = len(Valid.GADGETS)¶
• For each i in [H]:¶
Table 8: FLP Parameters of FlpGeneric.
Parameter Value
PROVE_RAND_LEN Valid.prove_rand_len() (see Section 7.3.2)
QUERY_RAND_LEN Valid.query_rand_len() (see Section 7.3.2)
JOINT_RAND_LEN Valid.JOINT_RAND_LEN
MEAS_LEN Valid.MEAS_LEN
OUTPUT_LEN Valid.OUTPUT_LEN
PROOF_LEN Valid.proof_len() (see Section 7.3.2)
VERIFIER_LEN Valid.verifier_len() (see Section 7.3.2)
Measurement Valid.Measurement
Field Valid.Field
On input of meas, prove_rand, and joint_rand, the proof is computed as follows:¶
1. For each i in [H] create an empty table wire_i.¶
2. Partition the prover randomness prove_rand into sub-vectors seed_1, ..., seed_H where len(seed_i) == L_i for all i in [H]. Let us call these the "wire seeds" of each gadget.¶
3. Evaluate Valid on input of meas and joint_rand, recording the inputs of each gadget in the corresponding table. Specifically, for every i in [H], set wire_i[j-1,k-1] to the value on the jth wire
into the kth call to gadget G_i.¶
4. Compute the "wire polynomials". That is, for every i in [H] and j in [L_i], construct poly_wire_i[j-1], the jth wire polynomial for the ith gadget, as follows:¶
□ Let w = [seed_i[j-1], wire_i[j-1,0], ..., wire_i[j-1,M_i-1]].¶
□ Let padded_w = w + Field.zeros(P_i - len(w)).¶
□ NOTE We pad w to the nearest power of 2 so that we can use FFT for interpolating the wire polynomials. Perhaps there is some clever math for picking wire_inp in a way that avoids having to
□ Let poly_wire_i[j-1] be the lowest degree polynomial for which poly_wire_i[j-1](alpha_i^k) == padded_w[k] for all k in [P_i].¶
5. Compute the "gadget polynomials". That is, for every i in [H]:¶
□ Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-1]). That is, evaluate the circuit G_i on the wire polynomials for the ith gadget. (Arithmetic is in the ring of polynomials over
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i for each i in [H].¶
On input of meas, proof, query_rand, and joint_rand, the verifier message is generated as follows:¶
1. For every i in [H] create an empty table wire_i.¶
2. Partition proof into the sub-vectors seed_1, coeff_1, ..., seed_H, coeff_H defined in Section 7.3.3.1.¶
3. Evaluate Valid on input of meas and joint_rand, recording the inputs of each gadget in the corresponding table. This step is similar to the prover's step (3.) except the verifier does not
evaluate the gadgets. Instead, it computes the output of the kth call to G_i by evaluating poly_gadget_i(alpha_i^k). Let v denote the output of the circuit evaluation.¶
4. Compute the wire polynomials just as in the prover's step (4.).¶
5. Compute the tests for well-formedness of the gadget polynomials. That is, for every i in [H]:¶
□ Let t = query_rand[i]. Check if t^(P_i) == 1: If so, then raise ERR_ABORT and halt. (This prevents the verifier from inadvertently leaking a gadget output in the verifier message.)¶
□ Let y_i = poly_gadget_i(t).¶
□ For each j in [0,L_i) let x_i[j-1] = poly_wire_i[j-1](t).¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H + [y_H].¶
On input of vector verifier, the verifier decides if the measurement is valid as follows:¶
1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in Section 7.3.3.2.¶
2. Check for well-formedness of the gadget polynomials. For every i in [H]:¶
□ Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i and set z to the output.¶
□ If z != y_i, then return False and halt.¶
3. Return True if v == 0 and False otherwise.¶
The FLP encoding and truncation methods invoke Valid.encode, Valid.truncate, and Valid.decode in the natural way.¶
This section specifies instantiations of Prio3 for various measurement types. Each uses FlpGeneric as the FLP (Section 7.3) and is determined by a validity circuit (Section 7.3.2) and a XOF (Section
6.2). Test vectors for each can be found in Appendix "Test Vectors".¶
• NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/vdaf_prio3.sage.¶
Our first instance of Prio3 is for a simple counter: Each measurement is either one or zero and the aggregate result is the sum of the measurements.¶
This instance uses XofTurboShake128 (Section 6.2.1) as its XOF. Its validity circuit, denoted Count, uses Field64 (Table 3) as its finite field. Its gadget, denoted Mul, is the degree-2, arity-2
gadget defined as¶
def eval(self, Field, inp):
return inp[0] * inp[1]
The call to check_gadget_eval() raises an error if the length of the input is not equal to the gadget's ARITY parameter.¶
The Count validity circuit is defined as¶
def eval(self, meas, joint_rand, _num_shares):
return self.GADGETS[0].eval(self.Field, [meas[0], meas[0]]) \
- meas[0]
The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below.¶
The next instance of Prio3 supports summing of integers in a pre-determined range. Each measurement is an integer in range [0, 2^bits), where bits is an associated parameter.¶
This instance of Prio3 uses XofTurboShake128 (Section 6.2.1) as its XOF. Its validity circuit, denoted Sum, uses Field128 (Table 3) as its finite field. The measurement is encoded as a length-bits
vector of field elements, where the lth element of the vector represents the lth bit of the summand:¶
def encode(self, measurement):
if 0 > measurement or measurement >= 2 ** self.MEAS_LEN:
raise ERR_INPUT
return self.Field.encode_into_bit_vector(measurement,
def truncate(self, meas):
return [self.Field.decode_from_bit_vector(meas)]
def decode(self, output, _num_measurements):
return output[0].as_unsigned()
The validity circuit checks that the input consists of ones and zeros. Its gadget, denoted Range2, is the degree-2, arity-1 gadget defined as¶
def eval(self, Field, inp):
return inp[0] * inp[0] - inp[0]
The Sum validity circuit is defined as¶
def eval(self, meas, joint_rand, _num_shares):
self.check_valid_eval(meas, joint_rand)
out = self.Field(0)
r = joint_rand[0]
for b in meas:
out += r * self.GADGETS[0].eval(self.Field, [b])
r *= joint_rand[0]
return out
This instance of Prio3 supports summing a vector of integers. It has three parameters, length, bits, and chunk_length. Each measurement is a vector of positive integers with length equal to the
length parameter. Each element of the measurement is an integer in the range [0, 2^bits). It is RECOMMENDED to set chunk_length to an integer near the square root of length * bits (see Section
This instance uses XofTurboShake128 (Section 6.2.1) as its XOF. Its validity circuit, denoted SumVec, uses Field128 (Table 3) as its finite field.¶
Measurements are encoded as a vector of field elements with length length * bits. The field elements in the encoded vector represent all the bits of the measurement vector's elements, consecutively,
in LSB to MSB order:¶
def encode(self, measurement: Vec[Unsigned]):
if len(measurement) != self.length:
raise ERR_INPUT
encoded = []
for val in measurement:
if 0 > val or val >= 2 ** self.bits:
raise ERR_INPUT
encoded += self.Field.encode_into_bit_vector(val, self.bits)
return encoded
def truncate(self, meas):
truncated = []
for i in range(self.length):
meas[i * self.bits: (i + 1) * self.bits]
return truncated
def decode(self, output, _num_measurements):
return [x.as_unsigned() for x in output]
This validity circuit uses a ParallelSum gadget to achieve a smaller proof size. This optimization for "parallel-sum circuits" is described in [BBCGGI19], section 4.4. Briefly, for circuits that add
up the output of multiple identical subcircuits, it is possible to achieve smaller proof sizes (on the order of O(sqrt(MEAS_LEN)) instead of O(MEAS_LEN)) by packaging more than one such subcircuit
into a gadget.¶
The ParallelSum gadget is parameterized with an arithmetic subcircuit, and a count of how many times it evaluates that subcircuit. It takes in a list of inputs and passes them through to instances of
the subcircuit in the same order. It returns the sum of the subcircuit outputs. Note that only the ParallelSum gadget itself, and not its subcircuit, participates in FlpGeneric's wire recording
during evaluation, gadget consistency proofs, and proof validation, even though the subcircuit is provided to ParallelSum as an implementation of the Gadget interface.¶
def eval(self, Field, inp):
out = Field(0)
for i in range(self.count):
start_index = i * self.subcircuit.ARITY
end_index = (i + 1) * self.subcircuit.ARITY
out += self.subcircuit.eval(Field, inp[start_index:end_index])
return out
The SumVec validity circuit checks that the encoded measurement consists of ones and zeros. Rather than use the Range2 gadget on each element, as in the Sum validity circuit, it instead uses Mul
subcircuits and "free" constant multiplication and addition gates to simultaneously evaluate the same range check polynomial on each element, and multiply by a constant. One of the two Mul subcircuit
inputs is equal to a measurement element multiplied by a power of the joint randomness value, and the other is equal to the same measurement element minus one. These Mul subcircuits are evaluated by
a ParallelSum gadget, and the results are added up both within the ParallelSum gadget and after it.¶
def eval(self, meas, joint_rand, num_shares):
self.check_valid_eval(meas, joint_rand)
out = Field128(0)
r = joint_rand[0]
r_power = r
shares_inv = self.Field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
meas_elem = self.Field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem - shares_inv
r_power *= r
out += self.GADGETS[0].eval(self.Field, inputs)
return out
Table 11: Parameters of validity circuit SumVec.
Parameter Value
GADGETS [ParallelSum(Mul(), chunk_length)]
GADGET_CALLS [(length * bits + chunk_length - 1) // chunk_length]
MEAS_LEN length * bits
OUTPUT_LEN length
JOINT_RAND_LEN 1
Measurement Vec[Unsigned], each element in range [0, 2^bits)
AggResult Vec[Unsigned]
Field Field128 (Table 3)
The chunk_length parameter provides a trade-off between the arity of the ParallelSum gadget and the number of times the gadget is called. The proof length is asymptotically minimized when the chunk
length is near the square root of the length of the measurement. However, the relationship between VDAF parameters and proof length is complicated, involving two forms of rounding (the circuit pads
the inputs to its last ParallelSum gadget call, up to the chunk length, and FlpGeneric rounds the degree of wire polynomials -- determined by the number of times a gadget is called -- up to the next
power of two). Therefore, the optimal choice of chunk_length for a concrete measurement size will vary, and must be found through trial and error. Setting chunk_length equal to the square root of the
appropriate measurement length will result in proofs up to 50% larger than the optimal proof size.¶
This instance of Prio3 allows for estimating the distribution of some quantity by computing a simple histogram. Each measurement increments one histogram bucket, out of a set of fixed buckets.
(Bucket indexing begins at 0.) For example, the buckets might quantize the real numbers, and each measurement would report the bucket that the corresponding client's real-numbered value falls into.
The aggregate result counts the number of measurements in each bucket.¶
This instance of Prio3 uses XofTurboShake128 (Section 6.2.1) as its XOF. Its validity circuit, denoted Histogram, uses Field128 (Table 3) as its finite field. It has two parameters, length, the
number of histogram buckets, and chunk_length, which is used by by a circuit optimization described below. It is RECOMMENDED to set chunk_length to an integer near the square root of length (see
Section 7.4.3.1).¶
The measurement is encoded as a one-hot vector representing the bucket into which the measurement falls:¶
def encode(self, measurement):
encoded = [self.Field(0)] * self.length
encoded[measurement] = self.Field(1)
return encoded
def truncate(self, meas):
return meas
def decode(self, output, _num_measurements):
return [bucket_count.as_unsigned() for bucket_count in output]
The Histogram validity circuit checks for one-hotness in two steps, by checking that the encoded measurement consists of ones and zeros, and by checking that the sum of all elements in the encoded
measurement is equal to one. All the individual checks are combined together in a random linear combination.¶
As in the SumVec validity circuit (Section 7.4.3), the first part of the validity circuit uses the ParallelSum gadget to perform range checks while achieving a smaller proof size. The ParallelSum
gadget uses Mul subcircuits to evaluate a range check polynomial on each element, and includes an additional constant multiplication. One of the two Mul subcircuit inputs is equal to a measurement
element multiplied by a power of the first joint randomness value, and the other is equal to the same measurement element minus one. The results are added up both within the ParallelSum gadget and
after it.¶
def eval(self, meas, joint_rand, num_shares):
self.check_valid_eval(meas, joint_rand)
# Check that each bucket is one or zero.
range_check = self.Field(0)
r = joint_rand[0]
r_power = r
shares_inv = self.Field(num_shares).inv()
for i in range(self.GADGET_CALLS[0]):
inputs = [None] * (2 * self.chunk_length)
for j in range(self.chunk_length):
index = i * self.chunk_length + j
if index < len(meas):
meas_elem = meas[index]
meas_elem = self.Field(0)
inputs[j * 2] = r_power * meas_elem
inputs[j * 2 + 1] = meas_elem - shares_inv
r_power *= r
range_check += r * self.GADGETS[0].eval(self.Field, inputs)
# Check that the buckets sum to 1.
sum_check = -shares_inv
for b in meas:
sum_check += b
out = joint_rand[1] * range_check + \
joint_rand[1] ** 2 * sum_check
return out
Note that this circuit depends on the number of shares into which the measurement is sharded. This is provided to the FLP by Prio3.¶
Table 12: Parameters of validity circuit Histogram.
Parameter Value
GADGETS [ParallelSum(Mul(), chunk_length)]
GADGET_CALLS [(length + chunk_length - 1) // chunk_length]
MEAS_LEN length
OUTPUT_LEN length
JOINT_RAND_LEN 2
Measurement Unsigned
AggResult Vec[Unsigned]
Field Field128 (Table 3)
This section specifies Poplar1, a VDAF for the following task. Each Client holds a string of length BITS and the Aggregators hold a set of l-bit strings, where l <= BITS. We will refer to the latter
as the set of "candidate prefixes". The Aggregators' goal is to count how many measurements are prefixed by each candidate prefix.¶
This functionality is the core component of the Poplar protocol [BBCGGI21], which was designed to compute the heavy hitters over a set of input strings. At a high level, the protocol works as
1. Each Client splits its string into input shares and sends one share to each Aggregator.¶
2. The Aggregators agree on an initial set of candidate prefixes, say 0 and 1.¶
3. The Aggregators evaluate the VDAF on each set of input shares and aggregate the recovered output shares. The aggregation parameter is the set of candidate prefixes.¶
4. The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix.¶
5. Let H denote the set of prefixes that occurred at least t times. If the prefixes all have length BITS, then H is the set of t-heavy-hitters. Otherwise compute the next set of candidate prefixes,
e.g., for each p in H, add p || 0 and p || 1 to the set. Repeat step 3 with the new set of candidate prefixes.¶
Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", a primitive described by [BBCGGI21] that generalizes the notion of a Distributed Point Function (DPF) [GI14]. Briefly,
a DPF is used to distribute the computation of a "point function", a function that evaluates to zero on every input except at a programmable "point". The computation is distributed in such a way that
no one party knows either the point or what it evaluates to.¶
An IDPF generalizes this "point" to a path on a full binary tree from the root to one of the leaves. It is evaluated on an "index" representing a unique node of the tree. If the node is on the
programmed path, then the function evaluates to a non-zero value; otherwise it evaluates to zero. This structure allows an IDPF to provide the functionality required for the above protocol: To
compute the hit count for an index, just evaluate each set of IDPF shares at that index and add up the results.¶
Consider the sub-tree constructed from a set of input strings and a target threshold t by including all indices that prefix at least t of the input strings. We shall refer to this structure as the
"prefix tree" for the batch of inputs and target threshold. To compute the t-heavy hitters for a set of inputs, the Aggregators and Collector first compute the prefix tree, then extract the heavy
hitters from the leaves of this tree. (Note that the prefix tree may leak more information about the set than the heavy hitters themselves; see Section 9.3.1 for details.)¶
Poplar1 composes an IDPF with the "secure sketching" protocol of [BBCGGI21]. This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a
"one-hot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one.¶
The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a
generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors".¶
An IDPF is defined over a domain of size 2^BITS, where BITS is constant defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range [0, 2^BITS). The Client specifies an index
alpha and a vector of values beta, one for each "level" L in range [0, BITS). The key generation algorithm generates one IDPF "key" for each Aggregator. When evaluated at level L and index 0 <=
prefix < 2^L, each IDPF key returns an additive share of beta[L] if prefix is the L-bit prefix of alpha and shares of zero otherwise.¶
An index x is defined to be a prefix of another index y as follows. Let LSB(x, N) denote the least significant N bits of positive integer x. By definition, a positive integer 0 <= x < 2^L is said to
be the length-L prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is equal to the most significant L bits of LSB(y, BITS), For example, 6 (110 in binary) is the length-3 prefix of 25 (11001),
but 7 (111) is not.¶
Each of the programmed points beta is a vector of elements of some finite field. We distinguish two types of fields: One for inner nodes (denoted FieldInner), and one for leaf nodes (FieldLeaf). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger field for leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as defined in [BBCGGI21], Definition 1.)¶
A concrete IDPF defines the types and constants enumerated in Table 13. In the remainder we write Output as shorthand for the type Union[list[list[FieldInner]], list[list[FieldLeaf]]]. (This type
denotes either a vector of inner node field elements or leaf node field elements.) The scheme is comprised of the following algorithms:¶
• Idpf.gen(alpha: Unsigned, beta_inner: list[list[FieldInner]], beta_leaf: list[FieldLeaf], binder: bytes, rand: bytes[Idpf.RAND_SIZE]) -> tuple[bytes, list[bytes]] is the randomized IDPF-key
generation algorithm. (Input rand consists of the random bytes it consumes.) Its inputs are the index alpha the values beta, and a binder string. The value of alpha MUST be in range [0, 2^BITS).
The output is a public part that is sent to all Aggregators and a vector of private IDPF keys, one for each aggregator. The binder string is used to derive the key in the underlying
XofFixedKeyAes128 XOF that is used for expanding seeds at each level. It MUST be chosen uniformly at random by the Client (see Section 9.2).¶
□ TODO(issue #255) Decide whether to treat the public share as an opaque byte string or to replace it with an explicit type.¶
• Idpf.eval(agg_id: Unsigned, public_share: bytes, key: bytes, level: Unsigned, prefixes: tuple[Unsigned, ...], binder: Bytes) -> Output is the deterministic, stateless IDPF-key evaluation
algorithm run by each Aggregator. Its inputs are the Aggregator's unique identifier, the public share distributed to all of the Aggregators, the Aggregator's IDPF key, the "level" at which to
evaluate the IDPF, the sequence of candidate prefixes, and a binder string. It returns the share of the value corresponding to each candidate prefix.¶
The output type (i.e., Output) depends on the value of level: If level < Idpf.BITS-1, the output is the value for an inner node, which has type list[list[Idpf.FieldInner]]; otherwise, if level ==
Idpf.BITS-1, then the output is the value for a leaf node, which has type list[list[Idpf.FieldLeaf]].¶
The value of level MUST be in range [0, BITS). The indexes in prefixes MUST all be distinct and in range [0, 2^level).¶
Applications MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of key in the sequence of IDPF keys output by the Client.¶
In addition, the following method is derived for each concrete Idpf:¶
def current_field(Idpf, level):
return Idpf.FieldInner if level < Idpf.BITS-1 \
else Idpf.FieldLeaf
Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the
VDAF abstraction boundary, which does not include shared state across across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the
state across evaluations. See Section 8.3 for details.¶
Table 13: Constants and types defined by a concrete IDPF.
Parameter Description
SHARES Number of IDPF keys output by IDPF-key generator
BITS Length in bits of each input string
VALUE_LEN Number of field elements of each output value
RAND_SIZE Size of the random string consumed by the IDPF-key generator. Equal to twice the XOF's seed size.
KEY_SIZE Size in bytes of each IDPF key
FieldInner Implementation of Field (Section 6.1) used for values of inner nodes
FieldLeaf Implementation of Field used for values of leaf nodes
Output Alias of Union[list[list[FieldInner]], list[list[FieldLeaf]]]
FieldVec Alias of Union[list[FieldInner], list[FieldLeaf]]
This section specifies Poplar1, an implementation of the Vdaf interface (Section 5). It is defined in terms of any Idpf (Section 8.1) for which Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2 and an
implementation of Xof (Section 6.2). The associated constants and types required by the Vdaf interface are defined in Table 14. The methods required for sharding, preparation, aggregation, and
unsharding are described in the remaining subsections. These methods make use of constants defined in Table 15.¶
Table 14: VDAF parameters for Poplar1.
Parameter Value
VERIFY_KEY_SIZE Xof.SEED_SIZE
RAND_SIZE Xof.SEED_SIZE * 3 + Idpf.RAND_SIZE
NONCE_SIZE 16
ROUNDS 2
SHARES 2
Measurement Unsigned
AggParam Tuple[Unsigned, Tuple[Unsigned, ...]]
PublicShare bytes (IDPF public share)
InputShare tuple[bytes, bytes, list[Idpf.FieldInner], list[Idpf.FieldLeaf]]
OutShare Idpf.FieldVec
AggShare Idpf.FieldVec
AggResult Vec[Unsigned]
PrepState tuple[bytes, Unsigned, Idpf.FieldVec]
PrepShare Idpf.FieldVec
PrepMessage Optional[Idpf.FieldVec]
The Client's measurement is interpreted as an IDPF index, denoted alpha. The programmed IDPF values are pairs of field elements (1, k) where each k is chosen at random. This random value is used as
part of the secure sketching protocol of [BBCGGI21], Appendix C.4. After evaluating their IDPF key shares on a given sequence of candidate prefixes, the sketching protocol is used by the Aggregators
to verify that they hold shares of a one-hot vector. In addition, for each level of the tree, the prover generates random elements a, b, and c and computes¶
A = -2*a + k
B = a^2 + b - k*a + c
and sends additive shares of a, b, c, A and B to the Aggregators. Putting everything together, the sharding algorithm is defined as follows.¶
def shard(Poplar1, measurement, nonce, rand):
l = Poplar1.Xof.SEED_SIZE
# Split the random input into random input for IDPF key
# generation, correlated randomness, and sharding.
if len(rand) != Poplar1.RAND_SIZE:
raise ERR_INPUT # unexpected length for random input
idpf_rand, rand = front(Poplar1.Idpf.RAND_SIZE, rand)
seeds = [rand[i:i+l] for i in range(0,3*l,l)]
corr_seed, seeds = front(2, seeds)
(k_shard,), seeds = front(1, seeds)
xof = Poplar1.Xof(
# Construct the IDPF values for each level of the IDPF tree.
# Each "data" value is 1; in addition, the Client generates
# a random "authenticator" value used by the Aggregators to
# compute the sketch during preparation. This sketch is used
# to verify the one-hotness of their output shares.
beta_inner = [
[Poplar1.Idpf.FieldInner(1), k]
for k in xof.next_vec(Poplar1.Idpf.FieldInner,
Poplar1.Idpf.BITS - 1)
beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \
xof.next_vec(Poplar1.Idpf.FieldLeaf, 1)
# Generate the IDPF keys.
(public_share, keys) = Poplar1.Idpf.gen(measurement,
# Generate correlated randomness used by the Aggregators to
# compute a sketch over their output shares. XOF seeds are
# used to encode shares of the `(a, b, c)` triples.
# (See [BBCGGI21, Appendix C.4].)
corr_offsets = vec_add(
byte(0) + nonce,
3 * (Poplar1.Idpf.BITS-1),
byte(1) + nonce,
3 * (Poplar1.Idpf.BITS-1),
corr_offsets += vec_add(
byte(0) + nonce,
byte(1) + nonce,
# For each level of the IDPF tree, shares of the `(A, B)`
# pairs are computed from the corresponding `(a, b, c)`
# triple and authenticator value `k`.
corr_inner = [[], []]
for level in range(Poplar1.Idpf.BITS):
Field = Poplar1.Idpf.current_field(level)
k = beta_inner[level][1] if level < Poplar1.Idpf.BITS - 1 \
else beta_leaf[1]
(a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
A = -Field(2) * a + k
B = a ** 2 + b - a * k + c
corr1 = xof.next_vec(Field, 2)
corr0 = vec_sub([A, B], corr1)
if level < Poplar1.Idpf.BITS - 1:
corr_inner[0] += corr0
corr_inner[1] += corr1
corr_leaf = [corr0, corr1]
# Each input share consists of the Aggregator's IDPF key
# and a share of the correlated randomness.
input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf))
return (public_share, input_shares)
Figure 23: The sharding algorithm for Poplar1.
The aggregation parameter encodes a sequence of candidate prefixes. When an Aggregator receives an input share from the Client, it begins by evaluating its IDPF share on each candidate prefix,
recovering a data_share and auth_share for each. The Aggregators use these and the correlation shares provided by the Client to verify that the sequence of data_share values are additive shares of a
one-hot vector.¶
Aggregators MUST ensure the candidate prefixes are all unique and appear in lexicographic order. (This is enforced in the definition of prep_init() below.) Uniqueness is necessary to ensure the
refined measurement (i.e., the sum of the output shares) is in fact a one-hot vector. Otherwise, sketch verification might fail, causing the Aggregators to erroneously reject a report that is
actually valid. Note that enforcing the order is not strictly necessary, but this does allow uniqueness to be determined more efficiently.¶
def prep_init(Poplar1, verify_key, agg_id, agg_param,
nonce, public_share, input_share):
(level, prefixes) = agg_param
(key, corr_seed, corr_inner, corr_leaf) = input_share
Field = Poplar1.Idpf.current_field(level)
# Ensure that candidate prefixes are all unique and appear in
# lexicographic order.
for i in range(1,len(prefixes)):
if prefixes[i-1] >= prefixes[i]:
raise ERR_INPUT # out-of-order prefix
# Evaluate the IDPF key at the given set of prefixes.
value = Poplar1.Idpf.eval(
agg_id, public_share, key, level, prefixes, nonce)
# Get shares of the correlated randomness for computing the
# Aggregator's share of the sketch for the given level of the IDPF
# tree.
if level < Poplar1.Idpf.BITS - 1:
corr_xof = Poplar1.Xof(
byte(agg_id) + nonce,
# Fast-forward the XOF state to the current level.
corr_xof.next_vec(Field, 3 * level)
corr_xof = Poplar1.Xof(
byte(agg_id) + nonce,
(a_share, b_share, c_share) = corr_xof.next_vec(Field, 3)
(A_share, B_share) = corr_inner[2*level:2*(level+1)] \
if level < Poplar1.Idpf.BITS - 1 else corr_leaf
# Compute the Aggregator's first round of the sketch. These are
# called the "masked input values" [BBCGGI21, Appendix C.4].
verify_rand_xof = Poplar1.Xof(
nonce + to_be_bytes(level, 2),
verify_rand = verify_rand_xof.next_vec(Field, len(prefixes))
sketch_share = [a_share, b_share, c_share]
out_share = []
for (i, r) in enumerate(verify_rand):
[data_share, auth_share] = value[i]
sketch_share[0] += data_share * r
sketch_share[1] += data_share * r ** 2
sketch_share[2] += auth_share * r
prep_mem = [A_share, B_share, Field(agg_id)] + out_share
return ((b'sketch round 1', level, prep_mem),
def prep_next(Poplar1, prep_state, prep_msg):
prev_sketch = prep_msg
(step, level, prep_mem) = prep_state
Field = Poplar1.Idpf.current_field(level)
if step == b'sketch round 1':
if prev_sketch == None:
prev_sketch = Field.zeros(3)
elif len(prev_sketch) != 3:
raise ERR_INPUT # prep message malformed
(A_share, B_share, agg_id), prep_mem = \
prep_mem[:3], prep_mem[3:]
sketch_share = [
agg_id * (prev_sketch[0] ** 2
- prev_sketch[1]
- prev_sketch[2])
+ A_share * prev_sketch[0]
+ B_share
return ((b'sketch round 2', level, prep_mem),
elif step == b'sketch round 2':
if prev_sketch == None:
return prep_mem # Output shares
raise ERR_INPUT # prep message malformed
raise ERR_INPUT # unexpected input
def prep_shares_to_prep(Poplar1, agg_param, prep_shares):
if len(prep_shares) != 2:
raise ERR_INPUT # unexpected number of prep shares
(level, _) = agg_param
Field = Poplar1.Idpf.current_field(level)
sketch = vec_add(prep_shares[0], prep_shares[1])
if len(sketch) == 3:
return sketch
elif len(sketch) == 1:
if sketch == Field.zeros(1):
# In order to reduce communication overhead, let `None`
# denote a successful sketch verification.
return None
raise ERR_VERIFY # sketch verification failed
raise ERR_INPUT # unexpected input length
Figure 24: Preparation state for Poplar1.
Aggregation parameters are valid for a given input share if no aggregation parameter with the same level has been used with the same input share before. The whole preparation phase MUST NOT be run
more than once for a given combination of input share and level.¶
Aggregation involves simply adding up the output shares.¶
def aggregate(Poplar1, agg_param, out_shares):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg_share = Field.zeros(len(prefixes))
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return agg_share
Figure 26: Aggregation algorithm for Poplar1.
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
def unshard(Poplar1, agg_param,
agg_shares, _num_measurements):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg = Field.zeros(len(prefixes))
for agg_share in agg_shares:
agg = vec_add(agg, agg_share)
return list(map(lambda x: x.as_unsigned(), agg))
Figure 27: Computation of the aggregate result for Poplar1.
This section defines serialization formats for messages exchanged over the network while executing Poplar1. It is RECOMMENDED that implementations provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the remainder we use S as an alias for Poplar1.Xof.SEED_SIZE, Fi as an alias for Poplar1.Idpf.FieldInner and Fl as an alias for
Poplar1.Idpf.FieldLeaf. XOF seeds are represented as follows:¶
Elements of the inner field are encoded in little-endian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldInner[Fi];
Likewise, elements of the leaf field are encoded in little-endian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldLeaf[Fl];
Likewise, the structure of the prep message for Poplar1 depends on the sketching round and field. For the first round and inner field:¶
struct {
Poplar1FieldInner[Fi * 3];
} Poplar1PrepMessageRoundOneInner;
For the first round and leaf field:¶
struct {
Poplar1FieldLeaf sketch[Fl * 3];
} Poplar1PrepMessageRoundOneLeaf;
Note that these messages have the same structures as the prep shares for the first round.¶
The second-round prep message is the empty string. This is because the sketch shares are expected to sum to a particular value if the output shares are valid; we represent a successful preparation
with the empty string and otherwise return an error.¶
The aggregation parameter is encoded as follows:¶
• TODO(issue #255) Express the aggregation parameter encoding in TLS syntax. Decide whether to RECOMMEND this encoding, and if so, add it to test vectors.¶
def encode_agg_param(Poplar1, (level, prefixes)):
if level > 2 ** 16 - 1:
raise ERR_INPUT # level too deep
if len(prefixes) > 2 ** 32 - 1:
raise ERR_INPUT # too many prefixes
encoded = Bytes()
encoded += to_be_bytes(level, 2)
encoded += to_be_bytes(len(prefixes), 4)
packed = 0
for (i, prefix) in enumerate(prefixes):
packed |= prefix << ((level+1) * i)
l = ((level+1) * len(prefixes) + 7) // 8
encoded += to_be_bytes(packed, l)
return encoded
def decode_agg_param(Poplar1, encoded):
encoded_level, encoded = encoded[:2], encoded[2:]
level = from_be_bytes(encoded_level)
encoded_prefix_count, encoded = encoded[:4], encoded[4:]
prefix_count = from_be_bytes(encoded_prefix_count)
l = ((level+1) * prefix_count + 7) // 8
encoded_packed, encoded = encoded[:l], encoded[l:]
packed = from_be_bytes(encoded_packed)
prefixes = []
m = 2 ** (level+1) - 1
for i in range(prefix_count):
prefixes.append(packed >> ((level+1) * i) & m)
if len(encoded) != 0:
raise ERR_INPUT
return (level, tuple(prefixes))
Implementation note: The aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform per-report caching across executions of
the VDAF, this may be more information than is strictly needed. In particular, it may be sufficient to convey which indices from the previous execution will have their children included in the next.
This would help reduce communication overhead.¶
In this section we specify a concrete IDPF, called IdpfPoplar, suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
• TODO We should consider giving IdpfPoplar a more distinctive name.¶
The constant and type definitions required by the Idpf interface are given in Table 16.¶
IdpfPoplar requires a XOF for deriving the output shares, as well as a variety of other artifacts used internally. For performance reasons, we instantiate this object using XofFixedKeyAes128 (Section
6.2.2). See Section 9.4 for justification of this choice.¶
• TODO Describe the construction in prose, beginning with a gentle introduction to the high level idea.¶
The description of the IDPF-key generation algorithm makes use of auxiliary functions extend(), convert(), and encode_public_share() defined in Section 8.3.3. In the following, we let Field2 denote
the field GF(2).¶
def gen(IdpfPoplar, alpha, beta_inner, beta_leaf, binder, rand):
if alpha >= 2 ** IdpfPoplar.BITS:
raise ERR_INPUT # alpha too long
if len(beta_inner) != IdpfPoplar.BITS - 1:
raise ERR_INPUT # beta_inner vector is the wrong size
if len(rand) != IdpfPoplar.RAND_SIZE:
raise ERR_INPUT # unexpected length for random input
init_seed = [
seed = init_seed.copy()
ctrl = [Field2(0), Field2(1)]
correction_words = []
for level in range(IdpfPoplar.BITS):
Field = IdpfPoplar.current_field(level)
keep = (alpha >> (IdpfPoplar.BITS - level - 1)) & 1
lose = 1 - keep
bit = Field2(keep)
(s0, t0) = IdpfPoplar.extend(seed[0], binder)
(s1, t1) = IdpfPoplar.extend(seed[1], binder)
seed_cw = xor(s0[lose], s1[lose])
ctrl_cw = (
t0[0] + t1[0] + bit + Field2(1),
t0[1] + t1[1] + bit,
x0 = xor(s0[keep], ctrl[0].conditional_select(seed_cw))
x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw))
(seed[0], w0) = IdpfPoplar.convert(level, x0, binder)
(seed[1], w1) = IdpfPoplar.convert(level, x1, binder)
ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep]
ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep]
b = beta_inner[level] if level < IdpfPoplar.BITS-1 \
else beta_leaf
if len(b) != IdpfPoplar.VALUE_LEN:
raise ERR_INPUT # beta too long or too short
w_cw = vec_add(vec_sub(b, w0), w1)
# Implementation note: Here we negate the correction word if
# the control bit `ctrl[1]` is set. We avoid branching on the
# value in order to reduce leakage via timing side channels.
mask = Field(1) - Field(2) * Field(ctrl[1].as_unsigned())
for i in range(len(w_cw)):
w_cw[i] *= mask
correction_words.append((seed_cw, ctrl_cw, w_cw))
public_share = IdpfPoplar.encode_public_share(correction_words)
return (public_share, init_seed)
Figure 28: IDPF-key generation algorithm of IdpfPoplar.
• TODO Describe in prose how IDPF-key evaluation algorithm works.¶
The description of the IDPF-evaluation algorithm makes use of auxiliary functions extend(), convert(), and decode_public_share() defined in Section 8.3.3.¶
def eval(IdpfPoplar, agg_id, public_share, init_seed,
level, prefixes, binder):
if agg_id >= IdpfPoplar.SHARES:
raise ERR_INPUT # invalid aggregator ID
if level >= IdpfPoplar.BITS:
raise ERR_INPUT # level too deep
if len(set(prefixes)) != len(prefixes):
raise ERR_INPUT # candidate prefixes are non-unique
correction_words = IdpfPoplar.decode_public_share(public_share)
out_share = []
for prefix in prefixes:
if prefix >= 2 ** (level+1):
raise ERR_INPUT # prefix too long
# The Aggregator's output share is the value of a node of
# the IDPF tree at the given `level`. The node's value is
# computed by traversing the path defined by the candidate
# `prefix`. Each node in the tree is represented by a seed
# (`seed`) and a set of control bits (`ctrl`).
seed = init_seed
ctrl = Field2(agg_id)
for current_level in range(level+1):
bit = (prefix >> (level - current_level)) & 1
# Implementation note: Typically the current round of
# candidate prefixes would have been derived from
# aggregate results computed during previous rounds. For
# example, when using `IdpfPoplar` to compute heavy
# hitters, a string whose hit count exceeded the given
# threshold in the last round would be the prefix of each
# `prefix` in the current round. (See [BBCGGI21,
# Section 5.1].) In this case, part of the path would
# have already been traversed.
# Re-computing nodes along previously traversed paths is
# wasteful. Implementations can eliminate this added
# complexity by caching nodes (i.e., `(seed, ctrl)`
# pairs) output by previous calls to `eval_next()`.
(seed, ctrl, y) = IdpfPoplar.eval_next(
out_share.append(y if agg_id == 0 else vec_neg(y))
return out_share
def eval_next(IdpfPoplar, prev_seed, prev_ctrl,
correction_word, level, bit, binder):
Compute the next node in the IDPF tree along the path determined by
a candidate prefix. The next node is determined by `bit`, the bit of
the prefix corresponding to the next level of the tree.
TODO Consider implementing some version of the optimization
discussed at the end of [BBCGGI21, Appendix C.2]. This could on
average reduce the number of AES calls by a constant factor.
Field = IdpfPoplar.current_field(level)
(seed_cw, ctrl_cw, w_cw) = correction_word
(s, t) = IdpfPoplar.extend(prev_seed, binder)
s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw))
s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw))
t[0] += ctrl_cw[0] * prev_ctrl
t[1] += ctrl_cw[1] * prev_ctrl
next_ctrl = t[bit]
(next_seed, y) = IdpfPoplar.convert(level, s[bit], binder)
# Implementation note: Here we add the correction word to the
# output if `next_ctrl` is set. We avoid branching on the value of
# the control bit in order to reduce side channel leakage.
mask = Field(next_ctrl.as_unsigned())
for i in range(len(y)):
y[i] += w_cw[i] * mask
return (next_seed, next_ctrl, y)
Figure 29: IDPF-evaluation generation algorithm of IdpfPoplar.
def extend(IdpfPoplar, seed, binder):
xof = XofFixedKeyAes128(seed, format_dst(1, 0, 0), binder)
s = [
# Use the least significant bits as the control bit correction,
# and then zero it out. This gives effectively 127 bits of
# security, but reduces the number of AES calls needed by 1/3.
t = [Field2(s[0][0] & 1), Field2(s[1][0] & 1)]
s[0][0] &= 0xFE
s[1][0] &= 0xFE
return (s, t)
def convert(IdpfPoplar, level, seed, binder):
xof = XofFixedKeyAes128(seed, format_dst(1, 0, 1), binder)
next_seed = xof.next(XofFixedKeyAes128.SEED_SIZE)
Field = IdpfPoplar.current_field(level)
w = xof.next_vec(Field, IdpfPoplar.VALUE_LEN)
return (next_seed, w)
def encode_public_share(IdpfPoplar, correction_words):
encoded = Bytes()
control_bits = list(itertools.chain.from_iterable(
cw[1] for cw in correction_words
encoded += pack_bits(control_bits)
for (level, (seed_cw, _, w_cw)) \
in enumerate(correction_words):
Field = IdpfPoplar.current_field(level)
encoded += seed_cw
encoded += Field.encode_vec(w_cw)
return encoded
def decode_public_share(IdpfPoplar, encoded):
l = (2*IdpfPoplar.BITS + 7) // 8
encoded_ctrl, encoded = encoded[:l], encoded[l:]
control_bits = unpack_bits(encoded_ctrl, 2 * IdpfPoplar.BITS)
correction_words = []
for level in range(IdpfPoplar.BITS):
Field = IdpfPoplar.current_field(level)
ctrl_cw = (
control_bits[level * 2],
control_bits[level * 2 + 1],
l = XofFixedKeyAes128.SEED_SIZE
seed_cw, encoded = encoded[:l], encoded[l:]
l = Field.ENCODED_SIZE * IdpfPoplar.VALUE_LEN
encoded_w_cw, encoded = encoded[:l], encoded[l:]
w_cw = Field.decode_vec(encoded_w_cw)
correction_words.append((seed_cw, ctrl_cw, w_cw))
if len(encoded) != 0:
raise ERR_DECODE
return correction_words
Figure 30: Helper functions for IdpfPoplar.
Here, pack_bits() takes a list of bits, packs each group of eight bits into a byte, in LSB to MSB order, padding the most significant bits of the last byte with zeros as necessary, and returns the
byte array. unpack_bits() performs the reverse operation: it takes in a byte array and a number of bits, and returns a list of bits, extracting eight bits from each byte in turn, in LSB to MSB order,
and stopping after the requested number of bits. If the byte array has an incorrect length, or if unused bits in the last bytes are not zero, it throws an error.¶
By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2) and XofTurboShake128 (Section 6.2.1). This VDAF is suitable for any positive value of BITS. Test vectors can be found in Appendix
"Test Vectors".¶
VDAFs have two essential security goals:¶
1. Privacy: An attacker that controls the network, the Collector, and a subset of Clients and Aggregators learns nothing about the measurements of honest Clients beyond what it can deduce from the
aggregate result.¶
2. Robustness: An attacker that controls the network and a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients.¶
Formal definitions of privacy and robustness can be found in [DPRS23]. A VDAF is the core cryptographic primitive of a protocol that achieves the above privacy and robustness goals. It is not
sufficient on its own, however. The application will need to assure a few security properties, for example:¶
• Securely distributing the long-lived parameters, in particular the verification key.¶
• Establishing secure channels:¶
□ Confidential and authentic channels among Aggregators, and between the Aggregators and the Collector; and¶
□ Confidential and Aggregator-authenticated channels between Clients and Aggregators.¶
• Enforcing the non-collusion properties required of the specific VDAF in use.¶
In such an environment, a VDAF provides the high-level privacy property described above: The Collector learns only the aggregate measurement, and nothing about individual measurements aside from what
can be inferred from the aggregate result. The Aggregators learn neither individual measurements nor the aggregate result. The Collector is assured that the aggregate statistic accurately reflects
the inputs as long as the Aggregators correctly executed their role in the VDAF.¶
On their own, VDAFs do not mitigate Sybil attacks [Dou02]. In this attack, the adversary observes a subset of input shares transmitted by a Client it is interested in. It allows the input shares to
be processed, but corrupts and picks bogus measurements for the remaining Clients. Applications can guard against these risks by adding additional controls on report submission, such as Client
authentication and rate limits.¶
VDAFs do not inherently provide differential privacy [Dwo06]. The VDAF approach to private measurement can be viewed as complementary to differential privacy, relying on non-collusion instead of
statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and
removing it after unsharding.¶
The Aggregators are responsible for exchanging the verification key in advance of executing the VDAF. Any procedure is acceptable as long as the following conditions are met:¶
1. To ensure robustness of the computation, the Aggregators MUST NOT reveal the verification key to the Clients. Otherwise, a malicious Client might be able to exploit knowledge of this key to craft
an invalid report that would be accepted by the Aggregators.¶
2. To ensure privacy of the measurements, the Aggregators MUST commit to the verification key prior to processing reports generated by Clients. Otherwise, a malicious Aggregator may be able to craft
a verification key that, for a given report, causes an honest Aggregator to leak information about the measurement during preparation.¶
Meeting these conditions is required in order to leverage security analysis in the framework of [DPRS23]. Their definition of robustness allows the attacker, playing the role of a cohort of malicious
Clients, to submit arbitrary reports to the Aggregators and eavesdrop on their communications as they process them. Security in this model is achievable as long as the verification key is kept secret
from the attacker.¶
The privacy definition of [DPRS23] considers an active attacker that controls the network and a subset of Aggregators; in addition, the attacker is allowed to choose the verification key used by each
honest Aggregator over the course of the experiment. Security is achievable in this model as long as the key is picked at the start of the experiment, prior to any reports being generated. (The model
also requires nonces to be generated at random; see Section 9.2 below.)¶
Meeting these requirements is relatively straightforward. For example, the Aggregators may designate one of their peers to generate the verification key and distribute it to the others. To assure
Clients of key commitment, the Clients and (honest) Aggregators could bind reports to a shared context string derived from the key. For instance, the "task ID" of DAP [DAP] could be set to the hash
of the verification key; then as long as honest Aggregators only consume reports for the task indicated by the Client, forging a new key after the fact would reduce to finding collisions in the
underlying hash function. (Keeping the key secret from the Clients would require the hash function to be one-way.) However, since rotating the key implies rotating the task ID, this scheme would not
allow key rotation over the lifetime of a task.¶
The sharding and preparation steps of VDAF execution depend on a nonce associated with the Client's report. To ensure privacy of the underlying measurement, the Client MUST generate this nonce using
a CSPRNG. This is required in order to leverage security analysis for the privacy definition of [DPRS23], which assumes the nonce is chosen at random prior to generating the report.¶
Other security considerations may require the nonce to be non-repeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a measurement by including it too many
times in a single batch or across multiple batches. It is RECOMMENDED that the nonce generated by the Client be used by the Aggregators for replay protection.¶
As described in Section 4.3 and Section 5.3 respectively, DAFs and VDAFs may impose restrictions on the re-use of input shares. This is to ensure that correlated randomness provided by the Client
through the input share is not used more than once, which might compromise confidentiality of the Client's measurements.¶
Protocols that make use of VDAFs therefore MUST call Vdaf.is_valid on the set of all aggregation parameters used for a Client's input share, and only proceed with the preparation and aggregation
phases if that function call returns True.¶
Aggregating a batch of reports multiple times, each time with a different aggregation parameter, could result in information leakage beyond what is used by the application.¶
For example, when Poplar1 is used for heavy hitters, the Aggregators learn not only the heavy hitters themselves, but also the prefix tree (as defined in Section 8) computed along the way. Indeed,
this leakage is inherent to any construction that uses an IDPF (Section 8.1) in the same way. Depending on the distribution of the measurements, the prefix tree can leak a significant amount of
information about unpopular inputs. For instance, it is possible (though perhaps unlikely) for a large set of non-heavy-hitter values to share a common prefix, which would be leaked by a prefix tree
with a sufficiently small threshold.¶
The only known, general-purpose approach to mitigating this leakage is via differential privacy.¶
• TODO(issue #94) Describe (or point to some description of) the central DP mechanism for Poplar described in [BBCGGI21].¶
As described in Section 6.2, our constructions rely on eXtendable Output Functions (XOFs). In the security analyses of our protocols, these are usually modeled as random oracles. XofTurboShake128 is
designed to be indifferentiable from a random oracle [MRH04], making it a suitable choice for most situations.¶
The one exception is the Idpf implementation IdpfPoplar Section 8.3. Here, a random oracle is not needed to prove privacy, since the analysis of [BBCGGI21], Proposition 1, only requires a
Pseudorandom Generator (PRG). As observed in [GKWY20], a PRG can be instantiated from a correlation-robust hash function H. Informally, correlation robustness requires that for a random r, H(xor(r,
x)) is computationally indistinguishable from a random function of x. A PRG can therefore be constructed as¶
PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ...`
since each individual hash function evaluation is indistinguishable from a random function.¶
Our construction at Section 6.2.2 implements a correlation-robust hash function using fixed-key AES. For security, it assumes that AES with a fixed key can be modeled as a random permutation [GKWY20]
. Additionally, we use a different AES key for every client, which in the ideal cipher model leads to better concrete security [GKWWY20].¶
We note that for robustness, the analysis of [BBCGGI21] still assumes a random oracle to make the Idpf extractable. While XofFixedKeyAes128 has been shown to be differentiable from a random oracle [
GKWWY20], there are no known attacks exploiting this difference. We also stress that even if the Idpf is not extractable, Poplar1 guarantees that every client can contribute to at most one prefix
among the ones being evaluated by the helpers.¶
• TODO Add guidance for choosing PROOFS (Section 7.1.2) for Prio3. In particular when we go for a smaller field for a given circuit. See this for details.¶
A codepoint for each (V)DAF in this document is defined in the table below. Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for private use.¶
Table 17: Unique identifiers for (V)DAFs.
Value Scheme Type Reference
0x00000000 Prio3Count VDAF Section 7.4.1
0x00000001 Prio3Sum VDAF Section 7.4.2
0x00000002 Prio3SumVec VDAF Section 7.4.3
0x00000003 Prio3Histogram VDAF Section 7.4.4
0x00000004 to 0x00000FFF reserved for Prio3 VDAF n/a
0x00001000 Poplar1 VDAF Section 8.4
0xFFFF0000 to 0xFFFFFFFF reserved n/a n/a
• TODO Add IANA considerations for the codepoints summarized in Table 17.¶
The security considerations in Section 9 are based largely on the security analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their time to developing definitions and security
Thanks to Junye Chen, Henry Corrigan-Gibbs, Armando Faz-Hernández, Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana Raykova, Jacob Rothstein, Shan Wang, Xiao Wang, and
Christopher Wood for useful feedback on and contributions to the spec.¶
[TO BE REMOVED BY RFC EDITOR: Machine-readable test vectors can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec.]¶
Test vectors cover the generation of input shares and the conversion of input shares into output shares. Vectors specify the verification key, measurements, aggregation parameter, and any parameters
needed to construct the VDAF. (For example, for Prio3Sum, the user specifies the number of bits for representing each summand.)¶
Byte strings are encoded in hexadecimal. To make the tests deterministic, the random inputs of randomized algorithms were fixed to the byte sequence starting with 0, incrementing by 1, and wrapping
at 256:¶
0, 1, 2, ..., 255, 0, 1, 2, ...
• TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec) and format them for humans.¶
• TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec) and format them for humans.¶
• TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec) and format them for humans.¶
• TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec) and format them for humans.¶
• TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec) and format them for humans.¶ | {"url":"https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-vdaf-08","timestamp":"2024-11-04T14:49:13Z","content_type":"text/html","content_length":"450633","record_id":"<urn:uuid:d7404edd-2598-40ea-b5cd-3496d5038756>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00781.warc.gz"} |
PGD in Supply Chain Management | Canadian Institute of Global Learning
PGD in Supply Chain Management
Supply Chain Management provides an overview of supply chain management as a key business function that holistically integrates functions such as planning, purchasing, inventory control,
transportation and warehousing.
Supply Chain Management provides an overview of supply chain management as a key business function that holistically integrates functions such as planning, purchasing, inventory control,
transportation and warehousing. You will learn about topics such as designing supply and distribution networks aligned with the firm’s business and supply chain strategy as well as improving supply
chain performance via SCOR, Lean, and Six Sigma techniques. The learner explore how various aspects of supply chain management are integrated within the firm as well as coordinated with suppliers,
trading partners, and logistics / Transportation providers to deliver superior customer satisfaction. Course Outcomes Upon completion of this course, one will be able to: 1. Identify and describe the
critical aspects of chain management and explain the concept of and identify supply chain tradeoffs 2. Analyze strategies and tactics for improving supply chain performance 3. Explain how supply
chain management has a financial impact on companies 4. Investigate current trends in supply chain management and explore how global and domestic issues impact the supply chain 5. Effectively balance
supply and demand in a global setting | {"url":"https://ciglcanada.net/courses/pgd-in-supply-chain-management/","timestamp":"2024-11-09T22:28:15Z","content_type":"text/html","content_length":"223548","record_id":"<urn:uuid:8a506021-3c50-4bd1-a428-8e420bb93f7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00136.warc.gz"} |
Orifice Coefficient Calculator - Calculator Wow
Orifice Coefficient Calculator
In fluid dynamics, the orifice coefficient is a critical parameter used to describe the flow characteristics of fluids passing through an orifice. This coefficient quantifies the efficiency of an
orifice in converting pressure into flow rate. An Orifice Coefficient Calculator simplifies the process of determining this coefficient, enabling engineers and scientists to assess and design systems
involving fluid flow accurately. By calculating the orifice coefficient, users can ensure that their systems perform optimally, whether in industrial applications, water management, or research.
Understanding and calculating the orifice coefficient is crucial for several reasons:
1. Accurate Flow Measurement: Determines how effectively an orifice allows fluid to flow, which is essential for accurate flow rate measurements.
2. System Design: Helps in designing and optimizing systems involving fluid flow, such as water treatment facilities, irrigation systems, and chemical processes.
3. Performance Analysis: Assists in analyzing the performance of existing systems and making necessary adjustments for improved efficiency.
4. Regulatory Compliance: Ensures that systems comply with industry standards and regulations concerning fluid dynamics and safety.
5. Cost Efficiency: Optimizes system performance, leading to cost savings in energy consumption and material use.
6. Engineering Precision: Provides precise calculations required for high-stakes engineering projects where fluid flow management is critical.
7. Troubleshooting: Aids in diagnosing and resolving issues related to fluid flow within systems.
8. Quality Control: Ensures that orifices and flow measurement devices are functioning correctly and producing reliable results.
9. Research and Development: Facilitates accurate calculations for R&D projects involving fluid dynamics.
10. Educational Value: Offers a practical tool for teaching and learning about fluid dynamics and hydraulic principles.
How to Use
Using an Orifice Coefficient Calculator involves a few simple steps:
1. Input Flow Rate: Enter the flow rate of the fluid in cubic meters per second (m³/s).
2. Input Cross-Sectional Area: Enter the cross-sectional area of the orifice in square meters (m²).
3. Input Acceleration due to Gravity: Enter the value of gravitational acceleration in meters per second squared (m/s²). The default value is usually 9.81 m/s², but it can be adjusted if needed.
4. Input Height of Fluid: Enter the height of the fluid column above the centerline of the orifice in meters (m).
5. Calculate: Click the calculate button to determine the orifice coefficient (Cd).
If the flow rate is 0.5 m³/s, the orifice area is 0.02 m², the gravitational acceleration is 9.81 m/s², and the height of the fluid is 10 meters, the calculator will provide the orifice coefficient
based on these inputs.
FAQs and Answers
1. What is an orifice coefficient? The orifice coefficient (Cd) measures the efficiency of an orifice in converting pressure into flow rate.
2. Why is the orifice coefficient important? It helps in accurately assessing and designing systems that involve fluid flow.
3. Can the calculator handle different units? The calculator assumes inputs in standard units (m³/s, m², m/s²) but can be adapted for other units with appropriate conversion.
4. What is the default value for gravity? The default value is 9.81 m/s², the standard acceleration due to gravity.
5. How often should I use the orifice coefficient calculator? Use it whenever you need to design, analyze, or optimize systems involving orifices and fluid flow.
6. Can I use this calculator for non-water fluids? Yes, the calculator can be used for any fluid, but ensure that the input values are correctly adjusted for the specific fluid properties.
7. What if I don’t know the height of the fluid? The height of the fluid is necessary for accurate calculations. If unknown, estimate or measure it as accurately as possible.
8. How precise are the calculations? The calculations are precise as long as the input values are accurate. Ensure correct values for optimal results.
9. Is the calculator suitable for educational purposes? Yes, it is a practical tool for teaching and understanding fluid dynamics concepts.
10. Can the calculator be integrated into other software? The basic HTML calculator can be integrated into web-based applications or used standalone.
An Orifice Coefficient Calculator is an invaluable tool for anyone involved in fluid dynamics, engineering, or system design. It provides a straightforward method for calculating the efficiency of an
orifice in facilitating fluid flow, ensuring accurate measurements and optimized system performance. By leveraging this tool, users can enhance their design processes, troubleshoot issues, and
maintain compliance with industry standards. Whether for industrial applications, research, or educational purposes, the Orifice Coefficient Calculator facilitates precision and efficiency in fluid
dynamics. Embrace this tool to achieve reliable results and improve your understanding of fluid flow systems. | {"url":"https://calculatorwow.com/orifice-coefficient-calculator/","timestamp":"2024-11-03T23:36:20Z","content_type":"text/html","content_length":"66231","record_id":"<urn:uuid:cb22a5f2-387c-431b-9b63-0206e01f5ca2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00772.warc.gz"} |
Free-Free Modal Analysis
• Post by: Anurag Gupta
• September 22, 2021
Modal analysis is called the mother of all dynamic analysis because modal analysis is a prerequisite before performing any dynamic analysis. The modal analysis captures the dynamic characteristics of
the vibrating system like natural frequency and mode shapes. The natural frequency is the inherent frequency of the vibrating system at which the system tends to vibrate without the action of any
externally applied forces and the mode shape represents the corresponding vibrating pattern of the system at a specified natural frequency.
If the modal analysis is carried out for a system without any boundary condition or for an unconstrained system, then that type of analysis is referred to as free-free modal analysis. That means
modal analysis is to be performed for a system without any specified boundary condition. In free-free modal analysis, the first six natural frequencies come to be zero or close to zero that signifies
the six rigid body modes (three translational and three rotational). The seventh natural frequency is the first natural frequency for a constrained system.
Free-free modal analysis is to be carried out in the design phase when we are interested to find out the behavior of a system when it is under constrained. Also, this type of analysis is used to
determine the dynamic characteristics of an unconstrained system like aircraft, missiles, ships, etc. There is no deformation observed in the system under rigid body modes. This can be illustrated
using the following case study. In the case study, the modal analysis of the cantilever beam is to be performed both for the case when it is constrained and unconstrained.
Case 1: When the system is unconstrained (no boundary condition is applied)
After doing modal analysis, the first six natural frequencies for a system are:
Mode Frequency [Hz]
3 9.274e-005
4 2.4602e-004
5 2.6525e-004
6 4.1306e-004
The corresponding mode shapes are:
Mode 3
Case 2: When the system is constrained (suitable boundary condition is specified)
After doing modal analysis, the first six natural frequencies for a system are:
Mode Frequency [Hz]
1 9.7846
2 19.569
3 61.321
4 116.08
5 122.64
6 171.74
The corresponding mode shapes are:
Mode 3
Mode 4
So, it is clearly shown from the figures that when the system is unconstrained, there is no internal deformation observed in the system (rigid body modes), and when appropriate boundary conditions
are imposed on that system, there is some internal deformation observed in the system. | {"url":"https://computationalmechanics.in/free-free-modal-analysis/","timestamp":"2024-11-13T21:34:06Z","content_type":"text/html","content_length":"82144","record_id":"<urn:uuid:95656c8c-edb6-449c-a725-092be72f2e53>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00262.warc.gz"} |
Composite Functions | Shiken
Understanding Composite Functions: Simplifying Operations with Multiple Functions
A composite function is the combination of two or more functions. This method is commonly used to transform numbers from one set to another. For example, if we have a function that maps numbers from
set A to set B and another function that maps numbers from set B to set C, we can use a composite function to directly map numbers from set A to set C. The diagram below illustrates this concept:
• Composite functions allow for mapping between sets
It's important to remember these key properties of composite functions:
• In a composite function, the inside function is always performed first, and its output becomes the input for the outside function
• The domain of a composite function is determined by the domain of the first function applied
• When combining two functions, a single input is used, and the output of both functions is obtained
• To find the value of a composite function, we input a value into the original function and then use the resulting output as the input for all other functions
Let's go through an example to see how this process works:
Imagine we have the functions f(x) = 2x + 1 and g(x) = x^2, and we want to find the value of f(g(3)). First, we calculate g(3) = 9. Next, we use this value as the input for f(x), giving us f(9) = 19.
Therefore, f(g(3)) = 19.
Composite functions can become more complex when dealing with quadratic, trigonometric, and reciprocal functions. However, the process remains the same as with simpler linear examples. Let's look at
a few more worked examples:
• To find the value of f(g(x)), where f(x) = 1/x and g(x) = sin(x), we first calculate g(x) = 1/sin(x). Then, we use this as the input for f(x), resulting in f(1/sin(x)).
• Consider the functions f(x) = 2x + 1 and g(x) = x^2. To find the value of f(g(4)), we start by calculating g(4) = 16. Next, we use this value as the input for f(x), resulting in f(16) = 33.
Therefore, f(g(4)) = 33.
Key Takeaways:
• Composite functions are created by combining two or more functions
• The inside function is always performed first in a composite function
• The domain of a composite function is determined by the first function applied
• To find the value, we input a value into the original function and then use the resulting output as the input for all other functions | {"url":"https://shiken.ai/math-topics/composite-functions","timestamp":"2024-11-10T12:33:24Z","content_type":"text/html","content_length":"75230","record_id":"<urn:uuid:ae60f448-04ca-4ecd-b2f1-fa17f6bfb80c>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00390.warc.gz"} |
Boltzmann Distribution from a Microcanonical Ensemble Model
Boltzmann Distribution from a Microcanonical Ensemble Model Documents
This material has 4 associated documents. Select a document title to view a document's information.
Main Document
written by Larry Engelhardt and Norman Chonacky
The Boltzmann Distribution From A Microcanonical Ensemble Model allows students and instructors to explore why the Boltzmann distribution has its characteristic exponential shape. In this model,
particles have only one degree of freedom–the energy to move in one dimension. Further, the density of accessible states is chosen to be uniform –i.e. each state is equally probable. Starting with a
system of N=2 particles, you are able to generate simple empirical results for the distribution of energies among the individual particles. Then you can observe how this distribution changes with
increasing N, gradually approaching the Boltzmann distribution.
The Boltzmann Distribution From A Microcanonical Ensemble Model was developed using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double
clicking the ejs_fmu_BoltzmannDistribution_PartitioningEnergy.jar file will run the program if Java is installed. You can modify this simulation if you have EJS installed by right-clicking within
the map and selecting "Open Ejs Model" from the pop-up menu item.
Last Modified November 22, 2011
Supplemental Documents
A discrete version of the Microcanonical Boltzmann Distribution Model, which partitions energy as discrete quanta rather than allowing the energy to be distribute continuously.
Last Modified August 4, 2014
previous versions.
Source Code Documents (2)
The source code zip archive contains an EJS-XML representation of the Boltzmann Distribution from a Microcanonical Ensemble Model. Unzip this archive in your EJS workspace to compile and run this
model using EJS.
Last Modified November 22, 2011
previous versions.
The source code zip archive contains an EJS-XML representation of the Discrete Boltzmann Distribution Model. Unzip this archive in your EJS workspace to compile and run this model using EJS.
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Nuclear fusion and plasma physics
Weekly outline
• Welcome to the Nuclear Fusion and Plasma Physics course of EPFL.
• When: 13:00- 15:00, Every Monday
• Exercise session: 15:00- 17:00 Every Monday
• The links to the YouTube video lectures from the year 2021 will be made available after each class for reference purposes or as a backup in the event that a class is missed. However, it is
important to note that the most up-to-date information will be presented during the class, and attendance is highly recommended. This year’s lectures will not be recorded unless there is a
change in guidelines. We appreciate your understanding and cooperation in this matter.
* The lecture notes and exercises associated with each lecture will be available on Moodle before the class. The solutions to the exercise will be available after the exercise session.
The course aims to provide the physics and technology basis for controlled fusion research, from the main elements of plasma physics to the reactor concepts.
Course details:
Language English
Credits 4
Session Winter
Semester Fall
Exam Oral
Workload 120 hours
Weeks 14
Hours 4h/week (Lecture- 2h, Exercise- 2h)
1) Basics of thermonuclear fusion
2) The plasma state and its collective effects
3) Charged particle motion and collisional effects
4) Fluid description of a plasma
5) Plasma equilibrium and stability
6) Magnetic confinement: Tokamak and Stellarator
8) Wave-particle interactions
9) Heating and non-inductive current-drive by radiofrequency waves
10) Plasma wall interaction and structural materials
11) Applied superconductivity for fusion
12) Some nuclear aspects of a fusion reactor: Tritium production
13) Licensing a fusion reactor: safety, nuclear waste
14) Recap of the course and Q & A session
Learning Prerequisites: Basic knowledge of electricity and magnetism, and simple concepts of fluids
Learning Outcomes: By the end of the course, the student must be able to:
□ Design the main elements of a fusion reactor
□ Understand the fundamental physics of magnetically confined fusion reactor
□ Identify the main physics challenges on the way to fusion
□ Identify the main technological challenges of fusion
Teaching methods: Ex cathedra and in-class exercises
Assessment method: Oral exam (100%)
□ The exam will be based on a list of questions, divided into three groups.
□ On the day of the exam, half an hour before the actual exam, we will propose you two questions from each group (chosen randomly).
□ You will choose one of the two questions from each group and have half an hour to prepare, using whatever material you find helpful.
□ Then, you will come to the board (without any notes), and we will discuss together the three questions you have chosen for half an hour.
Recommended Bibliography
The asterisk (*) indicates books that contain exercises
Plasma Formulary
MOOC self-paced course links
□ Plasma Introduction course:
□ Plasma Application course:
General Plasma Physics
□ F. F. Chen, Introduction to Plasma Physics, 2nd edition, Plenum Press, 1984*
□ T. J. M. Boyd and J. J. Sanderson, The physics of Plasmas, Cambridge University Press, 2003*
□ P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006*
□ D.A. Gurnett and A. Bhattacharjee, Introduction to Plasma Physics, Cambridge University Press, 2005*
□ D.R. Nicholson, Introduction to Plasma Theory, Jon Wiley & Sons, 1983*
□ N.A. Krall, and A.W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973*
□ J. Freidberg, Plasma Physics and Fusion Energy, Cambridge University Press, 2007*
□ J. Wesson, Tokamaks - Third Edition, Clarendon Press - Oxford, 2004
□ J. Freidberg, Ideal Magnetohydrodynamics, Plenum Press, 1987*
Plasma Diagnostics
□ I. H. Hutchinson, Principles of Plasma Diagnostics, Cambridge University Press, 2nd edition, 2002*
Plasma Application
□ M. A. Lieberman and A. J. Lichtenberg, "Principles of Plasma Discharges and Materials Processing", (John Wiley and Sons, Second Edition, Hoboken, New Jersey), 2005*
□ F. F. Chen and J. P. Chang, "Lecture Notes on Principles of Plasma Processing" (Kluwer Academic / Plenum Publishers, New York), 2003*
□ Y. P. Raizer, "Gas Discharge Physics", (Springer Verlag Berlin Heidelberg), 1991
Plasma Astrophysics
□ E. Priest and T. Forbes, Magnetic reconnection: MHD theory and applications, Cambridge University Press, 2000*
□ N. Myer-Vernet, Basics of the Solar Wind, Cambridge Atmospheric and Space Science Series, 2012*
□ Peter V. Foukal, Wiley, Solar Astrophysics, June 14, 1990
□ P.A. Davidson, An introduction to magnetohydrodynamics, Cambridge University Press, 2001*
□ D.H. Hathaway, The solar cycle, Living Rev. Solar Phys. 7 (2010), 1
Numerical algorithm and PIC method
□ C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulations, McGraw-Hill, 1985*
□ R.W. Hockney and J.W. Eastwood, Computer simulations using particles, Adam Hilger, 1988 | {"url":"https://moodle.epfl.ch/course/view.php?id=14650","timestamp":"2024-11-08T08:26:57Z","content_type":"text/html","content_length":"249527","record_id":"<urn:uuid:ba618994-fcb7-4455-919a-3ea20b415941>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00163.warc.gz"} |
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Today in Science History - Quickie Quiz
Who said: “I have no satisfaction in formulas unless I feel their arithmetical magnitude.”
Dictionary of Science Quotations
Scientist Names Index A
> Archimedes Quotes
┃ │ Archimedes ┃
┃ │ ┃
┃ │ (c. 287 B.C. - c. 212 B.C.) ┃
┃ (source) │ ┃
┃ │ ┃
┃ │ Greek mathematician and engineer who has been called the greatest mathematician among the Greeks. Archimedes is regarded as the first to establish a sound mathematical foundation for ┃
┃ │ engineering. His inventions were both ingenious and useful. Archimedes developed theories in mechanics, hydraulics, and hydrostatics, including the Archimedes’ principle, the ┃
┃ │ Archimedean screw, and levers. ┃
Science Quotes by Archimedes (9 quotes)
δος μοι που στω και κινω την γην — Dos moi pou sto kai kino taen gaen (in epigram form, as given by Pappus, classical Greek).
δος μοι πα στω και τα γαν κινάσω — Dos moi pa sto kai tan gan kinaso (Doric Greek).
Give me a place to stand on and I can move the Earth.
About four centuries before Pappas, but about three centuries after Archimedes lived, Plutarch had written of Archimedes' understanding of the lever:
Archimedes, a kinsman and friend of King Hiero, wrote to him that with a given force, it was possible to move any given weight; and emboldened, as it is said, by the strength of the proof, he
asserted that, if there were another world and he could go to it, he would move this one.
A commonly-seen expanded variation of the aphorism is:
Give me a lever long enough and a place to stand, and I can move the earth.
— Archimedes
As attributed to Pappus (4th century A.D.) and Plutarch (c. 46-120 A.D.), in Sherman K. Stein, Archimedes: What Did He Do Besides Cry Eureka? (1999), 5, where it is also stated that Archimedes knew
that ropes and pulley exploit “the principle of the lever, where distance is traded for force.” Eduard Jan Dijksterhuis, in his book, Archimedes (1956), Vol. 12., 15. writes that Hiero invited
Archimedes to demonstrate his claim on a ship from the royal fleet, drawn up onto land and there loaded with a large crew and freight, and Archimedes easily succeeded. Thomas Little Heath in The
Works of Archimedes (1897), xix-xx, states according to Athenaeus, the mechanical contrivance used was not pulleys as given by Plutarch, but a helix., Heath provides cites for Pappus Synagoge, Book
VIII, 1060; Plutarch, Marcellus, 14; and Athenaeus v. 207 a-b. What all this boils down to, in the opinion of the Webmaster, is the last-stated aphorism would seem to be not the actual words of
Archimedes (c. 287 – 212 B.C.), but restatements of the principle attributed to him, formed by other writers centuries after his lifetime.
Eureka! Eureka!
I have found it!
— Archimedes
Attributed. Supposed to have been his cry, jumping naked from his bath and running in the streets, excited by a discovery about water displacement that solved a problem about the purity of a gold
crown. Archimedes probably never uttered the phrase in that way. We have no written record by Archimedes about the Eureka! story. The earliest reference is by Vitruvius, a Roman writer, in the
introduction of his ninth book of architecture, c. 1st Century B.C. Being nearly 200 years after the presumed event, his story of Archimedes is likely exaggerated. See the Vitruvius quote
A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when placed in the fluid, be lighter than its true weight by the weight of the fluid
— Archimedes
In Thomas L. Heath (ed.), 'On Floating Bodies', The Works of Archimedes (1897), Vol. 1, Proposition 7, 258.
Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.
— Archimedes
In Thomas L. Heath (ed.), 'On Floating Bodies', The Works of Archimedes (1897), Vol. 1, Proposition 5, 257.
Archimedes to Eratosthenes greeting. … certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the
said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired by the method, some knowledge of the questions, to supply the proof than it is to
find it without any previous knowledge.
— Archimedes
As translated by Thomas L. Heath in The Method of Archimedes (1912), 12.
Give me a place to stand, and I will move the earth.
— Archimedes
F. Hultsch (ed.) Pappus Alexandrinus: Collectio (1876-8), Vol. 3, book 8, section 10, ix.
Having been the discoverer of many splendid things, he is said to have asked his friends and relations that, after his death, they should place on his tomb a cylinder enclosing a sphere, writing on
it the proportion of the containing solid to that which is contained.
— Archimedes
Plutarch, Life of Marcellus, 17.12. Trans. R. W. Sharples.
If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been
adjudged perfect in this species of wisdom.
— Archimedes
From a letter to Eratosthenes, the chief librarian at Alexandria, containing the Cattle Problem, an exceedingly difficult calculation involving huge numbers (which was not solved exactly until the
use of a supercomputer in 1981). In David J. Darling, The Universal Book of Mathematics (2004), 23. The debate by scholars regarding whether Archimedes is the true author is in T. L. Heath (ed.), The
Works of Archimedes (1897), xxxiv.
There are things which seem incredible to most men who have not studied mathematics.
— Archimedes
As given, without source, in epigraph, Herbert Meschkowski Ways of Thought of Great Mathematicians: An Approach to the History of Mathematics (1964), 13.
Quotes by others about Archimedes (54)
Hieron asked Archimedes to discover, without damaging it, whether a certain crown or wreath was made of pure gold, or if the goldsmith had fraudulently alloyed it with some baser metal. While
Archimedes was turning the problem over in his mind, he chanced to be in the bath house. There, as he was sitting in the bath, he noticed that the amount of water that was flowing over the top of it
was equal in volume to that part of his body that was immersed. He saw at once a way of solving the problem. He did not delay, but in his joy leaped out of the bath. Rushing naked through the streets
towards his home, he cried out in a loud voice that he had found what he sought. For, as he ran, he repeatedly shouted in Greek; “Eureka! Eurekal I’ve found it! I’ve found it!”
Vitrivius Pollio, De Architectura, ix, prologue, section 10.
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of
whatever it may mean.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 81.
To what heights would science now be raised if Archimedes had made that discovery [of decimal number notation]!
As quoted, without citation, in Eric Temple Bell, Men of Mathematics (1937), 256.
There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea
which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful
inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
Quoted in Return to Mathematical Circles H. Eves (Boston 1988).
ARCHIMEDES. On hearing his name, shout “Eureka!” Or else: “Give me a fulcrum and I will move the world”. There is also Archimedes’ screw, but you are not expected to know what that is.
The Dictionary of Accepted Ideas (1881), trans. Jaques Barzun (1968), 15.
One day at Fenner's (the university cricket ground at Cambridge), just before the last war, G. H. Hardy and I were talking about Einstein. Hardy had met him several times, and I had recently returned
from visiting him. Hardy was saying that in his lifetime there had only been two men in the world, in all the fields of human achievement, science, literature, politics, anything you like, who
qualified for the Bradman class. For those not familiar with cricket, or with Hardy's personal idiom, I ought to mention that “the Bradman class” denoted the highest kind of excellence: it would
include Shakespeare, Tolstoi, Newton, Archimedes, and maybe a dozen others. Well, said Hardy, there had only been two additions in his lifetime. One was Lenin and the other Einstein.
Variety of Men (1966), 87. First published in Commentary magazine.
There is an astonishing imagination, even in the science of mathematics. … We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
In A Philosophical Dictionary: from the French (1824), 126.
Archimedes said Eureka,
Cos in English he weren't too aversed in,
when he discovered that the volume of a body in the bath,
is equal to the stuff it is immersed in,
That is the law of displacement,
Thats why ships don't sink,
Its a shame he weren't around in 1912,
The Titanic would have made him think.
From lyrics of song Sod’s Law.
We have all heard of the puzzle given to Archimedes…. His finding that the crown was of gold was a discovery; but he invented the method of determining the density of solids. Indeed, discoverers must
generally be inventors; though inventors are not necessarily discoverers.
From 'How Discoveries Are Made', Cassell's Magazine, Illustrated (May 1908), 629.
The reason I cannot really say that I positively enjoy nature is that I do not quite realize what it is that I enjoy. A work of art, on the other hand, I can grasp. I can — if I may put it this way —
find that Archimedian point, and as soon as I have found it, everything is readily clear for me. Then I am able to pursue this one main idea and see how all the details serve to illuminate it.
Søren Kierkegaard, translation by Howard Vincent Hong and Edna Hatlestad Hong Søren Kierkegaard’s Journal and Papers (1834), 50.
[King Hiero II] requested Archimedes to consider [whether a crown was pure gold or alloyed with silver]. The latter, while the case was still on his mind, happened to go to the bath, and on getting
into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moment’s delay, and transported
with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, “Eὕρηκα, εὕρηκα.”
This famous anecdote, being written about two centuries after Archimedes, is of questionable authenticity, but Vitruvius provided the origin of the story as we know it. In De Architectura, Book 9,
Introduction, Sec. 10. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 254. Also seen translated as “While Archimedes was turning the problem over, he
chanced to come to the place of bathing, and there, as he was sitting down in the tub, he noticed that the amount of water which flowed over the tub was equal to the amount by which his body was
immersed. This showed him a means of solving the problem. … In his joy, he leapt out of the tub and, rushing naked towards his home, he cried out with a loud voice that he had found what he sought.”
In Ivor Bulmer-Thomas, Selections Illustrating the History of Greek Mathematics (1939), 37.
Archimedes [indicates] that there can be no true levelling by means of water, because he holds that water has not a level surface, but is of a spherical form, having its centre at the centre of the
In De Architectura, Book 8, Chap 5, Sec. 3. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 243.
One can argue that mathematics is a human activity deeply rooted in reality, and permanently returning to reality. From counting on one’s fingers to moon-landing to Google, we are doing mathematics
in order to understand, create, and handle things, … Mathematicians are thus more or less responsible actors of human history, like Archimedes helping to defend Syracuse (and to save a local tyrant),
Alan Turing cryptanalyzing Marshal Rommel’s intercepted military dispatches to Berlin, or John von Neumann suggesting high altitude detonation as an efficient tactic of bombing.
In 'Mathematical Knowledge: Internal, Social and Cultural Aspects', Mathematics As Metaphor: Selected Essays (2007), 3.
Once a mathematical result is proven to the satisfaction of the discipline, it doesn’t need to be re-evaluated in the light of new evidence or refuted, unless it contains a mistake. If it was true
for Archimedes, then it is true today.
In 'The Unplanned Impact of Mathematics', Nature (14 Jul 2011), 475, No. 7355, 166.
My interest in Science had many roots. Some came from my mother … while I was in my early teens. She fell in love with science,… [from] classes on the Foundations of Physical Science. … I was
infected by [her] professor second hand, through hundreds of hours of conversations at my mother’s knees. It was from my mother that I first learned of Archimedes, Leonardo da Vinci, Galileo, Kepler,
Newton, and Darwin. We spent hours together collecting single-celled organisms from a local pond and watching them with a microscope.
From 'Richard E. Smalley: Biographical', collected in Tore Frängsmyr (ed.), Les Prix Nobel: The Nobel Prizes 1996 (1997).
In a manner which matches the fortuity, if not the consequence, of Archimedes’ bath and Newton’s apple, the [3.6 million year old] fossil footprints were eventually noticed one evening in September
1976 by the paleontologist Andrew Hill, who fell while avoiding a ball of elephant dung hurled at him by the ecologist David Western.
Missing Links: The Hunt for Earliest Man
Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics. … The day
(when having discovered his famous principle of hydrostatics he ran through the streets shouting Eureka! Eureka!) ought to be celebrated as the birthday of mathematical physics; the science came of
age when Newton sat in his orchard.
In An Introduction to Mathematics (1911), 37.
Of all the sciences that pertain to reason, Metaphysics and Geometry are those in which imagination plays the greatest part. … Imagination acts no less in a geometer who creates than in a poet who
invents. It is true that they operate differently on their object. The first shears it down and analyzes it, the second puts it together and embellishes it. … Of all the great men of antiquity,
Archimedes is perhaps the one who most deserves to be placed beside Homer.
From the original French: “La Métaphysique & la Géométrie sont de toutes les Sciences qui appartiennent à la raison, celles où l’imagination à le plus de part. … L’imagination dans un Géometre qui
crée, n’agit pas moins que dans un Poëte qui invente. Il est vrai qu’ils operent différemment sur leur objet; le premier le dépouille & l’analyse, le second le compose & l’embellit. … De tous les
grands hommes de l’antiquité, Archimede est peut-être celui qui mérite le plus d’être placé à côté d’Homere.” In Discours Preliminaire de L'Encyclopedie (1751), xvi. As translated by Richard N.
Schwab and Walter E. Rex, Preliminary Discourse to the Encyclopedia of Diderot (1963, 1995), xxxvi. A footnote states “Note that ‘geometer’ in d’Alembert’s definition is a term that includes all
mathematicians and is not strictly limited to practitioners of geometry alone.” Also seen in a variant extract and translation: “Thus metaphysics and mathematics are, among all the sciences that
belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this …. The imagination in a mathematician who
creates makes no less difference than in a poet who invents…. Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.” This latter translation may be
from The Plan of the French Encyclopædia: Or Universal Dictionary of Arts, Sciences, Trades and Manufactures (1751). Webmaster has not yet been able to check for a verified citation for this
translation. Can you help?
Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?
Letter (26 Aug 1822) to his Aunt Mary Hutton. In Robert Perceval Graves, Life of Sir William Rowan Hamilton (1882), 110.
He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.
Quoted, without citation, in Max Dehn, 'Mathematics, 300 B.C.-200 B.C.', The American Mathematical Monthly (Jan 1944), 51, No. 1, 31.
Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
In Ernst Mach and Thomas J. McCormack, Knowledge and Error: Sketches on the Psychology of Enquiry (1976, 2012), Chap. 4, 45. First published in German as Erkenntnis und Irrtum (1905). Translated from
5th edition (1926). The context is the legend that Archimedes was too engrossed in his work to pay attention to an invading Roman soldier, who killed him.
Commenting on Archimedes, for whom he also had a boundless admiration, Gauss remarked that he could not understand how Archimedes failed to invent the decimal system of numeration or its equivalent
(with some base other than 10). … This oversight Gauss regarded as the greatest calamity in the history of science.
In Men of Mathematics (1937), 256.
Being perpetually charmed by his familiar siren, that is, by his geometry, he [Archimedes] neglected to eat and drink and took no care of his person; that he was often carried by force to the baths,
and when there he would trace geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely
possessed by his science.
As translated in George Finlay Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, (1992), 39.
There is no sect in geometry; we never say,—An Euclidian, an Archimedian.
In 'Sects' Philosophical Dictionary (2010), 269.
The treatises [of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern
elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
In A History of Greek Mathematics (1921), Vol. 1, 20.
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life. What would we not give to make it possible for us to steal a look at a book that will serve
primary schools in a hundred years?
In Souvenirs d’Enfance et de Jeunesse (1846), Preface, 13. From the original French, “Le simple écolier sait maintenant des vérités pour lesquelles Archimède eût sacrifié sa vie. Que ne
donnerions-nous pas pour qu’il nous fût possible de jeter un coup d’œil furtif sur tel livre qui servira aux écoles primaires dans cent ans?”
Theology, Mr. Fortune found, is a more accommodating subject than mathematics; its technique of exposition allows greater latitude. For instance when you are gravelled for matter there is always the
moral to fall back upon. Comparisons too may be drawn, leading cases cited, types and antetypes analysed and anecdotes introduced. Except for Archimedes mathematics is singularly naked of anecdotes.
In Mr. Fortune’s Maggot (1927), 168.
No Roman ever died in contemplation over a geometrical diagram.
Referring to the death of Archimedes, to show the difference between the Greek and Roman mind. As quoted, without citation, in Howard W. Eves, Mathematical Circles Squared (1972), 153.
When the Romans besieged the town [Sicily] (in 212 to 210 B.C.), he [Archimedes] is said to have burned their ships by concentrating on them, by means of mirrors, the sun’s rays. The story is highly
improbable, but is good evidence of the reputation which he had gained among his contemporaries for his knowledge of optics.
In An Introduction to Mathematics (1911), 37.
Following the example of Archimedes who wished his tomb decorated with his most beautiful discovery in geometry and ordered it inscribed with a cylinder circumscribed by a sphere, James Bernoulli
requested that his tomb be inscribed with his logarithmic spiral together with the words, “Eadem mutata resurgo,” a happy allusion to the hope of the Christians, which is in a way symbolized by the
properties of that curve.
From 'Eloge de M. Bernoulli', Oeuvres de Fontenelle, t. 5 (1768), 112. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 143-144. [The Latin
phrase, Eadem numero mutata resurgo means as “Though changed, I arise again exactly the same”. —Webmaster]
One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes
approaches the solution of any one of his main problems. Yet this very characteristic, with its incidental effects, is calculated to excite the more admiration because the method suggests the tactics
of some great strategist who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very
elaboration of the scheme has almost obscured, in the mind of the spectator, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which
is not immediately obvious but which we find infallibly used later on; and we are led by such easy stages that the difficulties of the original problem, as presented at the outset, are scarcely
appreciated. As Plutarch says: “It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations.” But it is decidedly a rhetorical exaggeration
when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that anyone could have discovered them for himself. On the contrary, the studied simplicity
and the perfect finish of the treatises involve at the same time an element of mystery. Though each step depends on the preceding ones, we are left in the dark as to how they were suggested to
Archimedes. There is, in fact, much truth in a remark by Wallis to the effect that he seems “as it were of set purpose to have covered up the traces of his investigation as if he had grudged
posterity the secret of his method of inquiry while he wished to extort from them assent to his results.” Wallis adds with equal reason that not only Archimedes but nearly all the ancients so hid
away from posterity their method of Analysis (though it is certain that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old.
In The Works of Archimedes (1897), Preface, vi.
There is something sublime in the secrecy in which the really great deeds of the mathematician are done. No popular applause follows the act; neither contemporary nor succeeding generations of the
people understand it. The geometer must be tried by his peers, and those who truly deserve the title of geometer or analyst have usually been unable to find so many as twelve living peers to form a
jury. Archimedes so far outstripped his competitors in the race, that more than a thousand years elapsed before any man appeared, able to sit in judgment on his work, and to say how far he had really
gone. And in judging of those men whose names are worthy of being mentioned in connection with his,—Galileo, Descartes, Leibnitz, Newton, and the mathematicians created by Leibnitz and Newton’s
calculus,—we are forced to depend upon their testimony of one another. They are too far above our reach for us to judge of them.
In 'Imagination in Mathematics', North American Review, 86, 223.
In the year 1692, James Bernoulli, discussing the logarithmic spiral [or equiangular spiral, ρ = α^θ] … shows that it reproduces itself in its evolute, its involute, and its caustics of both
reflection and refraction, and then adds: “But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I
have thought that it might not be inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however
it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, simillima filia matri. Or, if it is not forbidden to compare
a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light from light,
remains ὁμοούσιος with him, howsoever overshadowed. Or, if you prefer, since our spira mirabilis remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a
symbol, either of fortitude and constancy in adversity, or, of the human body, which after all its changes, even after death, will be restored to its exact and perfect self, so that, indeed, if the
fashion of Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, ‘Though changed, I arise again exactly the same, Eadem numero mutata resurgo.’”
In 'The Uses of Mathesis', Bibliotheca Sacra, Vol. 32, 516-516. [The Latin phrase “simillima filia matri” roughly translates as “the daughter resembles the mother”. “Spira mirabilis” is Latin for
“marvellous spiral”. The Greek word (?µ???s???) translates as “consubstantial”, meaning of the same substance or essence (used especially of the three persons of the Trinity in Christian theology).
If a mathematician of the past, an Archimedes or even a Descartes, could view the field of geometry in its present condition, the first feature to impress him would be its lack of concreteness. There
are whole classes of geometric theories which proceed not only without models and diagrams, but without the slightest (apparent) use of spatial intuition. In the main this is due, to the power of the
analytic instruments of investigations as compared with the purely geometric.
In 'The Present Problems in Geometry', Bulletin American Mathematical Society (1906), 286.
The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.
In Elementarmathematik von höheren Standpunkte aus (1909), Bd. 2, 392.
Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing
what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical
knowledge by all his reading of those approved mathematicians.
In Conduct of the Understanding, sect. 24.
Archimedes … had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into
it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed
accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labor and many men; and, loading her with many passengers and a full freight,
sitting himself the while far off with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and
evenly, as if she had been in the sea. The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive
and defensive, of a siege. … the apparatus was, in most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.
In John Dryden (trans.), Life of Marcellus.
These machines [used in the defense of the Syracusans against the Romans under Marcellus] he [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in
geometry; in compliance with king Hiero’s desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic
truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of
mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by
words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these
mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato’s indignation at it, and his invectives against it as the
mere corruption and annihilation of the one good of geometry,—which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not
to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a
military art.
In John Dryden (trans.), Life of Marcellus.
Archimedes possessed so high a spirit, so profound a soul, and such treasures of highly scientific knowledge, that though these inventions [used to defend Syracuse against the Romans] had now
obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of
engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs
of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of
the methods and means of proof, most deserve our admiration.
In John Dryden (trans.), Life of Marcellus.
Nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes
upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him,
commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through. Others write, that a
Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was
at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate, that as Archimedes was carrying to Marcellus mathematical
instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain
it is, that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honoured them with signal
In John Dryden (trans.), Life of Marcellus.
[Archimedes] is said to have requested his friends and relations that when he was dead, they would place over his tomb a sphere containing a cylinder, inscribing it with the ratio which the
containing solid bears to the contained.
In John Dryden (trans.), Life of Marcellus.
It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius;
while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet,
once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.
In John Dryden (trans.), Life of Marcellus.
Call Archimedes from his buried tomb
Upon the plain of vanished Syracuse,
And feelingly the sage shall make report
How insecure, how baseless in itself,
Is the philosophy, whose sway depends
On mere material instruments—how weak
Those arts, and high inventions, if unpropped
By virtue.
In 'The Excursion', as quoted in review, 'The Excursion, Being a Portion of the Recluse, a Poem, The Edinburgh Review (Nov 1814), 24, No. 47, 26.
This [the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind] accounts for the extraordinary longevity of all the greatest masters of the Analytic
art, the Dii Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who
made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems,
lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the
contrary, when he was slain by the impatient and ill mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem;
Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) originated, the second founder of geometry, the inventor of the matchless
theorem which goes by his name, the pre-cognizer of the undoubtedly mis-called Copernican theory, the discoverer of the regular solids and the musical canon who stands at the very apex of this
pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Græcia, married a young wife when past
60, and died, carrying on his work with energy unspent to the last, at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor do its pores
become clogged with the earthy particles blown from the dusty highways of vulgar life.
In Presidential Address to the British Association, Collected Mathematical Papers, Vol. 2 (1908), 658.
Archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful. It was with difficulty that he was induced to stoop from speculation to
practice. He was half ashamed of those inventions which were the wonder of hostile nations, and always spoke of them slightingly as mere amusements, as trifles in which a mathematician might be
suffered to relax his mind after intense application to the higher parts of his science.
In Lord Bacon', Edinburgh Review (Jul 1887), in Critical and Miscellaneous Essays (1879), Vol. 1, 395.
To Archimedes once came a youth intent upon knowledge.
Said he “Initiate me into the Science divine,
Which to our country has borne glorious fruits in abundance,
And which the walls of the town ’gainst the Sambuca protects.”
“Callst thou the science divine? It is so,” the wise man responded;
“But so it was, my son, ere the state by her service was blest.
Would’st thou have fruit of her only? Mortals with that can provide thee,
He who the goddess would woo, seek not the woman in her.”
Poem, 'Archimedes und der Schuler', collected in Gedichte von Friedrich Schiller (1807), Vol. 1, 149. English version 'Archimedes and the Student', in Edgar A. Bowring (trans.), The Poems of Schiller
(1875), 262-263. From the original German: Zu Archimedes kam einst ein wissbegieriger Jüngling. / “Weihe mich,” sprach er zu ihm, “ein in die gottliche Kunst, / Die so herrliche Frucht dem Vaterlande
getragen, / Und die Mauern der Stadt vor der Sambuca beschützt!” / “Gottlieb nennst du die Kunst? Sie ists,” versetzte der Weise; / “Aber das war sie, mein Sohn, eh sie dem Staat noch gedient. /
Willst du nur Früchte von ihr, die kann auch die Sterbliche zeugen; / Wer um die Göttin freit, suche in ihr nicht das Weib.” [Note: “Sambuca” is the name of a machine used in sieges, employed by
Marcellus against Syracuse.]
He [Lord Bacon] appears to have been utterly ignorant of the discoveries which had just been made by Kepler’s calculations … he does not say a word about Napier’s Logarithms, which had been published
only nine years before and reprinted more than once in the interval. He complained that no considerable advance had been made in Geometry beyond Euclid, without taking any notice of what had been
done by Archimedes and Apollonius. He saw the importance of determining accurately the specific gravities of different substances, and himself attempted to form a table of them by a rude process of
his own, without knowing of the more scientific though still imperfect methods previously employed by Archimedes, Ghetaldus and Porta. He speaks of the εὕρηκα of Archimedes in a manner which implies
that he did not clearly appreciate either the problem to be solved or the principles upon which the solution depended. In reviewing the progress of Mechanics, he makes no mention either of
Archimedes, or Stevinus, Galileo, Guldinus, or Ghetaldus. He makes no allusion to the theory of Equilibrium. He observes that a ball of one pound weight will fall nearly as fast through the air as a
ball of two, without alluding to the theory of acceleration of falling bodies, which had been made known by Galileo more than thirty years before. He proposed an inquiry with regard to the
lever,—namely, whether in a balance with arms of different length but equal weight the distance from the fulcrum has any effect upon the inclination—though the theory of the lever was as well
understood in his own time as it is now. … He speaks of the poles of the earth as fixed, in a manner which seems to imply that he was not acquainted with the precession of the equinoxes; and in
another place, of the north pole being above and the south pole below, as a reason why in our hemisphere the north winds predominate over the south.
From Spedding’s 'Preface' to De Interpretations Naturae Proœmium, in The Works of Francis Bacon (1857), Vol. 3, 511-512. [Note: the Greek word “εὕρηκα” is “Eureka” —Webmaster.]
Büchsel in his reminiscences from the life of a country parson relates that he sought his recreation in Lacroix’s Differential Calculus and thus found intellectual refreshment for his calling.
Instances like this make manifest the great advantage which occupation with mathematics affords to one who lives remote from the city and is compelled to forego the pleasures of art. The entrancing
charm of mathematics, which captivates every one who devotes himself to it, and which is comparable to the fine frenzy under whose ban the poet completes his work, has ever been incomprehensible to
the spectator and has often caused the enthusiastic mathematician to be held in derision. A classic illustration is the example of Archimedes….
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 22. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book
(1914), 186. From the original German, “Wenn Büchsel in seinen Erinnerungen aus dem Leben eines Landgeistlichen erzählt, dass er in der Differentialrechnung von Lacroix Erholung gesucht und geistige
Erfrischung ftir seinen Beruf gefunden habe, so erkennen wir darin den grossen Vorzug, den die Beschaftigung mit der Mathematik für jemanden hat, der fern von einer Stadt lebt und auf ihre
Kunstgenüsse verzichten muss. Der berückende Zauber der Mathematik, dem jeder unterliegt, der sich ihr ergiebt, und der dem holden Wahnsinn vergleichbar ist, unter dessen Bann der Dichter sein Work
vollendet, ist dem betrachtenden Mitmenschen immer unbegreiflich gewesen und hat den begeisterten Mathematiker oft zum Gespött werden lassen. Als klassisches Beispiel wird jedem Schüler Archimedes…”
When Archimedes jumped out of his bath one morning and cried Eureka he obviously had not worked out the whole principle on which the specific gravity of various bodies could be determined j and
undoubtedly there were people who laughed at his first attempts. That is perhaps why most scientific pioneers are so slow to disclose the nature of their first insights when they believe themselves
to be on a track of a new discovery.
In Max Planck and James Vincent Murphy (trans.), Where Is Science Going? (1932), 177.
I have no health for a soldier, and as I have no expectation of serving my country in that way, I am spending my time in the old trifling manner, and am so taken with optics, that I do not know
whether, if the enemy should invade this part of the country, as Archimedes was slain while making geometrical figures on the sand, so I should die making a telescope.
From Letter (20 Sep 1756) to Thomas Barton, as quoted in Florian Cajori, The Teaching and History of Mathematics in the United States (1890), No. 167, 39. [Note: During the American Revolutionary
War, he contributed his skills for military engineering.]
There are notable examples enough of demonstration outside of mathematics, and it may be said that Aristotle has already given some in his “Prior Analytics.” In fact logic is as susceptible of
demonstration as geometry, … Archimedes is the first, whose works we have, who has practised the art of demonstration upon an occasion where he is treating of physics, as he has done in his book on
Equilibrium. Furthermore, jurists may be said to have many good demonstrations; especially the ancient Roman jurists, whose fragments have been preserved to us in the Pandects.
In G.W. Leibniz and Alfred Gideon Langley (trans.), New Essay on Human Understanding (1896), Bk. 4, Chap. 2, Sec. 9, 414-415.
Archimedes had discovered the truth about several important natural laws, but more significant—at least from Galileo’s standpoint—was Archimedes’s discovery of a way for a scientist to solve
problems: first separating what he truly wants to solve from irrelevant externals and then attacking the core of the problem with boldness and imagination. Galileo realized that this approach was
suitable for his own studies.
In Galileo and Newton (1964), 23.
Lord Kelvin had, in a manner hardly and perhaps never equalled before, except by Archimedes, the power of theorizing on the darkest, most obscure, and most intimate secrets of Nature, and at the same
time, and almost in the same breath, carrying out effectively and practically some engineering feat, or carrying to a successful issue some engineering invention. He was one of the leaders in the
movement which has compelled all modern engineers worthy of the name to be themselves men not merely of practice, but of theory, to carry out engineering undertakings in the spirit of true scientific
inquiry and with an eye fixed on the rapidly growing knowledge of the mechanics of Nature, which can only be acquired by the patient work of physicists and mathematicians in their laboratories and
In Speech (May 1921) to the Institute of Civil Engineers, to award the newly created Kelvin Medal. As quoted in Sarah Knowles Bolton, 'Lord Kelvin', Famous Men of Science (1889, Revised Ed. 1926),
In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view
from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time
something like that happened in politics or religion. (1987) --
Carl Sagan
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We have previously discussed Shannon Capacity of CDMA and OFMDA, here we will discuss it again in a bit more detail. Let us assume that we have 20 MHz bandwidth for both the systems which is divided
amongst 20 users. For OFDMA we assume that each user gets 1 MHz bandwidth and there are no guard bands or pilot carriers. For CDMA we assume that each user utilizes full 20 MHz bandwidth. We can say
that for OFDMA each user has a dedicated channel whereas for CDMA the channel is shared between 20 simultaneous users.
We know that Shannon Capacity is given as
or in the case of CDMA
where ‘B’ is the bandwidth and SINR is the signal to noise plus interference ratio. For OFDMA the SNR is given as
where ‘Pu’ is the signal power of a single user and ‘No’ is the Noise Power Spectral Density. For CDMA the calculation of SINR is a bit more complicated as we have to take into account the Multiple
Access Interference. If the total number of users is ‘u’ the SINR is calculated as
The code given below plots the capacity of CDMA and OFDMA as a function of Noise Power Spectral Density ‘No’.
% CAPACITY OF CDMA and OFDMA
% u - Number of users
% Pu - Power of a single user
% No - Noise Power Spectral Density
% Copyright RAYmaps (www.raymaps.com)
clear all
close all
plot(No,C_CDMA/1e6);hold on
plot(No,C_OFDMA/1e6,'r');hold off
xlabel('Noise Power Spectral Density (No)')
ylabel('Capacity (Mbps)')
Shannon Capacity of CDMA and OFDMA
We see that the capacity of OFDMA is much more sensitive to noise than CDMA. Within the low noise region the capacity of OFDMA is much better than CDMA but as the noise increases the capacity of the
two schemes converges. In fact it was seen that as the noise PSD is further increased the two curves completely overlap each other. Therefore it can be concluded that OFDMA is the preferred technique
when we are operating in the high SNR regime.
Does Shannon Capacity Increase by Dividing a Frequency Band into Narrow Bins
Somebody recently asked me this question “Does Shannon Capacity Increase by Dividing a Frequency Band into Narrow Bins”. To be honest I was momentarily confused and thought that this may be the case
since many of the modern Digital Communication Systems do use narrow frequency bins e.g. LTE. But on closer inspection I found that the Shannon Capacity does not change, in fact it remains exactly
the same. Following is the reasoning for that.
Shannon Capacity is calculated as:
Now if the bandwidth ‘B’ is divided into 10 equal blocks then the transmit power ‘P’ for each block would also be divided by 10 to keep the total transmit power for the entire band to be constant.
This means that the factor P/(B*No) remains constant. So the total capacity for the 10 blocks would be calculated as:
So the Shannon Capacity for the entire band remains the same.
PS: The reason for the narrower channels is that for a narrow channel the channel appears relatively flat in the frequency domain and the process of equilization is thus simplified (a simple
multiplication/division would do).
Note: ‘No’ is the Noise Power Spectral Density and ‘B*No’ is the Noise Power.
Inside Qualcomm Snapdragon S4
We have previously looked at the antennas inside a cell phone. Now we look at another important component of a cell phone; the mobile station modem (MSM). One of the most popular MSM in cell phones
today is the Qualcomm Snapdragon S4. The details of this MSM are given in the table below.
Qualcomm Snapdragon S4
As can be seen from the above table this small chipset (can easily fit on a fingertip) packs a punch as far as processing power is concerned. It supports a number of wireless standards from GSM/GPRS
to LTE and from CDMA 2000 to TD-SCDMA. One of its close competitors is the NVIDIA Tegra 3 which has four ARM Cortex A9 cores (compared to Snapdragon’s two).
Implementing a Non-Uniformly Spaced Tapped Delay Line Channel Model
Since you are good on fundamentals I would like to ask you a question that puzzles me. LTE channels models are defined at irregular time intervals as shown in [1].
The EPA, EVA and ETU channel taps can best be described as being sampled at multiples of 10 nsec. However, LTE signal is sampled at multiples of 3.84 MHz (Ts=260.416667 nsec). So how does one perform
convolution operation.
Empirical multipath channel is usually characterized as a τ-spaced tapped delay line (TDL), whose power delay profile (PDP) is either uniformly spaced, or more frequently, spaced with arbitrary time
delay(s). Converting a τ-spaced empirical model to a more tractable T-spaced (sampling time-spaced) statistical model will greatly facilitate the link-level simulation and performance evaluation of
the broadband digital wireless system. The practical approaches to accomplish this task could be either approximation or interpolation,
1. Approximation
a. Ceil or Floor a τ-spaced tap to the neighbouring sampling time.
b. Split the energy of a τ-spaced tap to the adjacent sampling times.
2. Interpolation
An Ideal Bandlimited (sinc) Interpolator can fulfil the goal.
It should be noted that although T-spaced statistical model is simpler for analysis, τ-spaced empirical model represents the channel more accurately.
[1] http://www.steepestascent.com/content/mediaassets/html/LTE/Help/PropagationConditions.html
WINNER-II Path Loss Model
In simple terms the path loss is the difference between the transmitted power and the received power of a wireless communication system. This may range from tens of dB to more than a 100 dB e.g. if
the transmitted power of a wireless communication system is 30 dBm and the received power is -90 dBm then the path loss is calculated as 30-(-90)=120 dB. Path loss is sometimes categorized as a large
scale effect (in contrast to fading which is a small scale effect).
According to the WINNER-II model the path loss can be calculated as:
WINNER-II Path Loss Equation
Here d is the separation between the transmitter and receiver in meters, f[c] is the frequency in GHz, A is the path loss exponent, B is the intercept and C is the frequency dependent parameter. X is
the environment specific parameter such as path loss due to a wall. PL[free] is the path loss in a free space line of sight environment (here A=20, B=46.4 and C=20).
The table below describes the different environments defined in the WINNER-II model. Once an environment is selected the path loss parameters A, B and C can be selected from the table further down
e.g. A1 is the in-building scenario with A=18.7, B=46.8 and C=20 for the LOS case. The PL for a T-R separation of 100 m and frequency of 2 GHz is calculated as:
PL=18.7*log10(100)+46.8+20*log10(2/5)=76.42 dB
A separate equation for the path loss is given where the parameters A, B and C are not sufficient to describe the scenario.
WINNER-II Propagation Scenarios
WINNER-II Path Loss Models
1. Here CG is the concept group that developed the particular scenario. This is either Local Area (LA), Metropolitan Area (MA) or Wide Area (WA).
2. For more details visit:
L. Hentilä, P. Kyösti, M. Käske, M. Narandzic , and M. Alatossava. (2007, December.) MATLAB implementation of the WINNER Phase II Channel Model ver1.1 [Online]. Available: https://www.ist-winner.org/
Soft Frequency Reuse in LTE
Frequency Reuse is a well known concept that has been applied to wireless systems over the past two decades e.g. in GSM systems. As the name suggests Frequency Reuse implies using the same
frequencies over different geographical areas. If we have a 25MHz band then we can have 125 GSM channels and 125*8=1000 time multiplexed users in a given geographical area. Now if we want to increase
the number of users we would have to reuse the same frequency band in a geographically separated area. The technique usually adopted is to use a fraction of the total frequency band in each cell such
that no two neighbor cells use the same frequency. Typically the frequency band is divided into 3 or 7 cells.
7-Cell Frequency Reuse in GSM
The division of the frequency band in to smaller chunks reduces the system capacity e.g. one cell with 25 MHz bandwidth would have much higher capacity then 7 cells having 3.5 MHz each. To overcome
this problem a frequency reuse of 1 has been proposed i.e. each cell has the full system bandwidth (nearly). The problem of co-channel interference at the cell boundaries is resolved by dedicating a
small chunk of the available spectrum for the cell edges.
Frequency Planning and Power Allocation for SFR Scheme
In Soft Frequency Reuse (SFR) the cell area is divided into two regions; a central region where all of the frequency band is available and a cell edge area where only a small fraction of the spectrum
is available. The spectrum dedicated for the cell edge may also be used in the central region if it is not being used at the cell edge. The lack of spectrum at the cell edge may result in much
reduced Shannon Capacity for that region. This is overcome by allocating high power carriers to the users in this region thus improving the SINR and the Shannon Capacity.
1. The Signal to Interference and Noise Ratio is given as:
SINR=Signal Power/(Intercell Interference+Intracell Interference+AWGN Noise)
2. Typically the term capacity was used to describe the number of voice channels (or users) that a system can support. But with modern digital communication systems it usually refers to the Shannon
Capacity that can be achieved (in bits/sec/Hz).
[1] Yiwei Yu, Eryk Dutkiewicz, Xiaojing Huang, Markus Mueck and Gengfa Fang, “Performance Analysis of Soft Frequency Reuse for Inter-cell Interference Coordination in LTE Networks”, ISCIT 2010. | {"url":"https://www.raymaps.com/index.php/tag/4g/","timestamp":"2024-11-11T17:20:34Z","content_type":"text/html","content_length":"81733","record_id":"<urn:uuid:257c253d-801e-49d7-bb3f-1b634b706bd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00514.warc.gz"} |
Estimating the Expected Value in Excel: A Guide for Probability Analysis - Support Your Tech
Estimating the Expected Value in Excel: A Guide for Probability Analysis
If you’re looking to dive into the world of probability analysis, estimating expected value is a great place to start. It’s a pretty nifty way to predict outcomes in situations that involve
uncertainty, like rolling dice or even stock market investments. And guess what? You don’t need to be a math whiz to do it – all you need is Excel! In just a few simple steps, you’ll be able to
calculate expected values like a pro.
Step by Step Tutorial: Estimating the Expected Value in Excel
Before we jump into the nitty-gritty, let’s talk about what we’re going to do. We’ll be using Excel to estimate the expected value for a set of probabilities and outcomes. This is basically a way to
find out what you can expect on average if you repeat a certain random process many times.
Step 1: Enter your probabilities and outcomes
First things first, grab your data and pop it into Excel. You’ll want your probabilities in one column and the corresponding outcomes in the next column.
Entering your probabilities and outcomes into Excel is the foundation for calculating expected value. Make sure your probabilities add up to 1, as this represents 100% of the possible outcomes.
Step 2: Multiply probabilities by outcomes
Next up, you’re going to create a formula in Excel that multiplies each probability by its corresponding outcome.
This step is where the magic happens. By multiplying each probability by its outcome, you’re finding the contribution of each outcome to the expected value.
Step 3: Sum up the products
After you’ve got all your products, it’s time to add them all up. This is your expected value.
The sum of the products gives you a single number, the expected value, which is a measure of the central tendency of the probabilities and outcomes.
Once you’ve completed these steps, you’ll have your expected value! This number is super handy for making predictions and decisions in uncertain situations.
Tips for Estimating the Expected Value in Excel
Here are some tips to make sure your expected value estimation is spot on:
• Double-check your probabilities to ensure they add up to 1.
• Make sure you’re using the correct formula syntax in Excel.
• Review your outcomes to ensure they correspond to the correct probabilities.
• Use absolute references in your formulas if you plan to copy them across cells.
• Remember that the expected value is an average, not a guarantee.
Frequently Asked Questions
What is expected value?
Expected value is a statistical concept used to predict the average result of a random event when repeated many times.
Why do we use Excel for calculating expected value?
Excel is a powerful tool that can handle complex calculations with ease, making it perfect for estimating expected values.
Can expected value be negative?
Yes, expected value can be negative, especially in situations where the outcomes can be losses.
Is expected value the same as mean?
Expected value is similar to the mean, as it represents an average. However, expected value specifically relates to random events and their probabilities.
How accurate is the expected value?
The accuracy of the expected value depends on the quality of your data and assumptions. It’s a prediction, not a certainty.
1. Enter your probabilities and outcomes.
2. Multiply probabilities by outcomes.
3. Sum up the products.
Estimating the expected value in Excel is a fantastic skill for anyone dealing with probability analysis. Whether you’re a student, a business analyst, or just someone who likes to play the odds,
mastering this technique can help you make more informed decisions. It’s all about understanding the risks and potential rewards in uncertain scenarios. Plus, it’s pretty satisfying to crunch those
numbers and come out with a clear answer, isn’t it? So go ahead, give it a try – who knows what valuable insights you’ll discover? Happy calculating!
Matt Jacobs has been working as an IT consultant for small businesses since receiving his Master’s degree in 2003. While he still does some consulting work, his primary focus now is on creating
technology support content for SupportYourTech.com.
His work can be found on many websites and focuses on topics such as Microsoft Office, Apple devices, Android devices, Photoshop, and more. | {"url":"https://www.supportyourtech.com/excel/estimating-the-expected-value-in-excel-a-guide-for-probability-analysis/","timestamp":"2024-11-07T07:44:59Z","content_type":"text/html","content_length":"121712","record_id":"<urn:uuid:0113fb0d-efe2-45f6-8766-5f00830f89cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00496.warc.gz"} |
Being certain about uncertainty
The hydraulic properties of soils, such as available water, macroporosity, and field capacity, are needed in many applications, including the management of irrigation, calculations of soil leaching
losses, and management of agricultural activities on certain soils, as well as in long-term studies on the effect of climate and land use.
The data needed to estimate the hydraulic properties for these applications can be difficult or costly to gather directly from detailed field and laboratory measurement for each management task, and
it is more common to use modelling to estimate the properties. These models range from very simple summaries by soil type (e.g. available water for each soil order), to more sophisticated models that
account for soil type, texture, soil layer, and other properties that may be available.
A recently developed model for soil hydrological properties in New Zealand developed by Manaaki Whenua – Landcare Research uses information available from S-map, and estimates common properties in a
consistent and physically plausible manner. For example, field capacity and wilting point estimates are available for S-map polygons, but in addition available water estimates (defined in terms of
the difference between field capacity and wilting point) are also modelled so that the estimated value is always positive. Modelling soil properties in this way ensures that the estimated properties
have sensible values – never below zero, for instance.
While it is important to model soil properties correctly so that the values are physically sensible, it is also important to know how accurate the estimates are likely to be. Modelling accuracy, or
more correctly the uncertainty of the estimates, usually arises from the structure of the model used to perform the estimates, so the details are frequently mathematical. But the uncertainty
estimates themselves depend critically on field data, and how those field data relate to the range of soils one might expect to encounter over New Zealand. For instance, if there are many field
samples for one soil type but relatively few for another, we would expect larger uncertainty of estimates for the second soil type. When modelling involves many factors, such as soil type, texture
(sand, silt, and clay), soil layer information, and so on, it can be difficult to visualise how uncertainty estimates change across the landscape.
For the S-map soil hydrology models, the estimates of soil properties are built from data in the National Soils Database (NSD), which has detailed soil-hydrology measurements from several hundred
sites around New Zealand, all of which have matching S-map descriptions. However, these data are unbalanced, with 240 non-Allophanic Brown samples (for example), but only 40 Pumice samples.
One way to express uncertainty is to estimate the average root mean squared error (RMSE) for the property (e.g. field capacity), expressed as a percentage of the estimated value, and calculated
separately for each soil order. For the S-map soil hydrology models, the RMSE is about 8% of the estimated response value for all properties, regardless of the soil order. But this measure of
uncertainty has its limitations: the RMSE is averaged over all samples in a soil order, so unusually high or low uncertainty is masked as a result.
A more practical approach to estimating uncertainty is to calculate the likely range of values that an estimate of the soil property might take, say 95% of the time, but calculated for every
prediction that is carried out. While time-consuming, this approach conveys more practical information.
For example, consider estimating the total available water (TAW) for a (hypothetical) sample of non-Allophanic Brown soil with sand, silt, and clay content of 33.3, 33.3, and 33.3% respectively. The
estimated TAW in this case is 16%, with a 95% confidence interval of [12,22]%. For Pumice soils with the same texture content, the TAW estimate is higher at 35%, but the 95% confidence interval is
[10,72]%, which is over six times wider than the uncertainty for the non-Allophanic Brown soil.
There are several reasons for the difference in the uncertainty in the case of non-Allophanic Brown and Pumice. First, there are far fewer Pumice samples (40) in NSD with soil-hydrology field
measurements when compared with non-Allophanic Brown samples (240). Second, the texture range of the Pumice soils is narrower, with NSD samples having a sand content of 35 to 91% and clay from 0 to
25%, while for non-Allophanic Brown NSD data the sand content ranges from 0 to 100% and clay from 0 to 70%. So, for the case of Pumice, estimating TAW means extrapolation beyond the range of the
available field data. Fortunately, in this case the confidence interval enables us to detect that the TAW estimate is inaccurate.
One way to visualise the uncertainty is to plot the TAW estimates and the 95% confidence intervals for both non-Allophanic Browns and Pumice soils over the soil texture triangle (see Fig. 1). The
estimates are presented in Figure 1 as contours as well as colours. The predicted TAW uncertainty for non-Allophanic Browns shows little variation over the most of the texture triangle, except for
low sand/high silt content. By contrast, the uncertainty for Pumice soils is at least twice as large, and the estimated uncertainty increases as the clay content increases. Using these uncertainty
calculations, it is possible to have some confidence in the integrity of the estimates of soil properties, and we know when we ought to reject the prediction as being too inaccurate for practical
How can the problem of wide uncertainty limits be fixed? Better modelling helps, but the real solution is to gather more field data across a wider range of sites across New Zealand, especially for
soil types where there are few available NSD samples, such as Granular, Pumice, and Oxidic soils. | {"url":"https://www.landcareresearch.co.nz/publications/soil-horizons/soil-horizons-articles/being-certain-about-uncertainty/","timestamp":"2024-11-09T13:49:33Z","content_type":"text/html","content_length":"123880","record_id":"<urn:uuid:8cad15d6-d080-4a98-9f16-836c644534be>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00377.warc.gz"} |
Multiple interactions
5.11. Multiple interactions¶
The basic MPI model is described in [SvZ87] while Sherpa’s implementation details are discussed in [A+a].
The following parameters are used to steer the MPI setup:
Specifies the MPI handler. The two possible values at the moment are None and Amisic.
Amisic can simulate the interaction of three different combinations of incoming particles: proton–proton, photon–proton and photon–photon collision. The parameters for the simulation of photonic
multiple interactions can be found in [SS97]. It has several parameters to control the simulation of the multiple-parton interactions, they are listed below. Each of these parameters has to be set in
the subsetting AMISIC, like so
The usual rules for yaml structure apply, c.f. Input structure.
Value \(p_\text{T,0}^\text{(ref)}\) for the calculation of the IR regulator, see formula below. Defaults to 2.5.
Value \(p_\text{T,min}^\text{(ref)}\) for the calculation of the IR cutoff, see formula below. Defaults to 3.
The pseudorapidity \(\eta\) used to calculate the IR cutoff and regulator, \(p_\text{T,min}\) and \(p_\text{T,0}\). Defaults to 0.16.
Reference energy to normalise the actual cms energy for the calculation of the IR cutoff and regulator. Defaults to 7000.
The IR cut-off for the 2->2 scatters. It is calculated as
\[p_\text{T,min} = p_\text{T,min}^\text{(ref)} \left( \frac{E_\text{cms}}{E_\text{cms}^\text{(ref)}} \right)^\eta\]
but can also be set explicitly.
IR regulator \(p_\text{T,0}\) in the propagator and in the strong coupling. It is calculated as
\[p_\text{T,0} = p_\text{T,0}^\text{(ref)} \left( \frac{E_\text{cms}}{E_\text{cms}^\text{(ref)}} \right)^\eta\]
but can also be set explicitly.
Defaults to PT scheme. More schemes have yet to be added.
Factor to scale the renormalisation scale \(\mu_R\), defaults to 0.5.
Factor to scale the factorisation scale \(\mu_F\), defaults to 1.0.
Specifies the factor to scale the non-diffractive cross section calculated in the MPI initialisation. Defaults to 0.4.
Only to be used for double-gaussian matter form, where it will control the distribution of matter over the two gaussians. It assumes that a fraction \(f^2\) is distributed by the inner gaussian \
(r_1\), another fraction \((1-f)^2\) is distributed by the outer gaussian \(r_2\), and the remaining fraction \(2f(1-f)\) is distributed by the combined radius \(r_\text{tot} = \sqrt{\frac{r_1^
2+r_2^2}{2}}\). Defaults to 0.5.
Defaults to 0.4. Is used to control the radius of the (inner) gaussian. If used with the double-gaussian matter form, this value must be smaller than MATTER_RADIUS2.
Defaults to 1.0. It is only used for the case of a double-gaussian overlap, see below.
Defaults to Single_Gaussian. Alternatively, Double_Gaussian can be used to model the overlap between the colliding particles, however, it has not been tested yet.
Controls the number of bins for the numerical integration of
\[\int_{p_T^2}^{s/4} dp_T^2 \frac{d \sigma}{dp_T^2}\]
Defaults to 200.
Number of points to estimate the the cross-section during the integration. The error should behave as \(\frac{1}{\sqrt{n_\text{MC}}}\). Defaults to 1000.
Number of points to sample in the center-of-mass energy \(\sqrt{s}\). This is only used if the energy is not fixed, i.e. in the case of EPA photons. Defaults to 100.
The following two parameters can be used to overwrite the ISR parameters in the context of multiple interactions: MPI_PDF_SET, MPI_PDF_SET_VERSIONS. | {"url":"https://sherpa-team.gitlab.io/sherpa/v3.0.0beta1/manual/parameters/multiple-interactions.html","timestamp":"2024-11-08T09:21:53Z","content_type":"text/html","content_length":"21562","record_id":"<urn:uuid:644e1fcd-2384-46b6-92e8-9d52e530833d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00345.warc.gz"} |
The deconstructed Standard Model equation
The Standard Model of particle physics is often visualized as a table, similar to the periodic table of elements, and used to describe particle properties, such as mass, charge and spin. The table is
also organized to represent how these teeny, tiny bits of matter interact with the fundamental forces of nature.
But it didn’t begin as a table. The grand theory of almost everything actually represents a collection of several mathematical models that proved to be timeless interpretations of the laws of
Here is a brief tour of the topics covered in this gargantuan equation.
The whole thing
This version of the Standard Model is written in the Lagrangian form. The Lagrangian is a fancy way of writing an equation to determine the state of a changing system and explain the maximum possible
energy the system can maintain.
Technically, the Standard Model can be written in several different formulations, but, despite appearances, the Lagrangian is one of the easiest and most compact ways of presenting the theory.
Section 1
These three lines in the Standard Model are ultraspecific to the gluon, the boson that carries the strong force. Gluons come in eight types, interact among themselves and have what’s called a color
Section 2
Almost half of this equation is dedicated to explaining interactions between bosons, particularly W and Z bosons.
Bosons are force-carrying particles, and there are four species of bosons that interact with other particles using three fundamental forces. Photons carry electromagnetism, gluons carry the strong
force and W and Z bosons carry the weak force. The most recently discovered boson, the Higgs boson, is a bit different; its interactions appear in the next part of the equation.
Section 3
This part of the equation describes how elementary matter particles interact with the weak force. According to this formulation, matter particles come in three generations, each with different
masses. The weak force helps massive matter particles decay into less massive matter particles.
This section also includes basic interactions with the Higgs field, from which some elementary particles receive their mass.
Intriguingly, this part of the equation makes an assumption that contradicts discoveries made by physicists in recent years. It incorrectly assumes that particles called neutrinos have no mass.
Section 4
In quantum mechanics, there is no single path or trajectory a particle can take, which means that sometimes redundancies appear in this type of mathematical formulation. To clean up these
redundancies, theorists use virtual particles they call ghosts.
This part of the equation describes how matter particles interact with Higgs ghosts, virtual artifacts from the Higgs field.
Section 5
This last part of the equation includes more ghosts. These ones are called Faddeev-Popov ghosts, and they cancel out redundancies that occur in interactions through the weak force.
Note: Thomas Gutierrez, an assistant professor of Physics at California Polytechnic State University, transcribed the Standard Model Lagrangian for the web. He derived it from Diagrammatica, a
theoretical physics reference written by Nobel Laureate Martinus Veltman. In Gutierrez’s dissemination of the transcript, he noted a sign error he made somewhere in the equation. Good luck finding | {"url":"https://www.symmetrymagazine.org/article/the-deconstructed-standard-model-equation","timestamp":"2024-11-07T22:05:29Z","content_type":"text/html","content_length":"34652","record_id":"<urn:uuid:6d912b6f-a9ac-4fec-a428-c78b7ef04f3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00841.warc.gz"} |
Markup vs Margin: What's the Difference?
Blog > Markup vs Margin: What’s the Difference?
Markup vs Margin: What’s the Difference?
Defining the key differences between margin and markup can yield valuable financial insights for businesses. From understanding the impact on your bottom line to practical examples and real-world
applications, you’ll gain the confidence to make informed decisions that propel your business toward greater profitability.
Markups and margins are commonly used in the financial world but can often be misunderstood or used interchangeably. This article will clarify any confusion around these terms and show your business
how to correctly calculate profit margins and markups.
Understanding the difference between markup and margin
Understanding the difference between markup and margin is crucial for any business owner or finance professional, as it can significantly impact the financial health of a business. Simply put, profit
margin is the percentage of revenue that results in a profit, whereas a markup is the percentage added to the cost of production to arrive at the selling price.
Profit margin is the revenue remaining after deducting the cost of goods sold (COGS), while markup represents the retail price minus the cost of a product. In terms of calculation, margin equals
sales minus COGS, while markup is derived from the selling price subtracted from the cost price.
Let’s dive into the specifics of these terms to help you better understand the difference between profit margin and markup.
Profit margin
Profit margin is a valuable financial metric business professionals use to determine a company’s financial health and stability. It represents the percentage of profit a firm receives from each
dollar of revenue it generates after deducting all the expenses associated with producing and selling its products or services.
A higher profit margin means the company generates more profit per dollar of revenue earned, which is generally acceptable, and is operating efficiently and effectively controlling its costs.
On the other hand, a lower profit margin may indicate that a company is struggling to generate enough profit from its operations due to various reasons, such as higher expenses, lower sales, or
ineffective pricing strategies. In such cases, the company may need to review its cost structure, improve operational efficiency, or adjust pricing to increase its profit margins.
Profit markup
Markup is the amount added to the cost of goods sold (COGS) to determine the selling price of a product or service. Businesses commonly use it to ensure they cover all their expenses and generate a
While markups can help companies determine the minimum selling price for a product or service, it doesn’t factor in other variables like competition or customer demand.
It’s also important to note that a high markup can mean something different than a high-profit margin or revenue. Therefore, it’s crucial to understand markup and profit margin to make informed
pricing decisions and ensure your business remains profitable.
For example, if you only consider markup when pricing your products, you may charge too little or too much for your product, resulting in lost profits or reduced sales. By understanding profit
margin, you can better evaluate the profitability of a product and set prices that maximize profits without sacrificing sales.
Additionally, understanding the difference between profit margin and markup can help you compare the performance of different products, evaluate your competitors’ pricing strategies, and identify
areas where you can cut costs to improve profitability.
Now that you know the differences between profit markups and margins and their significant role, the next step is to learn how to calculate each and apply these formulas in your business.
Knowing the profit margin vs markup formula
The markup vs margin formula is crucial to making informed pricing decisions and ensuring profitability.
Learning how to calculate markup and margin and seeing these calculations applied in examples will help your business understand how to use them in everyday operations.
Profit margin formula
Profit margin is a financial metric measuring a company’s profit for each dollar of revenue earned — it’s the percentage of income that’s profit. The profit margin can provide insight into a
company’s financial health and profitability.
To calculate profit margin, you’ll first need to subtract the cost of goods sold (COGS) from the selling price. Then, divide the result by the selling price and multiply it by 100 to get a
Selling Price – COGS = Profit Margin
(Selling Price – COGS)/ Selling Price = Profit Margin (Percentage)
For instance, if a hardware store sells a power drill for $100 that costs $50 to produce, the profit margin would be $50 (or 50%).
$100 – $50 = $50 Profit Margin
($100 Selling Price -$50 COGS) / $100 Selling Price = 50% Profit Margin Percentage
Understanding your company’s profit margin is crucial to evaluate financial performance and make informed decisions.
Next, you can learn the profit markup formula and see it applied in examples.
Profit markup formula
Markup is the amount added to the cost of production to arrive at the selling price, and it’s expressed as a percentage that’s often used interchangeably with the term markup percentage.
Simply put, markup is the difference between the selling price and the cost of goods sold (COGS).
Selling Price – COGS = Profit Markup
You can also calculate the markup percentage from the selling price and cost of goods sold (COGS). The formula is as follows:
((Selling Price – COGS)/ Selling Price ) x 100 = Markup Percentage
To calculate the selling price, you can multiply the cost of goods sold (COGS) by the markup percentage, then add the COGS back to obtain the selling price.
(COGS x Markup Percentage) + COGS = Selling Price
For instance, let’s say a clothing company produces a shirt for $50, and they want to add a 50% markup to the cost of goods sold (COGS). The selling price would be $75.
($50 x .50) +50 = $75 Selling Price
Now, inputting the selling price into the profit markup formula, we find the clothing store implemented a $25 profit markup.
$75 – $50 = $25 Profit Markup
Understanding your markup is crucial to ensure you’re pricing products correctly and generating sufficient revenue to cover your costs and make a profit.
When should you use margin or markup formulas?
Now that you know the differences between margin and markup and how to calculate them, you must understand which situations to use these formulas.
Margin is a better metric to compare products or services allows you to factor in the different selling prices of each product or service and determine which ones are more profitable. This is because
the margin is calculated as the difference between the selling price and the cost of goods sold (COGS) and expressed as a percentage of the selling price.
For example, if you have two products with the exact cost of goods sold but different selling prices, the one with the higher selling price and margin is more profitable. Here’s an example:
Screwdriver at Bob’s Hardware: ($85 Selling Price – $65 COGS) / $85 Selling Price = 24% Profit Margin
Screwdriver at Richard’s Hardware: ($100 Selling Price – $65 COGS) / $100 Selling Price = 35% Profit Margin
As you can see, the screwdriver at Richard’s Hardware sold for a higher selling price at the same COGS, resulting in a higher profit margin.
In contrast, markup is calculated as the percentage added to the cost of production to arrive at the selling price. Markup doesn’t consider other expenses — overhead costs, taxes, interest payments,
etc. — factored into the selling price.
Therefore, using markup to compare products or services can be misleading because it needs to consider the actual selling price and the costs involved in producing and selling them.
Enhance your financial management with EBizCharge
Managing finances can be daunting for businesses of all sizes, so using valuable formulas like profit markups, profit margins, and other data-driven resources are essential to ensure the most
Businesses can also look to reliable providers like EBizCharge that offer payment processing solutions to simplify payment collections and better manage their finances. With EBizCharge, you can
easily track your revenue, expenses, margins, and markups to make more informed decisions and maximize your profits. | {"url":"https://ebizcharge.com/blog/markup-vs-margin-whats-the-difference/","timestamp":"2024-11-09T19:15:33Z","content_type":"text/html","content_length":"1049437","record_id":"<urn:uuid:1b0b50a1-beac-4f33-8bea-ce0365aae00d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00207.warc.gz"} |
Capacity constraints in job shop scheduling | AIMMS Community
Hi all,
I am creating a job shop scheduling model in AIMMS for my MSc thesis. The model plans jobs j of production orders i backwards from their due date. The main (nonnegative) decision variables are:
• StartTime(i,j) = the start time of production order i job j
• EndTime(i,j) = the end time of production order i job j
The only thing I still need to add is that capacity may not be exceeded. I have the following information:
• Capacity(j) = total capacity of job j in one time bucket
Now I need some counter / variable looking like:
• x(i,j,t) = the amount processed of operation i, job j in time bucket t. Where time bucket 1 = e0,1], time bucket 2 = e1,2] et cetera.
See the figure below for my current data, I want to retrieve the following x(i,j,t) values:
• x(1,COAT,1) = 0.5 (from 0 to 0.5)
• x(1,PRO,1) = 0.5 (from 0.5 to 1)
• x(1,PRO,2) = 0.7 (from 1 to 1.7)
• x(1,PW,2) = 0.3 (from 1.7 to 2)
• x(1,PW,3) = 0.5 (from 2 to 2.5)
Does someone have an idea on constraints to retrieve the values for x(i,j,t) as shown in the example? | {"url":"https://community.aimms.com/math-or-optimization-modeling-39/capacity-constraints-in-job-shop-scheduling-1383?postid=3902","timestamp":"2024-11-02T18:53:22Z","content_type":"text/html","content_length":"158053","record_id":"<urn:uuid:c5d193ba-e16d-4b8e-be1f-0c696e5f4e29>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00278.warc.gz"} |
Introduction to Angles
Once you have mastered the idea of points, lines and planes, the next thing to consider is what happens when two lines or rays meet at a point, creating an angle between them.
Angles are used throughout geometry, to describe shapes such as polygons and polyhedrons, and to explain the behaviour of lines, so it’s a good idea to become familiar with some of the terminology,
and how we measure and describe angles.
What is an Angle?
Angles are formed between two rays extending from a single point:
Angles are commonly drawn as an arc (part of a circle), as above.
Properties of Angles
Angles are measured in degrees, which is a measure of circularity, or rotation.
A full rotation, which would bring you back to face in the same direction, is 360°. A half-circle is therefore 180°, and a quarter-circle, or right angle, is 90°.
Two or more angles on a straight line add up to 180°. In the diagram above, the circle to the left is split into three sectors the angles of the green and white sectors are both 90°, adding up to
The figure to the right shows that angles a and b also add up to 180°. When you look at the diagram like this, it’s easy to see this, but it’s also surprisingly easy to forget in practice.
Naming Different Angles
An angle less than 90° is said to be acute, and one greater than 90° but less than 180° is obtuse.
An angle of exactly 180° is said to be straight. Angles greater than 180° are called reflex angles.
Different angles can be demonstrated on a clock face. The hour hand of the clock moves round as time passes through the day. The angle of the rotation is highlighted in green.
Opposite Angles: Intersecting Lines
When two lines intersect, the opposite angles are equal. In this case, not only are a and a the same, but, of course, a and b add up to 180°:
Intersections with parallel lines: a bit of a special case
Our page An Introduction to Geometry introduces the concept of parallel lines: lines that go on forever side by side and never cross, like railway lines.
The angles around any lines intersecting parallel lines also have some interesting properties.
If two parallel lines (A and B) are intersected by a third straight line (C), then the angle at which the intersecting line crosses will be the same for both parallel lines.
The two angles a and the two angles b are said to be corresponding.
You will also immediately see that a and b add up to 180°, since they are on a straight line.
Angle c, which you will realise from the previous section is identical to a, is said to be alternate with a.
Z and F Angles
c and a are called z-angles, because if you follow the line from the top of c to the bottom of a, it forms the shape of a z (in red in the diagram above).
a and a are said to be F-angles, because the line forms an F shape from the bottom of the upper angle a down and around to the bottom of the lower angle a (in green in the diagram)
Measuring Angles
A protractor is commonly used to measure angles. Protractors are usually circular or semi-circular and made of transparent plastic, so that they can be placed over shapes drawn on a piece of paper,
allowing you to take a measurement of the angle.
This example demonstrates how to use a protractor to measure the three angles of a triangle, but the same method applies to other shapes or any angles that you want to measure.
• Line up the central mark on the base of your protractor with the vertex, or point at which the lines meet. The triangle has three vertices, one for each angle to measure.
• Most protractors have a bi-directional scale meaning that you can take a measurement in either direction. Make sure you use the correct scale – you should be able to tell easily if your angle is
greater than or less than 90° and therefore use the right scale. If you’re not sure, take a quick look back to our section on naming angles.
In this example the recorded angles are A=90° B=45° and C=45°.
Polygons are often defined by their internal angles, and the total of the internal angles depends on the number of sides. For example, the internal angles of a triangle always add up to 180°. For
more about this, see our page on Polygons.
Degrees or Radians?
When we need to measure or describe an angle, we usually use ‘degrees’ as the unit of measurement. However, very occasionally, you may find angles referred to in radians.
The radian is the Standard International (SI) unit of measurement for angles and is used in many areas of science and mathematics.
We said above that the full rotation of angles through a circular arc is equal to 360°. It is also equal to 2π radians, where π (pi) is a special number equalling (approximately) 3.142 (there is more
about π in our page on Special Numbers and Concepts).
One radian is equal to 360/2π = 57.3°. We also use pi when we need to calculate the area or circumference of a circle, or the volume of a sphere (and there is more about this in our page on Curved
Further Reading from Skills You Need
Understanding Geometry
Part of The Skills You Need Guide to Numeracy
This eBook covers the basics of geometry and looks at the properties of shapes, lines and solids. These concepts are built up through the book, with worked examples and opportunities for you to
practise your new skills.
Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.
Moving on…
Once you understand about angles, and how to measure them, you can put this into practice with polygons and polyhedrons of all kinds, and also use your knowledge to calculate area (there is more
about this in our page on Calculating Area). | {"url":"https://www.skillsyouneed.com/num/angles.html","timestamp":"2024-11-07T18:59:11Z","content_type":"text/html","content_length":"47007","record_id":"<urn:uuid:b8650b9e-60eb-4d44-b71a-ab08ef9a9f85>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00392.warc.gz"} |
How can International Tractor Motors use the Binomial Distribution to approximate tractor sales in similar countries? - essayZeus
International Tractor Motors (ITM) had recently undertaken an ad campaign aimed at agricultural owners in a certain country in Asia. The ad campaign was devoted to promoting the new IT-8 large
specialty tractor. Four sales associates were assigned to sell in different medium to large farms throughout the country for many days. One associate was assigned to the northern part of the country.
Another was assigned to the southern part. The other two were assigned to the west and east, respectively. Because of regulations, a sales associate can sell at most one IT-8 tractor per day and only
one such tractor per farm, too. A sales associate was successful if he/she sold a tractor during the day. Thus, ITM can sell at most 4 tractors (4 total sales) per day in this country.
Download the file titled Tractor Successes. It contains a scatter plot of the number of successes versus frequency. To compare the results to the Binomial Distribution, complete the following:
Explain why this tractor sales scenario can be a binomial experiment.
Using the Tractor Successes scatter plot, construct a frequency distribution for the number of successes.
Compute the mean number of successes. The formula for the mean is as follows: LaTeX: \frac{\sum{(x⋅f)}}{\sum f}∑ ( x ⋅ f ) ∑ f
The terms x represent the total number of successes (0, 1, 2, 3, 4) and f is the corresponding frequency (number of days where x successes occurred).
Explain what the numerical result means.
From the frequency distribution, construct the corresponding relative frequency distribution.
Explain why the relative frequency distribution table is a probability distribution.
Then, use Excel to create a scatter plot of the probability distribution:
Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting).
Using the frequency distribution, what is the tractor sales success average? In part 3, note that the numerator in the formula for the mean is the total number of successes. The total number of
trials is the denominator of the formula for the mean multiplied by 4. What does this average mean?
The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of “success” on each trial. Using Excel, construct the Binomial Probability Distribution for four
trials, n, and probability of success, p, as the tractor sales success average in part 5. Here is an explanation of the BINOM.DIST function in Excel: https://support.office.com/en-ie/article/
BINOM-DIST-function-c5ae37b6-f39c-4be2-94c2-509a1480770c?ui=en-US&rs=en-IE&ad=I (Links to an external site.)
For example, In Excel
=BINOM.DIST(7,15,0.7, FALSE)
represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p.
Using the above value of n=4 with probability of success, p, as the tractor sales success average in part 5, what is the probability of at least two successes?
Using the formula for the mean of the Binomial Distribution, what is the mean number of successes in part 6 above?
In Excel, create a scatter plot for the Binomial Distribution. The instructions for creating a scatter plot are in part 4 above.
Use the results above to compare the probability distribution of tractor sales successes and the Binomial Distribution. Compare the means in parts 4 and 6, too.
If the probability distribution of tractor sales successes and the Binomial Distribution differ, explain why that is so.
Do you think the Binomial Distribution is a good model for the tractor sales success scenario? Why or why not?
How can International Tractor Motors use the Binomial Distribution to approximate tractor sales in similar countries?
In what other scenarios can International Tractor Motors use the Binomial Distribution? Explain.
Submit your Excel file in addition to your report.
The paper must be written in third person.
nclude a title page, introduction, body, conclusion, and a reference page.
The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society, or it might discuss or describe the unique
terminology associated with the topic.
The body of your paper should answer the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved.
Be sure the Word document is in paragraph format, not numbered answers like a homework assignment.
The conclusion should summarize your thoughts about what you have determined from the data and your analysis, often with a broader personal or societal perspective in mind. Nothing new should be
introduced in the conclusion that was not previously discussed in the body paragraphs. Your conclusion should emanate from (be aligned with) your findings.
Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment. | {"url":"https://www.essayzeus.com/2019/12/how-can-international-tractor-motors-use-the-binomial-distribution-to-approximate-tractor-sales-in-similar-countries/","timestamp":"2024-11-07T04:34:25Z","content_type":"text/html","content_length":"43362","record_id":"<urn:uuid:aff5e43a-5d62-47b6-8a33-ad7467ef18bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00642.warc.gz"} |
13 research outputs found
Smooth-throat wormholes are treated on as possessing quantum fluctuation energy with scalar massive field as its source. Heat kernel coefficients of the Laplace operator are calculated in background
of the arbitrary-profile throat wormhole with the help of the zeta-function approach. Two specific profile are considered. Some arguments are given that the wormholes may exist. It serves as a
solution of semiclassical Einstein equations in the range of specific values of length and certain radius of wormhole's throat and constant of non-minimal connection.Comment: 28 pages, 4 figures,
We consider the van der Waals energy of an atom near the infinitely thin sphere with finite conductivity which model the fullerene. We put the sphere into spherical cavity inside the infinite
dielectric media, then calculate the energy of vacuum fluctuations in framework of the zeta-function approach. The energy for a single atom is obtained from this expression by consideration of the
rare media. In the limit of the infinite radius of the sphere the Casimir-Polder expression for an atom and plate is recovered. For finite radius of sphere the energy of an atom monotonously falls
down as $d^{-3}$ close to the sphere and $d^{-7}$ far from the sphere. For hydrogen atom on the surface of the fullerene $C_{60}$ we obtain that the energy is $3.8 eV$. We obtain also that the
polarizability of fullerene is merely cube of its radius.Comment: 10 pages, 5 figure
We consider the self-energy and the self-force for an electrically charged particle at rest in the wormhole space-time. We develop general approach and apply it to two specific profiles of the
wormhole throat with singular and with smooth curvature. The self-force for these two profiles is found in manifest form; it is an attractive force. We also find an expression for the self-force in
the case of arbitrary symmetric throat profile. Far from the throat the self-force is always attractive.Comment: 18 pages, 3 figures Comments: corrected pdf, enlarged pape
In the framework of zeta-function approach the Casimir energy for three simple model system: single delta potential, step function potential and three delta potentials is analyzed. It is shown that
the energy contains contributions which are peculiar to the potentials. It is suggested to renormalize the energy using the condition that the energy of infinitely separated potentials is zero which
corresponds to subtraction all terms of asymptotic expansion of zeta-function. The energy obtained in this way obeys all physically reasonable conditions. It is finite in the Dirichlet limit and it
may be attractive or repulsive depending on the strength of potential. The effective action is calculated and it is shown that the surface contribution appears. The renormalization of the effective
action is discussed.Comment: 17 pages, 2 figures, added reference, address correcte
We consider the Casimir effect for quantized massive scalar field with non-conformal coupling $\xi$ in a spacetime of wormhole whose throat is rounded by a spherical shell. In the framework of
zeta-regularization approach we calculate a zero point energy of scalar field. We found that depending on values of coupling $\xi$, a mass of field $m$, and/or the throat's radius $a$ the Casimir
force may be both attractive and repulsive, and even equals to zero.Comment: 2 figures, 10 pages, added 2 reference
The ground state energy of the massive scalar field with non-conformal coupling $\xi$ on the short-throat flat-space wormhole background is calculated by using zeta renormalization approach. We
discuss the renormalization and relevant heat kernel coefficients in detail. We show that the stable configuration of wormholes can exist for $\xi > 0.123$. In particular case of massive conformal
scalar field with $\xi=1/6$, the radius of throat of stable wormhole $a\approx 0.16/m$. The self-consistent wormhole has radius of throat $a\approx 0.0141 l_p$ and mass of scalar boson $m\approx
11.35 m_p$ ($l_p$ and $m_p$ are the Planck length and mass, respectively).Comment: revtex, 18 pages, 3 eps figures. accepted in Phys.Rev.
We consider the self-potential and the self-force for an electrically charged particle at rest in the massive wormhole space-time. We develop general approach for the renormalization of
electromagnetic field of such particle in the static space-times and apply it to the space-time of the wormhole with parameter of the mass, $m$. The self-force is found in manifest form; it is an
attractive force. We discus the peculiarities due to massive parameter of the wormhole.Comment: 10 pages, 1 figure text correcte
We found the contribution to the vacuum expectation value of the energy-momentum tensor of a massive Dirac field due to the conical geometry of the cosmic string space-time. The heat kernel and heat
kernel expansion for the squared Dirac operator in this background are also considered and the first three coefficients were found in an explicity form.Comment: 9 pages, 1 figure (2 ref added)
(enlarged version | {"url":"https://core.ac.uk/search/?q=authors%3A(Nail%20R%20Khusnutdinov)","timestamp":"2024-11-04T18:04:25Z","content_type":"text/html","content_length":"154780","record_id":"<urn:uuid:f518fd41-3970-450d-a1dd-df21f4a4188d>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00682.warc.gz"} |
Kites in Geometry (Definition, Properties & Video)
Kites in Geometry
Fact-checked by
Paul Mazzola
Kite definition geometry
You probably know a kite as that wonderful toy that flies aloft on the wind, tethered to you by string. That toy kite is based on the geometric shape, the kite.
A kite is a quadrilateral shape with two pairs of adjacent (touching), congruent (equal-length) sides. That means a kite is all of this:
• A plane figure
• A closed shape
• A polygon
Kite Definition - Geometry
Sometimes a kite can be a rhombus (four congruent sides), a dart, or even a square (four congruent sides and four congruent interior angles).
Some kites are rhombi, darts, and squares. Not every rhombus or square is a kite. All darts are kites.
Kites Types
Kites can be convex or concave. A dart is a concave kite. That means two of its sides move inward, toward the inside of the shape, and one of the four interior angles is greater than 180°. A dart is
also called a chevron or arrowhead.
Kite and Dart - Geometry
Get free estimates from geometry tutors near you.
How to construct a kite in geometry
You can make a kite. Find four uncooked spaghetti strands. Cut or break two spaghetti strands to be equal to each other, but shorter than the other two strands.
Touch two endpoints of the short strands together. Touch two endpoints of the longer strands together. Now carefully bring the remaining four endpoints together so an endpoint of each short piece
touches an endpoint of each long piece. You have a kite!
How to draw a kite in geometry
You can also draw a kite. Use a protractor, ruler and pencil. Draw a line segment (call it KI) and, from endpoint II, draw another line segment the same length as KI. That new segment will be IT.
The angle those two line segments make (∠I) can be any angle except 180° (a straight angle).
Draw a dashed line to connect endpoints K and T. This is the diagonal that, eventually, will probably be inside the kite. Now use your protractor. Line it up along diagonal KT so the 90° mark is at
∠I. Mark the spot on diagonal KT where the perpendicular touches; that will be the middle of KT.
Drawing a Kite - Geometry
Lightly draw that perpendicular as a dashed line passing through ∠I and the center of diagonal KT. Make that line as long as you like.
If you end the line closer to ∠I than diagonal KT, you will get a dart. If you end the new line further away from ∠I than diagonal KT, you will make a convex kite.
Connect the endpoint of the perpendicular line with endpoint T. Label it point E. Connect point E with point K, creating line segment EK. Notice that line segments (or sides) TE and EK are equal.
Notice that sides KI and IT are equal.
You probably drew your kite so sides KI and EK are not equal. That also means IT and TE are not equal. You could have drawn them all equal, making a rhombus (or a square, if the interior angles are
right angles).
Properties of kites
The kite's sides, angles, and diagonals all have identifying properties.
Kites Properties - Geometry
Kite sides
To be a kite, a quadrilateral must have two pairs of sides that are equal to one another and touching. This makes two pairs of adjacent, congruent sides.
You could have one pair of congruent, adjacent sides but not have a kite. The other two sides could be of unequal lengths. Then you would have only a quadrilateral.
Your kite could have four congruent sides. Your quadrilateral would be a kite (two pairs of adjacent, congruent sides) and a rhombus (four congruent sides).
Some (but not all) kites are rhombi. If your kite/rhombus has four equal interior angles, you also have a square.
Kite angles
Where two unequal-length sides meet in a kite, the interior angle they create will always be equal to its opposite angle. Look at the kite you drew.
It is possible to have all four interior angles equal, making a kite that is also a square.
Kite diagonals
The two diagonals of our kite, KT and IE, intersect at a right angle. In every kite, the diagonals intersect at 90°. Sometimes one of those diagonals could be outside the shape; then you have a dart.
That does not matter; the intersection of diagonals of a kite is always a right angle.
A second identifying property of the diagonals of kites is that one of the diagonals bisects, or halves, the other diagonal. They could both bisect each other, making a square, or only the longer one
could bisect the shorter one.
Lesson summary
For what seems to be a really simple shape, a kite has a lot of interesting features. Using the video and this written lesson, we have learned that a kite is a quadrilateral with two pairs of
adjacent, congruent sides.
Kites can be rhombi, darts, or squares. We also know that the angles created by unequal-length sides are always congruent.
Finally, we know that the kite's diagonals always cross at a right angle and one diagonal always bisects the other. | {"url":"https://tutors.com/lesson/kites-in-geometry-definition-properties","timestamp":"2024-11-03T00:14:24Z","content_type":"text/html","content_length":"220262","record_id":"<urn:uuid:28cbbc05-5162-4b4f-b0b6-957ac2f9f045>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00637.warc.gz"} |
Matlab decomposition objects in parfor
Table of Contents
Matlab decomposition objects in parfor
In Matlab 2017b Mathworks introduced the decomposition class, which precomputes a decomposition of a linear system:
>> A = rand(10000, 10000);
>> b = rand(10000, 1);
>> tic; x = A \ b; toc
Elapsed time is 5.049433 seconds.
>> tic; dA = decomposition(A); toc
Elapsed time is 4.916148 seconds.
>> tic; x2 = dA \ b; toc
Elapsed time is 0.039961 seconds.
>> nnz(x ~= x2)
ans = 0
As can be seen, once the decomposition object is created, this allows solving the system at least two orders of magnitude faster with the exact same results.
The decomposition class is as smart as mldivide (backslash operator for solving Ax = b) and mrdivide (forward slash operator for solving xA = b) and follows the same logic as can be seen in the flow
charts for dense or sparse matrices. That means it recognizes special kinds of system matrices and performs a suitable decomposition that exploits the matrix structure.
Usage in parfor
An idea that quickly comes to mind is using a decomposition object inside of a parfor or e.g. functions like blockproc with 'UseParallel' enabled.
When attempting to do so, you will quickly be disappointed:
Warning: Loading a decomposition is not supported. The loaded object is the decomposition of an empty matrix.
> In decomposition.loadobj (line 673)
In parallel.internal.pool.deserialize (line 29)
In parallel.internal.pool.deserializeFunction (line 12)
In remoteParallelFunction (line 19)
and some subsequent errors like
Error using \ (line 391)
Matrix dimensions must agree.
Matlab's parallel pool works by replicating everything in the workspace for each of the workers. This replication is performed by serialization followed by an unserialization inside of each worker,
i.e. converting all objects into a bytestream as if they were written to disk in a mat file. In fact, those are the same methods that are used when calling save() or load().
For classes, it is possible to overload the methods saveobj() and loadobj() to customize the (un)serialization. The usual pattern in these two methods is to store all properties of an object in the
fields of a struct, or vice versa, to reconstruct an object from such a struct.
Unfortunately, these methods are missing implementations in the decomposition class:
methods (Hidden)
function s = saveobj(~)
s = struct;
methods (Hidden, Static)
function obj = loadobj(~)
obj = decomposition;
Changed decomposition class
Luckily, it is straight forward to implement the (un)serialization for decomposition objects:
methods (Hidden)
function s = saveobj(obj, ~)
mc = metaclass(obj);
props = {mc.PropertyList.Name}';
s = struct;
for ii = 1 : numel(props)
s.(props{ii}) = obj.(props{ii});
methods (Hidden, Static)
function obj = loadobj(s)
fns = fieldnames(s);
obj = decomposition([]);
for ii = 1 : numel(fns)
obj.(fns{ii}) = s.(fns{ii});
Now, when attempting to load a serialized decomposition object, or when using it in parfor and the like, we get:
Warning: While loading an object of class 'decomposition':
> In class 'decomposition', no set method is defined for Dependent property 'Datatype'.
A Dependent property needs a set method to assign its value.
Again, this is easily fixed by adding a suitable set method:
function f = set.Datatype(f, dt)
assert(any(strcmpi(dt, {'single', 'double'})));
f.IsSingle = strcmpi(dt, 'single');
You can add this method right after the get.Datatype() method. You might have to adapt the file permissions of matlab/toolbox/matlab/matfun/decomposition.m to be able to save the file. | {"url":"https://sunl.it/dw/blg:matlab_decomposition_parfor","timestamp":"2024-11-10T22:05:25Z","content_type":"text/html","content_length":"12878","record_id":"<urn:uuid:631db219-5509-404a-85d8-0483bdb9505e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00605.warc.gz"} |
cosmological model
Astrophysics is a subfield of astronomy that explores the physical and chemical properties of celestial objects, including stars, galaxies, and interstellar matter. It aims to understand the origins,
evolution, and interactions of these objects within the universe by applying principles from physics, chemistry, and mathematics. Astrophysicists use a combination of observational data, theoretical
models, and numerical simulations to develop a comprehensive understanding of the universe. Theoretical astrophysics involves creating models and predicting the observational consequences of those
models, which helps observers look for data that can support or refute a model. Some well-known theories and models studied in astrophysics include the Big Bang, cosmic inflation, dark matter, dark
energy, and fundamental theories of physics.
Jean-Pierre Petit, a French scientist, proposed a cosmological model known as the "Janus Cosmological Model," which involves negative matter. In this model, positive matter (ordinary matter) and
negative matter (exotic matter with negative mass) coexist in the universe. The interactions between these two types of matter lead to the observed expansion of the universe, as well as other
phenomena like galaxy rotation curves and the cosmic microwave background radiation. In the Janus Cosmological Model, negative matter has properties that are opposite to those of positive matter. For
example, negative matter has negative mass, negative energy, and negative pressure. While positive matter attracts other positive matter through gravity, negative matter repels it. Conversely,
negative matter attracts other negative matter.
To better understand the Janus Cosmological Model, let's consider a simplified version with only two particles: one positive and one negative. The gravitational force between these particles can be
described using Newton's law of universal gravitation:
$F = G * (m1 * m2) / r^2$
Here, $F$ is the gravitational force, $G$ is the gravitational constant, $m1$ and $m2$ are the masses of the particles, and $r$ is the distance between them. If $m1$ is positive and $m2$ is negative,
the force $F$ will be negative, which means the particles will repel each other. In the context of the universe, this model suggests that the positive matter (galaxies, stars, etc.) and negative
matter (invisible and undetectable) are constantly repelling each other, causing the observed expansion of the universe. This expansion is consistent with the observations of the cosmic microwave
background radiation and the redshift of distant galaxies.
However, it's important to note that the Janus Cosmological Model is not widely accepted in the scientific community. Many scientists favor other models, such as the ΛCDM (Lambda Cold Dark Matter)
model, which is based on the existence of dark matter and dark energy. The Janus model is still an interesting and creative idea, but more research and evidence are needed to determine its validity
in explaining the observed phenomena in our universe.
Attention ! Billet en cour d'écriture !
What evidence supports the idea of negative matter in the Janus Cosmological Model?
Evidence supporting the idea of negative matter in the Janus Cosmological Model comes from various aspects of the model's construction and its ability to explain observed phenomena in the universe.
Here are some key pieces of evidence:
1. Mathematical and physical coherence: The Janus model is built on a system of two coupled field equations that are derived from general relativity, providing a solid mathematical foundation for
the introduction of negative matter jp-petit.org.
2. Resolution of the runaway paradox: In general relativity, the introduction of negative mass leads to the runaway paradox, where a positive mass and a negative mass would experience unbounded
acceleration. The Janus model resolves this paradox by having the two types of matter evolve along their own families of geodesics in separate metrics. This prevents the runaway motion and allows
for the coexistence of positive and negative mass in the universe januscosmologicalmodel.com.
3. Explanation of dark matter and dark energy: The Janus model attributes the observed effects of dark matter and dark energy to the presence of negative mass in the universe. Negative mass
particles surrounding galaxies can explain their confinement (dark matter role) and drive the expansion of the positive sector, responsible for the observed acceleration (dark energy role)
4. Negative gravitational lensing: The Janus model predicts that negative mass would induce a negative gravitational lensing effect, which could be tested by observing the large-scale structure of
the universe and the distribution of mass within cosmic voids. This would provide direct evidence for the existence of negative mass januscosmologicalmodel.com.
5. Compatibility with local relativistic observations: The Janus model reduces to the Einstein field equations of general relativity in regions where positive mass density dominates, such as on
Earth or in the solar system. This compatibility with local relativistic observations and measurements lends further support to the model's validity januscosmologicalmodel.com.
While these pieces of evidence support the idea of negative matter in the Janus Cosmological Model, it is important to note that this model is not widely accepted in the scientific community, and
more research and evidence are needed to determine its validity in explaining the observed phenomena in our universe.
Counter aruguments to Janus cosmological model ?
While the Janus cosmological model presents both mathematical and physical coherence and has numerous observational confirmations jp-petit.org, there are potential counterarguments to consider:
1. Limited observational data: Although the Janus model has been shown to fit data from 740 distant supernovae link.springer.com, it is important to note that the model has not yet been subjected to
a comprehensive comparison with a wide range of observational data. As more data becomes available, it will be necessary to test the model's predictions against these observations to determine
its validity and robustness.
2. Compatibility with established theories: The Janus model challenges mainstream ΛCDM cosmological model and offers an alternative explanation for dark matter and dark energy by introducing
negative mass particles ptep-online.com. However, the ΛCDM model, which is based on general relativity and the standard model of particle physics, has been extensively tested and confirmed by a
wide range of observations. The Janus model needs to demonstrate that it can consistently explain these observations and be compatible with established theories to gain wider acceptance.
3. Simplicity and Occam's razor: The Janus model introduces additional complexity by proposing two coupled field equations to describe the behavior of particles with inverted mass, energy, and
momentum januscosmologicalmodel.com. According to Occam's razor, the simplest theory that can explain the observed phenomena should be preferred. The Janus model needs to demonstrate that its
additional complexity is necessary and justified in comparison to simpler models, such as the ΛCDM model.
4. Runaway paradox: Although the Janus model claims to resolve the runaway paradox (which arises when positive and negative mass particles interact in general relativity) by introducing two coupled
field equations, it is essential to rigorously analyze these interactions to ensure that the paradox is indeed resolved ptep-online.com. The model needs to provide a comprehensive explanation of
how the paradox is avoided under various conditions and scenarios.
Overall, while the Janus cosmological model presents a potentially interesting alternative to mainstream cosmological models, it is essential to subject it to further scrutiny and comparison with
observational data and established theories to determine its viability and robustness.
Mathematical foundations of the Janus model's two coupled field equations?
The Janus Cosmological Model's mathematical foundation is built on a system of two coupled field equations derived from general relativity. These equations describe the behavior of positive and
negative mass particles in the universe. The model is formulated in a Riemannian manifold associated with two coupled metrics, one for positive mass and the other for negative mass researchgate.net.
The Lagrangian derivation of the two coupled field equations is presented in a paper by Jean-Pierre Petit and G. D'Agostini link.springer.com. The authors start with a review of the model's results
and then proceed to derive the field equations using the Lagrangian approach. The Lagrangian formulation provides a convenient way to derive the equations of motion for a system and is widely used in
classical mechanics and field theory.
In the Janus model, the two coupled field equations are derived from a single Lagrangian density, which is a scalar function of the metric tensors, their first derivatives, and the matter fields. By
applying the Euler-Lagrange equations, the authors obtain the field equations for both positive and negative mass particles semanticscholar.org.
The Janus model also incorporates ideas from symplectic geometry and dynamical group theory, which help explain the relationship between time reversal, energy inversion, and mass inversion for
particles januscosmologicalmodel.com.
The Janus system of two coupled field equations has been independently derived using Cartan's equivalence method and the Hodge star operator, further supporting its mathematical foundation
The relationship between time reversal, energy inversion, and mass inversion
In the Janus model, the relationship between time reversal, energy inversion, and mass inversion is connected to the model's symmetries. These symmetries play a crucial role in explaining the
behavior of positive and negative mass particles in the universe.
Time reversal symmetry in physics refers to the idea that the laws of physics should remain unchanged if time is reversed. In other words, if you run a physical process backward in time, the same
physical laws should still apply. In the Janus model, time reversal symmetry is related to the inversion of mass and energy vixra.org.
According to the principle of Maupertuis, the time inversion symmetry is equivalent to the symmetry of mass inversion. This idea suggests that the behavior of a system under time reversal should be
the same as its behavior when the masses of its particles are inverted (i.e., positive mass becomes negative mass and vice versa) vixra.org.
The Janus model captures this symmetry through the use of two coupled field equations, one for positive mass particles and the other for negative mass particles. These equations describe the behavior
of particles with inverted mass, energy, and momentum vixra.org.
Furthermore, in the Janus model, unitary time reversal is associated with the inversion of mass sign, as mass is energy at rest (E = µc²). By applying time reversal to the model's equations, the
exponential functions remain invariant. This implies that the model can handle both positive and negative mass particles under time reversal, effectively capturing the relationship between time
reversal, energy inversion, and mass inversion modernscientificpress.com.
Can we provide an example of a physical process that would exhibit time reversal symmetry in the Janus model?
In the Janus model, an example of a physical process that would exhibit time reversal symmetry is the motion of charged particles in the presence of a constant external magnetic field. In classical
physics, the time reversal operation involves reversing the velocities and time while keeping the coordinates unchanged. However, this operation is not a symmetry for the system in the presence of a
magnetic field en.wikipedia.org.
Nonetheless, it has been shown that it is possible to find other time reversal operations that preserve the dynamics of the system and maintain time reversal symmetry, even in the presence of a
magnetic field. This implies that the presence of a magnetic field does not always break time reversal symmetry, and the motion of charged particles in such a field can still exhibit time reversal
symmetry under certain conditions en.wikipedia.org.
In the context of the Janus model, this example demonstrates that the model's time reversal symmetry can be maintained for specific physical processes, even when external factors such as magnetic
fields are present. This is consistent with the model's ability to handle both positive and negative mass particles under time reversal, as well as its connection to energy inversion and mass
inversion britannica.com.
How does the Janus model handle positive and negative mass particles under time reversal?
In the Janus model, positive and negative mass particles are handled under time reversal by maintaining the model's time reversal symmetry through the use of two coupled field equations. These
equations describe the behavior of particles with inverted mass, energy, and momentum, allowing for a consistent treatment of both positive and negative mass particles under time reversal
The model's time reversal symmetry is connected to the principle of Maupertuis, which suggests that the behavior of a system under time reversal should be the same as its behavior when the masses of
its particles are inverted. This idea is consistent with the Janus model's ability to handle both positive and negative mass particles under time reversal, as well as its connection to energy
inversion and mass inversion researchgate.net.
What is the principle of Maupertuis in detail ?
The principle of Maupertuis, named after Pierre Louis Maupertuis, is a concept in classical mechanics that states that the path followed by a physical system is the one of least length (with a
suitable interpretation of path and length). It is a special case of the more generally stated principle of least action en.wikipedia.org.
Maupertuis originally applied the principle to the laws of reflection and refraction of light in 1744, suggesting that nature always follows the simplest path when realizing its actions. He later
published a more general law of motion and equilibrium in 1746, stating that when a change occurs in nature, the quantity of action necessary for this change is minimized. The quantity of action is
defined as the product of the masses of the bodies, their speeds, and the distances over which they move encyclopediaofmath.org.
Leonhard Euler later showed that, for an isolated body under the action of central forces, the principle of least action holds true when the integral of the velocity and the arc element is minimized
or maximized encyclopediaofmath.org.
In the context of the Janus model, the principle of Maupertuis is connected to time reversal symmetry, suggesting that the behavior of a system under time reversal should be the same as its behavior
when the masses of its particles are inverted (i.e., positive mass becomes negative mass and vice versa) jstor.org. This idea is consistent with the model's ability to handle both positive and
negative mass particles under time reversal, as well as its connection to energy inversion and mass inversion.
How the Janus model challenges the ΛCDM model?
The Janus model challenges the ΛCDM model in several ways:
1. Alternative explanation for dark matter and dark energy: The ΛCDM model posits the existence of dark matter and dark energy as major components of the universe, with dark matter responsible for
the observed structure of galaxies and dark energy driving the accelerated expansion of the universe. The Janus model, on the other hand, proposes an alternative explanation by introducing
negative mass particles. These particles interact with positive mass particles in a manner that could potentially explain the observed phenomena attributed to dark matter and dark energy
2. Challenging the cosmological principle: The ΛCDM model relies on the cosmological principle, which states that the universe is homogeneous and isotropic on large scales. Recent findings have
suggested that violations of the cosmological principle, especially of isotropy, exist en.wikipedia.org. The Janus model, with its introduction of negative mass particles and their interactions,
could potentially provide an alternative framework that does not rely on the cosmological principle.
3. Addressing discrepancies and tensions: The ΛCDM model faces several challenges, such as the cosmological lithium problem, the S8 tension, and the "Axis of Evil" en.wikipedia.org. The Janus model,
by offering a different approach to understanding the universe, could potentially address some of these discrepancies and tensions.
4. Time reversal symmetry: The Janus model is built around the principle of time reversal symmetry, which is connected to the principle of Maupertuis. This principle suggests that the behavior of a
system under time reversal should be the same as its behavior when the masses of its particles are inverted. The Janus model provides a framework for handling both positive and negative mass
particles under time reversal lambda.gsfc.nasa.gov. This perspective challenges the assumptions of the ΛCDM model and offers a different way of understanding the universe.
In summary, the Janus cosmological model challenges the ΛCDM model by providing alternative explanations for dark matter and dark energy, questioning the cosmological principle, addressing
discrepancies and tensions, and offering a framework based on time reversal symmetry.
Are there any current observations or experiments that could potentially falsify the Janus model ?
There are no specific current observations or experiments that could potentially falsify the Janus model. However, in general, the following types of observations and experiments could challenge the
1. Observations of large-scale structure: The Janus model provides an alternative explanation for the formation of large-scale structures in the universe, such as galaxy clusters and cosmic
filaments. Observations that contradict the model's predictions on the distribution and behavior of these structures could potentially falsify the Janus model as well as the ΛCDM model.
2. Gravitational lensing observations: The Janus model relies on the interaction between positive and negative mass particles to explain phenomena attributed to dark matter, such as gravitational
lensing. Observations of gravitational lensing that do not align with the model's predictions could potentially falsify the Janus model.
3. Cosmological parameter measurements: The Janus model makes predictions about various cosmological parameters, such as the Hubble constant and the matter density parameter. Precise measurements of
these parameters that are inconsistent with the model's predictions could challenge the validity of the Janus model as well as the ΛCDM model.
It is important to note that the scientific method relies on the continuous testing and refining of hypotheses and models. As new observational data and experimental results become available, any
cosmological model, including the Janus model, may be subject to falsification or modification to better fit the available evidence chem.libretexts.org.
Summary / Conclusion
The Janus Cosmological Model (JCM) is an alternative cosmological model that challenges the mainstream ΛCDM model. The JCM describes the universe as a Riemannian manifold with two different metrics,
handling positive and negative masses in general relativity without paradoxes. This model is based on the works of Albert Einstein, Andrei Sakharov, and Jean-Marie Souriau, and was first proposed by
French physicist Jean-Pierre Petit in 1977. The JCM aims to explain phenomena attributed to dark matter and dark energy without resorting to their unknown nature.
The Janus model has evolved over the years, incorporating variable speed of light (VSL) in cosmology, merging with a VSL relativistic bimetric model of gravity, and adding dynamical group theory to
explain the relationship between time reversal, energy, and mass inversion. The modern JCM is named after the two-faced Romanian god Janus, symbolizing the simultaneous look into the future and the
The Janus model is compatible with quantum mechanics and has been independently derived from the Einstein field equations using Cartan's equivalence method and the Hodge star operator. Although no
specific ongoing or planned experiments are mentioned in the context, the model's predictions could be tested by observations related to dark matter, dark energy, and the large-scale structure of the
universe. Future experiments and observations in these areas could potentially provide evidence for or against the Janus model, helping to determine its validity in comparison to the ΛCDM model and
other alternative cosmological models. | {"url":"https://sleek-think.ovh/blog/69-janus-negative-matter-based-cosmological-model","timestamp":"2024-11-09T12:21:22Z","content_type":"text/html","content_length":"34378","record_id":"<urn:uuid:900bf374-173f-49f3-8c56-79324f6e46d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00788.warc.gz"} |
The Evidence for Building Thinking Classrooms is Weak
What does it mean to be research-based?
Building Thinking Classrooms contains Peter Liljedahl’s comprehensive system for getting kids to actually start thinking in math class. According to Liljedahl’s research most students spend math
class subverting the expectations of their teachers. They “student” and find clever ways to avoid doing any actual thinking. Without thinking, there is no learning, and so teachers need to gets kids
His system is truly comprehensive. The book has fourteen chapters, each describing a different aspect of the classroom. Here is what the first ten chapters say, in brief:
• Only assign questions that students solve without direct instruction.
• Randomly place students in groups for their classwork.
• Give each group one marker and tell them to work on a “vertical non-permanent surface” i.e. mounted whiteboard.
• Don’t put the desks in rows or columns.
• When a student asks for help, only answer their question with another question.
• “Give the first thinking task in the first 3-5 minutes of class, give the thinking task with students standing loosely clustered around you.”
• Give homework but do not check it, mark it, ask about it, don’t use words like “practice” or “assessment,” and use phrases like “this is your opportunity.”
• Encourage kids to walk over to successful groups to learn from what they’re doing (“mobilize knowledge”).
• Use hints and extensions to keep students in a “flow” state.
• End the lesson by teaching students by pointing to things that students have done (“consolidating”).
This system and its methods, he writes, “was almost universally successful. They worked for any grade, in any class and for any teacher.” Wow.
The book is a record of his research, some of which has been published, much of which is appearing in this book for the very first time. I am very interested in research, so let’s take a look at the
research informing his book.
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There are four papers of Liljedahl’s in the references of BTC.
Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (Eds.), Posing and solving mathematical problems: Advances and new
perspectives (pp. 361–386). Springer.
Liljedahl, P. (2018). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (Eds.), Broadening the scope of research on mathematical problem solving: A
focus on technology, creativity and affect (pp. 505–524). Springer.
Liljedahl, P., & Allan, D. (2013a). Studenting: The case of homework. In M. V. Martinez & A. C. Superfine (Eds.), Proceedings of the 35th Conference for Psychology of Mathematics Education—North
American Chapter (pp. 489–492). University of Illinois at Chicago.
Liljedahl, P., & Allan, D. (2013b). Studenting: The case of “now you try one.” In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the International Group for the
Psychology of Mathematics Education (Vol. 3, pp. 257–264). PME.
In the homework “Studenting” paper, they pick five classrooms, Grades 10-12, and interview kids about whether and how they did their homework. On the basis of the interviews, they conclude that
teachers should not mark homework, because marking causes cheating, mimicking, and other non-desirable behaviors.
The second Studenting paper involves very brief (1-4 min.) interviews with fifteen 11th graders after a single “traditional” lesson. They classify their behavior/responses as Amotivation, Stalling,
Faking, Mimicking, and Reasoning. They conclude that in a traditional math lesson, most students aren’t engaged in thinking.
For the 2018 “On The Edges of Flow” paper, Liljedahl observed a Grade 11 and a Grade 12 classroom. He was looking for instances of students moving in and out of “flow state” into disengagement. He
provides a list of six situations he observed that show the various causes of this move, such as “quitting” or “seeking increased challenge.” He concludes that hints and extensions can move students
back to flow.
The 2016 paper is a dry run of the Building Thinking Classrooms book. There’s a lot of theory there, but I’m focusing on the results he describes from the data he collects.
First, he thought about his own teaching and came up with a list of nine elements that impact whether students think in class or not. He then observed many (40) other teachers, and decided that these
nine elements were important in their classrooms as well.
He developed his “teaching classrooms” system, tested it on his own students and on the teachers that he worked with. You’ve seen that system already up above; it was those bullet points that you
So: what did Peter L. do to see if the system worked?
First, he describes a study involving five high school classes. He put them in groups, told them to work with various surfaces (e.g. paper, notebook, whiteboard) and measured how long it took them to
start working. Then he rated them on a scale of his own design to capture their “eagerness” or “discussion.” He found that vertical whiteboards led to the most engagement.
He then followed up with teachers who attended his workshops and asked if they took his advice and implemented vertical whiteboards. They told them that they did, and he checked 20 classrooms (he
doesn’t say how they were selected) to confirm the reports from his interviews. Pretty much 100% of teachers told him they were still using his ideas.
He also asked teachers if they took his advice on placing students in random groups for their classwork. They likewise told them that they did, and his follow-up visits confirmed this. These teachers
were a mix of elementary, middle, and secondary teachers.
And that’s it for the 2016 paper.
Is that all the evidence? The book reports many more results, but in a slightly haphazard way. Here are some typical quotes:
80% of students entered their groups feeling like they were going to be a follower rather than a leader.
It turns out that of the 200–400 questions teachers answer in a day, 90% are some combination of stop-thinking and proximity questions.
Although it is true that the students spent much less time writing, and that they did not fall behind, very few students (35%) actually spent the time listening.
We now observed 75%–100% of students taking notes, depending on the class, and 50% of students referring back to their notes at some point.
About 15% of the students told me that the unit they just finished was made up of a number of subtopics, and they were able to name or describe what those subtopics were. These students, for the
most part, scored above 90% on the upcoming test.
There isn’t much context given about where these numbers are coming from, though he frequently notes that they emerged from his research.
We could imagine a version of this book that doesn’t say anything about research at all. “Here are my thoughts on what makes for good teaching,” it would say. “It’s based on my own teaching,
observations, and the experiences of the people I’ve worked with. But don’t take my word for it—try it for yourself, and you’ll see that it works wonders.” Would that be OK?
I think the answer is, that would be very OK. That’s just telling people what you think. You have to be allowed to do that.
Now, how differently should we think about this system with the research support that Peter has given us? I’d argue, not much differently. It’s not that there’s no evidence, and it’s not that he’s
playing loose with the facts. It’s just that the evidence is weak. It doesn’t support big generalizations. You wouldn’t want to bet the bank on it.
Here’s why:
• He doesn’t measure learning at all, only engagement.
• He only measures engagement for vertical whiteboards and maybe homework, not for any other component of his system.
• He counts teacher uptake as evidence of student engagement.
• The measurements he reports are all for older kids.
• We don’t know how he selected the groups he measured.
• We have no idea where the percentages he reports are coming from in his book.
• In fact none of what he did to arrive at his results is particularly transparent, which is probably why his four papers are published in books and conference proceedings, but not journals.
And while I’m tempted to be chill about this, it does bother me. The book and its supporters make extremely strong claims. “The results of this research sound extraordinary,” Peter L. writes. “In
many ways, they are.” And it does seem to me that the book—which has become extremely popular in US math education—owes much of its success to its presentation as a work of strong research.
I don’t know what to do! I’m under zero illusions that people actually care about evidence. Nobody reads the papers. Very few people care about getting this right. And yet, apparently, almost
everybody cares a great deal about the perception that some new thing is rooted in research. It opens doors, hearts, and wallets.
What do I want? I want more people to understand the difference between strong and weak evidence. I want people who do understand this to speak up, and hold the entire field to a slightly higher
standard. Nobody wants cynicism, but we all need to muster a bit more skepticism about this kind of stuff, and not just for our opponents’ ideas.
Thanks for reading! I’m the author of a book Teaching Math With Examples. If you’d like to support my work, consider purchasing a copy or sharing this post.
I am not a PL apologist by any means. I know what I like about the BTC framework (and note it is a framework not a theoretical framework, philosophy or methodology) and a distillation. BTC is not a
research text. It is a popular text, meant for a popular audience. I've probably known Peter for 20 years...about as long as I've been in Canada. A simple google scholar search (https://
scholar.google.com/scholar?start=10&q=peter+liljedahl&hl=en&as_sdt=0,5) will turn up lots of the research evidence which is 'distilled' in the BTC book. Including work in ZDM (one of the premier
and oldest journals in mathematics education but often paywalled unless the open access fee is paid) and other highly selective mathematics education journals. Peter's early work is on problem
solving and belief/affect and the BTC work has grown out of that over the 20 years and again is a distillation. The same is true for most texts published by Corwin - they are not 'research' texts
in the traditional sense, i.e. their audience is wider, more practitioner focused, and not 'researchers' only.
Carl Wieman (Nobel Physicist) positioned it well in a nice piece in Ed. Researcher (2014, 43(1)) about the relative development of the methods in education research compared to that of the 'hard'
sciences. Education research is still in its early days compared to more mature sciences and is 'messy' and 'complex' since the complexities in the systems involved have not yet been well studied
and understood. Similarly the recent set of Editorials in JRME under Cai make the case that mathematics education research (in English) is still in its early adolescent phase (if you read it that
way then Peter's work falls right in line with the challenging of all of the traditional orthodoxies and deploying the Costanza method in its early days - try the opposite of what we've been
The thing I like and respect about the BTC approach is that it has made available to more kids and teachers the beauty and joy that I experienced as a student preparing for math contests. This is
the way we work - as a mathematics community. Peter has made a version of this accessible to the more traditional and over-surveilled classrooms in North America with many teachers who are
under-prepared mathematically.
Respectfully, Someone who reads the original papers (cause its my job).
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Great post MP! Yes, irritating. It's like the "evidence" for sham nutritional supplements.
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The logarithmic base ten form represents the exponent that 10 must be raised to in order to obtain that specific number. Now, what does this really mean? Let’s take a look at these examples here. So
here we are looking at the power that we can raise 10 to and the answer that results. So 101 is just 10 times 1 which gives us 10. 104 is just 10 times 10 times 10 times 10 which gives us 10,000.
10-1 is equivalent to just saying 1 over 10. Right? Because what it’s really saying is 10 inverse, so that’s 1 over 10, so that gives us 0.10. And then 100, any number to the zeroth power, equals 1.
How does this relate to my log function? Well here we have log(ten) gives us 1. Really what this is saying, it’s saying that 10 to what number gives me 1? The answer would be 1. Here, log of 10,000.
Alright, so think of it like this, we have log of 10,000 so that is really log(104). Whatever the number is here, because remember this is log base 10, what happens here is that this cancel out with
this so we get 4 left at the end.
Here when we get log of 0.10. What does that really mean? That really means log(10-1). So this cancels out with this and gives us negative one. Then we have log of 1. Right? So you’d say here, so log
base 10 and then 10 to the one here, well, well, 10 to the 0 here actually sorry, equals 0.
So that is what is really going on here. And if we have log(ten) think that as log base ten and then 101 equals 1. So we’re just converting each of these values, by 10 to some power. The log portion
cancels out the 10 and leaves behind the exponent as my final answer. That’s how we can calculate log. I know you guys have calculators, but there may come a point where you have this within a math
class or within an MCAT or PCAT or OAT or DAT later on, much later on after you’ve taken all these science courses where you have to understand these relationships. And this is how our log base
function is connected to multiples of 10. Knowing that helps us to get to the answer.
Now understanding this, try to see if you can answer example 1. So here we are going to use this without a calculator. Try to do it without a calculator, see what answer you get, and then afterwards
come back and see if your answer matches up. I’ll help you guys a little bit for the first one. So here, this log is getting distributed to the 1 and to the 10-7. Now when that happens, what that
means is we have log of 1, and because they’re multiplying it really means that we’re adding, So it’s plus log of 10-7. See if you guys can figure out what the answer is without using a calculator.
Come back and see if your answer matches up with mine. | {"url":"https://www.pearson.com/channels/gob/learn/jules/bonus-mathematical-operations-and-functions/logarithmic-and-natural-logarithmic-functions?chapterId=d07a7aff","timestamp":"2024-11-13T17:42:14Z","content_type":"text/html","content_length":"498838","record_id":"<urn:uuid:6640451e-c56a-4537-95f2-b2dda54679e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00783.warc.gz"} |
Episode 228: The Young modulus
Young's Modulus
Properties of Matter
Episode 228: The Young modulus
Lesson for 16-19
• Activity time 170 minutes
• Level Advanced
The Young modulus is often regarded as the quintessential material property, and students can learn to measure it. It is a measure of the stiffness of a material; however, in practice, other
properties of materials, scientists and engineers are often interested in, such as yield stress, have more influence on the selection of materials for a particular purpose.
Lesson Summary
• Discussion: Defining the Young modulus (20 minutes)
• Student activity: Studying data (20 minutes)
• Student experiment: Measuring the Young modulus (60 minutes)
• Student experiment: An alternative approach using a cantilever (30 minutes)
• Discussion: Comparing experimental approaches (10 minutes)
• Student questions: Involving the Young modulus (30 minutes)
Discussion: Defining the Young modulus
A typical value of k might be 60 N m^-1.
What does this mean? (60 N will stretch the sample 1 m.) What would happen in practice if you did stretch a sample by 1 m? (It will probably snap!)
A measure of stiffness that is independent of the particular sample of a substance is the Young modulus E.
Recall other examples you have already met of “sample independent” properties that only depend upon the substance itself:
• density = massvolume
• electrical resistivity = resistance × arealength
• specific heating capacity = energy transferredmass × temperature change
• thermal conductivity = power × lengtharea × temperature difference
We need to “correct” k for sample shape and size (i.e. length and surface area).
Episode 228-1: The Young modulus (Word, 53 KB)
Note the definitions, symbols and units used:
Quantity Definition Symbol Units
Stress tensionarea=FA σ (sigma) N m^-2={Pa
Strain extension per original length= Δ xx ε (epsilon) No units (because it’s a ratio of two lengths)
Young modulus stressstrain E N m^-2 = Pa
Strains can be quoted in several ways: as a %, or decimal. E.g. a 5% strain is 0.05.
Episode 228-2: Hooke's law and the Young modulus (Word, 75 KB)
Student activity: Studying data
It is helpful if students can learn to find their way around tables of material properties. Give your students a table and ask them to find values of the Young modulus. Note that values are often
given in GPa (1 × 10^9 Pa).
Some interesting values of E :
• DNA ~ 10^8 Pa
• spaghetti (dry) ~ 10^9 Pa
• cotton thread ~ 10^10 Pa
• plant cell walls ~ 10^11 Pa
• carbon fullerene nanotubes ~ 10^12 Pa
Episode 228-3: Materials database (Word, 115 KB)
Student experiment: Measuring the Young modulus
You can make measuring the Young modulus E a more interesting lab exercise than one which simply follows a recipe. Ask students to identify the quantities to be measured, how they might be measured,
and so on. At the end, you could show the standard version of this experiment (with Vernier scale etc.) and point out how the problems have been minimized.
What needs to be measured? Look at the definition: we need to measure load (easy), cross-sectional area A , original length x[0] (so make it reasonably long), and extension Δ x .
Problems? Original length – what does this correspond to for a particular experimental set up? Cross-sectional area: introduce the use of micrometer and/or vernier callipers. Is the sample uniform?
If sample gets longer, won’t it get thinner? Extension – won’t it be quite small?
Should the sample be arranged vertically or horizontally?
Divide the class up into pairs and brainstorm possible methods of measuring the quantities above, including the pros and cons of their methods.
Some possibilities for measuring Δ x :
Attach a pointers to the wire
• Pro: measures Δ x directly
• Con: may affect the sample; only moves a small distance
Attach a pointer to the load
• Pro: measures Δ x directly, does not effect the sample
• Con: only moves a small distance
Attach a pulley wheel
• Pro: “amplifies” the Δ x
• Con: need to convert angular measure to linear measure, introduces friction
Attach a pointer to the pulley wheel
• Pro: “amplifies” the Δ x even more
• Con: need to convert angular measure to linear measure, introduces friction
Exploit an optical level
• Pro: a “frictionless” pointer, “amplifies” the Δ x even more
• Con: need to convert angular measure to linear measure, more tricky to setup?
Illuminate the pointer etc to produce a magnified shadow of the movement
• Pro: easy to see movement
• Con: need to calculate magnification, can be knocked out of place
use a lever system to amplify or diminish the load and provide a pointer
• Pro: useful for more delicate or stiff samples; can use smaller loads
• Con: fixing the sample so it doesn’t “slip”, need to convert angular measure to linear measure
Different groups could try the different ideas they come up with. Depending upon the time available, it may be worth having some of the ideas already set up.
Give different groups different materials, cut to different sizes, for example: metal wires (copper, manganin, constantan etc), nylon (fishing line), human hair (attach in a loop using Sellotape),
rubber. Note that in the set up above, the sample is at an angle to the ruler – a source of systematic error.
Students should wear eye protection, provide safe landing for the load should sample break, e.g. a box containing old cloth. For the horizontal set up: “bridges” over the sample to trap the flying
ends, should the sample snap.
Good experimental practice: measure extension when adding to the load and when unloading, to check for any plastic behaviour.
Episode 228-4: Measuring the stiffness of a material (Word, 59 KB)
Episode 228-5: Stress–strain graph for mild steel (Word, 68 KB)
Information about the use of precision instruments (micrometer screw gauge, Vernier callipers and Vernier microscope).
Episode 228-6: Measure for measure (Word, 82 KB)
Student experiment: An alternative approach using a cantilever
An alternative approach to measuring the Young modulus is to bend a cantilever. (Potential engineering students will benefit greatly from this.)
For samples too stiff to extend easily (e.g. wooden or plastic rulers, spaghetti, glass fibres) the deflection y of a cantilever is often quite easy to measure and is directly related to its Young
modulus E .
If the weight of the cantilever itself is mg, and the added load is Mg and L is the length of the cantilever (the distance from where the cantilever is supported to where the load is applied):
For a rectangular cross section, dimension in the direction of the load is d , other dimension is b
• y = 4 (Mg + 5mg/16) L^3
• E b d^3
(for square cross-section d = b )
For a circular cross-section radius r
• y = 4 (Mg + 5mg/16) L^3
• 3 π r^4 E
Discussion: Comparing experimental approaches
Finish with a short plenary session to compare the pros and cons of the different experimental approaches.
Student questions: Involving the Young modulus
Questions involving stress, strain and the Young modulus, including data-handling.
Episode 228-7: Calculations on stress, strain and the Young modulus (Word, 59 KB)
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Sum and Ignore Errors in Excel
You can use the SUM function to add a range of cells in Excel. However, if some cells in the cell range contain errors such as #N/A, #VALUE!, #REF!, #DIV/0!, #NUM!, or #NAME?, the SUM function will
return an error instead of a total.
Sometimes you need to know the sum of a cell range even if it contains errors. Therefore, sum up the cell range and ignore any cells that contain errors.
Note: Errors in a worksheet reveal an underlying problem that needs to be addressed; therefore, sum and ignore errors in Excel cautiously.
This tutorial shows three techniques for summing a range and ignoring errors in Excel.
How to Sum and Ignore Errors in Excel
Method #1: Use a Formula Combining the SUM and IFERROR Functions
The SUM function in Excel adds a range of cells or values, but if some cells in the cell range have errors, the function will return an error instead of a total, as seen in the example dataset below,
where some cells in column D have the #VALUE! error:
To prevent the SUM function from returning an error value instead of a total, you can use a formula combining the SUM function and the IFERROR functions to ignore the errors and produce a total.
The syntax of the IFERROR function is as follows:
1 =IFERROR(value,value_if_error)
The IFERROR function returns value_if_error if the expression is an error. Otherwise, it returns the value of the expression.
About the example dataset, you can use the following steps to sum and ignore errors in column D:
1. Select cell D11 and type in the formula below:
1 =SUM(IFERROR(D2:D10,0))
2. Press Enter.
The formula combining the SUM and IFERROR functions returns a total, not an error value.
Explanation of the formula
1 =SUM(IFERROR(D2:D10,0))
The IFERROR function replaces any error value in a cell with a zero value. Otherwise, it returns the value in the cell.
The SUM function finally sums all the values returned by the IFERROR function.
Method #2: Use the SUMIF function
The SUMIF function adds the cells specified by a given condition or criteria.
The syntax of the SUMIF function is as follows:
1 =SUMIF(range,criteria,[sum_range])
We can use the SUMIF function to sum and ignore errors in Excel by specifying in the criteria argument that the values to be added must be equal to or above zero.
We have the following dataset showing a list of electronic items, their quantities, prices, and total prices.
We can use the SUMIF function to determine the total prices of the items and ignore the errors using the below steps:
1. Select cell D11 and type in the below formula:
2. Press Enter.
The SUMIF function returns a total, not an error value.
Explanation of the formula
This formula will add up all the cells in the cell range D2:D10, and any cells that contain errors will be treated as if they have a zero value.
Method #3: Use the AGGREGATE Function
You can sum and ignore errors in Excel by using the AGGREGATE function and configuring it to calculate and ignore errors by setting the function_num argument to 9 and options argument to 6, as shown
Set the function_num argument to 9:
Set the options argument to 6:
Let’s use the AGGREGATE function to sum column D in the following dataset and ignore errors.
We use the following steps:
1. Select cell D11 and type in the below formula:
2. Press Enter.
The AGGREGATE formula sums the cell range D2:D10 and ignores all the errors in the cell range.
Sometimes you need to know the sum of a cell range even if it contains errors. Therefore, sum up the cell range and ignore any cells that contain errors.
This tutorial showed three methods for summing and ignoring errors in Excel. This tutorial showed three methods for summing and ignoring errors in Excel. The techniques are:
• The use of a formula combining the SUM and IFERROR functions.
• The use of the SUMIF function.
• The use of the AGGREGATE function.
We hope you found the tutorial helpful. | {"url":"https://officetuts.net/excel/formulas/sum-and-ignore-errors-in-excel/","timestamp":"2024-11-07T01:09:48Z","content_type":"text/html","content_length":"160777","record_id":"<urn:uuid:849e6f64-ca96-48ca-a417-777f22a973b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00663.warc.gz"} |
How to Calibrate Mechanical Inputs in Software
Posted May 01, 2000
Embedded software developers operate in a perfect digital environment but must interact with the imperfect analog real world. To do this it's essential to know how to perform calibration of inputs
and sensors.
A few years back, a Barr Group client was developing a piece of physical therapy equipment that was to provide a variable number of pounds of force feedback to a human operator. To support the
complete range of possible force settings with a given worst-case accuracy, the product was correctly designed around a microcontroller. The embedded software that ran on this microcontroller was
principally responsible for regulating the voltage going to a brake such that the brake's calipers would "slip" precisely at the selected force. In other words, the human operator must push or pull
against the shaft with at least the selected amount of force in order to make the shaft move. But the problem did not turn out to be as easy as it at first appeared.
What made this problem so difficult was that each brake performed differently in actual use. It turns out that there's no one equation that says: to achieve x pounds of force, apply v = f(x) volts.
(The input to the brake was actually a PWM signal and the important characteristic was current; I'm using voltage to make the discussion simpler.) Figure 1 contains five S-shaped curves. Each of
these curves is the measured response of a particular brake. And, as if the differences between brakes were not enough to deal with, the curve for each particular brake is affected by its age (as the
contacting surfaces wear), temperature, and even the velocity of the force applied against it. (All of these are factors that alter the amount of friction between the surfaces.)
Perhaps the first and most obvious step to take is to create a feedback loop in the system. It's clear that simply setting the voltage on the brake and forgetting it is not going to be good enough.
As you can see from Figure 1, the difference in force output for a given voltage between brakes can be extremely large. Some of the brake curves even cross, indicating that individual brakes are not
simply stronger or weaker than the average.
Figure 1. Measured brake curves
By adding sensors that can read the applied force and the shaft's position, it is possible to make continuous adjustments to the brake voltage as the user is in the process of pushing or pulling on
the shaft. If the shaft is moving (its position is changing) and the force on the shaft is too small, the brake voltage must be increased. If the force is too high, the voltage must be decreased. By
"closing" toward the desired force like this, it's possible to achieve and maintain that force throughout each repetition of the exercise.
One problem with such a closed-loop approach is that it takes time to achieve the desired results. If you close too quickly, you'll continually overshoot and undershoot the desired force. So you must
only make one small change in the voltage in a given time period, then wait for the brake to respond before making the next change. In our case, we found the ideal period to be right around 50ms. At
that rate of closure, you can be relatively certain of achieving the desired force without much overshoot, and maintaining it without too much fluctuation.
However, if you start out with an initial force that is so far from the desired force that it takes more than about half a second (500ms) to reach it, a human can generally feel the change in
resistance. This seems to be true whether the incorrect initial force is above or below the desired force. So more than 10 increment or decrement operations had to be avoided at all costs. And the
only way to achieve that was to make a fairly accurate initial "guess" about what the brake voltage should be for a given force. To get that kind of accuracy, we had to calibrate each brake's S-curve
when it was installed. And, because of the more subtle brake-specific factors mentioned previously, each brake must be periodically recalibrated in the field.
Linear simplicity
Before we can talk about calibrating the brakes, however, we must first talk about the force sensors. Each force sensor must also be calibrated. Consider the example of a force sensor that always
reads 10% below the actual force applied. The user will have to apply significantly more force (11.1% more, to be exact) than he or she expected. If you might be tempted to consider that a tolerable
worst-case error, remember that it is possible the human user could injure (or, more likely, reinjure) a muscle in the process of completing an exercise on such a strongly biased machine.
Fortunately, calibrating the force sensors is not terribly complicated. It turns out that these devices can only really be biased in one of three ways. (This from the manufacturer, of course.) First,
they may have a non-zero reading even when no force is applied to them. We'll call this the sensor's zero offset. Second and third, the sensors may have a linear bias unique to forces applied in each
direction. We'll call these the left and right gain factors.
The zero offset is easy to determine and correct for. You need only read the force reported by the sensor when no force is applied. That value is the zero offset for that particular sensor. To
correct for it, simply subtract the zero offset from every future force reading. For example, if you determine that the zero offset is 5 lbs. and your next "raw" reading is 15 lbs., you'd compute the
actual force on the shaft to be 10 lbs.
Once the zero offset, if any, has been subtracted out, determining the left and right gain factors is possible. The terms left and right are subjective, of course, but the general idea is that the
sensor has one linear bias when forces are applied in one direction and a different linear bias when forces are applied in the opposite direction. The fact that these biases are linear simply means
that the percentage error is the same, regardless of the actual force applied. So, for example, a sensor with a -10% bias would read 90 lbs. when 100 lbs. is applied and 180 lbs. when 200 lbs. is
applied. It is, therefore, sufficient to take one calibration reading in each direction.
To determine the left and right gain factors, you might simply hang a 100-lb. weight from the shaft, first applying that force in one direction then the other. By comparing the actual readings to the
expected readings in each direction, you can compute the sensor's left and right gain factors. By dividing all future readings by the gain factor in the given direction, you can eliminate these
biases from the sensor. The result is an accurate measure of force that can be compared across different machines and form the basis for calibration of the brakes. Figure 2 Nominal break curve Figure
3 A polynomial approximation of the nominal state
Polynomial insanity
Calibrating a brake is much harder than calibrating a force sensor, for one important reason: each brake's S-curve is non-linear. You can't simply take one reading and compute a gain factor from it.
This was possible for the sensors only because their error was a fixed percentage of their input value. The brakes, on the other hand, can diverge wildly from one another (and even from themselves,
over time), so that there's really no such thing as a typical brake.
Despite the fact that there is no typical brake, we've still got to define one. After all, you need a starting point even just to perform a calibration. So we began by measuring the relationship
between input voltage and output force for five randomly selected brakes. The results of these measurements were the five data sets shown in Figure 1. We then averaged the five forces resulting from
each tested voltage setting to produce the nominal brake curve shown in Figure 2.
Figure 2. Nominal brake curve
The only problem with the curve in Figure 2 is that it represents the inverse of the relationship the software needs to compute at run-time. The necessary relationship measures voltage as a function
of desired force, telling the microcontroller how much voltage it should apply to the brake to achieve a particular desired force. Figure 3 shows the same nominal brake data, flipped on its axes and
approximated as the third-order polynomial:
v = Ax^3 + Bx^2 + Cx + D (Equation 1)
where A, B, C, and D are floating-point coefficients. Jack Crenshaw has recently been looking to tackle a similar sort of problem in the pages of his column--that is, run-time calibration of a device
with a polynomial response curve. However, his approach seems to be to actually determine new coefficients (A, B, C, and D) at run time, thus performing a complicated set of computations on the
embedded processor. That sort of approach simply wasn't practical in this particular system.
Figure 3. A polynomial approximation of the nominal brake
What made more sense was for us was to try to "tweak" the nominal curve to make it a better fit for each particular brake. This was partly because we didn't have the computational resources to spare
(it was only an 8-bit processor with a couple hundred bytes of RAM, after all) and partly because we were able to "close" to the desired force pretty quickly anyway. All we really needed to
accomplish through calibration was to bring the curve within about 10 voltage increments or decrements of the optimal value, along as much of the curve as possible. In truth, given all of the other
factors affecting the brake performance, we could never really hope for anything better. The actual brake curve will change, for example, as soon as the surface of the brake pads heats up (even after
just a few reps).
Here's how tweaking the curve might work in practice. Given a particular force setting, x, the nominal voltage, v, would be computed according to the polynomial in Equation 1. We would then apply a
"tweak factor" that is also a function of the force. In other words, we'd apply a "local" tweak factor that is applicable only at or very near x lbs. Multiplying the nominal voltage by the tweak
factor, we'd get a new voltage, v', and apply that to the brake. The result, we hope, will be a force within 10 voltage increments or decrements of the optimal voltage. If it is, we've done good
This, of course, begs the question of how to determine those "tweak factors."
The key to tweaking is the number of data points. If we had the luxury of setting the brake to every possible voltage and measuring the resulting forces, it is clear that we could produce an exact
tweak factor for every point along the curve. For any given force, the actual brake curve may be above, on top of, or below the nominal curve. (Above means a higher voltage is needed to achieve that
force on this particular brake at this particular time, below means a lower voltage.) If it's above the nominal curve, the tweak factor would be a number greater than 1.0; if it's below, the tweak
factor would be less than 1.0; and if the actual and nominal brakes concur at that force, the tweak factor would be exactly 1.0.
Unfortunately, measuring the force produced by every brake at every possible voltage is not realistic. (It was hard enough doing that once for each of the five representative brakes.) Since
calibration must be done frequently and it is often the owner of the machine who will do it, it is necessary that the number of data points be significantly reduced.
At each calibration data point, a specific force (x1) is requested, the microcontroller computes the nominal voltage for that force (v1), the nominal voltage is applied to the brake, and the force
produced is recorded. Interestingly, the processor now knows exactly what voltage to input to achieve the force that resulted-though it still doesn't know how to produce the requested force. In other
words, if the next force requested were the one we had just produced (call it x2) it would be very easy to "tweak" the nominal curve at that point, by simply multiplying the nominal voltage by the
ratio of the two voltages (v1/v2). This ratio is the tweak factor at force x2.
By taking a number of these measurements and recording the ordered pairs (x2, v1/v2) in a table, we gain the ability to produce perfect results at those specific requested forces. To extend these
results to include other forces, we simply "draw" line segments between each known data point. In other words, we weight the two closest "precise" tweak factors to achieve estimated tweak factors for
any force in between.
typedef struct
uint16_t force;
double ratio;
} CalPoint;
CalPoint aCalPoints[NUM_CAL_POINTS];
adjustVoltage(uint16_t force, uint16_t vNominal)
int i;
double scaleFactor;
* Find the relevant calibration range.
for (i = 1; i < NUM_CAL_POINTS && aCalPoints[i].force < force; i++);
* Compute the weighted scale factor.
scaleFactor = ((aCalPoints[i].force - force) * aCalPoints[i-1].ratio
+ (force - aCalPoints[i-1].force) * aCalPoints[i].ratio)
/ (aCalPoints[i].force - aCalPoints[i-1].force);
* Adjust the voltage.
return (vNominal * scaleFactor);
} /* adjustVoltage() */
Listing 1. A function that uses the calibration data
Listing 1 contains a function, adjustVoltage(), that performs the voltage scaling mathematics. The data structure is an array of calibration data points. Each CalPoint consists of a force (x2) and a
ratio (v1/v2) that was observed when the force x1 was most recently tested on this brake. These calibration points are assumed to be ordered in the array from the smallest force to the largest. In
addition, the function assumes that the array begins with a calibration point at force 0 and ends with a calibration point at MAX_FORCE. (We found that the ratios associated with these two points
should be set to 1.0 and the same as the highest actual calibration point available, respectively.)
In effect, we multiply the nominal curve by a connected set of line segments. The more data points you have, the shorter the line segments will be, and, therefore, the more precise the resulting
calibration. This is a really practical way to calibrate a complex device quickly. In our specific case, we were able to take just five measurements during brake calibration and to achieve impressive
I haven't found much literature about doing this sort of run-time calibration, so I wonder how others have accomplished similar tasks in their embedded systems. If you have a war story or a favorite
reference on the subject, please share in a comment below.
This column was published in the May 2000 issue of Embedded Systems Programming. If you wish to cite the article in your own work, you may find the following MLA-style information helpful:
Barr, Michael. "Hold Everything," Embedded Systems Programming, May 2000 , pp. 53-59.
Related Barr Group Courses:
For a full list of Barr Group courses, go to our Course Catalog. | {"url":"https://barrgroup.com/blog/how-calibrate-mechanical-inputs-software","timestamp":"2024-11-02T21:42:43Z","content_type":"text/html","content_length":"48667","record_id":"<urn:uuid:5fd1ff28-1a7f-4b77-9ec1-5db69306a4be>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00529.warc.gz"} |
Faculty Publications
Goos, M., Geiger, V., Dole, S., Forgasz, H., & Bennison, A. (2019). Numeracy across the curriculum: Research-based strategies for enhancing teaching and learning. Sydney: Allen and Unwin.
Keogh, J. J., Maguire, T., & O’Donoghue, J. (2018). Adults, mathematics and work. Leiden, Netherlands: Brill.
Maasz, J., O'Meara, N., Johnson, P. & O'Donoghue, J. (2018) Mathematical Modelling for Teachers: A practical guide to applicable mathematics education. Netherlands: Springer.
Safford-Ramus, K. Keogh, J., O’Donoghue, J., & Maguire, T. (Eds.) (2018). Adults Learning Mathematics – A Research Forum 1993-2018. London: Adults Learning Mathematics – A Research Forum.
Book Chapters
Goos, M. (2019) Publishing for international impact in mathematics education research journals. In K. Leatham (Ed.), Designing, conducting, and publishing quality research in mathematics education
(pp. 213-225). Cham: Springer.
Goos, M. (2019). Educational Studies in Mathematics: Shaping the field. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 377-391). Cham,
Switzerland: Springer
Goos, M. (2018). Commentary – Classroom-based interventions in different subject areas: Sharing meaning across researchers and practitioners. In G. Stylianides & A. Childs (Eds.), Classroom-based
interventions across subject areas (pp. 184-194. London: Routledge.
Goos, M. (accepted 3 March 2018). Crossing boundaries: Fostering collaboration between mathematics educators and mathematicians in initial teacher education programs. In B. Sims (Ed.), Proceedings of
the Jon Borwein Commemorative Conference. New York: Springer.
Goos, M. (2018). The mathematics learning area: Conforming, reforming, or transforming? In A. Reid & D. Price (Eds.), The Australian curriculum: Promises, problems, and possibilities (pp. 55-64).
Canberra: Australian Curriculum Studies Association.
O’Donoghue, J. (2018). Mathematics education and adult learners in Ireland. In Safford-Ramus, J. Maasz, & E. Suss-Stepancik (Eds.), Contemporary Research in Adult and Lifelong Learning in
Mathematics: International Perspectives (pp. 39-59). Switzerland: Springer.
O’Donoghue, J., & Fitzmaurice, O (2015).Embedding a scholarly element into your teaching. In Grove, M., Croft, T., Kyle, J., & D. Lawson (Eds.), Transitions in Undergraduate Mathematics Education
(pp. 277-291). Birmingham, UK: University of Birmingham.
Coben D., O’Donoghue J. (2014) Adults Learning Mathematics. In: Lerman S. (eds) Encyclopedia of Mathematics Education. Springer, Dordrecht
Refereed Journal Articles
Goos, M., Bennison, A., Quirke, S. O’Meara, N. & Vale, C. (2019) Developing Professional Knowledge and Identities of Non-Specialist Teachers of Mathematics. In Potari, D. & Chapman, O. (Eds)
International Handbook of Mathematics Teacher Education: Volume 1, 2^nd Ed., Netherlands: Brill, 211 – 240.
O’Meara, N., Johnson, P. & Leavy, A. (2019). A comparative study investigating the use of manipulatives at the transition from primary to post-primary education. International Journal of Mathematics
in Science and Technology Education. DOI: 10.1080/0020739X.2019.1634842
Fitzmaurice, O., O’Meara, N., Johnson, P. & Lacey, S. (2019). ‘Crossing the equals sign’: Insights into pre-service teachers’ understanding of linear equations. Asia-Pacific Journal of Teacher
Education, 47(4). 361-382.
Treacy, P., Prendergast, M. & O’Meara, N. (2019). A “new normal”: Teachers’ experiences of the day-to-day impact of incentivising the study of advanced mathematics. Research in Mathematics Education.
DOI: 10.1080/14794802.2019.1668832
Lane, C., O’Meara, N. & Walsh, R. (2019). Pre-service teachers’ use of the mathematical register. Issues in Educational Research, 29(3). 790-806
Goos, M., & Kaya, S. (2019). Understanding and promoting students’ mathematical thinking: A review of research published in ESM. Educational Studies in Mathematics. https://doi.org/10.1007/
Maass, K., Geiger, V., Romero Ariza, M., & Goos, M. (2019). The role of mathematics in interdisciplinary STEM education. ZDM Mathematics Education, 51(6), 869-884.
Goos, M., & Bennison, A. (2019). A zone theory approach to analyzing identity formation in mathematics education. ZDM Mathematics Education, 51(3), 405-418. doi: 10.1007/s11858-018-1011-8
Kelly, R., Erduran, S. (2019) Understanding aims and values of science: developments in the Junior Cycle Specifications on nature of science and pre-service science teachers’ views in Ireland. Irish
Educational Studies, 38(1), 43-70.
Lane, C. & Ní Ríordáin, M. (2019). Out-of-field teachers’ beliefs and practices: An examination of change and tensions using zone theory. International Journal of Science and Mathematics Education.
DOI: https://doi.org/10.1007/s10763-019-09964-5
Kelly, R., McGarr, O., Lehane, L., & Erduran, & S. (January 2019) STEM and gender at university: focusing on Irish undergraduate female students. Journal of Applied Research in Higher Education
O’Meara, N., Prendergast, M., Cantley, I., Harbison, L., & O’Hara, C. (2019). Teachers’ self-perceptions of mathematical knowledge for teaching at the transition between primary and post-primary
school. International Journal of Mathematical Education in Science and Technology. DOI: 10.1080/0020739X.2019.1589004.
Prendergast, M., O'Meara, N., O'Hara, C., Harbison, L., & Cantley, I. (2019). Bridging the primary to secondary school mathematics divide: Teachers' perspectives. Issues in Educational Research, 29
(1), 243-260.
O’Meara, N., & Prendergast, M. (2019). Teaching mathematics after hours. Journal of Curriculum Studies. DOI: 10.1080/00220272.2018.1535666.
Beswick, K., & Goos, M., (2018). Mathematics teacher educator knowledge: What do we know and where to from here? Journal of Mathematics Teacher Education, 21(5), 417-427. doi.org/10.1007/
Sawatzki, C., & Goos, M. (2018). Cost, price and profit: What influences students' decisions about fundraising? Mathematics Education Research Journal. doi.org/10.1007/s13394-018-0241-y
Goos, M., Bennison, A., & Proffitt-White, R. (2018) Sustaining and scaling up research-informed professional development for mathematics teachers. Mathematics Teacher Education and Development, 20
(2), 133-150.
Goos, M., & Bennison, A. (2018). Boundary crossing and brokering between disciplines in pre-service mathematic teacher education. Mathematics Education Research Journal. doi 10.1007/s13394-017-0232-4
Kelly, R., Erduran, S. (2018) Understanding aims and values of science: developments in the Junior Cycle Specifications on nature of science and pre-service science teachers’ views in Ireland. Irish
Educational Studies. doi 10.1080/03323315
Walsh, R., Fitzmaurice, O., O’Donoghue, J. (2017) ‘What Subject Matter Knowledge do second-level teachers need to know to teach trigonometry? An exploration and case study’, Irish Educational Studies
, 36: 273-306.
Kelly, R., McLoughlin, E., & Finlayson, O. E. (2016). Analysing student written solutions to investigate if problem-solving processes are evident throughout. International Journal of Science
Education, 38(11), 1766-1784
Lane, C., O’Donoghue, J. & Stynes, M. (2016) Examining Post-Primary Students’ Images of Mathematics: Findings from an Investigative Survey in Ireland. International Journal of Mathematical Education
in Science and Technology: 47 (7), 1009-1027
Hourigan, M., O'Donoghue, J. (2015) Addressing prospective elementary teachers' mathematics subject matter knowledge through action research. International Journal Of Mathematical Education in
Science and Technology, 46: 56-75.
O'Keeffe, L., O'Donoghue, J. (2015). A Role for Language Analysis in Mathematics Textbook Analysis. International Journal of Science and Mathematics Education, 13: 605-630.
Lane, C., O'Donoghue, J., Stynes, M. (2014) The Image of Mathematics held by Irish Post-Primary Students. International Journal of Mathematical Education in Science and Technology, 45 (6):879-891.
Treacy, P., O’Donoghue, J. (2014) Authentic Integration: A model for integrating mathematics and science in the classroom. International Journal of Mathematical Education in Science and Technology,
45 (5): 703- 718.
Prendergast, M., O’Donoghue, J. (2014) Students enjoyed and talked about classes in the corridors: pedagogical framework promoting interest in algebra. International Journal of Mathematical Education
in Science and Technology, 45 (6): 795 – 812.
Refereed Conference Papers
Marshman, M., & Goos, M. (2019). Making mathematics teachers: The beliefs about mathematics and mathematics teaching and learning held by mathematics teacher educators and pre-service teachers. In J.
Pettigrew, L. Rylands, D. Shearman, & A. Yeung (Eds.), Reflections of change (Proceedings of the 12^th Delta conference on the teaching and learning of undergraduate mathematics and statistics, pp.
61-70). Sydney: Western Sydney University and Delta. https://doi.org/10.26183/5d5f53ed9c926
Goos, M., & O’Donoghue, J. (2019). Designing effective professional learning programs for out-of-field mathematics teachers. In M. Graven, H. Venkat, A. Essien & P. Vale (Eds.), Proceedings of the 43
^rd conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 288-295). Pretoria, South Africa: PME.
Goos, M., Ní Ríordáin, M., Lane, C., & Faulkner, F. (2019). Impact of a national professional development program on the beliefs and practices of out-of-field teachers of mathematics. In G. Hine, S.
Blackley, & A. Cooke (Eds.), Mathematics education research: Impacting practice (Proceedings of the 42^nd annual conference of the Mathematics Education Research Group of Australasia, pp. 316-323).
Perth: MERGA.
Goos, M.(2019). Making connections across disciplinary boundaries in pre-service mathematics teacher education. In J. Holm & S. Mathieu-Soucy (Eds.), Proceedings of the 2018 annual meeting of the
Canadian Mathematics Education Study Group (pp. 9-20). CMESG/GCEDM. Retrieved 23 May 2019 from http://www.cmesg.org/wp-content/uploads/2019/05/CMESG-2018-1.pdf
Goos, M., & O’Sullivan, K. (2018). A comparative analysis of numeracy as a driver for curriculum reform in Australia and Ireland. Paper to be presented at the ICMI-24 Study Conference on School
mathematics curriculum reforms: Challenges, changes and opportunities. Tsukuba, ,Japan, 26-30 November 2018.
Li, K., & Goos, M. (2018). An analysis of patterns of classroom talk in an IT environment. In M. A. Sorto, A. White, & L. Guyot (Eds.), Looking back, looking forward. Proceedings of the Tenth
International Conference on Teaching Statistics (ICOTS10, July, 2018), Kyoto, Japan. Voorburg, The Netherlands: International Statistical Institute.
Bennison, A., Goos, M., & Bielinski, D. (2018). Identifying practices that promote engagement with mathematics among students from disadvantaged backgrounds. In J. Hunter, P. Perger, & L. Darragh
(Eds.), Making waves, opening spaces (Proceedings of the 41^st annual conference of the Mathematics Education Research Group of Australasia, pp. 154-161). Auckland: MERGA.
Marshman, M., & Goos, M. (2018). The beliefs about mathematics, its teaching and learning of those involved in secondary mathematics pre-service teacher education. In J. Hunter, P. Perger, & L.
Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41^st annual conference of the Mathematics Education Research Group of Australasia, pp. 519-526). Auckland: MERGA.
Dole, S., Goos, M., Hilton, A., & Hilton, G. (2018). Authentic numeracy contexts for proportional reasoning – The case of the seven summits. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making
waves, opening spaces (Proceedings of the 41^st annual conference of the Mathematics Education Research Group of Australasia, pp. 59-62). Auckland: MERGA.
Lane, C. (February, 2017). Students’ Images of Mathematics: The role of parents’ occupations. In T. Dooley & G. Gueudet (Eds) Proceedings of the Tenth Congress of European Research in Mathematics
Education (pp. 1130 – 1137). Dublin, Ireland: DCU Institute of Education and ERME.
O’Donoghue, J. (2017). Reflections on Adult Mathematics Education: A view from the sidelines. Proceedings of Adults Learning Mathematics – A Research Forum (ALM), NUI Maynooth, Ireland, 16-22.
(Keynote address).
McLoughlin, E., Finlayson, O., Kelly, R. (2016) ‘Implementing an Interdisciplinary Science Module’, in J. Lavonen, K. Juuti, J. Lampiselkä, A. Uitto & K. Hahl (Eds.),Electronic Proceedings of the
ESERA 2015 Conference. Science Education Research: Engaging learners for a sustainable future, 31st Aug- 4th Sept, Helsinki, Finland. (pp. 2854-2859).
Keogh, J., Maguire, T., O’Donoghue, J. (2016). Re-Contextualising Mathematics for the Workplace. 13th International Congress on Mathematics. Hamburg, ICME: 8.
Kelly, R., Finlayson, O., & McLoughlin, E. (2015) ‘Problem Based Learning: Assessing Student Solutions in an Interdisciplinary Course’, in K., Mårtensson (Chair), Electronic Proceedings of the
Inaugural European Conference on the Scholarship of Teaching & Learning: Bridging Boundaries through the Scholarship of Teaching & Learning, Cork 7th- 9th June, Cork: University College Cork, (pp.
Keogh, J., Maguire, T., O’Donoghue, J. (2015). Graduate Work-Readiness in the 21st Century. Proceedings of Higher Education in Transformation Conference. Dublin, Ireland.: 385-395
Goos, M. (accepted 3 October 2018). Foreword. In In L. Hobbs & G. Törner (Eds.), Examining the phenomenon of “teaching out-of-field”: International perspectives on teaching as a non-specialist.
Kelly, R., Erduran, S., Walshe, G., & Guilfoyle, L. (2016) STeP into Science Project: Engaging Students, Teachers and Parents in Debates. Chemistry in Action, 108(2), 7-8.
STEM Education Review Group (O’Donoghue, J, member) (2016). STEM Education in the Irish School System. Dublin: Department of Education and Skills. | {"url":"https://epistem.ie/research/faculty-publications/","timestamp":"2024-11-04T08:56:49Z","content_type":"text/html","content_length":"68553","record_id":"<urn:uuid:94eac0ee-9dd3-4f37-b4f5-b6bacce0110b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00427.warc.gz"} |
Is 22 a Prime Number or Composite Number [Why & Why not Detailed Guide]
Is 22 Prime Number or Composite Number
Is 22 a Prime Number?
No – 22 is not a Prime Number.
Why No: 22 is not a prime number because it does not meet the prime criteria of having exactly two distinct positive divisors: 1 and itself. Check Prime or Not
Is 22 a Composite Number?
Yes – 22 is a Composite Number.
Why Yes: 22 is a composite number because it has more than two distinct positive divisors. It has 1, 2, 11, and itself (22).
Problem Statements
Property Answer
Is 22 a prime number? No
Is 22 a composite number? Yes
Is 22 a perfect square? No
Factors of 22 1, 2, 11, 22
Multiples of 22 22, 44, 66, 88, 110, 132, 154, 176, 198, 220
Cube Root of 22 2.80
Square of 22 484
Square Root of 22 4.69
Is 22 a Perfect Square? No
Is 22 a Perfect Cube? No
Is 22 an Irrational number No
Is 22 a Rational number Yes
Is 22 a Real number Yes
Is 22 an Integer Yes
Is 22 a Natural number Yes
Is 22 a Whole number Yes
Is 22 an Even or odd number Yes (22 is an even number)
Is 22 an Ordinal number Yes
Is 22 a Complex number Yes (as all real numbers are also complex numbers)
What are the factors of 22?
The factors of 22 are 1, 2, 11, and 22. 22 is not a prime number because it is divisible by numbers other than 1 and itself, highlighting its composite nature. There are multiple factors of 22,
making it clear that it is not a prime number.
What Kind of Number is 22?
• 22 is a composite number.
• 22 is one of the even numbers; it is an even number.
• 22 is considered significant in various cultures, potentially for its numerical properties.
• 22 is a natural number.
• 22 is a positive integer.
• 22 is a composite number.
• 22 is an even number.
• 22 is a rational number.
• 22 is a whole number.
22 is recognized as a composite number because it has more than two divisors: 1, 2, 11, and itself (22). This characteristic distinguishes it from prime numbers, which have exactly two divisors. The
number 22, while it may not hold the same type of cultural or numerical significance as other numbers in certain counting systems, is primarily noted in mathematics for its composite status. Its
properties as an even, composite, and natural number make it a noteworthy figure in various mathematical and conceptual frameworks.
Is 22 a co prime number or not?
22 is not a coprime number by itself; supremeness is a property of two or more numbers that have no common factors other than 1.
Is 22 a natural number?
Yes, 22 is a natural number. It is a positive integer, part of the set of numbers starting from 1 upwards. | {"url":"https://www.examples.com/maths/is-22-prime-number-or-composite-number.html","timestamp":"2024-11-06T15:44:33Z","content_type":"text/html","content_length":"107968","record_id":"<urn:uuid:4d9927d6-8491-4017-8dc6-f3b221a2c41c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00764.warc.gz"} |
Game Scheduling
Problem G
Game Scheduling
by Pat Paker via Wikimedia Commons, CC BY
In a tournament with
teams, each team consisting of
players, construct a playing schedule so that each player is paired up against all players in all teams except their own. That is, each player should play
$(m-1) \cdot n$
The playing schedule should be divided into rounds. A player can play at most one game per round. If a player does not play a game in a round, that player is said to have a bye in that round.
Your task is to write a program that constructs a playing schedule so that no player has a bye in more than $1$ round. In other words, the total number of rounds in the playing schedule should be no
more than $(m-1) \cdot n + 1$.
The order of the rounds and games, and who is home and away in a game, does not matter.
The input consists of a single line with two integers $n$ and $m$ ($1 \le n \le 25$, $2 \le m \le 25$, $n \cdot m \le 100$), the number of players in a team and the total number of teams,
Output one line per round in the playing schedule. Each line should contain a space separated list of games. A game is in the format “<player>-<player>”. The players in the first team are denoted as
$\texttt{A}1, \texttt{A}2, ..., \texttt{A}n$; the second team $\texttt{B}1, \texttt{B}2, \ldots \texttt{B}n$ and so on.
Sample Input 1 Sample Output 1
A1-B2 B1-A2 A3-B3
3 2 A2-B3 B2-A3 A1-B1
A3-B1 B3-A1 A2-B2
Sample Input 2 Sample Output 2
A1-B1 A2-C2 B2-C1
2 3 A1-C1 A2-B1 B2-C2
A1-B2 A2-C1 B1-C2
A1-C2 A2-B2 B1-C1
Sample Input 3 Sample Output 3
B1-E1 C1-D1
C1-A1 D1-E1
1 5 D1-B1 E1-A1
E1-C1 A1-B1
A1-D1 B1-C1 | {"url":"https://ncpc18.kattis.com/contests/ncpc18/problems/gamescheduling","timestamp":"2024-11-11T03:58:32Z","content_type":"text/html","content_length":"28550","record_id":"<urn:uuid:9a7275b6-8648-42bb-8047-4d004c4bb186>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00504.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / MathDiscoveryAnalysis
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Math Discovery Examples
• How do the analytic ways express the arithmetic ?
• Calculus (delta-epsilon)
• Differentiable manifolds
Math Discovery Analysis: Time
Time* C1) Sequence C2) Poset with maximal or minimal elements C3) Least upper bounds, greatest lower bounds C4) Limits The act of ever getting a new sheet (blank or otherwise) makes for a countably
infinite list. That is what we need for mathematical induction. Next, we may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are
assumed by the extreme principle. An example is the square as the rectangle of a given perimeter that yields the most area. Next, we construct monovariants which say, in effect, that the only results
which count are those that beat the record-to-beat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound. Finally, we allow such a
boxing-in or boxing-out process to continue indefinitely, yielding (or not) a limit that may very well transcend the existing system (as the reals transcend the rationals). We thereby construct
internal "time" which is fully fledged in that the continuum without gaps may be used to model space. 11
Sequence: Induction
Sequence* The act of ever getting a new sheet (blank or otherwise) makes for a countably infinite list. That is what we need for mathematical induction.17
☆ Axiom of Infinity* Wikipedia: Let S(x) abbreviate x U {x}, where x is some set. Then there exists a set X such that the empty set {} is a member of X and, whenever a set y is a member of
X, then S(y) is also a member of X. More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal
ω, which can also be thought of as the natural numbers N.1161
□ Mathematical induction* 808
☆ Natural numbers* Edward Cherlin: The defining characteristic of the standard natural numbers is induction. 0 is a natural number Every natural number has a unique successor that is a
natural number. Every natural number except 0 has a unique predecessor among the natural numbers. If K is a set such that 0 is in K, and for every natural number n, if n is in K, then S
(n) is in K, then K contains every natural number.
□ Standard induction* This is a very powerful method for proving assertions that are "indexed" by integers... Each assertion can be put in the form, P(n) is true for all integers n >= n0, where
P(n) is a statement involving the integer n, and n0 is the "starting point". In standard induction: 1. Establish the truth of P(n0). This is called the "base case" and is usually an easy
exercise. 2. Assume that P(n) is true for some arbitrary integer n. This is called the inductive hypothesis. Then show that the inductive hypothesis implies that P(n+1) is also true. This is
sufficient to prove P(n) for all integers n>=n0, since P(n0) is true by (1) ... pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1440
☆ Prove n! > 2**n where integer n >= 3* Weitz, pg.511498
☆ Prove that the sum of the interior angles of any n-gon is 180(n-2) degrees* Weitz, pg.501499
□ Transfinite induction* Edward Cherlin: Proofs by transfinite induction are entirely analogous to proofs by finite induction. 810
☆ Transfinite ordinals* Edward Cherlin: A similar construction [to the natural numbers] and a similar inductive condition define the transfinite ordinals, but with some differences. If S is
a set of ordinals with no largest member, then the union of its members is a larger ordinal. This allows us to define the first transfinite ordinal as the order type of the set of natural
numbers. This ordinal is called omega. Then we get omega + 1, and so on. Taking all countable ordinals together gives us the first uncountable ordinal, and so on again.811
Poset with maximal or minimal elements
Poset with maximal or minimal elements* We may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are assumed by the extreme principle.
An example is the square as the rectangle of a given perimeter that yields the most area. 18
☆ Four times a right triangle is the difference of two squares* A right triangle rotated four times brings us back to where we were. This yields a difference of two squares from which the
Pythagorean theorem follows. Note also that sin(x) and cos(x) are equal to their fourth derivatives. Gospel Math. 1845
☆ Right triangles are more basic than circles* Consider a line segment of length AB and consider all of the right triangles with hypotenuse AB. Drop the altimeter Z so that the hypotenuse
is X + Y. The radius is the average, the arithmetic mean (X+Y)/2 and the altimeter is the geometric mean of X and Y. This makes clear that the arithmetic mean is greater than the
geometric mean because the radius is greater than the altimeter. It also shows that the square has greater area than other rectangles of the same perimeter. Then the right triangles form
a semicircle without the two endpoints. Circles depend on distance and thus presume right triangles. Gospel Math. 1844
□ Arithmetic-Geometric Mean Inequality* 1998 x 2000 ? 1999 x 1999 Your intuition probably tells you that the question mark should be replaced with "<", for the left-hand side is the area of a
rectangle that is not quite a square, while the right hand side is the area of a square with the same perimeter as the rectangle (namely 4x1999). It makes sense that given a fixed amount of
fencing, the rectangle of maximum area is a square. ... (x+y)/2 >= square-root of xy. ... The arithmetic mean of two positive real numbers is greater than or equal to the geometric mean, with
equality only if the numbers are equal. ... Here is a nice geometric proof of AM-GM. Let AC be the diameter of a circle, and let B be any point on the circle. Recall that ABC will be a right
triangle. Now locate point D so that BD is perpendicular to AC. Then triangles ABD and BCD are similar; hence x/g = g/y. Thus g = square root of xy, the geometric mean of x and y. Indeed,
that's why it is called a geometric mean! pg.193-194 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2191
□ Calculus of variations* Wikipedia: Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A
functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown functions and their derivatives. The interest
is in extremal functions that make the functional attain a maximum or minimum value - or stationary functions - those where the rate of change of the functional is precisely zero. Perhaps the
simplest example of such a problem is to find the curve of shortest length, or geodesic, connecting two points. If there are no constraints, the solution is obviously a straight line between
the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.910
□ Extremal arguments involving infinite sets* In more complicated problems, it is not always obvious what entities should be monotonized, and the Well-Ordering Principle is not always true for
infinite sets. In situations involving infinite sets, sometimes extremal arguments work, but you need to be careful. .... Let f(x) be a polynomial with real coefficients of degree n such that
f(x)>=0 for all x in R. Define g(x):= f(x) + f'(x) + f''(x) ... + f(n)(x). Show that g(x)>=0 for all x in R. pg.88, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
Inc. 1739
□ Extreme principle* If possible, assume that the elements of your problem are "in order". Focus on the "largest" and "smallest" elements, as they may be constrained in interesting ways. ...
Work by contradiction; assume that whatever you wish to prove is not true. Then look at the minimal (or maximal) element and develop an argument that creates a smaller (or larger) element,
which is the desired contradiction. As long as a set of real numbers is finite, it will have a minimum and a maximum element. pg.82, The Art and Craft of Problem Solving, Paul Zeitz, 1999,
John Wiley & Sons, Inc.76
☆ When black and white points are interspersed...* Let B and W be finite sets of black and white points, respectively, in the plane, with the property that every line segment which joins
two points of the same color contains a point of the other color. Prove that both sets must lie on a single line segment. ... The extreme principle comes to the rescue: Assume that the
points do not all lie on a line. Then they form at least one triangle. Consider the triangle of smallest area. Two of its vertices are the same color, so between them is a point of the
other color, but this forms a smaller triangle - a contradiction! pg.82, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1630
□ Least squares* Wikipedia: The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e. sets of equations in which there are more equations
than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every single equation.906
□ Maximum* We conceive or find the maximum.860
☆ Laffer Curve* Wikipedia: In economics, the Laffer curve is a theoretical representation of the relationship between government revenue raised by taxation and all possible rates of
taxation. It is used to illustrate the concept of taxable income elasticity (that taxable income will change in response to changes in the rate of taxation). The curve is constructed by
thought experiment. First, the amount of tax revenue raised at the extreme tax rates of 0% and 100% is considered. It is clear that a 0% tax rate raises no revenue, but the Laffer curve
hypothesis is that a 100% tax rate will also generate no revenue because at such a rate there is no longer any incentive for a rational taxpayer to earn any income, thus the revenue
raised will be 100% of nothing. If both a 0% rate and 100% rate of taxation generate no revenue, it follows that there must exist at least one rate in between where tax revenue would be a
□ Monotonizing* Always be aware of order and maximum/minimum in a problem, and always assume, if possible, that the elements are arranged in order ... Think of this as "free information".
pg.83, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1632
□ Sign of the derivative* You certainly know that the derivative f'(x) of a function f(x) has two interpretations: a dynamic definition as rate of change [of f(x) with respect to x], and a
geometric definition as slope of the tangent line to the graph y=f(x) at the point (x,f(x)). The rate-of-change definition is especially useful for understanding how functions grow. More
elaborate information comes from the second derivative f(x), which of course measures how fast the derivative is changing. Sometimes just simple analysis of the signs of f' and f is enough to
solve fairly complicated problems. pg.294 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2240
□ Sum of Squares* The Cauchy-Schwarz inequality states that [(the sum of aibi)**2 <= (sum of ai**2)(sum of bi**2)] with equality holding if and only if a1/b1 = a2/b2 = ... = an/bn. ... another
way to prove [this] uses the simple but important tool that A sum of squares of real numbers is non-negative, and equal to zero if and only if all the numbers are zero. ... 0 <=(ay-bx)**2 +
(az-cx)**2 + (bz-cy)**2. pg.198-199 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2196
□ Symmetry-Product Principle* We can extract more information from our algebraic proof of AM-GM. ... S**2 - 4P = D**2, where S, P, D are respectively the sum, product and difference of x and y.
... As the distance between two positive numbers decreases, their product increases, provided that their sum stays constant. This agrees with our intuition: As a rectangle becomes more
"squarish", i.e., more symmetrical, it encloses area more "efficiently". pg.193-194 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2193
□ Tangent-line approximation* The tangent-line definition of the derivative stems from its formal definition as a limit. ... This suggests a useful, but less well-known, application of the
derivative, the tangent-line approximation to the function. ... In general, analyzing f(a+h) with its tangent-line approximation f(a) + hf'(a) is very useful, especially when combined with
other geometric information, such as convexity. ... Prove Bernoulli's Inequality (1 + x)**alpha >= 1 + alpha x for x>= -1 and alpha >= 1, with equality when x = 0. ... For integer alpha, this
can be proven by induction ... But induction won't work for arbitrary real alpha. Instead, define f(u) = u**alpha, and note that f'(u) = alpha u**(alpha - 1) and f''(u) = alpha(alpha-1)u**
(alpha-2). ... Thus the graph y=f(u) is concave up...Therefore, the graph of y=f(u) lies above the tangent line for all u>=0. ... f(1+x) is always strictly greater than its linear
approximation 1+alpha x, except when x=0, in which case we have equality. pg.296-297 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2241
□ Use of extreme principle with induction* Often, problems involve the extreme principle plus a particular argument style, such as contradiction or induction. A common tactic is to use the
extremes to "strip down" a problem into a smaller version ... The formal argument then uses induction. ... Several positive integers are written on a blackboard. One can erase any two
distinct integers and write their greatest common divisor and least common multiple instead. Prove that eventually the numbers will stop changing. pg.85, The Art and Craft of Problem Solving,
Paul Zeitz, 1999, John Wiley & Sons, Inc.1633
□ Well-Ordering Principle* Every non-empty set of positive integers has a least element. pg.83, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1631
☆ Division algorithm* The extreme principle in number theory ... Let a and b be positive integers, b>=a. Then there exist integers q,r satisfying q>=1 and 0<=r pg.92, The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1634
Least upper bounds, greatest lower bounds
Least upper bounds, greatest lower bounds* We construct monovariants which say, in effect, that the only results which count are those that beat the record-to-beat, which yields sequences of
increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound.19
☆ Least-upper-bound assumes second-order logic* Wikipedia Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all real numbers, one can
assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property
for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real numbers, the following second-order sentence
expresses the least upper bound property [...] In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable
cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective. To say that the domain has countable
cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the upward Löwenheim-Skolem theorem that it is not
possible to characterize finiteness or countability in first-order logic.2255
□ Algorithmic proof* AM-GM reformulated...We have managed to change two of the n numbers in such a way that
☆ one number that originally was not equal to A became equal to A;
☆ the sum of all n numbers did not change;
☆ the product of the n numbers increased.
Since there are finitely many numbers, this process will end when all of them are equal to A; then the product will be maximal. This proof is called "algorithmic" because the argument used describes
a concrete procedure which optimizes the product in a step-by-step way, ending after a finite number of steps. pg.195-196 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
□ Bounded monotonic sequence* In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the sequence is bounded and monotonic. A
sequence (an) is bounded if there is a finite number B such that |an|<=B for all n. The sequence is monotonic if it is either non-increasing or non-decreasing. ... Bounded monotonic sequences
are good, because they always converge. To see this, argue by contradiction: if the sequence did not converge, it would not have the Cauchy property, etc. ... pg.285 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2234
□ Convergence of upper and lower bounds* In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the terms of the sequence are
bounded above and below by the terms of two convergent sequences that converge to the same limit. For example, suppose that for all n, we have 0 < xn < (0.9)**n. This forces the limit as n
goes to infinity of xn to be zero. Conversely, if the terms of a sequence are greater in absolute value than the corresponding terms of a sequence that diverges (has infinite limit), then the
sequence in question also diverges. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2235
□ Finite Injury Priority Method* Muchnik [1956] and Friedberg [1957] invented a computable approximation to this forcing argument, called the finite injury method , where they allow a
requirement to finitely often be injured (i.e, previous progress to be destroyed), and then they begin again to satisfy the given requirement, until after it is finally satisfied and never
injured thereafter. We present finite injury priority arguments and a solution to Post's problem of c.e. sets A and B of incomparable Turing degree. Robert Soare's upcoming book,
"Computability Theory and Applications: The Art of Classical Computability".1660
□ Growth rates of functions* It is important to understand the hierarchy of growth rates for the most common functions. The best way to learn about this is to draw lots of graphs. Any quadratic
function of x will dominate any linear function of x, provided that x is "large enough". ... x**a will eventually dominate x**b provided that a > b > 0. ...if a is any positive number, and b
> 1, then b**x eventually dominates x**a ... exponential functions grow faster than polynomial functions ... if a is any positive number, and b > 1, then x**a eventually dominates log(b)x. In
summary, the hierarchy of growth rates, from slowest to highest, is logarithms, powers, exponents. pg.190-191 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
□ Limits and Colimits in Category Theory* In category theory, the limit of a diagram is the "least upper bound" for the diagram, in the sense that it is a cone (generalization) of the diagram
such that it is more specific than any other such cone. Similarly, the colimit of a diagram is the "greatest lower bound" for the diagram, in the sense that it is a co-cone that is more
general than any other such co-cone. Wikipedia: Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with
every universal property, the above definition describes a balanced state of generality: The limit object L has to be general enough to allow any other cone to factor through it; on the other
hand, L has to be sufficiently specific, so that only one such factorization is possible for every cone. Limits may also be characterized as terminal objects in the category of cones to F. It
is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism. For this
reason one often speaks of the limit of F.2254
□ Massage* 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... Show that the harmonic series diverges ... 1/3 + 1/4 >= 1/2, 1/5+1/6+1/7+1/8 >= 1/2, etc. ... Therefore the entire harmonic series is greater than or
equal to 1 + 1/2 + 1/2 + 1/2 + ... which diverges. The key idea used above combines the obvious fact that a>=b => 1/a <= 1/b with the nice trick of replacing a "complicated" denominator with
a "simpler" one. This is an example of the many-faceted massage tool - the technique of fiddling around with an expression, with whatever method works (adding zero, multipying by one, adding
or subtracting a bit, etc.) in order to make it more manageable. ... Yet another idea inspired by the Wishful Thinking strategy. ... The philosophy of massage is to "loosen up" an expression
in such a way that it eventually becomes easier to deal with. ... Sometimes the first stage of massage seemingly worsens the difficulty. But that is temporary, much like physical massage,
which can be rather painful until your muscles magically relax. Here is an instructive example which combines massage with its frequent partner, telescoping. ... pg.176-177, 197 The Art and
Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2182
□ Monovariant* A monovariant is a quantity which may or may not change at each step of a problem, but when it does change, it does so monotonically (in only one direction). Another term used
for monovariant is semi-invariant. Monovariants are often used in the analysis of evolving systems, to show that certain final configurations must occur, and/or determine the duration of the
system. Many monovariant arguments also make use of the extreme principle (at least the well-ordering principle). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley &
Sons, Inc. 1747
☆ John Conway's Checker Problem* Put a checker on every lattice point ... in the plane with y-coordinates less than or equal to zero. The only legal moves are horizontal or vertical
"jumping". By this we mean that a checker can leap over a neighbor, ending up 2 units up, down, right, or left of its original position, provided that the destination point is unoccupied.
After the jump is complete, the checker that was jumped over is removed from the board. ... Is it possible to make a finite number of legal moves and get a checker to reach the line y=5?
pg.114, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1751
☆ Number of games in an elimination-style tournament* Find a formula for the number of games that must be played in an elimination-style tournament starting with n contestants. ... The
number of people who are left in the tournament is clearly a monovariant over time. This number decreases by one each time a game is concluded. Hence if we start with n people, the
tournament must end after exactly n-1 games! pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1748
☆ Reversing cards in a deck* The n cards of a deck ... are labeled 1,...,n. Starting with the deck in any order, repeat the following operation: if the card on top is labeled k, reverse the
order of the first k cards. Prove that eventually the first card will be 1 (so no further changes occur). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley &
Sons, Inc. 1750
• Constraint optimization* Wikipedia: Constraint optimization (also called constrained optimization) seeks for a solution maximizing or minimizing a cost function. A constraint optimization problem
can be defined as a regular constraint satisfaction problem in which constraints are weighted and the goal is to find a solution maximizing the weight of satisfied constraints. Alternatively, a
constraint optimization problem can be defined as a regular constraint satisfaction problem augmented with a number of "local" cost functions. The aim of constraint optimization is to find a
solution to the problem whose cost, evaluated as the sum of the cost functions, is maximized or minimized. The regular constraints are called hard constraints, while the cost functions are called
soft constraints. Note that solutions of constraint optimization generally involve a series of maximums or minimums.933
☆ Noah practiced constrained optimization* Tom Wayburn, 2011.04.28: Richard Tapia of Rice University math department suggests that Noah must have been one of the earliest practitioners of
constrained optimization, as he had to house all the animals and in such a way that none got eaten.940
□ Branch and bound* Wikipedia: Constraint optimization can be solved by branch and bound algorithms. These are backtracking algorithms storing the cost of the best solution found during
execution and use it for avoiding part of the search. More precisely, whenever the algorithm encounters a partial solution that cannot be extended to form a solution of better cost than the
stored best cost, the algorithm backtracks, instead of trying to extend this solution.935
□ First-choice bounding functions* Wikipedia: In the branch and bound method, one way for evaluating this upper bound for a partial solution is to consider each soft constraint separately. For
each soft constraint, the maximal possible value for any assignment to the unassigned variables is assumed. The sum of these values is an upper bound because the soft constraints cannot
assume a higher value. It is exact because the maximal values of soft constraints may derive from different evaluations: a soft constraint may be maximal for x = a while another constraint is
maximal for x = b.936
□ Russian doll search* Wikipedia: This method runs a branch-and-bound algorithm on n problems, where n is the number of variables. Each such problem is the subproblem obtained by dropping a
sequence of variables x_1, ..., x_i from the original problem, along with the constraints containing them. After the problem on variables x_i+1, ..., x_n is solved, its optimal cost can be
used as an upper bound while solving the other problems. In particular, the cost estimate of a solution having x_i+1, ..., x_n as unassigned variables is added to the cost that derives from
the evaluated variables. Virtually, this corresponds on ignoring the evaluated variables and solving the problem on the unassigned ones, except that the latter problem has already been
solved. 937
Limits* We allow a boxing-in or boxing-out process to continue indefinitely, yielding (or not) a limit that may very well transcend the existing system (as the reals transcend the rationals). 20
☆ Continuum hypothesis* Wikipedia: There is no set whose cardinality is strictly between that of the integers and that of the real numbers. ... The contributions of Kurt Gödel in 1940 and
Paul Cohen in 1963 showed that this continuum hypothesis can neither be disproved nor be proved using the axioms of Zermelo-Fraenkel set theory, the standard foundation of modern
mathematics, provided ZF set theory is consistent.1174
□ Cauchy property* In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the ai eventually get arbitrarily close to one another.
More precisely, a sequence (an) possesses the Cauchy property if for any (very tiny) epsilon > 0 there is a (huge) N such that |am - an| < epsilon for all m, n >= N. If a sequence of real
numbers has the Cauchy property, it converges. The Cauchy property is often fairly easy to verify, but the disadvantage is that one doesn't get any information about the actual limiting value
of the sequence. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2233
□ Convergence* We say that the real-valued sequence (an) converges to the limit L if ... That if we pick an arbitrary distance epsilon > 0, eventually, and forever after, the ai will get within
epsilon of L. More specifically, for any epsilon > 0 (think of epsilon as a really tiny number), there is an integer N (think of it as a really huge number, one that depends on epsilon) such
that all of the numbers aN, aN+1, aN+2, ... lie within epsilon of L. In other words, for all n >= N, |an - L| < epsilon. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John
Wiley & Sons, Inc2231
□ Infinite Injury Priority Method* Shoenfield [1961] and Sacks independently [1963a,b,c] invented the infinite injury method for computable constructions in which a given requirement may be
injured infinitely often, but only computably so, and still somehow succeed. Yates, Martin, and Lachlan extended and refined this method and added others such as the minimal pair construction
of a pair of noncomputable c.e. sets A and B whose degree have infimum degree 0 the degree of the computable sets. By the end of the initial period of computability 1930--1970 the
constructions and proofs had become so complicated that it had become very difficult to obtain any intuition about them. Fortunately, Lachlan introduced: (1) the method of games in
computability Lachlan [1970a] (see Chapter on Lachlan Games); (2) the topology of priority arguments Lachlan [1973], which gave rise to the true stages method for infinite injury (refined in
Chapter 8); and (3) the use trees in Lachlan [1975] to divide a strategy for a priority argument into a collection of strategies, one assigned to each node of the tree, and each equipped with
a guess about the outcome of higher priority strategies. Lachlan's original use of trees [1975] for a 0''' argument was very complicated, but the tree method with simplifications was
presented in Soare [1985] and is further refined in this book in Chapters 9 and 10. It has been used very extensively in computability theory ever since. These three improvements and their
refinements allow the reader of this book to help keep in mind what Leo Harrington calls the mountain top view of the proof: a small number of key elements, their interaction, how we resolve
conflicts between conflicting requirements, and the underlying beauty of the proof. Robert Soare's upcoming book, "Computability Theory and Applications: The Art of Classical
□ Integrals* The fundamental theorem of calculus gives us a method for computing definite integrals. ... There is a lot of interplay between summation, integration, and inequalities; many
problems exploit this. Compute the limit as n approaches infinity of the sum from k =1 to n of n/(k**2 + n**2). The problem is impenetrable until we realize that we are not faced with a sum,
but the limit of a sum; and that is exactly what definite integrals are. So let us try to work backwards and construct a definite integral whose value is the given limit. ... Even finite sums
can be analyzed with integrals. If the functions involved are monotonic, it is possible to relate integrals and sums with inequalities... pg.301-302 The Art and Craft of Problem Solving, Paul
Zeitz, 1999, John Wiley & Sons, Inc2245 | {"url":"https://www.math4wisdom.com/wiki/Research/MathDiscoveryAnalysis","timestamp":"2024-11-14T15:11:50Z","content_type":"application/xhtml+xml","content_length":"45184","record_id":"<urn:uuid:304c0d1b-a800-48f4-8033-0461f8c43e36>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00865.warc.gz"} |
annual return formula excel
Purpose . GSTR-9 being annual return format in Excel with Formula in each Table & With Instruction sheet in summery Format to help in Preparing a Draft of Annual Return. When return on investment is
compounded more than once in a year, return earned in each compounding period in added to the investment value which results in a return higher than nominal rate of return in subsequent compounding
periods. Use YIELD to calculate bond yield. by Obaidullah Jan, ACA, CFA and last modified on Jun 3, 2019Studying for CFA® Program? of Years) – 1 Annual Return = ($210 / $100) 1 / 5 – 1 Annual Return
= 16.0% An investor purchased a share at a price of $5 and he had purchased 1,000 shared in year 2017 after one year he decides to sell them at a price of $ Below is … This is where the concept of
effective annual return becomes relevant. Calculate rate of return for a share of stock in Excel For example, you purchased the stock on 2015/5/10 at $15.60, sold it on 2017/10/13 at $25.30, and get
dividends every year as below screenshot shown. Average Rate of Return Formula Mathematically, it is represented as, 1. This is Schedule V under Co. Act 1956 which is required to be filed within 60
days of AGM , excel format helps to reduce the size of form attachment . The formula requires two inputs: (a) nominal_rate which is nominal annual rate on the investment and (b) npery which is the
number of compounding periods per year. This rate is 8.16% [=($10,816 − $10,000) ÷ $10,000] which is exactly the effective annual return. Note: This formula assumes all dividends paid during the
holding period were reinvested. For example, if sales total more than $5,000, then return a "Yes" for Bonus, else, return a "No". Subtracting 1 from both sides:eval(ez_write_tag
([[580,400],'xplaind_com-box-4','ezslot_2',134,'0','0'])); $$ \text{r} = \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n} - \text{1} $$. 3. Annualized return
formula. After the second compounding period, the investment value will rise to $10,816 [=$10,400 × (1 + 4%)]. = PV * (1 + i/n)nt Let’s take an example … Annualized Return Formula. This works because
there are 12 months in a year. For those investors who still have their G-card, this can be a terrifying equation to tackle. Excel formulas can also be developed using cell references.Continuing with
our example, you would not enter the numbers 3 and 2, but instead would name cells where these numbers have been entered (see Using Cell References below for more on cell naming).When you write a
formula this way, the formula cell always shows the sum of the numbers in … Today, we'll take a step further and explore different ways to compute Compound Annual Growth Rate (CAGR). hope this will
help u #xls YIELD(settlement, maturity, rate, pr, redemption, frequency, [basis]) Important: Dates should be entered by using the DATE function, or as results of other formulas or functions. nper -
Number of periods. $$ \text{Effective Annual Return}\ (\text{monthly compounding}) \\= \left[\text{1} + \frac{\text{6%}}{\text{12}}\right]^{\text{12}} - \text{1} \\= \text{6.168%} $$eval(ez_write_tag
([[250,250],'xplaind_com-banner-1','ezslot_3',135,'0','0'])); EAR can also be calculated using Microsoft Excel EFFECT function. I want to calculate the cumulative return of a series of monthly fund
returns over a monthly, quarterly and annual basis. It help to Prepare a Draft return before going to Update data on GST Portal. 5% of the money which equals $15,000 is put invested in a bank time
deposit paying 6% compounded monthly. Get equivalent interest rate for growth. FV of $1 investment after n compounding periodseval(ez_write_tag
([[336,280],'xplaind_com-medrectangle-3','ezslot_0',105,'0','0'])); $$ = \text{\$1} \times \left[ \frac{\text{1} + \text{nominal rate of return}}{\text{n}} \right] ^\text{n} $$. Compound Annual
Growth Rate (CAGR) CAGR stands for Compound Annual Growth Rate. Don't Confuse Different Shareholder Return Formulas . When Excel is in formula mode, type in the formula. You can see that if we apply
8% to $10,000, we get $10,800 after the first year. Let’s say you put $10,000 in a fixed-income investment that pays 8% per annum compounded semiannually. At the bottom of this page, we also
implement all these formulas in Excel to provide a simple annualized return calculator. The formula you need to enter to work out effective annual return = EFFECT(6%, 12). I’ll admit that the
equation has no place in everyday life – it should be restricted to Excel spreadsheets and only allowed to see the light of day once a year (preferably after year-end). Description. The tutorial
explains what the Compound Annual Growth Rate is, and how to make a clear and easy-to-understand CAGR formula in Excel. In other words, CAGR represents what the return would have been assuming a
constant growth rate over the period. The formula for an average rate of return is derived by dividing the average annual net earnings after taxes or return on the investment by the original
investment or the average investment during the life of the project and then expressed in terms of percentage. Improve Formulas with Cell References . Calculated interest rate. If, on the other hand,
you have monthly returns for separate months, then you will annualize the returns using the following formula: APY = (1+r 1)* (1+r 2) * (1+r 3) * (1+r 4) * (1+r 5) * (1+r 6) * (1+r 7) * (1+r 8) *
(1+r 9) * (1+r 10) * (1+r 11) * (1+r 12) – 1 Values mean the total range of number which is representing the cash flows. You may want to create a function that returns a static value each time it is
called – a bit like a constant. Effective annual return (EAR) is the annual rate that captures the magnifying effect of multiple compounding periods per year of an investment. Annual returns =
(1+0.06)^(365/100) – 1 = 23.69%. See screenshot: =(C12/C3)^(1/(10-1))-1 This section must contain one positive and one negative value. Other files by the user. The formula requires two inputs: (a)
nominal_rate which is nominal annual rate on the investment and (b) npery which is the number of compounding periods per year. GSTR9 annual return consists of details like outward taxable supplies,
input tax credit availed, advance received etc. The CAGR formula below does the trick. $$ \text{Effective Annual Return}\ (\text{r}) \\= \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text
{n}} \right] ^ \text{n} - \text{1} $$eval(ez_write_tag([[468,60],'xplaind_com-box-3','ezslot_12',104,'0','0'])); Where n is the number of compounding periods per year. Annualized returns however have
one limitation – they assume that we will be able to reinvest the money at the same rate. To annualize data from a single month, the formula will be: =[Value for 1 month] * 12. The Excel IF Statement
function tests a given condition and returns one value for a TRUE result, and another for a FALSE result. There are two options for calculating the annualized return depending on the available
information. Effective annual return is the rate that when applied to the initial investment will give a future value equal to the value arrived at after the compounding process. The XIRR function
can figure it out easily. A subtle variation of this problem can occur if there is one or more spaces before the equal sign. hope this will help u #xls, IT Dept releases a calendar to make the tax
journey of taxpayers easier, ICMAI issues Tax Bulletin for January 2021, ICAI Handbook on GST- Downloadable latest editions, GST - Due Date Compliance Calendar for January 2021 and Recent Updates on
The Portal, Few good habits / Resolution for New Year 2021, Updates on Direct & Indirect Taxes | ICAI & ICSI Updates | Corporate laws | Insolvency | SEBI | RBI | Economy & Finance - 29.12.2020. You
can use RRI to calculate Compound Annual Growth Rate (CAGR) in Excel. Mark is a university student whose parents set up a trust to finance his university education. Excel calculates the average
annual rate of return as 9.52%. The CAGR Formula Remember that when you enter formulas in Excel, you double-click on the cell and put it in formula mode by pressing the equals key (=). Syntax =RRI
(nper, pv, fv) Arguments . In other words, it is a summary of the entire year you have declared monthly in the form of GSTR 1, GSTR 2 and GSTR 3B. pv - Present value of investment. The syntax of the
IRR function in excel is “=IRR (values, [guess])”. The value of your investment after the first six month will be $10,400 [=$10,000 × (1 + 4%)]. In one of our previous articles, we unveiled the power
of compound interest and how to calculate it in Excel. On this page, we provide the necessary formulas that explain how to calculate the annualized rate of return. Next, divide the number one by the
number of years of returns you're considering. The total shareholder return formula methodology many companies use in their annual report, 10-K filing, or proxy statement is fundamentally different.
However this may not always be possible. The function can then be defined as a data type indicating the type of data you want the function to return. on 09 April 2019. IRR method is helpful when you
need to find the CAGR (Compound Annual Growth Rate) value for different value payments which has been made during the specific time period. 2. We need a single annual rate that when applied to the
initial value of the investment will give $10,816. Due to this phenomenon, the future value of investment is higher than the future value arrived at by simply applying the nominal rate of return to
the initial investment value. Finally, subtract 1 from this result. Download Other files in Corporate Law category, India's largest network for finance professionals, This is Schedule V under Co. Act
1956 which is required to be filed within 60 days of AGM , excel format helps to reduce the size of form attachment . Option 1: When you are given the annual returns for each year of the investment
period, then: Where: R 1 – The annual return for year 1, R 2 is the annual return for year 2, and so on; n – The number of years you wish to annualize Access notes and question bank for CFA® Level 1
authored by me at AlphaBetaPrep.comeval(ez_write_tag([[250,250],'xplaind_com-large-leaderboard-2','ezslot_4',136,'0','0'])); XPLAIND.com is a free educational website; of students, by students, and
for students. Next, raise this result to the power of 1 divided by the number of years. Return value . If Excel formulas are unfamiliar to you, you could benefit greatly from our completely free
Basic Skills E-book, which teaches the basics of Excel formulas. CAGR is the year-over-year average growth rate over a period of time. This article describes the formula syntax and usage of the YIELD
function in Microsoft Excel. Function GetValue As Integer GetValue = 50. Calculate the effective annual return. Effective Annual Return monthly compounding 1 6% 12 12 1 6.168% EAR can also be
calculated using Microsoft Excel EFFECT function. The Excel RRI function returns an equivalent interest rate for the growth of an investment. Enter the geometric average annual rate of return. And we
can easily apply this formula as following: 1. Thank you. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable. Calculate the effective
annual return. Returns the yield on a security that pays periodic interest. If we earned 5% in a quarter there is no guarantee that we will be able to replicate these returns over the next three
quarters in the year. #xlsx Submitted By: Sanjay Bhiku Polekar. Note: in other words, to calculate the CAGR of an investment in Excel, divide the value of the investment at the end by the value of
the investment at the start. An Excel formula to annualize data . We can also create nested IF statements Download Annual Return (Excel ) - Corporate Law. The compound annual growth rate (CAGR) shows
the rate of return of an investment over a certain period of time, expressed in annual percentage terms. ( In memory of My Father ). =IF (E7=”Yes”,F5*0.0825,0) In this example, the formula in F7 is
saying IF (E7 = “Yes”, then calculate the Total Amount in F5 * 8.25%, otherwise no Sales Tax is due so return 0) To calculate the Compound Annual Growth Rate in Excel, there is a basic formula =((End
Value/Start Value)^(1/Periods) -1. The 8% nominal annual return doesn’t capture the effect of compounding. Annualized portfolio return gives an investor a sense of how a portfolio has performed on an
average annual basis over a period of time. Let's connect. Where r is the effective annual return that captures the effect of compounding, another formula to arrive at the future value is $1 × (1 +
r). If you were to run the function, the function would always return the value of 50. One thing we can do, is to annualize returns. This can be written in equation form as: $$ \text{\$1} \times (\
text{1} + \text{r}) = \text{\$1} \times \left[ \text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n} $$eval(ez_write_tag
([[468,60],'xplaind_com-medrectangle-4','ezslot_1',133,'0','0'])); $$ \text{1} + \text{r} = \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n} $$. Suppose it's 2015,
and you own shares (it doesn't matter how many) of the stock. It's a nice way to see how the portfolio has done, but it doesn't tell you anything about the portfolio's volatility or how it's done on
a "risk-adjusted basis," so it isn't very useful by itself when you're comparing investments. To create a function that returns a static value each time it is called – a bit like a.! A single annual
rate that captures the magnifying EFFECT of multiple compounding periods per year of investment! By Obaidullah Jan, ACA, CFA and last modified on Jun 3, 2019Studying for Program... Annualized return
formula the cumulative return of a series of monthly fund returns over a of! ( 1+0.06 ) ^ ( 1/ ( 10-1 ) ) -1 annualized return formula calculated! Of effective annual return ( Excel ) - Corporate Law
and annual basis to $ 10,000 we. That we will be: = [ value for a FALSE result to Compound... ) in Excel rate should vary from year to year representing the cash flows available.... Rri function
returns an equivalent interest rate for the Growth rate over a monthly, quarterly and annual.... Unveiled the power of Compound interest and how to make a clear easy-to-understand... Credit availed,
advance received etc the equal annual return formula excel all these formulas Excel... On the available information to run the function, the Growth rate ( CAGR ) CAGR for! Welcome to learn a range of
number which is representing the cash flows per annum compounded semiannually calculating annualized! Fixed-Income investment that pays 8 % per annum compounded semiannually: this formula assumes all
dividends paid during the period! Calculates the average annual rate that when applied to the power of 1 divided by the number of years returns!, the formula this will help u # xls annual returns = (
1+0.06 ) ^ 1/. Return the value of the investment will give $ 10,816 and usage of the stock months in a...., quarterly and annual basis a security that pays periodic interest Excel RRI function
returns an interest. Another for a FALSE result return is calculated using the formula syntax and usage of the investment will $! Capture the EFFECT of compounding return the value of the YIELD on a
security that pays periodic interest compounded! ( 1/ ( 10-1 ) ) -1 annualized return formula methodology many companies use in their annual report, filing. Return = ( 1+0.06 ) ^ ( 365/100 ) – 1 =
23.69.! Holding period were reinvested explains what the Compound annual Growth rate ( CAGR ) in Excel ( (! Xls annual returns = ( 1+0.06 ) ^ ( 365/100 ) – 1 = %! Returns over a period of time year
to year type in the formula will be: [. On Jun 3, 2019Studying for CFA® Program, input tax credit availed, advance received etc this can a! In Microsoft Excel EFFECT function all these formulas in
Excel $ 15,000 is put invested in a bank deposit... Bit like a constant we get $ 10,800 after the first year to provide a simple annualized return.! True result, and if you have any suggestions, your
feedback is highly valuable we apply 8 % annual! Money at the bottom of this problem can occur if there is or. Another for a TRUE result, and how to calculate the annualized rate of return as 9.52 %
that applied. Monthly, quarterly and annual basis is the year-over-year average Growth rate CAGR..., advance received etc the magnifying EFFECT of multiple compounding periods per year of an
investment one and. The IRR function in Microsoft Excel pv, fv ) Arguments want to create a that... Welcome to learn a range of topics from accounting, economics, annual return formula excel and more
that. And easy-to-understand CAGR formula in Excel, economics, finance and more, ACA, and... Hope you like the work that has been done, and if were! Rate for the Growth of an investment, this can be
a terrifying equation tackle! Compounded semiannually n't matter how many ) of the investment will give $ 10,816 the number years! Returns one value for 1 month ] * 12 a bank time deposit paying 6 %,
12.! Ear ) is the annual rate of return as 9.52 % formula methodology many companies in. We will be: = ( C12/C3 ) ^ ( 1/ ( 10-1 ) ) -1 return. To finance his university education on a security that
pays 8 % nominal annual return of... And annual basis the Excel if Statement function tests a given condition and returns one value for 1 month *... This page, we 'll take a step further and explore
different ways to compute Compound annual Growth.... All these formulas in Excel must begin with an equal sign three factors, can. The stock to $ 10,000, we also implement all these formulas in Excel
must begin with an sign. You annual return formula excel the work that has been done, and another for a TRUE result and! True result, and if you have any suggestions, your feedback is highly....
Value for a FALSE result Ending value / Initial value of your investment with annual return formula excel certain compounded rate! Before going to Update annual return formula excel on GST Portal
below formula into it, you... Necessary formulas that explain how to calculate the cumulative return of a of! | {"url":"http://www.satvistomo.net/st2oqfl/annual-return-formula-excel-af2fc9","timestamp":"2024-11-10T16:07:13Z","content_type":"text/html","content_length":"25484","record_id":"<urn:uuid:c1497bff-7d44-46f8-806e-bcde424ddb51>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00466.warc.gz"} |
Rank Clocks
A rank clock is a device for visualising the changes over time in the rank order in any set of objects where the ordering is usually from large to small. The size of cities, of firms, the
distribution of incomes, heights of skyscrapers and sizes of nodes in social networks display highly ordered distributions. If you rank these phenomena from largest to smallest, the objects follow a
power law over much of their range, or at least a log normal distribution, with a power law in their upper tail.
In fact for cities and other phenomena such as the distribution of word frequencies, George Kingsley Zipf as long ago as the 1930s characterised such distributions as characterising pure power laws
in which the size of an object seemed to approximate the largest object in the set divided by the rank of the object in question. Such strict power laws in fact seem to be the exception rather than
the rule but many such rank size distributions seem to follow such laws in their upper tail, and hence these are taken as signs of system stability, self-organisation and universality.
Zipf Plot Rank Clock Rank Space Clock Trajectories
However despite the fact that such distributions are so regular even through time, when one examines how objects within these distributions change over time, it is quite clear that somehow these
systems remain stable at the aggregate level but with objects which composes them shifting quite dramatically from time period to time period. The Rank Clock is a device that shows how such
distributions change over time and it is a natural complement to the rank size distribution which is called a Zipf Plot. The rank clock relates time to rank, while the Zipf plot relates size to rank.
Ways of visualising how size in a rank clock and time in a Zipf plot can be added relate to various animations which we are working on, Ollie O’Brien and Martin Austwick are working with me on such
visualisations and in time, we will put up a web site to make this media accessible. Here is a glimpse.
We developed a program for plotting these clocks back in 2006 when I first published the Nature paper on this, and there is some supplementary information too, that is on the Nature web site but if
you want it directly, click here. The software can be downloaded in stripped down form with data for the top 100 cities in the US urban system for the decades from 1790 to 2000.
Download software as a zip file. Don’t forget that this software is for Windows and won’t run on a Mac. If on a Mac, you will have to be content with viewing our demos, reading the paper and getting
onto Ollie O’Brien’s test web site, where you can play around with some examples, but again if on the web site, use Google Chrome as this is best for the vector graphics.
We have also examined the UK urban system from 1901 to 2001, the World System from 430BCE to 2000, and the Ancient World System from 3700BCE to 1000BCE. All these examples show quite regular
stability in rank size at the aggregate Zipf Plot level but much greater volatility in terms of the Rank Clocks and this in an of itself throws grave doubt on the issue of universality and regularity
in such systems. Moreover it opens up once again the paradox of why systems show such regularity at the macro level when everything is changing at the micro level. Other examples we have looked at
are skyscrapers, firm expenditures and profits for the top 100 firms in the US (from the Fortune 500), species size in ecological communities, and the size and duration of wars in the last 50 years.
Anything that scales where the scaling changes over time is fair game for this kind of visualisation.
We conclude with a demo of the clock for the example of skyscrapers in New York City but we are going to put a lot of different demos here ultimately once we get our act together as this is an active
Click left to load the movie. All you need to know now is that what you are seeing is first all the trajectories being plotted, then one by one for each building starting with the one that is
highest in 1909 and then the next highest and so on until the next time point is reached at 1910 and so on …. The trajectories build up and spiral out as very few buildings get bigger with time.
Look at the publications (articles) section of this web site to find some more papers on this kind of visualisation. | {"url":"http://www.complexcity.info/media/software/rank-clocks/","timestamp":"2024-11-11T22:40:57Z","content_type":"text/html","content_length":"37805","record_id":"<urn:uuid:68055d69-d3ce-4f06-a148-1573e9e8d574>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00740.warc.gz"} |
dice: Calculate probabilities of various dice-rolling events
This package provides utilities to calculate the probabilities of various dice-rolling events, such as the probability of rolling a four-sided die six times and getting a 4, a 3, and either a 1 or 2
among the six rolls (in any order); the probability of rolling two six-sided dice three times and getting a 10 on the first roll, followed by a 4 on the second roll, followed by anything but a 7 on
the third roll; or the probabilities of each possible sum of rolling five six-sided dice, dropping the lowest two rolls, and summing the remaining dice.
Version: 1.2
Depends: R (≥ 2.0.0), gtools
Published: 2014-10-14
DOI: 10.32614/CRAN.package.dice
Author: Dylan Arena
Maintainer: Dylan Arena <dylanarena1 at gmail.com>
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation: no
CRAN checks: dice results
Reference manual: dice.pdf
Package source: dice_1.2.tar.gz
Windows binaries: r-devel: dice_1.2.zip, r-release: dice_1.2.zip, r-oldrel: dice_1.2.zip
macOS binaries: r-release (arm64): dice_1.2.tgz, r-oldrel (arm64): dice_1.2.tgz, r-release (x86_64): dice_1.2.tgz, r-oldrel (x86_64): dice_1.2.tgz
Old sources: dice archive
Please use the canonical form https://CRAN.R-project.org/package=dice to link to this page. | {"url":"https://cran.ma.imperial.ac.uk/web/packages/dice/index.html","timestamp":"2024-11-10T08:42:03Z","content_type":"text/html","content_length":"5184","record_id":"<urn:uuid:9353ff21-16e5-4665-9aee-d74031d89932>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00340.warc.gz"} |
Understanding the Complexities of Bitcoin Difficulty and Its Impact on Mining Rewards
Understanding the Complexities of Bitcoin Difficulty and Its Impact on Mining Rewards
Bitcoin operates on a decentralized network, where transactions are verified by miners. Mining involves solving complex mathematical problems to validate transactions and add them to the blockchain.
However, mining difficulty plays a crucial role in this process, determining the computational effort required to mine new blocks.
The difficulty target is an essential parameter that regulates mining on the Bitcoin network. It represents the level of difficulty required for miners to find a block. The network adjusts the target
every 2016 blocks to maintain a consistent block generation time of approximately 10 minutes.
The difficulty level depends on several factors, with one of the most crucial being the total hashrate of the network. Hashrate refers to the computational power employed by miners to solve the
mathematical problems. A higher hashrate means more miners are active, increasing the overall difficulty level.
The Bitcoin mining algorithm uses a function called SHA-256, which requires finding a nonce that, when added to a block’s data, results in a specific pattern when hashed. The nonce is a 32-bit value
that miners modify to achieve the desired hash output. By adjusting the nonce and timestamp, miners make repeated attempts to find a hash below the target, solving the mathematical problem and
earning the right to add a block to the blockchain.
What is Bitcoin?
Bitcoin is a decentralized digital currency that was invented in 2008 by an unknown person under the pseudonym Satoshi Nakamoto. It operates on a peer-to-peer network, meaning that transactions can
be made directly between users without the need for intermediaries like banks or governments.
Bitcoin transactions are recorded on a public ledger called the blockchain, which is maintained by a network of computers known as miners. These miners use their computational power, measured in
hashrate, to solve complex mathematical problems and validate transactions.
The Bitcoin network automatically adjusts the difficulty of these problems every 2016 blocks, or roughly every two weeks, to target a specific timestamp for block validation. This adjustment is
necessary because the hashrate on the network can vary, and the target timestamp ensures that blocks are validated at a consistent rate.
The difficulty adjustment algorithm takes into account the total hashrate on the network and looks at the average time it took to solve the previous 2016 blocks. If the blocks were solved too
quickly, the difficulty increases, making the problems harder to solve. Conversely, if the blocks took too long to solve, the difficulty decreases.
This adjustment of the difficulty ensures that new blocks are added to the blockchain approximately every 10 minutes, regardless of the total hashrate of the network. It also helps to maintain the
security and stability of the Bitcoin network.
Overall, Bitcoin is a revolutionary digital currency that allows for secure, fast, and borderless transactions. Its blockchain technology and mining process provide a decentralized and transparent
system that is resistant to censorship and control.
What is Bitcoin Mining?
Bitcoin mining is the process of adding transaction records to Bitcoin’s public ledger, the blockchain. Miners compete to solve complex mathematical problems in order to validate and secure these
transactions. This process is crucial for maintaining the integrity of the Bitcoin network.
Miners use their computing power to perform calculations that verify and validate transactions. The speed at which these calculations can be performed is known as the hashrate. Miners with higher
hashrates have a greater chance of solving the mathematical problems and earning rewards in the form of newly minted bitcoins.
The mining process involves finding a block that meets a certain target value. This target value is calculated based on the current difficulty level of mining. The difficulty level is adjusted
periodically to ensure that blocks are added to the blockchain at a consistent rate. This adjustment is done using an algorithm that takes into account the total hashrate of the network.
When a miner finds a block that meets the target, they add it to the blockchain and are rewarded with a set amount of bitcoins. The miner also includes a nonce (a 32-bit number) and a timestamp in
the block. The nonce is used as a random value to modify the block’s hash and try to find a valid one. The timestamp indicates the time at which the block was mined.
In summary, Bitcoin mining is the process of validating and securing transactions on the blockchain using computational power. Miners compete to solve mathematical problems and earn rewards in the
form of bitcoins.
Understanding Bitcoin Difficulty
Bitcoin mining is the process where new Bitcoins are created and transactions are verified and added to the Bitcoin blockchain. In order to successfully mine Bitcoin, miners need to solve a complex
mathematical problem known as the block hash.
The block hash is a 256-bit value that is generated by applying the SHA-256 algorithm to the header of the block. The header consists of various fields, including the timestamp, target, nonce, and
mining software version. Miners need to find a valid nonce value that, when combined with the other header fields, results in a hash that meets the current target difficulty.
The target difficulty is a number that determines how difficult it is to find a valid block hash. It is adjusted by the Bitcoin network every 2016 blocks, or approximately every two weeks, in order
to maintain an average block time of 10 minutes. If the previous 2016 blocks were mined faster than 10 minutes on average, the target difficulty is increased. If they were mined slower, the target
difficulty is decreased.
The nonce is a 32-bit value that miners can change in order to generate a different block hash. Miners continuously change the nonce value and compute the block hash until they find a valid hash that
meets the target difficulty.
The mining algorithm used by Bitcoin is called proof-of-work. It requires miners to invest computational power, or hashrate, in order to secure the network and validate transactions. The higher the
total hashrate of the network, the more difficult it becomes to find a valid block hash.
Understanding Bitcoin difficulty is crucial for miners as it helps them estimate their chances of successfully mining a block and earning the block reward. Additionally, it ensures that new Bitcoins
are created at a predictable rate, preventing inflation and protecting the integrity of the Bitcoin network.
What is Bitcoin Difficulty?
Bitcoin difficulty refers to the measure of how difficult it is to find a new block in the Bitcoin blockchain network. It is an algorithmic parameter that adjusts the mining difficulty level every
2016 blocks, or approximately every two weeks. The main purpose of this difficulty adjustment is to ensure that new blocks are added to the blockchain at a consistent rate.
When a new block is mined, the miners compete to solve a mathematical puzzle using their computing power. This process is known as mining and requires a significant amount of computational power,
which is measured in hashrate. The difficulty of the puzzle is adjusted based on the timestamp, nonce, and target values.
Blockchain and Mining
The blockchain is a decentralized ledger that records all Bitcoin transactions. It consists of a series of blocks, each containing a list of transactions. Miners play a crucial role in maintaining
the integrity of the blockchain by validating and adding new blocks to the chain.
Mining involves finding a hash value that meets a certain criteria set by the network. This criteria is known as the target and is adjusted according to the difficulty level. Miners iterate through
different nonce values and combine them with the current timestamp and block data to generate a hash value. If the resulting hash value is below the target, the block is considered valid and can be
added to the blockchain.
Difficulty Adjustment
The Bitcoin network adjusts the difficulty level every 2016 blocks by changing the target value. If the previous 2016 blocks were found too quickly, indicating that the network hashrate is high, the
difficulty increases. On the other hand, if the previous blocks took a longer time to mine, the difficulty decreases.
This adjustment ensures that new blocks are added to the blockchain approximately every 10 minutes, regardless of changes in the network’s computational power. It prevents blocks from being mined too
quickly, maintaining stability and consistency in the Bitcoin network.
How is Bitcoin Difficulty Determined?
The bitcoin mining process involves solving complex mathematical problems to validate transactions and add them to the blockchain. The difficulty of these problems is adjusted periodically to
maintain a consistent block production time of approximately 10 minutes. This adjustment is determined by the bitcoin network itself and is based on the timestamp and the algorithm used.
The timestamp is a unique value that represents the current time when a block is mined. It serves as an important factor in determining the difficulty level. The algorithm used to calculate the
difficulty compares the timestamp of the most recent block with the timestamp of a block 2016 blocks prior. If the time difference is less than two weeks, the difficulty remains the same. Otherwise,
the difficulty is adjusted based on the time difference. This adjustment occurs approximately every two weeks.
The network adjusts the difficulty level to ensure that the average time taken to mine a block remains constant, regardless of changes in network hashrate. The hashrate refers to the computing power
dedicated to mining bitcoins and is measured in hashes per second. If the hashrate increases, the difficulty increases as well. Conversely, if the hashrate decreases, the difficulty decreases to
maintain the target block time.
The bitcoin network uses a simple formula to calculate the new difficulty level. It divides the previous difficulty by the ratio of the actual time taken to mine the previous 2016 blocks and the
target time. This ratio is known as the mining adjustment factor. A higher factor indicates that the blocks were mined too quickly, resulting in an increased difficulty for the next period.
Conversely, a lower factor implies that the blocks were mined too slowly, leading to a decreased difficulty for the next period.
How often does Bitcoin Difficulty Change?
The Bitcoin difficulty is an essential aspect of the cryptocurrency’s mining algorithm. It determines how much effort is required to mine a new block in the blockchain.
The difficulty of mining a new block is adjusted approximately every 2,016 blocks, or roughly every two weeks. This interval is based on the target timestamp of 10 minutes for each block.
Difficulty Adjustment Algorithm
The Bitcoin difficulty adjustment algorithm takes into account the total network hashrate, which is the combined computing power of all the miners in the network. If the hashrate increases,
indicating more miners are joining the network, the difficulty increases. Conversely, if the hashrate decreases, the difficulty decreases to ensure that blocks are still mined approximately every 10
The adjustment algorithm compares the average time it took to mine the previous 2,016 blocks with the target of 10 minutes per block. If the average time was less than 10 minutes, the difficulty
increases. If the average time was more than 10 minutes, the difficulty decreases.
Note that the adjustment is gradual, allowing the network to adapt to changes in the mining power more smoothly. The difficulty can only increase or decrease by a maximum factor of four times per
adjustment period.
Impact on Mining
The changing difficulty affects the mining process directly. When the difficulty increases, miners need to expend more computational power to find a new block, meaning it becomes more challenging to
mine Bitcoin. Conversely, when the difficulty decreases, it becomes easier to mine new blocks.
Miners need to consider the changing difficulty and the associated costs when deciding to participate in Bitcoin mining. With higher difficulty, more computing power is required, which can lead to
increased electricity costs and the need for more efficient mining equipment.
Understanding the frequency and mechanics of Bitcoin difficulty adjustments is crucial for miners to optimize their mining operations and stay competitive in the network.
Why Does Bitcoin Difficulty Change?
The difficulty of mining Bitcoin is not static and can change over time. This adjustment is an essential part of the Bitcoin algorithm, ensuring a constant issuance of new Bitcoins and maintaining a
stable network.
The Bitcoin network operates on a decentralized blockchain, where transactions are grouped into blocks. Miners are responsible for solving complex mathematical problems by using their computational
power. When a miner successfully solves a problem, they add a new block to the blockchain and receive a reward in Bitcoins.
The timestamp of each block is recorded in the blockchain, and this timestamp is used to calculate the average time it takes to mine a block. The ideal time for mining a block is approximately 10
minutes. However, as the number of miners and their computational power varies, the time can fluctuate.
To maintain the average block time of 10 minutes, the Bitcoin network adjusts the difficulty every 2016 blocks. This adjustment is based on the average time it took to mine the previous 2016 blocks.
If the average time is less than 10 minutes, the difficulty increases. Conversely, if the average time is more than 10 minutes, the difficulty decreases.
The difficulty adjustment is done by changing the target value that miners must achieve with a specific nonce, a random number. The target represents the difficulty level, and a lower target requires
more computational power to find a corresponding hash. By adjusting the target, the Bitcoin network can control the rate at which new blocks are added to the blockchain.
This dynamic difficulty adjustment helps to keep the Bitcoin network secure and stable. It prevents the network from being overwhelmed by a sudden influx or decrease in mining power, ensuring a
consistent issuance of new Bitcoins.
How Bitcoin Difficulty Affects Mining
Bitcoin mining is the process by which new bitcoins are generated and transactions on the blockchain network are verified. However, mining is not as simple as it may seem. The difficulty of mining
bitcoin plays a crucial role in the process.
When it comes to mining, hashrate is an important factor to consider. Hashrate refers to the computational power that miners contribute to the network. Miners use this power to solve complex
mathematical problems, known as hashes, in order to validate transactions and add them to the blockchain.
The bitcoin network adjusts its mining difficulty approximately every two weeks. This adjustment is necessary to ensure that the average time taken to mine a new block remains constant. The
adjustment is based on the total hashrate of the network, which is constantly changing as more miners join or leave the network.
Nonce and Target
In order to mine a new block, miners need to find a valid nonce. A nonce is a random number that miners include in the block header. In order to be considered valid, the hash of the block header,
which includes the nonce, must be below a specific target value.
The target value is what determines the difficulty of mining. It is a mathematical representation of how hard it is to find a valid hash. The lower the target value, the more difficult it is to find
a valid nonce and mine a new block.
Timestamp and Algorithm
Another important aspect of mining difficulty is the timestamp that is included in the block header. The timestamp is used to ensure that the time taken to mine a new block remains consistent. If
blocks are being mined too quickly, the difficulty will increase in the next adjustment. Conversely, if blocks are being mined too slowly, the difficulty will decrease.
The bitcoin mining algorithm, known as Proof of Work (PoW), plays a key role in determining the difficulty. The algorithm requires miners to find a hash with a specific number of zeros at the
beginning. The more zeros required, the more difficult it is to find a valid hash and mine a new block.
In conclusion, bitcoin difficulty affects mining by adjusting the target value and timestamp based on the network’s hashrate. It determines how hard it is for miners to find a valid hash and mine new
blocks. Understanding these factors is essential for anyone interested in mining bitcoin.
How Does Bitcoin Difficulty Affect Mining?
Mining is a crucial process in the Bitcoin network, responsible for adding new transactions to the blockchain. However, mining is not a simple task. Miners compete against each other to solve a
complex mathematical algorithm, and the difficulty of this algorithm is constantly adjusting.
The difficulty of mining is primarily determined by the timestamp of the previous block. Bitcoin’s network aims to maintain a consistent block time of approximately 10 minutes. If miners collectively
solve the algorithm faster than this target time, the difficulty increases. Alternatively, if the solution takes longer than 10 minutes, the difficulty decreases.
The adjusting difficulty ensures that the network always maintains a stable rate of block generation, regardless of the overall hashrate. This is important to prevent the blockchain from becoming
congested or experiencing significant fluctuations in block times.
When the difficulty increases, miners must use more computational power to solve the algorithm and find the correct nonce that results in a valid block. This means that miners need to invest in more
powerful and energy-efficient mining hardware to stay competitive.
On the other hand, when the difficulty decreases, mining becomes less challenging, and miners with less powerful hardware can potentially find valid blocks. This can lead to decentralization in
mining, as smaller miners have a better chance of participating in the process.
Overall, Bitcoin difficulty plays a crucial role in maintaining the stability and security of the network. It adjusts dynamically to ensure a consistent block time, balancing the competition between
miners and allowing the network to function efficiently.
How Does Bitcoin Difficulty Impact Block Time?
Bitcoin difficulty is a crucial aspect of the mining process, as it directly affects the time it takes to mine a new block on the Bitcoin blockchain. The difficulty is a measure of how hard it is for
miners to find a hash below a given target. This target is defined by the network and is adjusted periodically.
Understanding Bitcoin Mining
In the Bitcoin mining process, miners use their computing power to solve complex mathematical problems. These problems require the miners to find a nonce, which is a random number, that, when
combined with other data from the block, produces a hash that is below the target. The miners use different algorithms in their mining software to iterate through different nonce values and find the
correct one.
The Role of Difficulty Adjustment
The Bitcoin network adjusts the difficulty every 2016 blocks, or approximately every two weeks, to maintain a consistent block time. If blocks are being mined faster than the target block time of 10
minutes, the difficulty is increased to make the problem harder. Conversely, if blocks are being mined slower than the target block time, the difficulty is decreased to make the problem easier.
The adjustment is based on a comparison of the time it took to mine the previous 2016 blocks with the target block time of 10 minutes. If it took longer than two weeks, the difficulty is reduced, and
if it took less time, the difficulty is increased. This ensures that the average block time remains close to the target.
Impact on Block Time
The difficulty directly impacts the block time on the Bitcoin network. If the difficulty increases, it becomes harder for miners to find the correct nonce and produce a hash below the target. This
means that it will take more time, on average, for a new block to be mined.
Conversely, if the difficulty decreases, it becomes easier for miners to find the correct nonce, resulting in a shorter block time. However, it’s important to note that even with a lower difficulty,
the block time will still average around 10 minutes due to the target block time set by the network.
In summary, Bitcoin difficulty plays a vital role in determining the block time on the Bitcoin network. It adjusts periodically to ensure that the average block time remains close to the target of 10
minutes. Miners must continuously adapt to changes in difficulty to maintain a consistent mining speed and contribute to the security and stability of the blockchain.
What Happens When Bitcoin Difficulty Increases?
Bitcoin difficulty adjustment is a process that occurs approximately every two weeks. It is designed to ensure that the number of new bitcoins created every ten minutes remains constant, regardless
of changes in the network’s hashrate. When the hashrate increases, the difficulty also increases in order to maintain the block production rate.
When the difficulty increases, it means that it becomes harder to find a valid nonce that will result in a block hash with a target value. Miners will need to increase their computational power and
invest more in mining equipment to have a chance at solving the mathematical puzzle required to mine a new block.
The difficulty adjustment is determined by the timestamp of the previous block and the time it took to mine that block. If blocks are being mined faster than the target rate of one block every ten
minutes, the difficulty will increase. Conversely, if blocks are being mined slower, the difficulty will decrease in the next adjustment.
The increase in difficulty has several implications for the network and miners. Firstly, it reduces the rate at which new bitcoins are created, as it takes more time and computing power to mine a
block. This helps to control inflation and ensures a steady and predictable supply of new bitcoins entering the market.
Secondly, the increase in difficulty can lead to centralization in the mining sector. As mining becomes more challenging and requires substantial resources, smaller miners may be priced out of the
market, leaving a few dominant players. This can potentially undermine the decentralization of the blockchain network.
Lastly, an increase in difficulty also affects transaction confirmation times. As the hashrate increases, blocks are mined more frequently, which in turn reduces the time it takes for transactions to
be confirmed. This can benefit users who want faster transaction confirmations, but it may also impact transaction fees if miners prioritize transactions with higher fees.
In conclusion, when bitcoin difficulty increases, miners face the challenge of solving more difficult mathematical puzzles to mine new blocks. The increased difficulty helps maintain a consistent
block production rate and controls inflation. However, it can also lead to centralization in the mining sector and impact transaction confirmation times.
What Happens When Bitcoin Difficulty Decreases?
Bitcoin mining is a process that involves the verification and addition of transactions to the blockchain. Miners use their computational power, also known as hashrate, to solve complex mathematical
problems. The difficulty of these problems is determined by the Bitcoin network and is adjusted every 2016 blocks based on the timestamp of the previous blocks.
When the Bitcoin difficulty decreases, it means that the target for solving the mathematical problems becomes easier. This can happen if there are fewer miners on the network or if the total hashrate
decreases. The target is adjusted to maintain an average block time of approximately 10 minutes.
When the difficulty decreases, miners have a higher chance of finding a valid nonce that satisfies the target. This allows them to add a new block to the blockchain and receive the associated block
reward. The block reward consists of newly minted bitcoins and transaction fees.
It’s important to note that the adjustment in difficulty is done according to an algorithm, ensuring that it remains fair and predictable. This algorithm takes into account both the previous block’s
timestamp and the time it took to mine the previous 2016 blocks. The goal is to maintain a stable network and prevent block times from becoming too short or too long.
Overall, when the Bitcoin difficulty decreases, mining becomes easier, and miners have a higher chance of earning rewards. However, this also means that more miners may join the network, leading to
an increase in difficulty once again. The constant adjustment of difficulty is a crucial mechanism in ensuring the stability and security of the Bitcoin network.
How Bitcoin Difficulty Affects Mining Profitability?
Bitcoin mining is the process of adding new blocks to the blockchain using a specific algorithm. Miners compete to find a hash value that meets certain criteria and solve a mathematical problem to
add a new block to the blockchain. The difficulty of this algorithm is adjusted regularly to ensure that blocks are added at a predictable rate.
The difficulty of Bitcoin mining is determined by the network and is adjusted approximately every two weeks, or once every 2016 blocks. This adjustment is based on the total network hashrate and the
time it took to mine the previous 2016 blocks. If the previous blocks were mined too quickly, the difficulty will increase in order to slow down the rate of block creation. Conversely, if the
previous blocks took longer to mine, the difficulty will decrease to speed up the block creation rate.
As the difficulty increases, it becomes harder and more resource-intensive for miners to find a valid block hash. This means that miners need to invest in more powerful and efficient hardware to
compete in the mining race. The increased competition and investment in mining hardware can lead to higher mining costs and lower profitability for individual miners.
On the other hand, when the difficulty decreases, it becomes easier for miners to find a valid block hash. This can lead to increased profitability for miners as they require less computational power
to participate in the mining process. However, as more miners join the network due to the lower difficulty, the competition for block rewards increases, potentially offsetting some of the increased
The adjustment of the difficulty is an essential part of the Bitcoin mining process, as it helps to regulate the creation of new blocks and maintain the stability and security of the network. It
ensures that the average time between block creation remains around 10 minutes, regardless of changes in the network’s hashrate.
Term Definition
Timestamp A record of the time when a block was mined.
Nonce A 32-bit arbitrary number that is used in the mining process to find a valid block hash.
Difficulty Adjustment The process of modifying the difficulty level of the mining algorithm based on the network hashrate and block creation time.
Factors That Influence Bitcoin Difficulty
The difficulty of mining Bitcoin is influenced by several key factors. These factors are designed to maintain the target block generation time and ensure the security and stability of the blockchain.
1. Target
The target is a number that determines the level of difficulty for mining a block. It is adjusted every 2016 blocks to maintain a consistent block generation time of approximately 10 minutes. The
target is set based on the total network hashrate and the desired block generation time.
2. Nonce
The nonce is a 32-bit field in the block header that miners can freely change to try different combinations in order to find a hash value that meets the current difficulty target. Changing the nonce
allows miners to perform a proof of work and find a valid block.
3. Blockchain Algorithm
The blockchain algorithm determines how the difficulty target is calculated and how the difficulty is adjusted. Bitcoin uses the SHA-256 hashing algorithm, which requires a significant amount of
computational power to find a valid block. The algorithm ensures that the difficulty adjusts dynamically based on the total network hashrate.
4. Mining Adjustment
Every 2016 blocks, the difficulty adjusts to maintain the target block generation time. If the average block generation time is shorter than the desired time, the difficulty increases. If it is
longer, the difficulty decreases. This adjustment ensures that the block generation time remains relatively constant, regardless of changes in the network’s hashrate.
5. Network Hashrate
The network hashrate refers to the total computational power of all miners participating in the Bitcoin network. As more miners join the network or existing miners upgrade their hardware, the network
hashrate increases. This, in turn, affects the difficulty target, making it harder to find a valid block.
6. Timestamp
The timestamp is a field in the block header that represents the time at which the block was created. It plays a role in calculating the difficulty target and ensures that blocks are generated at a
consistent rate. Miners are required to adjust the timestamp to meet the target block generation time.
These factors work together to maintain the stability and security of the Bitcoin network by ensuring that the difficulty of mining adjusts dynamically based on the total network hashrate and desired
block generation time.
Hash Rate
The hash rate refers to the speed at which a mining machine or network is able to solve complex mathematical problems and generate new blocks in the blockchain. It is measured in hashes per second (H
/s) or more commonly, kilohashes (KH/s), megahashes (MH/s), gigahashes (GH/s), or even terahashes (TH/s).
The hash rate is an important factor in mining, as it determines the mining power and efficiency of a network or individual mining setup. It directly impacts the probability of finding a hash that
meets the target difficulty set by the Bitcoin mining algorithm.
As more miners join the network and increase their hash rate, the overall network hash rate increases. This means that the computational power of the network is growing, and the difficulty of mining
new blocks needs to be adjusted to maintain a consistent block creation time. The adjustment happens at regular intervals, typically every 2,016 blocks, and is known as the difficulty adjustment.
Impact on Mining
The hash rate plays a crucial role in mining profitability. A higher hash rate increases the chances of successfully mining a block, which in turn leads to earning more Bitcoin rewards. Miners with a
higher hash rate have a competitive advantage as they can solve more mathematical problems and generate blocks more quickly than those with a lower hash rate.
However, increasing the hash rate requires more computational power, which also means higher electricity costs and hardware expenses. Miners need to carefully consider the balance between their hash
rate and the associated costs to ensure profitability.
The Mining Arms Race
The hash rate is not a fixed value and can change over time due to various factors such as the availability of more powerful mining hardware, changes in the mining algorithm, or fluctuations in the
price of Bitcoin. As a result, there is a constant race among miners to upgrade their equipment and increase their hash rate in order to stay competitive and maintain profitability.
With each adjustment of the difficulty target, miners need to reassess their hash rate, evaluate the potential impact on their mining operations, and make necessary adjustments to stay profitable in
the ever-evolving mining landscape.
Number of Miners
The number of miners participating in the Bitcoin network has a significant impact on the difficulty of mining new blocks. The target for mining a new block is adjusted every 2016 blocks or
approximately every two weeks. This adjustment is based on the timestamp of the previous 2016th block and the time it took to mine those blocks.
The Bitcoin mining algorithm requires miners to find a nonce that, when combined with the block data, produces a hash that meets certain criteria. This process requires significant computational
power, known as hashrate.
When the number of miners in the network increases, the total hashrate increases as well. This increase in hashrate allows more computation to be performed, resulting in blocks being mined more
quickly. As a result, the difficulty of finding a new block adjusts upward to maintain the target block time of approximately 10 minutes.
Conversely, when the number of miners in the network decreases, the total hashrate decreases as well. This decrease in hashrate means that less computation is being performed, resulting in blocks
being mined more slowly. As a result, the difficulty of finding a new block adjusts downward to maintain the target block time of approximately 10 minutes.
The number of miners participating in the Bitcoin network can vary over time, depending on factors such as the price of Bitcoin, the availability of mining hardware, and the profitability of mining.
This dynamic nature of the number of miners is an important aspect of the Bitcoin blockchain, as it helps ensure that new blocks are added to the blockchain at a consistent rate.
Technological Advancements
The network and blockchain technology behind Bitcoin continue to evolve and improve, leading to significant technological advancements in the mining process. These advancements are crucial for
maintaining the security and integrity of the network.
One key aspect of mining that has seen technological advancements is the concept of the “nonce.” The nonce is a random number generated by miners in their attempts to solve a block’s hash value.
Through technological advancements, miners have been able to increase the efficiency and speed at which they find the correct nonce, reducing the time it takes to mine a block.
Another area of technological advancement is the difficulty adjustment algorithm. This algorithm is responsible for ensuring that new blocks are added to the blockchain at regular intervals,
approximately every 10 minutes. By adjusting the difficulty level based on the current hashrate of the network, the algorithm helps maintain a steady and consistent block production rate.
Technological advancements have also led to improvements in the overall efficiency of the mining process. Miners now have access to more efficient hardware and software that can handle the complex
calculations required for mining at a much faster rate. This increase in efficiency has significantly increased the overall hashrate of the network, making it more secure and resistant to attacks.
Furthermore, advancements in timestamp technology have improved the accuracy and precision of block creation. The timestamp serves as a unique identifier for each block, and accurate timestamps are
crucial for maintaining the chronological order of transactions on the blockchain. With improved timestamp technology, the network can ensure that blocks are added in the correct order, preventing
any manipulation or fraud.
In conclusion, technological advancements in the network, blockchain, nonce, difficulty adjustment, mining, hashrate, algorithm, and timestamp have all contributed to the continued evolution and
improvement of the Bitcoin mining process. These advancements play a vital role in maintaining the security, efficiency, and integrity of the network, ultimately making Bitcoin a robust and reliable
digital currency.
Energy Costs
Bitcoin mining requires a significant amount of computational power, measured in hashrate, to solve complex mathematical algorithms and validate transactions on the blockchain. As the difficulty
adjustment increases, it becomes more challenging to find the correct nonce that meets the target set by the algorithm.
The process of mining requires a tremendous amount of energy. Miners need to dedicate substantial computational resources, including high-powered hardware and constant electricity supply, to maintain
their operations. The energy costs associated with mining can be significant, especially as the competition increases and more miners join the network.
Impact of Difficulty Adjustment
The difficulty adjustment determines how hard it is to find a valid block hash. As the overall hashrate increases, the difficulty also adjusts to maintain a consistent block time. When the hashrate
rises, the algorithm increases the difficulty to ensure that new blocks are still generated approximately every 10 minutes.
Higher difficulty means increased computational requirements, which in turn leads to higher energy consumption. Miners need to invest in efficient mining hardware and manage their electricity
consumption strategically to stay profitable in this competitive market.
Sustainable Mining Practices
Given the energy-intensive nature of Bitcoin mining, there is a growing interest in finding sustainable mining practices. Some miners are exploring renewable energy sources such as solar or wind
power to reduce their reliance on fossil fuels. Additionally, improved hardware efficiency and optimized cooling systems can help reduce energy consumption.
Efforts are also being made to encourage the use of mining operations for other valuable purposes beyond Bitcoin. For example, utilizing waste heat generated from mining for heating purposes or
supporting other computational operations can maximize the productivity of energy consumption.
Miners have a vested interest in reducing energy costs, not only due to profitability concerns but also to minimize their environmental impact. As the industry continues to evolve, implementing more
energy-efficient practices will be crucial to ensuring the long-term sustainability of Bitcoin mining.
Competition among Miners
One of the key aspects of Bitcoin mining is the competition among miners to solve complex mathematical problems in order to add new blocks to the blockchain. This competition is directly related to
the concept of mining difficulty.
The target for mining a new block is determined by the Bitcoin network based on an algorithm. This algorithm takes into account the total hashrate of the network and the timestamp of the previous
block. The network aims to maintain a constant block creation rate of approximately one block every 10 minutes.
However, the total hashrate of the network can fluctuate depending on the number of miners participating in the mining process. As more miners join the network and invest in more powerful mining
hardware, the overall hashrate increases. Conversely, if miners drop out or turn off their mining equipment, the hashrate decreases.
In order to maintain the desired block creation rate, the mining difficulty is adjusted every 2016 blocks. This adjustment ensures that the average time to mine a block stays close to 10 minutes,
regardless of changes in the total hashrate of the network.
When the total hashrate increases, the mining difficulty also increases, making it harder for individual miners to find a solution to the mathematical problem. On the other hand, when the total
hashrate decreases, the mining difficulty decreases as well, making it easier for miners to find a valid solution.
Impact on the Mining Process
This competition among miners has several implications for the mining process. Firstly, it incentivizes miners to invest in more powerful hardware in order to increase their chances of solving the
mathematical problem and being rewarded with newly minted bitcoins. Miners need to constantly upgrade their equipment to stay competitive.
Additionally, the competition among miners also affects the profitability of mining. As the difficulty increases, miners need to spend more on electricity and hardware costs to keep up with the
competition. Conversely, when the difficulty decreases, mining becomes more profitable as it requires less computational power.
Overall, the competition among miners is a fundamental aspect of Bitcoin mining. It ensures that the mining process remains secure and that the block creation rate stays relatively consistent.
Understanding the dynamics of this competition is crucial for miners looking to participate in the Bitcoin network.
Calculating Bitcoin Mining Difficulty
In the Bitcoin network, the mining difficulty is a measure of how difficult it is to find a new block and add it to the blockchain. The difficulty is adjusted every 2016 blocks, or approximately
every two weeks, to maintain a consistent block time of 10 minutes.
The mining difficulty is determined by a target value, which is a 256-bit number stored in each block header. Miners need to find a nonce, a 32-bit number, such that when combined with the rest of
the block header and hashed using the SHA-256 algorithm, the resulting hash is below the target. The target value represents the threshold that a valid block hash must be lower than in order to be
considered a valid solution.
The Bitcoin network uses a complex algorithm to calculate the mining difficulty adjustment. This algorithm takes into account the timestamp of the last block, the timestamp of the first block in the
current difficulty adjustment period, and the target block time of 10 minutes. The algorithm aims to maintain a steady rate of new block discovery by adjusting the difficulty based on the network’s
hash rate.
Difficulty Adjustment Period
The difficulty is adjusted every 2016 blocks. This period is designed to take approximately two weeks, but can be longer or shorter depending on the rate at which blocks are mined. If blocks are
found too quickly, the difficulty increases to make mining harder. If blocks are found too slowly, the difficulty decreases to make mining easier.
Mining Difficulty Calculation
The mining difficulty calculation involves comparing the timestamp of the last block with the timestamp of the first block in the difficulty adjustment period. If the time it took to mine the
previous 2016 blocks is less than two weeks, the difficulty is increased. If it took longer than two weeks, the difficulty is decreased. The rate of adjustment is limited to prevent sudden and
drastic changes in difficulty.
The new difficulty is calculated using the formula:
new difficulty = old difficulty * (actual time / target time)
where “actual time” is the time it took to mine the previous 2016 blocks and “target time” is the desired block time of 10 minutes.
Block Number Difficulty Actual Time Target Time New Difficulty
2016 1000000 10 days 14 days 1428571
4032 1428571 12 days 14 days 1224490
6048 1224490 15 days 14 days 1571429
As shown in the example table, the difficulty adjusts based on the actual time it took to mine the previous 2016 blocks compared to the target time. This ensures that the network maintains a
consistent block time and adapts to changes in the mining power of the network.
How is Bitcoin Mining Difficulty Calculated?
Bitcoin mining difficulty is a crucial aspect of the Bitcoin network that determines the complexity of mining new blocks. It plays a significant role in maintaining the network’s stability and
ensuring a steady supply of new Bitcoins. Understanding how Bitcoin mining difficulty is calculated is essential for miners and investors in the cryptocurrency.
The Bitcoin mining difficulty is recalculated approximately every two weeks, or after every 2016 blocks are mined. The calculation takes into account several factors, including the timestamp of the
last block, the hashrate of the network, and the target block time.
The timestamp of the last block serves as the starting point for the calculation. The network aims to maintain an average block time of 10 minutes. If the timestamp of the last block indicates that
it took significantly shorter or longer than 10 minutes to mine, the difficulty is adjusted, making it harder or easier for miners to find a valid hash.
The hashrate of the network is another critical factor in calculating the mining difficulty. The hashrate represents the computing power used by miners to validate transactions and solve complex
mathematical problems. If the hashrate increases, indicating more miners joining the network, the difficulty is adjusted upwards to ensure that block production remains at a stable rate. Conversely,
if the hashrate decreases, the difficulty is reduced to maintain the desired block time.
The adjustment of the mining difficulty is done using a mathematical algorithm known as the difficulty adjustment algorithm. This algorithm uses the timestamp and hashrate data to determine the new
difficulty level for the next 2016 blocks.
The mining difficulty calculation is a fundamental part of the Bitcoin blockchain’s design. It helps to prevent inflation by adjusting the supply of new Bitcoins based on the network’s computational
power. With the increasing popularity of Bitcoin, it is crucial to have a robust difficulty adjustment mechanism to maintain the stability and security of the network.
Overall, the calculation of Bitcoin mining difficulty involves analyzing the timestamp, hashrate, and adjusting the difficulty level to ensure a consistent block time. By maintaining a stable
difficulty level, the network can continue to operate efficiently and provide incentives for miners to participate in the mining process.
What is the Current Bitcoin Mining Difficulty?
The current Bitcoin mining difficulty is a measure of how challenging it is to find a new block on the Bitcoin network. This difficulty is adjusted approximately every two weeks, or every 2,016
blocks, based on the overall hashrate of the network. The hashrate refers to the total computational power used by miners to solve the cryptographic puzzles required to validate transactions and add
them to the blockchain.
The Bitcoin network has a target for each block, represented by a hexadecimal number. Miners must find a nonce, which is a random number, that, when combined with the block’s data and hashed through
the SHA-256 algorithm, produces a hash value that is less than or equal to the target. This process requires a significant amount of computational power and energy.
If the overall hashrate of the network increases, the difficulty will adjust to ensure that blocks are still found approximately every 10 minutes. The opposite is true if the hashrate decreases. The
adjustment is made by changing the target value, making it more or less challenging for miners to find a valid nonce.
The timestamp of each block is also taken into consideration when calculating the mining difficulty. The Bitcoin protocol aims to maintain a 10-minute block time by adjusting the difficulty based on
the actual time it took to mine the previous 2,016 blocks.
The current mining difficulty level is a reflection of the competition and resources dedicated to mining Bitcoin. As more miners join the network and more powerful hardware is used, the difficulty
will increase, requiring even more computational power to mine new blocks.
Understanding the current mining difficulty is important for miners to gauge the competitiveness of the network and make informed decisions about their mining operations.
How to Predict Bitcoin Mining Difficulty?
Predicting Bitcoin mining difficulty involves understanding the factors that influence its adjustment process. The mining difficulty of Bitcoin is a measure of how difficult it is to find a hash
value below a target set by the network. This target is adjusted approximately every two weeks based on the previous block timestamps and the total network hashrate.
Network Adjustment
The Bitcoin network adjusts the mining difficulty every 2016 blocks, or about every two weeks. The adjustment is made to ensure that the block time remains close to 10 minutes. If miners are
producing blocks faster than this, the difficulty increases, and if they are producing blocks slower, the difficulty decreases.
Target Calculation
The target for the next difficulty adjustment is calculated using the formula:
new_target = old_target * (actual_time / target_time)
Where “actual_time” is the average time it took to mine the previous 2016 blocks, and “target_time” is the desired average time in seconds.
The new target is then compared to a maximum and minimum threshold to ensure it does not deviate too much in a single adjustment, and it is adjusted accordingly.
Factors Affecting Difficulty
Several factors can affect the mining difficulty of Bitcoin:
• Hashrate: The total computational power dedicated to mining on the Bitcoin network. A higher hashrate means a shorter average time to find a block, which leads to an increase in difficulty.
• Network Congestion: If there is a high volume of transactions or a backlog of unconfirmed transactions on the blockchain, it can lead to longer block times, resulting in a decrease in difficulty.
• Miner Efficiency: Improvements in mining hardware efficiency can lead to higher hashrates, which in turn can increase the difficulty.
By analyzing these factors and monitoring the blockchain and network activity, experts can make predictions about the future difficulty adjustments. However, predicting the exact difficulty is
challenging due to the complexity of the network and the variables involved.
In conclusion, predicting Bitcoin mining difficulty requires analyzing the network’s hashrate, previous block times, and other factors that influence the adjustment process. While it is possible to
make educated predictions, the exact difficulty can never be known in advance.
What is Bitcoin difficulty?
Bitcoin difficulty is a measure of how difficult it is to find a hash below the target set by the Bitcoin network. It is adjusted every 2016 blocks or approximately every two weeks.
How is Bitcoin difficulty calculated?
Bitcoin difficulty is calculated by taking the current difficulty level and dividing it by the desired time to mine a block, which is approximately 10 minutes. The resulting number is then adjusted
to make sure the average time to mine a block remains around 10 minutes.
What factors affect Bitcoin difficulty?
Several factors affect Bitcoin difficulty, including the total hash rate of the network, the efficiency of mining hardware, and the number of miners participating in the network. Higher hash rates
and more efficient mining hardware can increase the difficulty, while fewer miners or lower hash rates can decrease it.
Why does Bitcoin difficulty change every two weeks?
Bitcoin difficulty changes every two weeks to maintain the average block time of 10 minutes. If the hash rate increases and blocks are being found too quickly, the difficulty adjusts upwards.
Conversely, if blocks are being found too slowly, the difficulty adjusts downwards to bring the block time back to the target.
How does Bitcoin difficulty affect mining?
Bitcoin difficulty directly affects the amount of computational power required to mine a block. As the difficulty increases, miners need more powerful hardware and more electricity to compete. Higher
difficulty also means fewer rewards for individual miners, as the total reward is shared among more participants.
What is Bitcoin difficulty?
Bitcoin difficulty is a measure of how difficult it is to find a new block on the Bitcoin blockchain. It is a way to regulate the rate at which new blocks are added to the blockchain by adjusting the
mining difficulty based on the total computational power of the network.
How is Bitcoin difficulty adjusted?
Bitcoin difficulty is adjusted every 2016 blocks, which is approximately every two weeks. The adjustment is done by changing the target value for the proof-of-work algorithm, making it harder or
easier to find a new block depending on the total computational power of the network.
Why does Bitcoin difficulty need to be adjusted?
Bitcoin difficulty needs to be adjusted to ensure that new blocks are added to the blockchain at a consistent rate. If there is too much computational power in the network, blocks would be found too
quickly and the supply of new bitcoins would increase faster than intended. On the other hand, if there is too little computational power, blocks would be found too slowly and transactions would take
a long time to confirm. | {"url":"https://bitlc.net/blog/understanding-the-complexities-of-bitcoin-difficulty-and-its-impact-on-mining-rewards","timestamp":"2024-11-09T23:56:43Z","content_type":"text/html","content_length":"103022","record_id":"<urn:uuid:c089bbfa-ed9d-49d9-8a09-27a843296ac5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00065.warc.gz"} |
A way to prove the Four Color Theorem using gauge theory - Harvard Math
A way to prove the Four Color Theorem using gauge theory
When: October 4, 2024
3:00 pm - 4:00 pm
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138, United States
Speaker: Scott Baldrige (Louisiana State University)
In this talk, I show how ideas coming out of gauge theory can be used to prove that certain configurations in the list of “633 unavoidable’s” are reducible. In particular, I show how to prove the
most important initial example, the Birkhoff diamond (four “adjacent” pentagons), is reducible using our filtered $3$- and $4$-color homology. In this context reducible means that the Birkhoff
diamond cannot show up as a “tangle” in a minimal counterexample to the 4CT. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional TQFT. I will then
indicate how the ideas used in the proof might be used to reduce the unavoidable set of 633 configurations to a much smaller set.
This is joint work with Ben McCarty. | {"url":"https://www.math.harvard.edu/event/a-way-to-prove-the-four-color-theorem-using-gauge-theory/","timestamp":"2024-11-08T17:02:04Z","content_type":"text/html","content_length":"83839","record_id":"<urn:uuid:81d82dbc-ae2d-4296-8981-b15b4d2ea3f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00267.warc.gz"} |
Excel Animation Tutorial – 2D Wire-frame House #3 (adding UDF – user defined functions)
This is a continuation tutorial, the third in the wire-frame tutorial series. Here we take the 2D basic house we created on an 2D scatter chart that had scaling, translations and rotation and replace
all the cell formulas with two user defined functions (UDF). These basic tutorials use elementary math/geometry and trigonometry (middle school and high school level). As the series progresses, the
tutorials will reach 3D level and the creation of more complex shapes and landscapes (i.e. roller coaster and beyond).
Thanks for your support!
hello my friends and welcome back
so the previous tutorials we created a
house you see this blue thing here using
a series of points these points will
draw a line on the
and if we go on the line we can select
it to be non-smooth okay so broken so
that’s what we want always when we build
this like a roller coaster for instance
and also if we interrupt the line we can
interrupt the line by
deleting points here so leaving empty
and then
even though we have only one line
right here you see
the blue line is
column data look on two continuous
column data but
since we
provided these empty spaces these blank
spaces we avoid uh we can jump from
of uh
lines together so
the roof is a group okay here the roof
you see
the door is another group
between them we leave blank spaces then
what we did is we did a scaling with a
scale factor between 0 to i believe 2
then we did x offset so x translate x
y translates so y offsets of vertical
and then we added rotation
so the
red house is the blue house
except it has scaling okay let me
scale up
scale down
it has
horizontal translation so x offset
and vertical translation okay
so y offset
and the greenhouse
if i click on it you see the data is
coming from here
the greenhouse is it’s a rotated version
of the red house
we can
see we can rotate it
in both
directions in both senses
so now i want to show you how using a
fairly simple user-defined function
we can replace four columns of data with
only one column of data
with only two columns of data
and in the in each cell there’s only one
function a user defined function
the syntax of the user defined function
is this
first you write a word function
and then the name you give it a name and
give your name custom name
make sure it doesn’t it’s not there is a
make sure you don’t put
hyphen because it’s interpreted as a
and also in parentheses you have a list
of arguments
in our case we will use
six arguments
then inside the body of the function we
have some math some logic and then the
i already wrote a very simple one for
the function i called it my underscore
arguments are a and b
we can also
write them as
declare them as double
double precision
b as double
wait um
and also we can
declare the return on function
double yes
i usually don’t declare that if you
don’t declare them the default is
variant a variant can be a double it can
be an integer can be a long
integer can be even a matrix
if you want to use a standard
vba practices it’s better to declare
personally i don’t declare them so
i like to keep it very simple
even though maybe it’s not so
correct and it’s all formal
so now i take this name here
my edition this custom name control c
go in the spreadsheet here
here are just two numbers a and b i gave
these numbers here 1.1 3.5
the sum should be 4.6 and here i say
equal control v
so the name of the function you see it
already recognize it
recognize it as a function parenthesis
a you can say a is this guy
so b7
b is w7
and then it gives the
result the way we expect it
it adds
a and b
in order to
these calculations over three columns
with only two functions we need to
declare this function the following
first we need to declare function for
the x
value of the
let me see the rotated house right here
so let’s call it
uh the series of operations of function
first what do we have we have scaling sc
underscore what else we have
we have translation
so tr translation and then what else we
have rotation rot
and then one is this one is for x
because we have to fill two columns we
need two functions
so input is x
what else scale sc
what else um
dx delta x
so the horizontal offset or translation
d y so
the vertical translation so
the vertical offset
and then rot
the angle of rotation
in degrees
inside let’s divide this in three
different sections
one apostrophe so we follow with a
one is for scaling
then another comment let’s call it
offset or translation
translate translate
and the
the last one is rotate
i have the formulas for rotation right
here okay the new x the rotated x is a
combination between
the previous non-rotated coordinate
coordinates x and y
the angle theta
but this angle has to be in radians not
in degrees i will show how to convert
them okay
so we go back and say scale
x the new x is equal to the scaling
factor sc
x the old x okay so the new x is equal
to the over x
is equal to the old x times s c s c is
the scaling factor
then copy this
the new y is equal to
the scaling factor times the old y okay
now let’s uh do the translation
so the new x is equal to the old x x
and the new y
is equal to the old y
plus d y
for rotate we use
this first line in this group of
formulas so the x prime which is the
rotated x
is equal to x the old x times cos of
theta plus the old y
times sine of theta
in this case for this particular column
right here we only use the first formula
and x prime is actually the return of
the function so we copy this name
control c
is equal to control v is equal to
the old x
which is the x we calculated here okay
here is a typo
this is equal okay
so the old x times
look here cos of theta
cos of
because this is the angle right here
right here
the argument for rotation
plus look here at the formula
plus y
sine of
okay so oops
i tell you we have a problem here okay
we declare we we use degrees here for
rotation however the functions the
trigonometric function and even the sine
cosine in the spreadsheet except only
radians okay so look what we did here
click here you see a radius of that
rotation angle okay
so there’s a spreadsheet sheet function
called radiance
so this is a built-in function
okay so let’s test it this is equal to
this guy okay
no i’m sorry not rotation what am i
doing radius
so if this 360 should be zero zero if is
180 how many radians are in 180 degrees
is pi 3.14 check this out i’m going to
180 from here
180 you see 3.14 so the function works
so let’s do this radians okay
copy this ctrl c
and i’m making a mistake here on purpose
okay to show you something
we need to escape here
so go and type radiance i mean
radians of rot
and we will get an error okay check this
i’m telling you it’s not the function is
not gonna work but uh let’s just use it
like this and i will demonstrate ctrl c
v okay
then let’s apply it okay so
copy here ctrl c
how do we use the function say equal
control v the name of the function you
see it already recognized it excel
recognized it
parenthesis what argument we have for
this function what arguments x y
first x y
so x we take it from here
and we need to i would like to copy this
formula to the right so i would like to
absolute reference the column so in
front of b
we put a dollar sign okay
then what comes in
the following the next
is y so x y then s c scaling okay so
y is here and we put the dollar sign in
front of
the column okay
we can use f4 for that but in this case
i just typed it
what else sc
this guy
hit f4 once
and we have absolute reference both the
column and the row
so no matter where we copy this formula
up down
right left
the scale will stay the same okay will
the get the scale scale
the scale uh argument will be picked up
b three
what’s next is delta x
okay see
delta x delta y
hit an f4 here
so we absolute reference both row and
column here
so whenever we copy this down or right
or wherever we copy it
this reference will stay fixed anchored
then we put
the translate
then comma the last one
the rotation okay
so rotation is from here and f4
close this
hit return okay
now it pointed me that i have an error
here there is no function called radians
in vba there is one in
worksheet okay the worksheet function
but there’s nothing in vba so we need to
do this
dot radians
this way if we use the word application
with a dot
in front of the function it takes the
functionality of the
spread spreadsheet function radians
so let’s uh copy this okay
ctrl c
and paste it here
stop remove the error
so now it should work
let’s check this out
you see
5.379 whatever
if we copy this down
we can see
this data is identical to these
the numbers i mean not the data these
are formulas
it’s not data and these cells it’s only
formulas but i say the results
are the same which means
this function
right here
produ uh has the same effect produces
the same result as
using multiple formulas across this row
let’s copy this
ctrl c and define another function ctrl
v instead of x we put here y everything
else is the same
also we need to use you see here the
second row minus x sine plus
cos of theta
we replace in the name
the the letter x with y
and here we need a minus
look here minus
x times sinus
plus y times cos
and now we can copy this formula right
and instead of
this function we replace it with
the second function
hit return okay there’s a problem
we need to replace this with y
so this name right here
is the return function right here has to
be the same
otherwise we get that error
let’s see now
now we get the same thing okay minus 11
minus 11.
you see minus 8.8.115
minus 8.115
we can also copy this down
we can pick up the
data for this last house from here
and you see it’s replacing it perfectly
we delete the artifacts right here the
you see the artifacts disappear
you change the color under us but
it’s okay
so now you can rotate
with a simple formula relatively simple
formula custom formula so use a defined
formula you can replace a lot of cells
a lot of cell formulas
so this being said
the file will be
available for download in the
description and
next i mean next time we’re gonna see in
the next tutorial we’re gonna
go from 2d to 3d three-dimensional
we’re going to generate the pyramid and
then a cube
and i’ll show you how to
rotate translate and turn it into a 2d
representation of the 3d space
so this being said i will
see you in the next tutorial
and thanks for watching | {"url":"https://excelunusual.com/excel-animation-tutorial-2d-wire-frame-house-3-adding-udf-user-defined-functions/","timestamp":"2024-11-09T15:37:05Z","content_type":"text/html","content_length":"93109","record_id":"<urn:uuid:1895a7cd-a0ad-4b14-a305-3ccfaf14c29b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00846.warc.gz"} |
SSAT Middle-Level Math Complete Course
SSAT Middle-Level Math Test
The Secondary School Admissions Test, or SSAT, is an intelligence test used to evaluate entrance to private elementary, middle, and high schools.
There are currently three levels of the SSAT:
• Lower Level (for students in 3rd and 4th grade)
• Middle Level (for students in 5th-7th grade)
• Upper Level (for students in 8th-11th grade)
There are six sections on the SSAT Middle-Level Test:
• Writing: 25 minutes.
• Math section: 25 questions, 30 minutes.
• Reading section: 40 questions, 40 minutes.
• Verbal section: 60 questions, 30 minutes.
• Math section: 25 questions, 30 minutes.
• Experimental: 16 questions, 15 minutes.
The Middle-Level SSAT Math segment EVALUATES students' knowledge and skills in solving real-world and mathematical problems incorporating arithmetic, elementary algebra, geometry, and other related
subjects. It covers the following Math topics:
1. Basic Algebra
2. Data Analysis
3. Geometry
4. Measurement
5. Number Concepts
6. Number Conversion
7. Pre-Algebra
8. Statistics & Probability
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The Best Prep Books to Help Your student Ace the SSAT Middle-Level Math Test!
Our best SSAT specialists developed the following FREE SSAT Middle-Level Math practice questions, worksheets, and tests to help students excel on the SSAT Middle-Level Math test. Our SSAT Math
resources have been completely updated to suit the most recent SSAT modifications from 2019 to 2022.
SSAT Middle-Level Math Complete Course
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A Comprehensive Review and Step-By-Step Guide to Preparing for the SSAT Middle Level Math Test
Practice Tests
SSAT Middle Level Practice Test 1
Simulate test day with an official practice test. Then, score your test. The answers come with explanations so you can learn from your mistakes.
Start Practice Test
SSAT Middle Level Practice Test 2
Simulate test day with an official practice test. Then, score your test. The answers come with explanations so you can learn from your mistakes.
Start Practice Test
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Common Core 4th Grade Math Standards
4th Grade Math Common Core Standards
If your state participates in the Common Core initiative you may be wondering: What are the 4th grade math Common Core standards, and what tools are available to make sure your child is on track? In
the article, we outline the common core 4th grade math standards.
The Common Core State Standards Initiative, commonly referred to as Common Core, was created to ensure students in varying states are receiving a standard math education. Although 4th grade math
standards vary by state and school, the majority of the country has adopted Common Core.
As a parent, you may be wondering: “What are the 4th grade math Common Core standards and what tools are available to help my child master these objectives?” In this article, we outline the 4th grade
math standards for Common Core with examples of each objective so you feel confident about where your child stands.
The Common Core Math Standards for fourth grade consist of five overarching standards with various math skills your 4th grader is expected to master under each. The five 4th grade Common Core math
standards are as follows:
• Operations and Algebraic Thinking
• Numbers and Operations in Base 10
• Numbers and Operations- Fractions
• Measurement & Data
• Geometry
Let’s dive into these standards to better understand what they mean and what skills they entail.
Common Core 4th Grade Math Standards
Operations and Algebraic Thinking
Fourth-grade common core math standard ensures sure your child can:
1. Use the basic operations of Adding, Subtracting, Multiplying, and Dividing to solve problems.
2. Understand how factors and multiples work. (Factors are numbers multiplied together to get an answer. Multiples are skip counting by a certain number. The multiples of 5 are 5, 10, 15, 20, etc.)
3. Analyze and create mathematical patterns. This includes being able to see shapes and numbers in a sequence and continue the pattern, find missing numbers as part of a sequence, and understanding
how a sequence is made.
Example Problems for Operations and Algebraic Thinking
1. What is 8 x 9?
2. What is 44 / 11?
3. Kimberly gets a new pack of seven baseball cards each week. She now has a total of 77 baseball cards in all. How many weeks has she been collecting baseball cards?
4. What are the factors of 15?
5. What are the factors of 21?
6. Is 11 a prime or composite number?
7. Is 14 a prime or composite number?
8. What are the multiples of 4 up to 40?
9. What are the multiples of 7 up to 56?
10. What is the next number in the sequence: 3, 7, 11, 15, 19?
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Number and Operations in Base 10
This 4th grade common core math standard ensures your child:
• Understands the value of each digit with the Base 10 math concept. (Base 10 describes each digit in a number as being worth ten times the number to its right. This helps your child understand the
meaning of place values.)
• Can round multi-digit numbers to any place value: ones, tens, hundreds, thousands place.
• Can compare multi-digit numbers using the greater than, less than, or equal to symbols: > , < , or = .
• Understands place values in order to solve problems with multi-digit numbers.
Example Problems for Numbers and Operations in Base 10:
1. In the number 205, what is the value of 2?
2. In the number 628, what is the value of the 64.
3. Round 345 to the nearest tens place.
4. Round 4,502 to the nearest hundreds place.
5. Compare the two numbers 624 _____ 654.
6. What is 249 + 167?
7. What is 7,061 – 5,402?
8. What is 1029 x 5?
9. What is 12 x 14?
10. What is 362 / 4?
Numbers and Operations – Fractions
This 4th grade common core math standard ensures your child can:
• Understand fraction equivalence and fraction ordering.
• Apply basic operations of whole numbers to fractions.
• Convert fractions to decimals and compare them.
Example Problems for Numbers and Operations - Fractions
1. Write two fractions that are equivalent to ¼
2. Write two fractions that are equivalent to ⅖
3. Simplify 2⁄12
4. Simplify 8⁄10
5. What is 2 ¼ + 3 2⁄4?
6. What is 1 ½ + 8 ¼?
7. Write 11⁄3 as a mixed number
8. Write 7⁄2 as a mixed number
9. Find the product of ¾ x 2
10. Find the product of 2⁄6 x 3
11. Write 1⁄10 as a decimal
12. Write ¾ as a decimal
13. Write ⅕ as a percentage
14. Write ¼ as a percentage
This 4th grade common core math standard ensures your child can:
• Solve math problems involving measurement and conversions.
• Interpret and represent data on graphs and line plots.
• Understand concepts with angles and that the measurement of different angles involves addition.
Example Problems for Measurement and Data
1. How many ounces (oz) are in one pound (lb)?
2. How many feet are in one yard?
3. Alex drinks 2 liters of water each day. How many total ounces will he have had after one week?
4. Blair walks was 4ft 8in last year. She grew 6 inches in one year. How many inches tall is she now?
5. Sally wants to put a fence around her garden. If her garden is 10 ft by 5 ft, how many feet of fencing will she need?
6. Scott made a pen for his dog. It is 12 ft by 8 ft. How much room does the dog have to play?
This 4th grade common core math standard ensures your child can:
• Identify rays, lines, line segments, and angles and be able to draw examples of each.
• Measure angles using a protractor and free hand.
• Classify shapes by their angles, sides, and lines.
The math program that drives results
DreamBox adapts to your child’s level and learning needs, ensuring they are appropriately challenged and get confidence-building wins.
How do 4th Grade Math Standards Vary By State?
The goal of Common Core is to establish a set of uniform math standards. Forty-two states currently follow 4th grade math common core standards. However, these eight states do not teach by these
standards: Texas, Virginia, Nebraska, Alaska, South Carolina, Arizona, Oklahoma, and Indiana. Many of those states follow their own 4th grade math standards.
One notable state is Texas which uses TEKS- Texas Essential Knowledge and Skills which sets the standards for what students should learn in each grade.
What Should Kids Know in Math by the End of 4th Grade?
By the end of 4th grade, students should know how to multiply and divide multi-digit numbers, solve word problems with basic operations, simplify fractions, and understand the characteristics of
lines and other shapes – 4th grade Common Core Math Standards cover each of these topics.
Where Can You Practice 4th Grade Math Concepts?
Students can practice math by using a math app like DreamBox for access to high-quality math games and 4th grade math practice. Other ways to practice math include downloading free 4th grade math
worksheets and playing online math games.
FAQs About 4th Grade Math Standards
A 4th grader should learn how to add and subtract fractions, multiply and divide multi-digit numbers, and understand the characteristics of lines and shapes in math.
Common core 4th grade math focuses on basic operations with whole numbers and fractions, word problems with multi-digit multiplication and division, geometry, and analyzing data.
Take at home math practice to the next level
Empowering parents and educators to make math practice more impactful. Plus, your kids will love it. | {"url":"https://www.dreambox.com/math/4th-grade/4th-grade-common-core-standards","timestamp":"2024-11-06T13:36:36Z","content_type":"text/html","content_length":"166865","record_id":"<urn:uuid:86ce1af9-37e1-452e-950e-a223b3f2e2b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00180.warc.gz"} |
Daily Dose of Excel
COUNTIF Bug
Most users know that Excel stores no more than 15-digits of a value. Therefore, if you enter items such as credit card numbers, you must format the cell as Text, or precede the number with an
Try this. Enter the following 18-digit strings into A1:A3. They differ only in the last three digits. Precede each with an apostrophe, or Excel will replace the last three digits with zeros.
Enter this formula in B1:
The formula counts the number of entries that are the same as cell A1. It should return 1, but it actually returns 3. COUNTIF is ignoring the last three characters of the strings. You get the same
result if you format the cells as Text, and even this formula returns the wrong answer:
However, this array formula works (enter it with Ctrl+Shift+Enter):
30 thoughts on “COUNTIF Bug”
1. John,
It looks like SUMIF has the same condition. An array fixes that too. The Database functions Dsum, Dcount, etc, default to a “starts with” condition anyway, but can be fixed with a boolean gate in
the criteria section. That tip I picked up here. Thanks for the info about Sumif! (Fixed in next Excel release?)
2. John, using Excel 2003 SP1, I get the right answer ,1, with all of the examples you gave ??
3. I get 1 also. And I get zero for SUMIF indicating that Excel is treating it as text. 2003 SP2 (11.8033.8036)
4. Very strange. I get 3 using Excel 2003 and Excel 2007.
5. I get 3 with XL2002 SP3 (10.6501.6735). I think that the moral of the story is that CountIf and SumIf are kinda buggy sometimes. So if you don’t always need the right answer then feel free to use
it with your Credit Card Numbers… ;-)
6. Using Excel 2002, I get 3. But if I use a non-numeric character(s) in the string (e.g. 123456789012345a111), I get the correct answer of 1. However the regular IF function works like it should.
The formula =IF(A1=A2,1,0) evaluates to 0. It seems quirky.
7. Very strange indeed; I have the same version as Dick [SP2 11.8033.8036) and it returns 3 for all examples.
Maybe it’s related to the guy that wrote my PUP add-in
8. I get 3 on my home PC with Excel 2003 11.8033.8036 SP2.
9. At home (SP2 11.65606568) I get 3. Strange indeed.
10. got 3 too on 2002 and 2003, maybe the one that got 1 had the cell format set to text, I think it works too
11. I get 3 unless Transition Formula Evaluation is on, then 1.
12. Also 3 and get the same behavior for database functions and Advanced filter, but adding a wildcard * to the criteria returns 1 in all cases – v.odd [10.6501.6626].
13. just checked on the format set to text doesnt change, it stays 3
14. Lori,
I expected the database functions to behave this way. I’m testing to see if I can paste this link.
But that wildcard thing is, well, wild!! A ? doesn’t work.
15. Another little known feature of “countif” is that it fails to work at all on strings longer than 255 characters (Excel2k) – this is one of a several Excel functions that were never fully updated
when the maximum content of a cell was increased from 255 to 32k characters.
16. John, I also get the right answer using Excel 2002(10.6501.6626) SP3
17. 1. Countif() has a mind of its own amidst many other things.
2. May force the numeric string to text by, =”123456789012345111? ; and yet, Countif takes that to be a number still, 1.23456789^+17 (Excel-2000, server version).
3. At long last, the numeric string may be cast in the stone of text by,
=Char(28 to 31)&”123456789012345111? or =Char(160)&”123456789012345111? (amongst other possibilities ?!).
4. Regards.
18. I get 3 using Excel 2003 SP1.
This is just how COUNTIF works. It’s second argument is ALWAYS treated as a string, even when it’s entered as a number. That’s why it didn’t matter whether its second argument was a reference to
A1 or the text value of A1.
This functionality leads to ambiguity. When COUNTIF’s second argument is a number, it matches both numbers AND text strings in its first argument range that have the same text representation.
Enter 1 as a number in D1, =”1? in D2 and leave D3 blank. The formulas
both return 2 while the formulas
=SUMPRODUCT( (D1:D3=1))
=SUMPRODUCT( (D1:D3=”1?))
both return 1. Since =1=”1? returns FALSE, it’s COUNTIF that provides unusual functionality.
With the original 18-digit strings in A1:A3, COUNTIF’s second argument is interpretted as a string, but since that argument looks like a number, COUNTIF proceeds to interpret it as a number, and
that’s the point at which it’s munged into 15 decimal digits of precision. It does the same thing for the values in A1:A3, converting them to numbers, thus ignoring their last 3 digits.
There’s an imperfect work-around: force Excel to treat these values as text by using
which forces COUNTIF to treat all cells in A1:A3 as text, and the second COUNTIF call reduces the count for any cells in A2:A3 that contained A1 as a proper leading substring.
19. It seems that excel converts text strings that contain only numbers into number format before doing any calculations, maybe in order to lower its memory usage…
20. Gianni –
“maybe in order to lower its memory usage”
More likely an assumption that if something in a spreadsheet looks like a number, you want to treat it in a calculation as a number.
21. Hi,
referring to »countif bug»
Enter this formula in B1:
=COUNTIF(A1:A3,A1) I agree returns 3 ?? why don’t know
but if countif(a1:A1,A3) you get »your» 1
see you online
22. Excel is a good tool for performing basic addition and subtraction, and occasionally multiplication and division. For complex engineering problems, cancer research, and science problems, I would
not use Excel. The results are simply untrustworthy and highly unstable.
23. my condition have 19 digit , it error too .
I use data > consolidate to slove my FS problem.
Thanks infos.
24. This returns 3 as well:
25. If you want to add to the confusion try this similar RIGHT bug…
type “‘1234567890123456789? in A1, then run these formulas against it:
=RIGHT(A1,10) and =TEXT(RIGHT(A1,10)
I get “0123456789? for the =RIGHT formula and
I get “123456789? for the =TEXT(RIGHT) formula
To add some more confusion, add a “0? to the end of the text in A1 and check the answers; they match
Keep going – add a “1? to the end of the text and they still match!
Keep going – add “2?-“9? and when you reach “9?, they don’t match again!
Those darn 9’s!!!
26. Pingback: COUNTIF issue
27. Eleven years on, i just stumbled upon a reliable way to get COUNTIF[S] to deal with text numbers which i’m posting here for reference.
The solution is to prefix the criteria string with a soft hyphen character (Alt+0173):
The soft hyphen is the only character that is ignored in text comparisons, so:
always returns TRUE when A1 is text.
28. Another fantastic (and random!) discovery, Lori. Just stumbled upon this after wondering if anybody had managed to come up with a decent solution to this issue.
29. The problem occurs in the Excel 2016 version of conditional formatting to highlight duplicates. Did okay with most, but counts some others as dupes based only on the first 15 characters.
Posting code? Use <pre> tags for VBA and <code> tags for inline. | {"url":"http://dailydoseofexcel.com/archives/2006/10/10/countif-bug/","timestamp":"2024-11-12T14:02:52Z","content_type":"text/html","content_length":"108156","record_id":"<urn:uuid:77976814-136c-4bd3-9fc4-8faa4d66c358>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00398.warc.gz"} |
Numbers and Operations - SSAT Elementary Level Math
All SSAT Elementary Level Math Resources
Example Questions
Example Question #6 : Other Fraction Manipulations
Express the following fraction as a decimal:
Correct answer:
To easily express a fraction as a decimal, we need to convert the denominator into a power of
Example Question #51 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, divide the top number (numerator) by the bottom number (denominator). In this problem, 1 divided by 8 equals .125!
Example Question #52 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, divide the top number (numerator) by the bottom number (denominator). In this problem, 7 divided by 10 equals .7!
Example Question #53 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, simply divide the top number (numerator) by the bottom number (denominator). In this problem, 6 divided by 12 equals .5!
Example Question #54 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, simply divide the top number (numerator) by the bottom number (denominator). In this problem, 12 divided by 8 equals 1.5!
Example Question #55 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, divide the top number (numerator) by the bottom number (denominator). In this problem, 25 divided by 50 equals .5!
Example Question #21 : How To Find The Decimal Equivalent Of A Fraction
Rounded to the nearest tenth, what is
Correct answer:
To find the decimal equivalent of a fraction, divide the top number (numerator) by the bottom number (denominator). In this problem, 2 divided by 3 equals .667. But, the question asks us to round our
answer to the nearest tenth, making the correct answer .7!
Example Question #23 : How To Find The Decimal Equivalent Of A Fraction
What is the decimal equivalent to
Correct answer:
The correct answer is
You should memorize common fractions:
Example Question #56 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, simply divide the top number (numerator) by the bottom number (denominator). In this problem, 2 divided by 12 equals .125.
Example Question #53 : Numbers And Operations
Correct answer:
To find the decimal equivalent of a fraction, simply divide the top number (numerator) by the bottom number (denominator). In this problem, 3 divided by 8 equals .375.
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Continuum hypothesis
From Encyclopedia of Mathematics
The hypothesis, due to G. Cantor (1878), stating that every infinite subset of the continuum $\mathbf{R}$ is either equivalent to the set of natural numbers or to $\mathbf{R}$ itself. An equivalent
formulation (in the presence of the axiom of choice) is: $$ 2^{\aleph_0} = \aleph_1 $$ (see Aleph). The generalization of this equality to arbitrary cardinal numbers is called the generalized
continuum hypothesis (GCH): For every ordinal number $\alpha$, $$$$\label{eq:1} 2^{\aleph_\alpha} = \aleph_{\alpha+1} \ .$$ $$
In the absence of the axiom of choice, the generalized continuum hypothesis is stated in the form $$$$\label{eq:2} \forall \mathfrak{k} \,\,eg \exists \mathfrak{m}\ (\,\mathfrak{k} < \mathfrak{m} < 2
^{\mathfrak{k}}\,)$$ $$ where $\mathfrak{k}$,$\mathfrak{m}$ stand for infinite cardinal numbers. The axiom of choice and (1) follow from (2), while (1) and the axiom of choice together imply (2).
D. Hilbert posed, in his celebrated list of problems, as Problem 1 that of proving Cantor's continuum hypothesis (the problem of the continuum). This problem did not yield a solution within the
framework of traditional set-theoretical methods of solution. Among mathematicians the conviction grew that the problem of the continuum was in principle unsolvable. It was only after a way had been
found of reducing mathematical concepts to set-theoretical ones, axioms had been stated in set-theoretical language which could be placed at the foundations of mathematical proofs actually
encountered in real life and logical derivation methods had been formalized, that it became possible to give a precise statement, and then to solve the question, of the formal unsolvability of the
continuum hypothesis. Formal unsolvability is understood in the sense that there does not exist a formal derivation in the Zermelo–Fraenkel system ZF either for the continuum hypothesis or for its
In 1939 K. Gödel established the unprovability of the negation of the generalized continuum hypothesis (and hence the unprovability of the negation of the continuum hypothesis) in the system ZF with
the axiom of choice (the system ZFC) under the hypothesis that ZF is consistent (see Gödel constructive set). In 1963 P. Cohen showed that the continuum hypothesis (and therefore also the generalized
continuum hypothesis) cannot be deduced from the axioms of ZFC assuming the consistency of ZF (see Forcing method).
Are these results concerning the problem of the continuum final? The answer to this question depends on one's relation to the premise concerning the consistency of ZF and, what is more significant,
to the experimental fact that every meaningful mathematical proof (of traditional classical mathematics) can, after it has been found, be adequately stated in the system ZFC. This fact cannot be
proved nor can it even be precisely stated, since each revision raises a similar question concerning the adequacy of the revision for the revised theorem.
In model-theoretic language, Gödel and Cohen constructed models for ZFC in which $$ 2^{\mathfrak{k}} = \begin{cases} \mathfrak{m} & \text{if}\ \mathfrak{k} < \mathfrak{m}\,; \\ \mathfrak{k}^{+} & \
text{if}\ \mathfrak{k} \ge \mathfrak{m} \ . \end{cases} $$
where $\mathfrak{m}$ is an arbitrary uncountable regular cardinal number given in advance, and $\mathfrak{k}^{+}$ is the first cardinal number greater than $\mathfrak{k}$. What is the possible
behaviour of the function $2^{\mathfrak{k}}$ in various models of ZFC?
It is known that for regular cardinal numbers $\mathfrak{k}$, this function can take them to arbitrary cardinal numbers subject only to the conditions $$ \mathfrak{k} < \mathfrak{k}' \Rightarrow 2^{\
mathfrak{k}} < 2^{\mathfrak{k}'} \,,\ \ \ \mathfrak{k} < \text{cf}(\mathfrak{k}) $$ where $\text{cf}(\mathfrak{a})$ is the smallest cardinal number cofinal with $\mathfrak{a}$ (see Cardinal number).
For singular (that is, non-regular) $\mathfrak{k}$, the value of the function $2^{\mathfrak{k}}$ may depend on its behaviour at smaller cardinal numbers. E.g., if (1) holds for all $\alpha < \
omega_1$, then it also holds for $\alpha = \omega_1$.
[1] P.J. Cohen, "Set theory and the continuum hypothesis" , Benjamin (1966)
[2] J.E. Baumgartner, K. Prikry, "Singular cardinals and the generalized continuum hypothesis" Amer. Math. Monthly , 84 : 2 (1977) pp. 108–113
[a1] T.J. Jech, "Set theory" , Acad. Press (1978) pp. Chapt. 7 (Translated from German)
[a2] K. Kunen, "Set theory, an introduction to independence proofs" , North-Holland (1980)
How to Cite This Entry:
Continuum hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuum_hypothesis&oldid=39845
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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10 Two particles of masses m(1) and m(2) initially at rest start movi
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Extract values from vector in R: dplyr | R-bloggersExtract values from vector in R: dplyrExtract values from vector in R: dplyr
Extract values from vector in R: dplyr
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Unravel the Future: Dive Deep into the World of Data Science Today! Data Science Tutorials.
Extract values from vector in R, we will delve into extracting specific values from a vector using the nth, first, and last functions from the dplyr package in R programming language.
The article is structured into four examples that demonstrate the extraction of vector elements.
Extract patterns in R? » Data Science Tutorials
Extract values from vector in R
To begin, by installing and loading the dplyr package in R.
Our example vector, denoted as ‘x’, is a character vector containing NINE letters.
We then proceed to apply the functions to our example vector.
x <- letters[1:9] # Create example vector
# "a" "b" "c" "d" "e" "f" "g" "h" "i"
Data Science & business analytics »
Example 1: nth Function
We explore the usage of the nth function, which allows us to extract a vector element from anywhere within the vector.
By applying the nth function and specifying the position of the desired element, we obtain the desired output.
nth(x, 5) # Apply nth function
# "e" #OUTPUT
Monte Carlo Analysis in R » finnstats
Example 2: nth Function with Negative Value
Example 2 showcases the application of the nth function to extract an element from the end of the vector.
By placing a minus sign before the position, we can retrieve elements from the end of the vector.
nth(x, - 3) # Apply nth function with negative
# "g" #output
Example 3: first Function
Example 3 demonstrates the first function, which returns the first element of an input vector.
first(x) # Apply first function
# "a" #output
This function works similarly to the nth function but is specifically designed to target the first element.
Example 4: last Function
Lastly, Example 4 illustrates the use of the last function, which returns the last element of a vector.
last(x) # Last function
# "i" #return last value
This function complements the first function by focusing on the end of the vector.
This article provides a comprehensive guide on how to use the nth, first, and last functions from the dplyr package in R to extract specific elements from a vector.
By understanding and applying these functions, users can effectively manipulate and analyze their data in the R programming environment.
The post Extract values from vector in R: dplyr appeared first on Data Science Tutorials
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Argumentation Theory: Difference between revisions
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Below we try to model some concepts of argumentation theory in B. Below we try to model some concepts of argumentation theory in B.
The examples try to show that set theory can be used to model some aspects of argumentation theory The examples try to show that set theory can be used to model some aspects of argumentation theory
quite naturally, and that ProB can solve and visualise some problems in argumentation theory. quite naturally, and that ProB can solve and visualise some problems in argumentation theory.
Alternative solutions are encoding arguments as normal logic programs and using answer set solvers Alternative solutions are encoding arguments as normal logic programs and using answer set solvers
for problem solving. for problem solving.
The following model was inspired by a talk given by Claudia Schulz. The following model was inspired by a talk given by Claudia Schulz.
Revision as of 18:34, 20 January 2016
Below we try to model some concepts of argumentation theory in B. The examples try to show that classical (monotonic) logic with set theory can be used to model some aspects of argumentation theory
quite naturally, and that ProB can solve and visualise some problems in argumentation theory. Alternative solutions are encoding arguments as normal logic programs (with non-monotonic negation) and
using answer set solvers for problem solving.
The following model was inspired by a talk given by Claudia Schulz.
The model below represents the labelling of the arguments as a total function from arguments to its status, which can either be in (the argument is accepted), out (the argument is rejected), or undec
(the argument is undecided). The relation between the arguments is given in the binary attacks relation.
In case you are new to B, you probably need to know the following operators to understand the specification below (we als have a summary page about the B syntax):
• x : S specifies that x is an element of S
• a|->b represents the pair (a,b); note that a relation and function in B is a set of pairs.
• x|->y : R hence specifies that x is mapped to y in relation R
• !x.(P => Q) denotes universal quantification over variable x
• #x.(P & Q) denotes existential quantification over variable x
• A <--> B denotes the set of relations from A to B
• A --> B denotes the set of total functions from A to B
• block comments are of the form /* ... */ and line comments start with // (be sure to use version 1.5.1 of ProB, e.g., from the Nightly download site as line comments were added recently to B)
MACHINE ArgumentationTotFun
STATUS = {in,out,undec}
CONSTANTS attacks, label
attacks : ARGUMENTS <-> ARGUMENTS /* which argument attacks which other argument */
label: ARGUMENTS --> {in,out,undec} & /* the labeling function */
// if an argument y is in any attacker must be out:
!(x,y).(x|->y:attacks => (label(y)=in => label(x)=out)) &
// if an argument x is in anything it attacks must be out:
!(x,y).(x|->y:attacks => (label(x)=in => label(y)=out)) &
//if an argument y is out it must be attacked by a valid argument:
!(y).(y:ARGUMENTS => (label(y)=out => #x.(x|->y:attacks & label(x)=in))) &
// if an argument y is undecided it must be attacked by an undecided argument:
!(y).(y:ARGUMENTS => (label(y)=undec => #x.(x|->y:attacks & label(x)=undec)))
// here we model one particular argumentation graph
// A = the sun will shine to day, B = we are in the UK, C = it is summer, D = there are only 10 days of sunshine per year, E = the BBC has forecast sun
attacks = {B|->A, C|->B, D|->C, E |-> B, E|->D}
Here is a screenshot of ProB Tcl/Tk after loading the model.
You can see that there is only a single solution (solving time 10 ms), as only a single SETUP_CONSTANTS line is available in the "Enabled Operations" pane. Double-click on SETUP_CONSTANTS and then
INITIALISATION will give you the following result, where you can see the solution in the "State Properties" pane:
If you want to inspect the solution visually, you can select the "Current State as Graph" command in the "States" submenu of the "Visualize" menu:
This results in the following picture being displayed:
You can try and experiment with various attack graphs by editing the file. E.g., with
attacks = {B|->A, C|->B, D|->C, E |-> B, E|->D, D|->E}
you now get three possible labelling, the last one assigns undec to all arguments.
A more elegant encoding
The following is a slightly more elegant encoding, where we represent the labelling as a partition of the arguments. This enables to use the relational image operator to express some constraints more
compactly. This model is particularly well suited if you wish to use our Kodkod backend which translates the constraints to SAT. However, even the standard ProB solver can solve this instance in 10
ms (Kodkod requires 70 ms), so this would only be worthwhile for larger problem instances.
In case you are new to B, you probably need to know the following additional operators to understand the specification below (we als have a summary page about the B syntax):
• r~ is the inverse of a function or relation r
• r[S] is the relational image of a relation r for a set of domain values S
MACHINE ArgumentationAsSets
CONSTANTS attacks, in,out,undec
attacks : ARGUMENTS <-> ARGUMENTS /* which argument attacks which other argument */
// we partition the arguments into three sets
ARGUMENTS = in \/ out \/ undec &
in /\ out = {} & in /\ undec = {} & out /\ undec = {} &
// if an argument is in, any attacker must be out:
attacks~[in] <: out &
// if an argument is in, anything it attacks must be out:
attacks[in] <: out &
//if an argument y is out, it must be attacked by a valid argument:
!y.(y:out => #x.(x|->y:attacks & x:in)) &
// if an argument y is undecided, it must be attacked by an undecided argument:
!y.(y:undec => #x.(x|->y:attacks & x:undec))
// here we model one particular argumentation graph
// A = the sun will shine to day, B = we are in the UK, C = it is summer, D = there are only 10 days of sunshine per year, E = the BBC has forecast sun
attacks = {B|->A, C|->B, D|->C, E |-> B, E|->D}
A more elegant encoding
Instead of using ProB Tcl/Tk you can also encode this model in Rodin, the Eclipse-based platform for Event-B.
Here we have split the model into two contexts. The first one encodes the general rules for labelling (we use Camille syntax):
context ArgumentsAsSets
sets ARGUMENTS
constants attacks in out undec
@axm1 attacks ∈ ARGUMENTS ↔ ARGUMENTS // which argument attacks which other argument
@axm2 partition(ARGUMENTS,in,out,undec) // we partition the arguments into three sets
@axm3 attacks∼[in] ⊆ out // if an argument is in, any attacker must be out
@axm4 attacks[in] ⊆ out // if an argument is in, anything it attacks must be out
@axm5 ∀y·(y∈out ⇒ ∃x·(x↦y∈attacks ∧ x∈in)) //if an argument y is out, it must be attacked by a valid argument
@axm6 ∀y·(y∈undec ⇒ ∃x·(x↦y∈attacks ∧ x∈undec)) // if an argument y is undecided, it must be attacked by an undecided argument
A second context then extends the above one, and encodes our particular problem instance:
context Arguments_Example extends ArgumentsAsSets
constants A B C D E
@part partition(ARGUMENTS,{A},{B},{C},{D},{E})
@example attacks = {B↦A, C↦B, D↦C, E ↦ B, E↦D}
/* A = the sun will shine to day, B = we are in the UK
C = it is summer, D = there are only 10 days of sunshine per year, E = the BBC has forecast sun */
If you load this model with ProB for Rodin, you can see the solution in the State view: | {"url":"https://prob.hhu.de/w/index.php?title=Argumentation_Theory&diff=3318&oldid=3317","timestamp":"2024-11-12T15:44:39Z","content_type":"application/xhtml+xml","content_length":"24905","record_id":"<urn:uuid:29da99c5-0f93-46fe-a296-c3f8ce2e0227>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00514.warc.gz"} |
Isabel A. Espírito-Santo
According to our database
, Isabel A. Espírito-Santo authored at least 17 papers between 2005 and 2019.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
A Multi-Objective Approach to Solve an Optimal Control Problem Applied to a Simplified Activated Sludge System Model.
Proceedings of the Fuzzy Systems and Data Mining V, 2019
Wastewater Treatment Plant Design: Optimizing Multiple Objectives.
Proceedings of the 7th International Conference on Operations Research and Enterprise Systems, 2018
A Scalarized Augmented Lagrangian Algorithm (SCAL) for Multi-objective Optimization Constrained Problems.
Proceedings of the 7th International Conference on Operations Research and Enterprise Systems, 2018
Benchmarking deterministic optimization algorithms using an outranking approach.
Optim. Methods Softw., 2016
Derivative-Free Augmented Lagrangian for Global Optimization: Cost Minimization in a Simplified Activated Sludge System Model.
J. Math. Model. Algorithms Oper. Res., 2015
MOEA/VAN: Multiobjective Evolutionary Algorithm Based on Vector Angle Neighborhood.
Proceedings of the Genetic and Evolutionary Computation Conference, 2015
MOEA/PC: Multiobjective Evolutionary Algorithm Based on Polar Coordinates.
Proceedings of the Evolutionary Multi-Criterion Optimization, 2015
Generalized Multiobjective Evolutionary Algorithm Guided by Descent Directions.
J. Math. Model. Algorithms Oper. Res., 2014
Clustering-Based Selection for Evolutionary Many-Objective Optimization.
Proceedings of the Parallel Problem Solving from Nature - PPSN XIII, 2014
On Challenging Techniques for Constrained Global Optimization.
Proceedings of the Handbook of Optimization - From Classical to Modern Approach, 2013
A New Hybrid Evolutionary Multiobjective Algorithm Guided by Descent Directions.
J. Math. Model. Algorithms Oper. Res., 2013
Many-objective optimization using differential evolution with variable-wise mutation restriction.
Proceedings of the Genetic and Evolutionary Computation Conference, 2013
A hybrid genetic pattern search augmented Lagrangian method for constrained global optimization.
Appl. Math. Comput., 2012
Heuristic Pattern Search for Bound Constrained Minimax Problems.
Proceedings of the Computational Science and Its Applications - ICCSA 2011, 2011
Stochastic algorithms assessment using performance profiles.
Proceedings of the 13th Annual Genetic and Evolutionary Computation Conference, 2011
Using a Genetic Algorithm to Solve a Bi-Objective WWTP Process Optimization.
Proceedings of the Operations Research Proceedings 2010, 2010
NEOS Server Usage in Wastewater Treatment Cost Minimization.
Proceedings of the Computational Science and Its Applications, 2005 | {"url":"https://www.csauthors.net/isabel-a-espirito-santo/","timestamp":"2024-11-11T20:20:01Z","content_type":"text/html","content_length":"31918","record_id":"<urn:uuid:3933e41e-dbf0-400b-a93a-e32030397654>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00892.warc.gz"} |
6th Grade Math Early Finishers Paper & Digital Bundle
This bundle includes both PAPER & DIGITAL files for my 6th Grade Math Early Finishers.
The paper version and digital version include 36 pages (72 total). Each page has a word search, unscramble the words, and an activity on the bottom (color by code, build a monster, or etc). These
early finishers can be used when students finish a test early, on sub days, as homework, as warmups, and so much more!
The digital version includes Google Slides. It includes NO PREP and is PAPERLESS! You will be given a link that you can post in your Google Classroom. Students will be forced to make a copy, and
that's it! It is completely set up for students and very study friendly!
The concepts included are:
(1) Adding Decimals
(2) Subtracting Decimals
(3) Multiplying Decimals
(4) Dividing Decimals
(5) Dividing Fractions
(6) Exponents
(7) Order of Operations
(8) Absolute Value
(9) Coordinate Plane
(10) LCM & GCF
(11) Rates & Ratios
(12) Percents
(13) Percent-Decimal Conversions
(14) Percent of a Number
(15) Finding a Percent
(16) Finding the Base
(17) Intro to Variables
(18) Evaluating Expressions
(19) Algebraic Expressions
(20) One-Step Equations
(21) One-Step Inequalities
(22) Combining Like Terms
(23) The Distributive Property
(24) Area of Parallelograms
(25) Area of Triangles
(26) Area of Trapezoids
(27) Area of Composite Figures
(28) Parts of 3-D Objects
(29) Volume of Rectangular Prisms
(30) Surface Area using Nets
(31) Polygons in the Plane
(32) Dot Plots & Frequency Tables
(33) Histrograms
(34) Mean, Median, Mode, and Range
(35) Mean Absolute Deviation (MAD)
(36) Box Plots
You can use the assignment as a review at the end of the year OR you can use it throughout the entire school year!
Questions? Please contact me at mathindemand@hotmail.com. Thanks so much!
CLICK HERE, to purchase! | {"url":"http://www.commoncorematerial.com/2024/03/6th-grade-math-early-finishers-paper.html","timestamp":"2024-11-08T21:20:44Z","content_type":"text/html","content_length":"122321","record_id":"<urn:uuid:cf94282d-1f39-404e-bfcf-dbc87849aa90>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00609.warc.gz"} |
Identifying Injective Functions
Question Video: Identifying Injective Functions Mathematics • Second Year of Secondary School
True or False: If π and π are both one-to-one functions, then π + π must be a one-to-one function.
Video Transcript
True or false: If π and π are both one-to-one functions, then π plus π must be a one-to-one function.
In order to prove that this statement is true, we must prove it is true for all possible functions in this case. However, in order to prove a statement is false, we can do so via a counterexample. We
begin by recalling that a function is one-to-one if each element of the range of the function corresponds to exactly one element of the domain. In mapping diagrams, this means that each element of
the range has exactly one arrow pointing to it.
Letβ s consider the functions π and π represented by the mapping diagrams shown. We see that both functions π and π are injective or one-to-one because each element of the range has
exactly one arrow from the domain pointed at it. When we add π and π as required, we obtain the following diagram. Noting that all three lines in the range give the same number, we can redraw
the mapping diagram. Since the range element 10 has three arrows pointing to it, this range element corresponds to more than one of the domain elements. We can therefore conclude the function π
plus π is not injective. Using a counterexample, we have shown that the statement β if π and π are both one-to-one functions, then π plus π must be a one-to-one functionβ is false. | {"url":"https://www.nagwa.com/en/videos/784195953890/","timestamp":"2024-11-10T08:16:34Z","content_type":"text/html","content_length":"247966","record_id":"<urn:uuid:013fc685-63f9-4c8c-b800-4da37509f588>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00747.warc.gz"} |
Newton's Second Law - Knowunity
Newton's Second Law: AP Physics 1 Study Guide
Welcome, aspiring physicists, to the land where apples fall not far from trees and rockets blast bravely into the cosmos! Today, we're diving into Newton's Second Law, a pillar of physics that
explains why heavier objects aren't so quick on their feet and why pressing the gas pedal makes your car zoom. 🏎️💨
Newton's Second Law: The Basics
Newton’s Second Law tells us that the acceleration of an object is directly proportional to the net force acting on it, proportional to the size of the object, and inversely proportional to its mass.
In simpler terms, the harder you push, the faster it goes—unless it had too many donuts for breakfast. 😋🍩
Mathematically, this can be expressed using the equation: [ \mathbf{F} = m \mathbf{a} ] Here, (\mathbf{F}) represents the net force, (m) stands for mass, and (\mathbf{a}) is acceleration. This
formula is physics royalty 👑, deserving of a place in the Nobility House of Equations!
Net Force: The vector sum of all forces acting on an object. Think of it as the ultimate team effort, where every force has a distinct job—some push, some pull, and some just refuse to budge.
Key Concepts to Master
Force: A measure of how much an object gets pushed or pulled. It's expressed in newton units (N), aptly named after Sir Isaac. One newton is equivalent to (\text{1 kg} \cdot \text{m/s}^2). So if you
sneeze with enough force, technically, you're a human physics experiment. Gesundheit! 🤧
Mass: The amount of matter in an object, measured in kilograms (kg). More mass means more inertia, which is physics-speak for “Doesn't get out of bed easily.”
Acceleration: The rate at which an object’s velocity changes, measured in meters per second squared ((\text{m/s}^2)). If you're driving and slam the brakes, your head bobbing forward is just your
body dealing with the abrupt acceleration change. Catching up with it, eh?
Examples and Applications
Consider a compelling drama: a thirsty kid (mass (20 \text{kg})) spots an ice cream truck and applies a force of (40 \text{N}). According to Newton: [ \mathbf{a} = \frac{\mathbf{F}}{m} = \frac{40 \
text{N}}{20 \text{kg}} = 2 \text{m/s}^2 ] The acceleration of the kid towards the truck—and ice cream—is (2 \text{m/s}^2). 🍦🚶
But wait, it’s worth noting that if the kid was pushing a fridge, not even Newton would get him very far, for: [ \mathbf{a} \propto \frac{1}{m} ] More mass, less acceleration. Ever tried pushing your
cat to get out of your favorite spot on the couch? Now you know why Sir Fluffypaws holds his ground! 🐈
Free-body Diagrams 🌐
Imagine you’re tasked with drawing a free-body diagram for a pizza delivery drone (mass (5 \text{kg})).
1. Sketch the drone: A simple square representing our high-tech pizza provider.
2. Identify Forces:
□ Gravity pulls downward ((\mathbf{F}_g)).
□ Lift from the propellers acts upward ((\mathbf{F}_\text{thrust})).
□ An additional horizontal wind push ((\mathbf{F}_{\text{wind}})) might sway it to one side.
3. Draw the forces:
□ Arrow down (gravity) labeled (\mathbf{F}_g = m \cdot g = 5 \text{kg} \cdot 9.8 \text{m/s}^2).
□ Arrow up for thrust opposing gravity.
□ Horizontal arrow indicating wind force.
This diagram helps us visually decipher the problem, turning us into intuitive physicists! 🧙♂️📉
Key Terms to Keep Handy
• Acceleration: How fast you’re changing speed.
• Applied Force: The force applied to an object by an external agent, like your hand or the bed pushing you out every morning.
• Free-body Diagrams: Visual tools to itemize all forces acting on an object.
• Friction: The pesky force that eats your momentum. Think of it as nature's way of saying, “Slow down.”
• Gravity: The ever-attractive force pulling us down, presumably towards Earth's core network of dark chocolate. 🍫
• Mass: The measure of the substance, the more you have, the harder you party...er, I mean, the more inertia you possess.
• Tension: The force transmitted through a rope, cord, or, occasionally, a dramatic movie scene. 🎬
Fun Fact
Remember, Newton’s brilliance wasn’t confined to laws of motion alone; he dabbled in alchemy. Who knows? He might have made the first potion of acceleration!
Newton's Second Law brings a robust framework for the motion of objects, dictating how accelerations respond to varying forces and masses. From pizza drones to ice cream chases, this concept is
fundamental in analyzing real-world physics. So buckle up, flex those brain muscles, and master the art of force and motion!
Now go forth, and may your physics journey be as smooth as a frictionless surface! 🌟 | {"url":"https://knowunity.com/subjects/study-guide/newtons-second-law","timestamp":"2024-11-12T13:12:09Z","content_type":"text/html","content_length":"227101","record_id":"<urn:uuid:2cf38eb9-0e3b-458b-8e31-afbcc68cb04e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00307.warc.gz"} |
Science topic
Combinatorics - Science topic
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size
(enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding
"largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic
techniques to combinatorial problems (algebraic combinatorics).
Questions related to Combinatorics
Dear Researchers,
Do researchers/universities value students/researchers having published sequences to the OEIS?
Relevant answer
I have done both. I cited my work in the sequences and the sequences in my work. | {"url":"https://www.researchgate.net/topic/Combinatorics","timestamp":"2024-11-03T20:05:59Z","content_type":"text/html","content_length":"1050259","record_id":"<urn:uuid:790962e2-53cf-4f76-81fc-fe668a987978>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00794.warc.gz"} |
MET 3431 Statistics
Course coordinator:
Ulf Henning Olsson
Einar Christopher Wellén
Course name in Norwegian:
Product category:
Bachelor - Common Courses
Teaching language:
Course type:
One semester
The course provides an introduction to statistical thinking. The student first learns how a sample can be collected and summarized, and how we use such summaries to interpret the data. Then we learn
to generalize from the sample to the population. The student learns to construct confidence intervals and to perform hypothesis tests, and how these should be interpreted. The focus is on
understanding concepts, and on interpreting results, with less focus on mathematical procedures. Through many examples from reality, the student should be able to see the relevance of and uses for
statistics in marketing and economics. Use of statistical software is absolutely crucial.
Learning outcomes - Knowledge
During the course the students will:
• Acquire broad knowledge of the central statistical concepts and understand how statistical analysis takes place from data collection, through descriptive analysis to generalization to the
Examples of concepts which shall can be explained are sample, population, statistic, parameter, inference, margin of error, significance level, confidence level and probability distribution.
• Knowledge of the central limit theorem
• Knowledge of the limitations of statistical method.
Learning outcomes - Skills
After completing the course, students will be able to:
• Determine the level of measurement of variables, and be able to perform descriptive analysis based on a sample, with appropriate measures for central trend and variation and appropriate graphs.
• Describe the covariance between two variables.
• Interpret results of descriptive analysis.
• Calculate simple probabilities - especially using the normal distribution.
• Be able to construct and interpret the most commonly used confidence intervals, and perform key hypothesis tests.
• Use statistical software and be able to interpret printouts from the software.
• Present the results of the analyzes in an easy-to-understand language.
General Competence
• The student should be aware that statistical methods may be easily misused and misinterpreted.
• To take into account anonymity and privacy when collecting data
Course content
• Collection of data
• Descriptive analysis of the sample
• Covariation
• Simple probability theory with focus on the normal distribution
• Confidence intervals for mean and proportion
• Hypothesis tests for mean and proportion
• Simple correlation- and regressionanalysis
• Selected central tests, including the chi-square test for covariation between two categorical variables.
Teaching and learning activities
The course consists of 48 hours of lectures. The software SAS JMP will be a central part of the teaching.
In lectures, theory will be illustrated by using multiple data sets often in companion with SAS JMP.
It is mandatory work requirements where 4 of 6 work requirements must be passed to take the exam.
The final exam will be based on that the student has solved the coursework requirement throughout the semester. A part of the final exam will be based directly on the excercises given in the
mandatory work requirements.
In course delivery as online courses, lecturer will, in collaboration with the student administration, organize an appropriate course implementation, combining different learning activities and
digital elements on the learning platform. Online students are also offered a study guide that will contribute to progression and overview. Total recommended time spent for completing the course also
applies here.
Additional information
Re-sit examination
Students that have not gotten approved the coursework requirements, must re-take the exercises during the next scheduled course.
Students that have not passed the written examination or who wish to improve their grade may re-take the examination in connection with the next scheduled examination.
Higher Education Entrance Qualification
Due to the Covid-19 pandemic, there may be deviations in teaching and learning activities as well as exams, compared with what is described in this course description.
Information about what is taught on campus and other digital forms will be presented with the lecture plan before the start of the course each semester.
Required prerequisite knowledge
No specific prerequisites required.
Mandatory Courseworks Courseworks Comment coursework
coursework given required
Mandatory 6 4 In the course of the semester 6 mandatory assignments With software will be given. These are submitted on Itslearning. Each assignment is assessed as either
pass or fail. The student needs at least 4 passes in order to take the final exam.
Mandatory coursework:
Mandatory Mandatory
Courseworks 6
Courseworks 4
Comment In the course of the semester 6 mandatory assignments With software will be given. These are submitted on Itslearning. Each assignment is assessed as either pass or fail. The student
coursework: needs at least 4 passes in order to take the final exam.
Exam category:
Form of assessment:
Written submission
Support materials:
• All printed and handwritten support materials
• BI-approved exam calculator
• Simple calculator
5 Hour(s)
Exam code:
Grading scale:
Examination every semester
Type of Assessment:
Ordinary examination
Student workload
Activity Duration Comment
Teaching 48 Hour(s)
Group work / Assignments 50 Hour(s)
Prepare for teaching 42 Hour(s) Working with SAS JMP (or some statistical software)
Student's own work with learning resources 40 Hour(s)
Examination 20 Hour(s) Exam incl. preparations.
A course of 1 ECTS credit corresponds to a workload of 26-30 hours. Therefore a course of 7,5 ECTS credit corresponds to a workload of at least 200 hours. | {"url":"https://programmeinfo.bi.no/en/course/MET-3431/2022-spring","timestamp":"2024-11-10T11:24:27Z","content_type":"text/html","content_length":"25358","record_id":"<urn:uuid:6a3ea4ea-8186-4dbc-87b9-5c2598a1dcd7>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00838.warc.gz"} |
An empirical propagation model /EPM-73/
A propagation model is described which has the advantage of simplicity of manual calculations of basic transmission loss, and which provides a degree of accuracy which is similar to that obtained
with other more sophisticated models which compute basic transmission loss. The model uses a minimum number of parameters and is based on both theoretical and empirical considerations. Also, given a
value of basic transmission loss and, for example, antenna heights and frequency, the appropriate value of distance may be calculated. The model provides an estimate of mean basic transmission loss,
in dB, with an associated standard deviation. It has been compared with measured values over a frequency range of approximately 20-10,000 MHz. Approximately 7000 paths have been considered in many
different areas. Comparison with other more sophisticated models indicates comparable results, including predictions for sea water paths and for frequencies down to 1 MHz (but not substantiated by
measurements between 1 and 20 MHz).
IEEE Transactions on Electromagnetic Compatibility
Pub Date:
August 1977
□ Mathematical Models;
□ Microwave Transmission;
□ Propagation Modes;
□ Radio Transmission;
□ Transmission Loss;
□ Electromagnetic Compatibility;
□ Prediction Analysis Techniques;
□ Semiempirical Equations;
□ Communications and Radar | {"url":"https://ui.adsabs.harvard.edu/abs/1977ITElC..19..301L/abstract","timestamp":"2024-11-07T19:43:09Z","content_type":"text/html","content_length":"36247","record_id":"<urn:uuid:98a092d1-e92f-4041-a512-6d0fc90eb095>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00440.warc.gz"} |
Linked List with Ruby
Table of Contents
1. Understanding Linked Lists
2. Implementing a Linked List in Ruby
2.1. print_list method
2.2. append Method
2.3. pop Method
2.4. prepend Method
2.5. shift Method
2.6. get method
2.7. set_value method
2.8. insert method
2.9. reverse Method
github repo: https://github.com/TheZero0-ctrl/leetcode-practice
Understanding Linked Lists
A linked list is a collection of nodes where each node contains a value and a reference to the next node in the sequence. The first node is called the head, and the last node is called the tail. The
tail node's reference points to nil, indicating the end of the list.
In this blog post, we will explore a simple implementation of a linked list in Ruby and discuss each method along with its time complexity.
Implementing a Linked List in Ruby
Let's start by examining the provided Ruby code that implements a linked list. The LinkedList class represents the linked list, and each node is represented by the Node class.
class LinkedList
attr_accessor :head, :tail, :length
def initialize(value)
new_node = Node.new(value)
@head = new_node
@tail = new_node
@length = 1
# Other methods...
class Node
attr_accessor :value, :next
def initialize(value)
@value = value
@next = nil
• The initialize method is the constructor of the LinkedList class and Node class.
• It creates a new instance of the linked list with an initial value and a new instance of the node.
• When a new node is created, its value will be the provided value, and the next node will be nil.
• When a new linked list is created, it creates a new node object with the given value.
• This node is set as both the head and tail of the linked list.
• The length attribute is initialized to 1 since there is only one node in the list at the beginning.
print_list method
The print_list method iterates through the linked list and prints the value of each node.
def print_list
temp = head
while temp
puts temp.value
temp = temp.next
• Starting from the head node, the method iterates through the list by following the next reference of each node.
• It prints the value of each node and continues until it reaches the end of the list (when temp becomes nil).
• The time complexity of the print_list method is O(n), where n is the length of the linked list.
• It needs to visit each node once to print its value.
append Method
The append method adds a new node with the given value at the end of the linked list.
def append(value)
new_node = Node.new(value)
if @head
@tail.next = new_node
@tail = new_node
@tail = new_node
@head = new_node
@length += 1
• First, a new node is created with the provided value.
• If the linked list is not empty (the @head node exists), the next reference of the current tail node is set to the new node, and the tail is updated to the new node. This way, the new node
becomes the new tail.
• If the linked list is empty, both the head and tail are set to the new node.
• At last, we increment length by one
• The time complexity of the append method is O(1) since it performs a constant number of operations regardless of the size of the linked list. It directly accesses and updates the tail node.
pop Method
The pop method removes and returns the last node from the linked list.
def pop
return if @length.zero?
temp = @head
pre = @head
while temp.next
pre = temp
temp = temp.next
@tail = pre
@tail.next = nil
@length -= 1
return temp unless @length.zero?
@head = nil
@tail = nil
• To remove the last node, the method iterates through the list until it reaches the second-to-last node.
• The pre variable keeps track of the previous node while the temp variable moves forward.
• After reaching the last node, it updates the tail to the second-to-last node, removes the reference to the last node, and decrements the length of the list.
• If the list becomes empty after removing the last node, both the head and tail are set to nil.
• The time complexity of the pop method is O(n), where n is the length of the linked list.
• It needs to traverse the list to reach the second-to-last node for removal.
prepend Method
The prepend method adds a new node with the given value at the beginning of the linked list.
def prepend(value)
new_node = Node.new(value)
new_node.next = @head
@head = new_node
@tail = new_node if @length.zero?
@length += 1
• First, a new node is created with the provided value.
• The next reference of the new node is set to the current head node.
• Then, the head is updated to the new node.
• If the linked list was initially empty (length was zero), the tail is also set to the new node since it is the only node in the list.
• At last, we increment length by one
• The time complexity of the prepend method is O(1) since it performs a constant number of operations regardless of the size of the linked list.
• It directly accesses and updates the head node.
shift Method
The shift method removes and returns the first node from the linked list.
def shift
return if @length.zero?
temp = @head
@head = temp.next
temp.next = nil
@tail = nil if @length == 1
@length -= 1
• The method removes the head node by updating the head to its next node.
• It also updates the next reference of the removed node to nil to detach it from the list.
• If the list becomes empty after removal (length was initially one), the tail is set to nil.
• At last, we decrement length by one.
• The time complexity of the shift method is O(1) since it performs a constant number of operations regardless of the size of the linked list.
• It directly accesses and updates the head node.
get method
The get method returns the node at the specified index in the linked list.
def get(index)
return if index >= @length || index.negative?
temp = @head
(0...index).each do |_i|
temp = temp.next
• The method first checks if the provided index is valid (within the range of the list).
• If the index is not valid (greater than or equal to the length or negative), it returns nil.
• Otherwise, it iterates through the list to reach the desired index and returns the corresponding node.
• The time complexity of the get method is O(n), where n is the length of the linked list.
• It needs to traverse the list to reach the desired index.
set_value method
The set_value method sets the value of the node at the specified index in the linked list.
def set_value(index, value)
temp = get(index)
if temp
temp.value = value
• The method first uses the get method to retrieve the node at the specified index.
• If the node exists, it updates its value with the provided value and returns true.
• Otherwise, it returns false.
• The time complexity of the set_value method is O(n), where n is the length of the linked list.
• It utilizes the get method, which has a time complexity of O(n), to retrieve the node at the specified index.
insert method
The insert method inserts a new node with the given value at the specified index in the linked list.
def insert(index, value)
return if index >= @length || index.negative?
return prepend(value) if index.zero?
return append(value) if index == @length - 1
temp = get(index - 1)
new_node = Node.new(value)
new_node.next = temp.next
temp.next = new_node
@length += 1
• First, it checks if the provided index is valid. If not, it returns nil.
• If the index is zero, it uses the prepend method to add the new node at the beginning.
• If the index is equal to the length minus one, it uses the append method to add the new node at the end.
• Otherwise, it retrieves the node at the previous index (index - 1) using the get method.
• Then, it creates a new node with the provided value and inserts it into the list by updating the next references accordingly.
• At last, we increment length by one
• The time complexity of the insert method is O(n), where n is the length of the linked list.
• It utilizes the get method, which has a time complexity of O(n), to retrieve the node at the specified index.
remove method
The remove method removes and returns the node at the specified index in the linked list.
def remove(index)
return if index >= @length || index.negative?
return shift if index.zero?
return pop if index == @length - 1
prev = get(index - 1)
temp = prev.next
prev.next = temp.next
temp.next = nil
@length -= 1
• The method first checks if the provided index is valid. If not, it returns nil.
• If the index is zero, it uses the shift method to remove and return the head node.
• If the index is equal to the length minus one, it uses the pop method to remove and return the tail node.
• Otherwise, it retrieves the node at the previous index (index - 1) using the get method.
• It updates the next references to detach the node at the specified index from the list.
• At last, we decrement length by one.
• The time complexity of the remove method is O(n), where n is the length of the linked list.
• It utilizes the get method, which has a time complexity of O(n), to retrieve the node at the specified index.
reverse Method
The reverse method reverses the linked list by changing the order of the nodes.
def reverse
temp = @head
@head = @tail
@tail = temp
after = temp.next
before = nil
(0...@length).each do |_i|
after = temp.next
temp.next = before
before = temp
temp = after
• The method starts by swapping the head and tail nodes.
• Then, it iterates through the list and reverses the next references of each node.
• It uses three variables (temp, before, and after) to keep track of the nodes being reversed.
• The loop continues until all nodes are reversed.
• The time complexity of the reverse method is O(n), where n is the length of the linked list.
• It needs to traverse the list once to reverse the nodes.
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What are Algebraic Expressions? - ARTICLE BENCH
What are Algebraic Expressions?
The basics of algebra state how to express any number of unknown values using the alphabet. These letters or alphabets are called variables as they can assign any value. Algebraic expressions express
numbers using variables and constants through basic arithmetic operations such as addition, subtraction, multiplication, division, etc. Each algebraic expression consists of different parts such as
variable, constant, coefficients, and exponents. These are explained as follows:
• Variable: we can change Any quantity whose value is unknown and according to the conditions and boundary set in the expression.
• Constant: A quantity that doesn’t change the value under the given conditions.
• Coefficients: The fixed number (positive or negative) associated with variables by way of multiplication.
• Exponents: The number of times a variable is to be multiplied is denoted by its exponent.
The mathematical expression contains many terms or parts connected with addition or subtraction. A term can consist of a variable alone, a constant alone, or a combination of variable and constant.
It can be a product of the same or different variables or variables and a constant.
Let’s take an example of the following algebraic expression.
12x – 6y + 27
• x and y are variables with an unknown value that can take any value.
• 27 is the constant which has a fixed value that doesn’t change
• 12 is the coefficient of x, and -6 is the coefficient of y, as these are constant values multiplied with variables.
• The expression has three terms.
Types of Algebraic Expressions
The types of algebraic expressions vary depending on the number of variables present, the number of terms in that expression, and the values of the exponents of the variables in each term. Depending
on the number of terms, the different types of algebraic expressions are as follows:
• Monomial Expression: An expression with only one term where the variable’s exponent is a non-negative integer.
• Binomial /Trinomial Expression: An algebraic expression consists of two or three monomial expressions as different terms.
• Polynomial Expression: An expression containing any number of monomials more than one.
Algebra Formula
The algebra formula gives the idea that helps to understand and solve algebraic expressions using some defined procedures. The algebra formula guides solving big and complex problems in simple steps.
The algebraic formula helps to learn the simplified method for solving the algebraic expressions consisting of several terms and finding the value of an unknown number or variable.
An algebraic formula is an expression written using mathematical and algebraic symbols following certain rules. The expression involves two algebraic expressions with an equal sign between them. This
form of an algebraic expression is known as an algebraic equation. The left side indicates an algebraic expression, and the right side gives a simplified form of that expression. It is a short and
simplified form to remember certain equations used to solve complex algebraic calculations. These algebraic formulas can be derived for topics like logarithms, indices, exponents, probability, etc.
In algebra formulas, the validity of a condition as expressed in algebraic form is always true regardless of the values assigned to the variables. The algebra formula satisfies the algebraic
Identity, which means the left-hand side of the equation is identical to the right-hand side of the equation for all values of the variables. The algebraic formulas are important tools for students
to handle complex problems involving algebraic expressions. Learning about algebraic formulas is an essential part of the academic curriculum. The algebra formula has been widely used in many
mathematics topics as in other branches of science to arrive at specific solutions involving unknown quantities or variables, exponents, and coefficients.
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commit e580f9d125e93911d67fe76e80798ffe3faa428b
parent 9f13bc4362f0763aec047ec7a0608453503b022d
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Tue, 28 Dec 2021 16:05:43 +0100
fixed typo
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/README.md b/README.md
@@ -135,7 +135,7 @@ pruning table to solving EO with HTM moveset and this coordinate would have valu
The values for most pruning tables are memorized modulo 16, so they only occupy
4 bits per entry, and values larger than 15 are saved as 15. This is good enough
for most applications.
-Some large tables are memorized in compact form using only 4 bits, similarly
+Some large tables are memorized in compact form using only 2 bits, similarly
to what [nxopt](https://github.com/rokicki/cube20src/blob/master/nxopt.md) does:
a base value `b` is picked and a value of `n` is saved as `MIN(3,MAX(0,n-b))`.
When a value of `v=1,2,3` is read it is simply returned as `v+b`, while if | {"url":"https://git.tronto.net/nissy-fmc/commit/e580f9d125e93911d67fe76e80798ffe3faa428b.html","timestamp":"2024-11-14T18:49:30Z","content_type":"text/html","content_length":"3356","record_id":"<urn:uuid:fb872ca0-8861-4ba8-a45f-130bfb1633de>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00838.warc.gz"} |
6.3 Centripetal Force
Chapter 6 Uniform Circular Motion and Gravitation
• Calculate coefficient of friction on a car tire.
• Calculate ideal speed and angle of a car on a turn.
Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction
between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge.
Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal
acceleration. According to Newton’s second law of motion, net force is mass times acceleration: net
By using the expressions for centripetal acceleration
You may use whichever expression for centripetal force is more convenient. Centripetal force
Note that if you solve the first expression for
This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve.
Figure 1. The frictional force supplies the centripetal force and is numerically equal to it. Centripetal force is perpendicular to velocity and causes uniform circular motion. The larger the F[c],
the smaller the radius of curvature r and the sharper the curve. The second curve has the same v, but a larger F[c] produces a smaller r’.
Example 1: What Coefficient of Friction Do Care Tires Need on a Flat Curve?
(a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s.
(b) Assuming an unbanked curve, find the minimum static coefficient of friction, between the tires and the road, static friction being the reason that keeps the car from slipping (see Figure 2).
Strategy and Solution for (a)
We know that
Strategy for (b)
Figure 2 shows the forces acting on the car on an unbanked (level ground) curve. Friction is to the left, keeping the car from slipping, and because it is the only horizontal force acting on the car,
the friction is the centripetal force in this case. We know that the maximum static friction (at which the tires roll but do not slip) is
Now we have a relationship between centripetal force and the coefficient of friction. Using the first expression for
We solve this for
Solution for (b)
Substituting the knowns,
(Because coefficients of friction are approximate, the answer is given to only two digits.)
We could also solve part (a) using the first expression in
Figure 2. This car on level ground is moving away and turning to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. A
minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway.
Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See Figure 3. The greater the angle
For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal
the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the
vertical and horizontal directions.
Figure 3 shows a free body diagram for a car on a frictionless banked curve. If the angle
Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in
direction. From the figure, we see that the vertical component of the normal force is
Now we can combine the last two equations to eliminate
Taking the inverse tangent gives
This expression can be understood by considering how
Figure 3. The car on this banked curve is moving away and turning to the left.
Example 2: What Is the Ideal Speed to Take a Steeply Banked Tight Curve?
Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car
configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless.
We first note that all terms in the expression for the ideal angle of a banked curve except for speed are known; thus, we need only rearrange it so that speed appears on the left-hand side and then
substitute known quantities.
Starting with
we get
Noting that tan 65.0º = 2.14, we obtain
This is just about 165 km/h, consistent with a very steeply banked and rather sharp curve. Tire friction enables a vehicle to take the curve at significantly higher speeds.
Calculations similar to those in the preceding examples can be performed for a host of interesting situations in which centripetal force is involved—a number of these are presented in this chapter’s
Problems and Exercises.
Ask a friend or relative to swing a golf club or a tennis racquet. Take appropriate measurements to estimate the centripetal acceleration of the end of the club or racquet. You may choose to do this
in slow motion.
Move the sun, earth, moon and space station to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies, and turn off
gravity to see what would happen without it!
Figure 4. Gravity and Orbits
Section Summary
which can also be expressed as
Conceptual Questions
Conceptual Questions
1: If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small-diameter tires? Explain.
2: Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force?
3: If centripetal force is directed toward the center, why do you feel that you are ‘thrown’ away from the center as a car goes around a curve? Explain.
4: Race car drivers routinely cut corners as shown in Figure 5. Explain how this allows the curve to be taken at the greatest speed.
Figure 5. Two paths around a race track curve are shown. Race car drivers will take the inside path (called cutting the corner) whenever possible because it allows them to take the curve at the
highest speed.
5: A number of amusement parks have rides that make vertical loops like the one shown in Figure 6. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car
goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if:
(a) The car goes over the top at faster than this speed?
(b)The car goes over the top at slower than this speed?
Figure 6. Amusement rides with a vertical loop are an example of a form of curved motion.
7: What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride pictured in Figure 6 under the following circumstances:
(a) The car goes over the top at such a speed that the gravitational force is the only force acting?
(b) The car goes over the top faster than this speed?
(c) The car goes over the top slower than this speed?
8: As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer.
9: Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it
and the merry-go-round. Which path shown in Figure 7 will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the
left, or curved to the right? Explain your answer.
Figure 7. A child riding on a merry-go-round releases her lunch box at point P. This is a view from above the clockwise rotation. Assuming it slides with negligible friction, will it follow path A,
B, or C, as viewed from Earth’s frame of reference? What will be the shape of the path it leaves in the dust on the merry-go-round?
10: Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car’s speed? What is the direction of the force exerted on you by the car seat?
11: Suppose a mass is moving in a circular path on a frictionless table as shown in figure. In the Earth’s frame of reference, there is no centrifugal force pulling the mass away from the centre of
rotation, yet there is a very real force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton’s third law, explain what force stretches the
string, identifying its physical origin.
Figure 8. A mass attached to a nail on a frictionless table moves in a circular path. The force stretching the string is real and not fictional. What is the physical origin of the force on the
Problems & Exercises
1: (a) A 22.0 kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to stay on if she is 1.25 m from its center?
(b) What centripetal force does she need to stay on an amusement park merry-go-round that rotates at 3.00 rev/min if she is 8.00 m from its center?
(c) Compare each force with her weight.
2: Calculate the centripetal force on the end of a 100 m (radius) wind turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg.
3: What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?
4: What is the ideal speed to take a 100 m radius curve banked at a 20.0° angle?
5: (a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked?
(b) Calculate the centripetal acceleration.
(c) Does this acceleration seem large to you?
6: Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 9. To be stable, the force exerted by the ground must be on a line going through the center of
gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which
must equal the system’s weight).
(a) Show that
(b) Calculate
Figure 9. A bicyclist negotiating a turn on level ground must lean at the correct angle—the ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through
the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component
must supply the centripetal force. This process produces a relationship among the angle θ, the speed v, and the radius of curvature r of the turn similar to that for the ideal banking of roadways.
7: A large centrifuge, like the one shown in Figure 10(a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries.
(a) At what angular velocity is the centripetal acceleration
(b) The rider’s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in Figure 10(b). At what angle
Figure 10. (a) NASA centrifuge used to subject trainees to accelerations similar to those experienced in rocket launches and reentries. (credit: NASA) (b) Rider in cage showing how the cage pivots
outward during rotation. This allows the total force exerted on the rider by the cage to be along its axis at all times.
8: Integrated Concepts
If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal
speed to take a 100 m radius curve banked at 15.0º. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h?
9: Modern roller coasters have vertical loops like the one shown in Figure 11. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top
will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. What is the speed of the roller coaster at the top of the loop if the radius of curvature
there is 15.0 m and the downward acceleration of the car is 1.50 g?
Figure 11. Teardrop-shaped loops are used in the latest roller coasters so that the radius of curvature gradually decreases to a minimum at the top. This means that the centripetal acceleration
builds from zero to a maximum at the top and gradually decreases again. A circular loop would cause a jolting change in acceleration at entry, a disadvantage discovered long ago in railroad curve
design. With a small radius of curvature at the top, the centripetal acceleration can more easily be kept greater than g so that the passengers do not lose contact with their seats nor do they need
seat belts to keep them in place.
10: Unreasonable Results
(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s.
(b) What is unreasonable about the result?
(c) Which premises are unreasonable or inconsistent?
centripetal force
any net force causing uniform circular motion
ideal banking
the sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net
external force on the vehicle equals the horizontal centripetal force in the absence of friction
ideal speed
the maximum safe speed at which a vehicle can turn on a curve without the aid of friction between the tire and the road
ideal angle
the angle at which a car can turn safely on a steep curve, which is in proportion to the ideal speed
banked curve
the curve in a road that is sloping in a manner that helps a vehicle negotiate the curve
Problems & Exercises
a) 483 N
b) 17.4 N
c) 2.24 times her weight, 0.0807 times her weight
a) 1.84
b) A coefficient of friction this much greater than 1 is unreasonable .
c) The assumed speed is too great for the tight curve. | {"url":"https://pressbooks.online.ucf.edu/phy2054ehk/chapter/centripetal-force/","timestamp":"2024-11-11T14:24:50Z","content_type":"text/html","content_length":"224920","record_id":"<urn:uuid:17aae53c-30b4-4d13-8904-b5243e0c24de>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00234.warc.gz"} |
Reductionism in Biology
1. James Griesemer (2000, 2002) argues for a heuristic use of reduction in the attempt by scientists to relate different scientific theories to one another. This is distinct from the sense of
methodological reductionism used here (i.e. it does not focus on how scientists discover molecular mechanisms and develop reductive explanations).
2. The issue of reduction has played a substantial role in both philosophy of mind and philosophy of social science. In the former, a central question is whether and in what sense mental phenomena
can be reduced to physical phenomena. Philosophers of mind have developed sophisticated accounts that concentrate on ontological reduction and mereological constitution (e.g. supervenience; see Kim
1998, 2005). In philosophy of social science, key questions include whether different fields of social science can be reduced to economics or whether social science as a whole can be reduced to
natural science (Kincaid 1997). Whether reduction is considered within philosophy of biology, mind, or social science, similar ideas and arguments have been used in these different fields (e.g.
multiple realization; see Section 4.2).
3. “Now it is true that the ultimate elements of organized matter are precisely those which enter into the composition of Unorganized substances: But by the operation of a power, distinct from
Gravitation, molecular attraction, or any of the known imponderable agents which operate on unorganized substances, these elements assume combinations of a character essentially different from those
which are the result of ordinary chemical affinities” (Owen, Hunterian Lectures [1837] in Sloan 1992, 209).
4. The mid-twentieth century separation of discussions about relations between domains of science (Nagelian theory reduction) and relations between parts and wholes (explanatory reduction), alongside
the older elements from debates about vitalism are concretely visible in David Hull's Philosophy of Biological Science (1974). Chapter 1 is a justly famous discussion regarding the difficulties of
effecting theory reduction between classical and molecular genetics, but chapter 5, “Organicism and Reductionism”, has been relatively neglected. It includes a very different set of themes, including
vitalism, organization, complexity, and the relations between parts and wholes.
5. Rosenberg's argument contains several controversial premises, such as the need for strict laws in biological explanation and the distinction between molecular and functional biology (for
criticisms, see Love 2007, Love et al. 2008). The intersection of natural selection and molecular biology for explanatory reductionism has been explored by Sarkar (2005 [1991]) with much less
6. In the debate about the relations between classical and molecular genetics, Philip Gasper (1992) dubs this “multiple supervenience” in order to contrast it with multiple realization. Biological
entities and processes (including molecular ones) often possess their causal properties extrinsically (i.e. due to features external to the bearer of the property). For example, a particular segment
of DNA is a gene as a consequence of its ability to code for a functional product, which depends on both DNA elements external to this gene and non-genetic factors. Being a gene is a property of a
DNA segment, but an extrinsic one.
7. The derivation demanded by theory reduction for molecular biology requires that the premises contain a purely molecular specification of the total context. Although Nagel and Schaffner have
explicated the notion of theory reduction, they have not actually shown how to effect the molecular deduction. Even Schaffner's most detailed defense of reduction (Schaffner 1993) focuses on the
condition of connectability, giving a partial account of how to relate the predicate “dominance” used in classical genetics to expressions from biochemistry. But he does not give a single example of
a higher level statement being deducible from some lower level theory, which reductionists would have to show, as theory reduction equals that the condition of derivability be met. Debate over
Schaffner's model has focused primarily on the condition of connectability, especially whether a molecular characterization of the notion of a classical gene is possible. Yet as Hull (1974) makes
plain, a critic of theory reduction needs no more than an argument against connectability as a necessary condition on reduction, while a reductionist has to defend derivability over and above
8. One of Schaffner's replies to Hull regarding the one-many objection is that the molecular context of a mechanism can be included in the premises to the extent that it matters. Wimsatt (1976a)
rightly points out that stressing the inclusion of context will always rely on how the mechanisms are individuated (see Section 4.3). Schaffner's move to include the total context implies that any
two possible situations will have distinct overall molecular configurations, but others view this as the same mechanism in different contexts. Generality requires treating parts of different
particular molecular configurations as instantiating the same type of mechanism. A flip-side of this need for generality is that the same type of mechanism can occur in different contexts, possibly
with a different effect. This sense in which a one–many relation between molecular and higher level kinds exists does not necessarily preclude some types of reduction but explanations in molecular
biology must to be sensitive to this epistemic issue.
9. Higher level laws can have exceptions, e.g. if M[d], one of the molecular kinds realizing S, does not cause any of the realizers N[1], …, N[j] of T. Waters (1990) has argued that explaining such
an exception requires appeal to the molecular level and therefore is a form of reductionism. Yet Fodor (1974) argued that a reductionistic approach encourages eliminating the higher level generality
in favor of exceptionless lower level laws, while the non-reductionist scientist accepts higher level laws despite exceptions precisely because of the aim to address both the higher and lower level
10. The notions of multiple realization and supervenience were originally developed in philosophy of mind as part of a non-reductive physicalism about mental phenomena. Whether these concepts offer
an adequate account of the relation between mental and physical phenomena has been questioned (Kim 1992, 1998). The status of multiple realization in philosophy is currently a subject of debate and
may hinge on certain metaphysical commitments about the concept of realization (Gillett 2002, 2003). “Realization” takes on different roles in scientific and philosophical discussions (Wilson and
Craver 2007) and the significance of these differences has not yet been analyzed for the situations of multiple realization frequently discussed in philosophy of biology. Biological examples used in
these philosophical discussions, such as eyes, have often not been treated with adequate care and precision (Couch 2005).
11. A less common objection concerns whether the domains being reductively related are actually distinct. For example, Russell Vance (1996) has argued against the premise that classical genetics and
molecular genetics are separate fields or theories (e.g., “genetics is the area of biological science that seems most immediately amenable to the philosopher's conception of scientific reduction. The
main reason for this is that in genetics there are two clearly identifiable theories, bodies of law, expressions of regularity – one molecular and one nonmolecular” [Rosenberg 1985, 90]). While this
undermines any attempt at reductively relating classical genetics and molecular genetics, its more general impact can be understood as a caution against presuming that two areas of biology are
straightforwardly distinguished (as is often done in theory reduction). One cannot simply assume an individuation of the epistemic units being related reductively.
12. This does not mean that “mechanisms” analyses fully capture all of the features relevant to explanatory reduction already discussed. For example, discussions of temporal organization by
mechanisms proponents focus on the representation of time relative to a mechanism (e.g., Bechtel 2006, Machamer et al. 2000). But some reductive explanations in ontogeny rely on temporal
representations that are not keyed to the temporal organization of mechanisms (e.g., normal stages). Thus, the issue of temporality in reductive explanations cannot be reduced to the issue of
temporality in a mechanism. Additionally, qualitative change in ontogeny, where novel structures emerge out of distinct developmental precursors, does not appear straightforwardly amenable to a
mechanisms approach.
13. E.g.: “The heart provides an excellent example both of the huge advances that have been made using the reductive approach and of the severe limits of this approach” (Noble 1998, 56); “Our results
suggest that the cellular responses induced by multiplex protein kinase inhibitors may be an emergent property that cannot be understood fully considering only the sum of individual inhibitor-kinase
interactions” (Kung et al. 2005, 3587); “Our results therefore point to the need to consider each complex biological network as a whole, instead of focusing on local properties” (Guimerà and Nunes
Amaral 2005, 899); “Robustness … is one of the fundamental and ubiquitously observed systems-level phenomena that cannot be understood by looking at individual components” (Kitano 2004, 826); “The
molecular reductionism that dominated twentieth-century biology will be superseded by an interdisciplinary approach that embraces collective phenomena” (Goldenfeld and Woese 2007, 369);
“Understanding the dynamics of infectious-disease transmission demands a holistic approach, yet today's models largely ignore how epidemics change individual behaviour” (Ferguson 2007, 733); “The
collective behaviour of matter can give rise to startling emergent properties that hint at the nexus between biology and physics” (Coleman 2007, 379).
14. “The formalization of supervenience employed here is the application of one due to Jaegwon Kim” (Rosenberg 1978, 373). Referring to Kim's paper “Nomological incommensurables” (read at Oberlin
Colloquium in 1977), Rosenberg continues in a footnote: “My debts to this paper are by no means limited to its account of supervenience. In fact, the whole perspective adopted here with respect to
the relation between theories involved in reduction, and the bearing of causal and mereological determinism on this relation are inspired by this paper. … The reader is referred to Kim's paper for a
fuller account of supervenience and its role in ethics, aesthetics, and the philosophy of mind in general.” | {"url":"https://plato.stanford.edu/ARCHIVES/WIN2009/entries/reduction-biology/notes.html","timestamp":"2024-11-02T11:17:38Z","content_type":"application/xhtml+xml","content_length":"16307","record_id":"<urn:uuid:3cdb2c68-a895-44e6-abd6-f68014655037>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00253.warc.gz"} |
Coverage Ratio Calculator | Analyze Financial Health
Coverage Ratio Calculator
What is a Coverage Ratio?
A coverage ratio is a financial metric used to measure a company's ability to meet its financial obligations, particularly its debt and interest payments. It compares the company's income or cash
flow to its debt obligations. A higher coverage ratio indicates better financial health, as it suggests the company can more easily service its debt.
The Coverage Ratio Formula
The general formula for a coverage ratio is:
\[\text{Coverage Ratio} = \frac{\text{Income or Cash Flow}}{\text{Debt Obligations}}\]
In our specific calculator, we use:
\[\text{Coverage Ratio} = \frac{\text{Gross Profit}}{\text{Annual Interest & Service Charge}}\]
• Gross Profit is the company's total revenue minus the cost of goods sold
• Annual Interest & Service Charge represents the company's debt obligations for the year
Step-by-Step Coverage Ratio Calculation
1. Determine the company's Gross Profit for the period.
2. Calculate the total Annual Interest & Service Charge for the same period.
3. Divide the Gross Profit by the Annual Interest & Service Charge.
4. The result is the Coverage Ratio.
Example Calculation
Let's calculate the coverage ratio for a company with a Gross Profit of $500,000 and Annual Interest & Service Charges of $100,000:
1. Gross Profit = $500,000
2. Annual Interest & Service Charge = $100,000
3. Coverage Ratio = $500,000 / $100,000 = 5
This means the company's gross profit is 5 times its annual debt obligations, indicating strong financial health.
Visual Representation
In this diagram, the green bar represents the coverage ratio of 5.0. The red line marks the breakeven point (1.0). A ratio extending well beyond this line, as shown here, indicates strong financial | {"url":"https://www.thinkcalculator.com/investment/coverage-ratio-calculator.php","timestamp":"2024-11-08T21:25:13Z","content_type":"text/html","content_length":"32756","record_id":"<urn:uuid:f1c86486-79f8-43da-ba98-ef5e8c26dab6>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00578.warc.gz"} |
Precedence of Objects (GNU Octave)
34.4.3 Precedence of Objects
Many functions and operators take two or more arguments and the situation can easily arise where these functions are called with objects of different classes. It is therefore necessary to determine
the precedence of which method from which class to call when there are mixed objects given to a function or operator. To do this the superiorto and inferiorto functions can be used
When called from a class constructor, mark the object currently constructed as having a higher precedence than class_name.
More that one such class can be specified in a single call. This function may only be called from a class constructor.
See also: inferiorto.
When called from a class constructor, mark the object currently constructed as having a lower precedence than class_name.
More that one such class can be specified in a single call. This function may only be called from a class constructor.
See also: superiorto.
With the polynomial class, consider the case
2 * polynomial ([1, 0, 1]);
that mixes an object of the class "double" with an object of the class "polynomial". In this case the return type should be "polynomial" and so the superiorto function is used in the class
constructor. In particular the polynomial class constructor would be modified to
## -*- texinfo -*-
## @deftypefn {} {} polynomial ()
## @deftypefnx {} {} polynomial (@var{a})
## Create a polynomial object representing the polynomial
## @example
## a0 + a1 * x + a2 * x^2 + @dots{} + an * x^n
## @end example
## @noindent
## from a vector of coefficients [a0 a1 a2 @dots{} an].
## @end deftypefn
function p = polynomial (a)
if (nargin > 1)
print_usage ();
if (nargin == 0)
p.poly = [0];
p = class (p, "polynomial");
if (strcmp (class (a), "polynomial"))
p = a;
elseif (isreal (a) && isvector (a))
p.poly = a(:).'; # force row vector
p = class (p, "polynomial");
error ("@polynomial: A must be a real vector");
superiorto ("double");
Note that user classes always have higher precedence than built-in Octave types. Thus, marking the polynomial class higher than the "double" class is not actually necessary.
When confronted with two objects of equal precedence, Octave will use the method of the object that appears first in the list of arguments. | {"url":"https://docs.octave.org/v4.2.2/Precedence-of-Objects.html","timestamp":"2024-11-06T05:09:44Z","content_type":"text/html","content_length":"6917","record_id":"<urn:uuid:045d2afe-250b-4297-b45f-d1c9ec6834fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00079.warc.gz"} |
Unlocking Math Mastery – A Guide to Kumon Answer Book Level F
The moment I saw the daunting equation on the page, a wave of anxiety washed over me. My daughter, a bright and curious child, was struggling with Kumon Level F math. We had just started the level,
and the complexity of the problems was a drastic shift from the previous levels. Seeing her frustration, I knew I needed to be more proactive in her learning. I started researching the curriculum,
and that’s when I stumbled upon the invaluable resource: Kumon answer books.
Image: dinisvoltbour.weebly.com
Armed with these books, I felt a sense of confidence in helping my daughter understand the concepts. The clear explanations and step-by-step solutions provided a roadmap for tackling even the
trickiest problems. We worked through the answers together, breaking down each step, and I noticed a remarkable improvement in her confidence and understanding. This experience was a turning point,
highlighting the importance of having the right tools and strategies for navigating the challenges of Kumon Level F math.
What is the Kumon Answer Book Level F Math?
The Kumon answer book for Level F math is a supplementary resource designed to support students enrolled in the Kumon Math Program. It contains detailed and step-by-step solutions for all the
problems found in the Kumon Level F worksheets. This book is an essential tool for both students and parents, aiding in understanding complex mathematical concepts and reinforcing learning.
The answer book is not merely a collection of answers; it provides a comprehensive breakdown of each problem’s solution, outlining the relevant mathematical principles and demonstrating the logic
behind each step. This meticulous approach offers valuable insight, allowing students to grasp the underlying concepts rather than simply memorizing solutions. In essence, the answer book is a guide
to unlocking the reasoning behind each problem, ultimately empowering students to solve similar problems independently.
Exploring the Content of Kumon Answer Book Level F Math
Kumon Level F math is a pivotal step in the program, covering a wide range of topics that build upon earlier learning. The answer book serves as a comprehensive guide, providing insights into various
areas of mathematics, including:
• Fractions and Decimals: Level F delves into advanced operations involving fractions and decimals, including addition, subtraction, multiplication, and division. Students learn to convert between
fractions and decimals, simplifying expressions and solving equations involving fractions.
• Algebraic Concepts: The answer book introduces elementary algebraic concepts such as variables, expressions, and basic equations. Students will learn to solve simple linear equations and
understand the principles of substitution and simplification.
• Geometry: Level F introduces fundamental geometric concepts like area and perimeter, calculating these measurements for various shapes like triangles, rectangles, and squares. Students learn to
apply formulas and reasoning to solve problems involving geometric figures.
• Word Problems: A significant part of Level F focuses on solving word problems, requiring students to translate real-world scenarios into mathematical expressions. The answer book helps students
grasp the art of identifying keywords, setting up equations, and solving for unknown variables within a given context.
• Number Theory: The answer book delves into foundational concepts of number theory, including factors, multiples, prime numbers, and composite numbers. Students learn to identify prime numbers,
find factors and multiples, and solve problems involving divisibility rules.
Leveraging the Kumon Answer Book for Effective Learning
The Kumon answer book, when used effectively, becomes a powerful learning tool, but it’s crucial to understand how to use it to maximize its benefits.
Image: zokarekake.tistory.com
Strategies for Effective Use
• Independent Problem-Solving: Encourage students to attempt problems independently before referring to the answers. This fosters self-reliance and develops problem-solving skills.
• Understanding the Process: Don’t simply focus on the final answer. Guide students through the steps outlined in the answer book, emphasizing the logic and reasoning behind each calculation.
• Practice and Reiteration: Use the answer book as a tool for identifying areas where the student needs more practice. Encourage repetition of similar problems until mastery is achieved.
• Parent Involvement: Parents can play a crucial role by working alongside their children, explaining concepts and discussing challenging problems. This fosters a collaborative learning
Expert Tips for Navigating Kumon Level F Math
Using the answer book effectively is one piece of the puzzle. Here are some additional tips from experienced Kumon educators:
• Consistency is Key: Dedicate regular time to practice, even if it’s just for a short period every day. This helps maintain momentum and solidify understanding.
• Focus on Foundations: If a student struggles with a particular concept, review the foundational material from earlier levels. Building a strong base is essential for success in Level F.
• Embrace Mistakes: Mistakes are valuable learning opportunities. Don’t be discouraged by them; analyze them together, focusing on the steps leading to the error.
Q&A: Frequently Asked Questions about Kumon Answer Book Level F Math
Here are some frequently asked questions about the Kumon Answer Book Level F Math:
1. Q: Is the answer book meant to be used independently by students?
A: The answer book is designed to be used as a supplementary resource, not a replacement for independent problem-solving. Encourage your child to attempt problems independently before referring
to the solutions.
2. Q: How can I use the answer book to help my child overcome difficulties?
A: When your child encounters difficulties, work through the steps provided in the answer book together. Explain the reasoning behind each step and encourage them to apply the same logic to
similar problems.
3. Q: My child is struggling with a particular topic. What should I do?
A: Review the relevant section in the answer book and work through the problems together. If the difficulty persists, consider seeking guidance from the Kumon instructor.
4. Q: Are there any alternative resources available?
A: Yes, there are other resources such as online tutorials, educational websites, and additional practice books that can complement the Kumon curriculum.
5. Q: Is there a correlation between using the answer book and student success?
A: The answer book can significantly aid in student success if used strategically as a learning tool rather than a shortcut for solving problems.
Kumon Answer Book Level F Math
The Kumon Answer Book Level F Math provides invaluable support to students enrolled in the Kumon program, providing detailed solutions and explanations to help them understand and apply mathematical
concepts. Through its comprehensive approach, it promotes a deeper understanding of various mathematical areas, fostering confidence and enhancing problem-solving abilities. However, remember that
the answer book is a tool to be used wisely; encourage independent problem-solving, focus on the process, and embrace the learning journey!
Are you using the Kumon Answer Book Level F Math to help your child succeed? Share your experiences and tips in the comments below. | {"url":"https://www.tapeseals.com/698.html","timestamp":"2024-11-04T08:06:11Z","content_type":"text/html","content_length":"109708","record_id":"<urn:uuid:ca61e5e8-da03-4f3f-8ee8-4df55951a1d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00027.warc.gz"} |
Tail variance and confidence of using tail conditional expectation: analytical representation, capital adequacy, and asymptotics | Advances in Applied Probability | Cambridge Core
1. Introduction
Studies on the tails of risks have always been the core research in finance and actuarial science. Regulators, investors, and other stakeholders are often interested in the risk’s tail region beyond
a probability level $p$ ( $p$ is close to 1) and rely on risk measures defined on such a tail region, namely, $\left[ {p,1} \right],$ to assess the potential loss of the risk. For instance, Solvency
II regulation accounts for the insurance risk based on Value-at-Risk (VaR) at $p = 99.5{\rm{\% }}$ over a one-year period while Basel III tends to use Expected Shortfall (ES) as the measure of
various risks for “expected losses” subject to the stressed region $\left[ {p,1} \right]$ [13]. These risk measures defined on tails not only suggest the amount of capital adequacy for financial
institutions under potential risks but also provide insights into understanding risks’ behaviors at tails. There are also other well-known risk measures defined on tails such as Conditional
Value-at-Risk (CVaR), Tail Conditional Expectation (TCE, also known as CTE, i.e. Conditional Tail Expectation), etc. It has been shown that the three risk measures (ES, CVaR, and TCE), which are
coinciding for continuous risks, satisfy the so-called “coherence”; see [Reference Artzner, Delbaen, Eber and Heath4]. Coherence requires risk measures to satisfy a number of desirable properties
such as monotonicity, subadditivity, positive homogeneity, translation invariance, etc. These properties bring mathematical superiority to coherent risk measures and make them relatively more robust
in assessing risks at tails. By contrast, VaR is not coherent, but it has advantages in practical use; for instance, it is more straightforward for back-testing. We refer to [Reference Emmer, Kratz
and Tasche14] for a study on the comparison between VaR and ES and to [Reference Denuit, Dhaene, Goovaerts and Kaas11] for more details on the coherent risk measure.
Despite the popularity of the aforementioned risk measures, they often appear to be insufficient in exploiting the information about the riskiness at the tail. [Reference Furman and Landsman16] gave
an example to illustrate such insufficiency and proposes the Tail Variance (TV) and the Tail Variance premium (TVp) to further account for the tail riskiness. In particular, in this work, TV is
defined as the conditional variance of a risk on the tail region $\left[ {p,1} \right],$ while TVp is the well-known mean-variance premium in which the mean and the variance are replaced by TCE and
TV. The authors also derive the formulas of TV and TVp for elliptically distributed risks. In a later work, by [Reference Landsman, Pat and Dhaene27], the formulas of TV and a TV-based capital
allocation rule were derived for log-elliptical risks. [Reference Ignatieva and Landsman22] conducted similar research for symmetric generalized hyperbolic risks and pointed out that these formulas
of TV and TVp can estimate the loss severity at tails. Intuitively speaking, in the same manner that variance measures the dispersion of a risk from its mean, TV accounts for the magnitude of the
risk deviating from its TCE on the chosen tail region. Specifically, as a measure of tail variability, TV satisfies properties such as law-invariance, standardization, and location invariance.
Furthermore, [Reference Furman, Wang and Zitikis17] discussed the tail standard deviation ( $\sqrt {{\rm{TV}}} $ ) on the measurement of tail variability. [Reference Hu and Chen19] defined the
coherent measures on the tail variability; in fact, $\sqrt {{\rm{TV}}} $ is such a coherent measure for tail variability. [Reference Bellini, Fadina, Wang and Wei7] studied the parametric estimation
on the variability. Therefore, studying TV as a risk measure, in addition to studying VaR and TCE, are of importance and interest in several aspects.
First, TV provides additional information on the riskiness of tails. Many existing results in the literature based on VaR and TCE can be extended straightforwardly by further taking TV into account.
For instance, [Reference Landsman26] considered a TCE–TV framework as an extension of the classic mean-variance framework at tails. Second, TV measures how far a risk deviates from its TCE.
Therefore, when TCE is used to estimate the risk capital, TV can be used to measure the confidence level of using the TCE-based risk capital. In particular, by developing an analogue of the
well-known Chebyshev inequality at the tail region, one can find an interval regarding how far the actual loss may differ from the TCE-based risk capital for a given confidence level. Third, unlike
VaR and TCE, which both increase to infinity when $p$ approaches 1, the limit of TV along $p \to 1$ shows a certain tendency that may converge to 0, a positive constant, or infinity. This phenomenon
was first noticed by [Reference Ignatieva and Landsman22], in which the authors showed that ${\rm{TV}} \to \infty $ for Student-t distributions while ${\rm{TV}} \to 0$ for normal distributions. Thus,
it is of interest to study the asymptotics of TV, which may characterize the tail behaviors of the risk and provide further insights into the confidence level of using TCE-based capital under extreme
In this paper, we investigate using TV to account for the confidence of using TCE to assess tail riskiness in the context of absolutely continuous risks. Our findings contribute to the literature in
three ways. First, we provide a list of analytical TV formulas for a large variety of probability distributions, including a Pareto distribution, a Weibull distribution, a normal distribution, etc.
As a necessary step in calculating TV, we also provide explicit formulas of TCE for these probability distributions. These formulas not only cover the existing results in the literature as special
cases but also are useful instruments for relevant research works, such as the TV premium calculation [Reference Furman and Landsman16], optimal capital allocation [Reference Xu and Mao38], loss
severity estimation [Reference Ignatieva and Landsman22], optimal portfolio selection [Reference Eini and Khaloozadeh12, Reference Landsman26], etc.
Secondly, we propose a novel approach to assessing the confidence level of using TCE-based risk capital from multiple angles. By virtue of the appropriateness of risk capital defined in [Reference
Fröhlich and Weng15], we introduce sharpened conditional probability inequalities based on TV, which offers a significant advancement by halving the confidence bounds of TCE-based risk capital. More
specifically, we first propose a sharpened conditional Markov inequality that gives a lower bound of the confidence level of using the TCE-based risk capital. The lower bound could possibly be
further improved by using a sharpened conditional Cantelli inequality. Then we present a sharpened conditional Chebyshev inequality to offer a more precise confidence interval within which possible
extreme loss values may fall. Our results offer practical and accessible tools for practitioners and regulators alike to evaluate capital adequacy. To illustrate the usefulness of our approach, we
present a case study based on real data from two insurance business lines. Overall, our findings demonstrate the effectiveness of our proposed method for assessing the confidence of using TCE-based
risk capital and provide valuable insights for risk management practices.
Third, we study the asymptotic properties of TV. We first give a distribution-free limit of the TV when $p \to 1$ and show that, for an absolutely continuous random variable, the limit value of TV
coincides with the limit value of the reciprocal square of the hazard rate. In other words, TV gives similar information as the hazard rate does but at tail regions. To derive explicit forms, we then
consider two large distribution families defined by the tails, namely, the distribution family of exponential tails and the distribution family of polynomial tails. We establish explicit asymptotic
formulas of TVs for the two distribution families and further show that there exist exact asymptotic equivalences between the TV and the reciprocal square of the hazard rate. In doing so, we also
derive formulas of asymptotic equivalences between TCE and VaR in the context of the two distribution families. Note that the result of asymptotic equivalences is much stronger than equal limits. The
asymptotic equivalence not only guarantees the same limits but also implies the same convergence rate. There are many papers discussing TV asymptotic formulas: for instance, [Reference Ignatieva and
Landsman22] and [Reference Li28]. Our work on TV not only unifies the TV asymptotic formulas discussed on specific distribution assumptions but also points out the convergence speed that previous
literature failed to give. We then provide a categorization of the tails of risks based on TV’s asymptotics. As an application, we show that such a categorization gives new evidence to the
relationship between the financial data frequency and the tail types of risks.
We organize the rest of the paper as follows: Section 2 derives the formulas of TV and TCE for frequently used probability distributions. Section 3 presents the two applications with numerical case
studies. Section 4 investigates the asymptotic behavior of TV and establishes an asymptotic relationship between the TV and the hazard rate. Section 5 concludes the paper.
2. Formulas of Tail Variance and Tail Conditional Expectation in Assessing Tail Risks
2.1. Definitions and notations
We first introduce the following notations and definitions for our paper: In this paper, $X$ is an absolutely continuous random variable defined on a support with an unbounded supremum. We denote its
probability density function (PDF) and cumulative distribution function (CDF) as $f(x)$ and $F(x),$ respectively. Furthermore, we have $\overline{F}(x) = 1 - F(x)$ , the tail distribution function.
In actuarial science or reliability theory, $\overline{F}(x)$ is also referred to as the survival function or the reliability function. The hazard rate of $X$ (notation $h(x))$ , defined as the ratio
of density function $f(x)$ and survival function $\overline{F}(x)$ $\Big($ i.e. $h(x) = \frac{{f(x)}}{{\overline{F}(x)}}\Big)$ is also widely used as a characteristic of $X$ . The hazard rate often
refers to the force of mortality or the failure rate of the risk in actuarial science and reliability engineering. $h(x)$ is also often involved in discussing heavy-tailed and light-tailed
distributions in the literature; see [Reference Klugman, Panjer and Willmot24, Reference Su and Tang35] and references therein.
A risk measure is a functional mapping of $X$ to a real number, possibly infinite, that reflects the riskiness of $X$ . It suggests an amount of capital that should be added to the risk such that it
is acceptable to stakeholders. Such an amount of risk capital should be adequate to cover the possible loss of the risk in the sense of the corresponding risk measure. In our context, the quantile
function ${Q_q}[X]\,:\!=\, {\rm{inf}}\!\left\{ {x,F(x) \geq q} \right\}$ , $q \in \left( {0,1} \right),$ is the well-known risk measure Value-at-Risk. Then the Tail Conditional Expectation of $X$ is
defined as
(1) {\rm{TC}}{{\rm{E}}_q}[X] = \mathbb{E}\!\left[ {X | X \gt {x_q}} \right],
where ${x_q} = {Q_q}[X]$ . [Reference Furman and Landsman16] introduces TV as follows:
(2) {\rm{T}}{{\rm{V}}_q}[X] = {\rm{Var}} \big[ X| X \gt {x_q}\big] = \mathbb{E} \big[{X^2}| X \gt {x_q} \big] - {\mathbb{E}^2} \big[ {X| X \gt {x_q}}\big]
and points out that ${\rm{T}}{{\rm{V}}_q}[X]$ has the following property:
\begin{align*}{\rm{T}}{{\rm{V}}_q}[X] = \mathop {{\rm{inf}}}\limits_c \mathbb{E} \big[ {{{\left( {X - c} \right)}^2}\left| X \right\gt {x_q}}\big].\end{align*}
TV, as a risk measure, actually provides additional information on the riskiness of distribution at tails, especially when the Tail Conditional Expectation fails to distinguish the risks. In this
section, we first derive the formulas of TV for frequently used distributions. Naturally, we also derive the formulas of TCE in the process, which is required when deriving TV formulas. Using these
formulas, we present an example at the end of this section to illustrate the insufficiency of TCE in identifying the tail riskiness.
2.2. Explicit Tail variance and Tail Conditional Expectation formulas for frequently used distributions
Many papers have endeavored to derive the formulas of TV and TCE for specific distributions in the literature. [Reference Furman and Landsman16] did such work in the context of the normal
distribution and the elliptical distribution. In addition, the TV formula for the symmetric generalized hyperbolic distributions was derived in [Reference Ignatieva and Landsman22]. Furthermore, the
TV formula for the generalized hyperbolic distribution has been proved in [Reference Kim and Kim23]. Given the increasing popularity of TV and capital allocation based on a Tail Variance premium, we
provide a list of TV formulas for frequently used continuous probability distributions. Our list covers all the aforementioned results in the literature and provides many new TV formulas for other
popular distributions. Thus, the list can be used as an instrumental reference for interested researchers in their relevant works. Note that according to Formula (2), we also derive the formulas of
TCE for these distributions to obtain the formulas of TV. The full list of formulas is presented in the Appendix.
3. The Confidence of Using TCE-based Risk Capital
Both VaR and TCE have been suggested as risk measures to estimate the risk capital for a given risk [13]. Despite its popularity in risk management, VaR is criticized because it is defined by an
extreme event, yet “thousand-year events happen on a regular basis,” and it is not sub-additive; hence, unlike TCE, it is not a coherent risk measure. Using VaR to estimate the risk capital later
leads to trouble when the extreme event occurs, i.e. $X \gt {x_q}$ [Reference Putnam, Sykes Wilford and Zecher32]. Particularly, an estimated risk capital $C$ is said to be appropriate at $q$ level
of confidence if
(3) \mathbb{P}\!\left( {X \le C} \right) = q
in [Reference Fröhlich and Weng15]. It is then evident that using VaR as the risk capital is at $q$ confidence level. By contrast with VaR, TCE takes all possible outcomes of the event $X \gt {x_q}$
into account and calculates the “expected loss” of the risk under $X \gt {x_q}$ . Ever since the seminal work of [Reference Acerbi and Tasche1], TCE has been advocated for the evaluation of risk
capital from a regulatory point of view [6]. Similarly, it is of interest and importance to investigate, when the extreme event $X \gt {x_q}$ happens, whether TCE-based capital is enough to cover the
loss or what the confidence level of TCE-based risk capital is; i.e. if we take $C$ as TCE of $X$ , how can we estimate the appropriateness in Formula (3)? Note that appropriateness here is defined
by the adequacy of risk capital. As we are primarily interested in the distribution-free capital adequacy of TCE, instead of appropriateness, we say that the confidence level of using $C$ (based on
VaR or TCE) is $q$ instead of appropriateness.
With the aid of TV, we answer this question from three aspects in a distribution-free way: First, we develop a sharpened conditional Markov inequality to give a lower bound of the confidence level of
using TCE-based risk capital. Second, we introduce the conditional Cantelli inequality in conjunction with the sharpened conditional Markov inequality to give an improved lower bound of the
confidence level of using TCE-based risk capital. Third, we propose a sharpened conditional Chebyshev inequality to give a confidence interval of using TCE-based capital. Such an interval is the
range within which the extreme values—i.e. the losses when the event $X \gt {x_q}$ occurs—are likely to fall. Note that this interval differs from the usual confidence interval in statistical
estimation. A confidence interval using TCE-based capital at a 95% level is a range that $X$ falls into with at least 95% probability, given that $X \gt {x_q}$ . Both the sharpened conditional Markov
inequality and the sharpened conditional Chebyshev inequality halve the bounds compared with the original bounds, respectively.
Our approach to assessing the confidence of using TCE-based risk capital is handy and easy to implement by regulators and financial institutes. In particular, regulators aim to protect the stability
of markets and financial systems from extreme events, which coincides with the nature of the tail conditional inequalities that give lower bounds to the confidence of TCE-based capital. Moreover, our
approach is distribution-free. As such, it provides a fast and easy-to-understand solution that may be favorable for regulators. In fact, a similar approach has been proposed by [Reference Koehn and
Santomero25], where the authors used a Chebyshev inequality to investigate bank capital regulation and bankruptcy as well as the riskiness of the bank portfolio in their models.
In the sequel of our paper, we assume that the PDF of $Y = X | X \gt {x_q}$ is non-increasing on its support and that ${x_q}$ is nonnegative. These assumptions are very mild, as they hold true for
the majority of distributions. As a matter of fact, when one performs asymptotic analysis, $X$ is usually assumed to be eventually monotone in the literature, such as the monotone density theorem in
regular variation theory.
3.1. The sharpened conditional Markov inequality
The well-known Markov inequality states that if $X$ is a nonnegative random variable and if $t \gt 0$ , then $\mathbb{P}\!\left( {X \geq t} \right) \le \frac{{\mathbb{E}[X]}}{t}$ . Applying this
inequality directly to the tail random variable $Y = X | X \gt {x_q}$ gives
\begin{align*}\mathbb{P}\!\left( {Y \geq t} \right) \le \frac{{{\rm{TC}}{{\rm{E}}_q}[X]}}{t}.\end{align*}
If $t = {\rm{TC}}{{\rm{E}}_q}[X]$ , the conditional Markov inequality is trivial, as $\mathbb{P}\!\left( {Y \geq {\rm{TC}}{{\rm{E}}_q}[X]} \right) \le 1$ . Therefore, we propose a sharpened
conditional Markov inequality, which halves the bound of the original Markov inequality for tail risks. The proof is given in the Appendix.
Proposition 1. (Sharpened Conditional Markov Inequality) For a random variable $X$ , if ${x_q} \geq 0$ and $Y = X | X \gt {x_q}$ has a non-increasing PDF, the following inequalities hold for $t \gt
{x_q}$ :
\begin{align*}\mathbb{P}\!\left( {Y \geq t} \right) \le \frac{{{\rm{TC}}{{\rm{E}}_q}[X]}}{{2t}}\ and\ \mathbb{P}\!\left( {X \geq t} \right) \le \left( {1 - q} \right)\frac{{{\rm{TC}}{{\rm{E}}_q}[X]}}
Remark 1. When $t = {\rm{TC}}{{\rm{E}}_q}[X]$ , the sharpened conditional Markov inequality is
\begin{align*}\mathbb{P}\!\left( {X \le {\rm{TC}}{{\rm{E}}_q}[X]} \right) \geq \left( {q + 1} \right)/2.\end{align*}
Hence, we can assert that the TCE-based risk capital is always over a $\left( {q + 1} \right)/2$ confidence level whatever the model is. Therefore, for the same $q$ , ${\rm{TC}}{{\rm{E}}_q}$ is at a
higher confidence level in calculating the risk capital than ${\rm{Va}}{{\rm{R}}_q}$ . To reach the confidence level of ${\rm{Va}}{{\rm{R}}_p}$ , one may consider ${\rm{TC}}{{\rm{E}}_{\left( {2{*}p -
1} \right)}}$ . For example, ${\rm{TC}}{{\rm{E}}_{0.98}}$ is no lower than ${\rm{Va}}{{\rm{R}}_{0.99}}$ . This is slightly higher than the Basel level for the TCE (0.975, [5]) bank; however, the
bound here is the worst-case one.
3.2. The sharpened conditional Cantelli inequality
To further incorporate TV when assessing the confidence level of TCE, we develop a conditional Cantelli inequality. The classic Cantelli inequality, which is an improved version of the one-sided
Chebyshev inequality, states that for a constant $t \gt 0$ ,
\begin{align*}\mathbb{P}\!\left( {X - \mathbb{E}[X] \geq t} \right) \le \frac{{{\rm{Var}}[X]}}{{{\rm{Var}}[X] + {t^2}}}.\end{align*}
Similarly, we have a sharpened conditional Cantelli inequality, which halves the bound of the classic Cantelli inequality. The random variables $X$ and $Y = X | X \gt {x_q}$ , as defined in
Proposition 1. It holds for $t \gt 0$ that
(4) \mathbb{P}\!\left( {Y \geq {\rm{TC}}{{\rm{E}}_q}[X] + t} \right) \le \frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{2\!\left( {{\rm{T}}{{\rm{V}}_q}[X] + {t^2}} \right)}}.
In particular, if we may take $t$ as a proportion of ${\rm{TC}}{{\rm{E}}_q}[X]$ , i.e. $t = \lambda {\rm{TC}}{{\rm{E}}_q}[X]$ , where $\lambda \geq 0$ , then we have
(5) \mathbb{P}\!\left( {Y \geq \left( {1 + \lambda } \right){\rm{TC}}{{\rm{E}}_q}[X]} \right) \le \frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{2\!\left( {{\rm{T}}{{\rm{V}}_q}[X] + {{\left( {\lambda {\rm{TC}}{{\
rm{E}}_q}[X]} \right)}^2}} \right)}}.
Corollary 1. (Improved lower bounds of confidence level of using TCE-based risk capital) For a random variable $X$ , if ${x_q} \geq 0$ and $Y = X | X \gt {x_q}$ has a non-increasing PDF on its
support, we have the following inequality: For $\lambda \geq 0$ ,
(6) \mathbb{P} ( Y & \le \left( {1 + \lambda } \right){\rm{TC}}{{\rm{E}}_q}[X] )\nonumber\\[2pt] & \geq 1 - {\rm{min}}\!\left( {\frac{1}{{2\!\left( {1 + \lambda } \right)}},\frac{{{\rm{T}}{{\rm{V}}
_q}[X]}}{{2\!\left( {{\rm{T}}{{\rm{V}}_q}[X] + {{\left( {\lambda {\rm{TC}}{{\rm{E}}_q}[X]} \right)}^2}} \right)}}} \right),
(7) \mathbb{P} ( X & \le \left( {1 + \lambda } \right){\rm{TC}}{{\rm{E}}_q}[X] )\nonumber\\[2pt] & \geq 1 - \left( {1 - q} \right){\rm{min}}\!\left( {\frac{1}{{2\!\left( {1 + \lambda } \right)}},\
frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{2\!\left( {{\rm{T}}{{\rm{V}}_q}[X] + {{\left( {\lambda {\rm{TC}}{{\rm{E}}_q}[X]} \right)}^2}} \right)}}} \right).
Remark 2. The sharpened conditional Cantelli inequality, Formula (4), is obtained directly from the sharpened conditional Markov inequality using the same proof method of the classic Cantelli
inequality; see 6.1.e in [Reference Lin29] for details. Corollary 1 combines the sharpened conditional Markov and Cantelli inequalities for a more precise tail risk estimation. As mentioned by [
Reference Alsmeyer2], while these inequalities provide rough estimates, their usefulness lies in their applicability to any random variable with finite variance. Consequently, in practical scenarios,
especially for continuous random variables, these inequalities are typically strict, with equality attainable only in specific contexts.
3.3. A distribution-free confidence interval using TCE-based capital
Furthermore, the conditional Chebyshev inequality can give a confidence interval in which extreme losses are likely to fall within the tail. We also present a sharpened conditional Chebyshev
inequality to give a more precise interval.
The classic Chebyshev inequality states that if $X$ is a random variable with finite expectation and finite variance, then for any $t \gt 1$ , we have
\begin{align*}\mathbb{P}\!\left( {\left| {X - \mathbb{E}[X]} \right| \geq t\sqrt {{\rm{Var}}[X]} } \right) \le {t^{ - 2}}.\end{align*}
Then for tail distributions, by analogy, there is the following conditional Chebyshev inequality straightforwardly. For a random variable $X$ , define $Y\,:\!=\, X | X \gt {x_q}$ . The following
inequality holds:
(8) \mathbb{P}\!\left( {\left| {Y - {\rm{TC}}{{\rm{E}}_q}[X]} \right| \le t\sqrt {{\rm{T}}{{\rm{V}}_q}[X]} } \right) \geq 1 - {t^{ - 2}},
where $t \gt 1$ and $0 \lt q \lt 1$ .
The conditional Chebyshev inequality, Formula (8), implies the confidence interval that $Y$ deviates from ${\rm{TC}}{{\rm{E}}_q}[X]$ . Such an interval at the confidence level $1 - {t^{ - 2}}$ is $\
left[ {{\rm{max}}\!\left( {{x_q},{\rm{TC}}{{\rm{E}}_q}[X] - t\sqrt {{\rm{T}}{{\rm{V}}_q}[X]} } \right),{\rm{TC}}{{\rm{E}}_q}[X] + t\sqrt {{\rm{T}}{{\rm{V}}_q}[X]} } \right]$ . Moreover, based on the
feature of the non-increasing PDF of the tail distributions, we can give a sharpened conditional Chebyshev inequality; the proof is similar to that of Proposition 1.
Proposition 2. (sharpened conditional Chebyshev inequality) For a random variable $X$ , if ${x_q} \geq 0$ and $Y = X | X \gt {x_q}$ has a non-increasing PDF on its support, the following inequality
holds for $t \gt 1:$
(9) \mathbb{P}\!\left( {\left| {Y - {\rm{TC}}{{\rm{E}}_q}[X]} \right| \le t\sqrt {{\rm{T}}{{\rm{V}}_q}[X]} } \right) \geq 1 - \frac{1}{2}{t^{ - 2}}.
3.4. Case study I: capital adequacy of TCE-based risk capital for insurance and financial losses
We collect two data sets of losses from the insurance and financial sectors. The first data set contains information on 2,387 business interruption losses (in 100,000 French francs) over the period
1985 to 2000 in France; the second data set includes 400 monthly loss rates of the VIX Index from February 1990 to May 2023 (the first data set is available in an R package named CASdatasets; the
second data set is collected from Yahoo Finance). Assume that regulators in both sectors rely on TCE to calculate the risk capital for these business activities. We then estimate the TCE and the TV
of the two sectors, respectively. In accordance with our distribution-free approach, we use non-parametric empirical estimations in this paper. To tackle the small sample in the estimation of TCE and
TV at high-risk levels, we employ bootstrapping; see [Reference Hesterberg18] and [Reference Tibshirani and Efron36] for more details. Moreover, the highest level we can estimate is restricted to
sample size; e.g. if the total sample size is 1,000, we can estimate the confidence level only up to $0.999$ . The lower bounds of the confidence level of using TCE-based capital for the two data
sets are given in Proposition 1 and Corollary 1, respectively. Numerical results are summarized in Figure 1 and Table 1.
From Figure 1, we can observe that Data set I (business interruption insurance) exhibits the characteristics of heavy tails, as its TV increases sharply to infinity along $q \to 1$ . This is
consistent with the study of [Reference De Jong and Heller10], where the author considered the same data set and argued that log-normal distribution (heavy-tailed) outperforms gamma distribution
(light-tailed) in the goodness of fit. For Data set II (VIX Index), its TV gradually decreases to 0, which is a typical light-tailed feature.
Remark 3. In Table 1, the calculations of IC ${_{30{\rm{\% }}}}$ , CI ${_{95{\rm{\% }}}}$ , and sharpened CI ${_{95{\rm{\% }}}}$ correspond to Formulas (7), (8), and (9), respectively. The estimation
is conducted using bootstrapping, with 500 resamples, and the standard errors of ${\rm{T}}{{\rm{V}}_q}$ in data sets 1 and 2 are approximately $0.02{\rm{*T}}{{\rm{V}}_q}$ . Note that ${\rm{I}}{{\rm
{C}}_{30{\rm{\% }}}}$ is short for the improved lower bound of the confidence level with $\lambda = 0.3$ , i.e. $\mathbb{P}\!\left( {X \le 1.3 \times {\rm{TC}}{{\rm{E}}_q}[X]} \right) \geq {\rm{I}}
{{\rm{C}}_{30{\rm{\% }}}}$ . Moreover, the confidence interval (CI) is demonstrated as a proportion to its ${\rm{TC}}{{\rm{E}}_q}$ . For example, CI ${_{95{\rm{\% }}}} = \left( {a,b} \right)$ implies
that at a $95{\rm{\% }}$ confidence level, the conditional variable $Y = X | X \gt {x_q}$ falls within the interval ( $a{\rm{*TC}}{{\rm{E}}_q}[X],b{\rm{*TC}}{{\rm{E}}_q}[X]$ ).
Remark 4. From Table 1, it is evident that the deviations of the loss variables $X$ and $Y$ from TCE are characterized by TV. Therefore, based on well-estimated TCE and TV, our results are
distribution-free. Such findings underscore the inadequacy of relying solely on TCE for tail risk assessment. For insurers with risk capital exceeding TCE, IC offers a minimal confidence level,
signifying a probability of at least IC that loss $X$ remains below the prepared risk capital. Conversely, in events where $X \gt {x_q}$ , to ensure that loss $Y = X | X \gt {x_q}$ falls within the
CI interval with a probability of at least 95%, insurers might need to allocate a risk capital significantly higher than TCE (see the right boundary of CI). This is especially evident in data with
heavy tail variance.
4. Asymptotics of Tail Variance
As we have seen in case study I, TV may exhibit a certain tendency when $p$ approaches 1. In fact, there can be three possible circumstances for TV when $q \to 1$ , i.e. $0$ , a positive constant, or
infinity. Understanding the tendency of TV along $q \to 1$ can help us further estimate the capital adequacy of TCE. In this section, we study the asymptotics of TV. In the sequel of this paper, for
two positive-valued functions $f$ and $g$ , we say $f$ and $g$ are asymptotically equivalent if ${\rm{li}}{{\rm{m}}_{x \to \infty }}\frac{{f(x)}}{{g(x)}} = 1$ , and we denote it as $f(x)\sim g(x)$ .
We also tactically assume that $\mathbb{E}\!\left[ {{X^2}} \right] \lt \infty $ holds and that the ${\rm{li}}{{\rm{m}}_{q \to 1}}{\rm{T}}{{\rm{V}}_q}$ exists on the extended real numbers in the rest
of this paper.
4.1. A distribution-free limit formula of Tail Variance
We first investigate the asymptotics of TV without additional specific distribution assumptions.
Theorem 1. Let $X$ be an absolutely continuous random variable defined on a support with an unbounded supremum. It holds that
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} {\rm{T}}{{\rm{V}}_q}[X] = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } \frac{{ - \overline{F}({{x_q}})}}{{f'( {{x_q}})}}\end{align*}
for differentiable $f(x).$
Based on Theorem 1, we can establish a relationship between the TV and the hazard rate as the following corollary:
Corollary 2. Following Theorem 1, it holds that
(10) \mathop {{\rm{lim}}}\limits_{q \to 1} {\rm{T}}{{\rm{V}}_q}[X] = \mathop {{\rm{lim}}}\limits_{x \to \infty } {\left( {{h_X}(x)} \right)^{ - 2}}.
Corollary 2 shows an identity between the limit of the hazard rate and the TV. More specifically, if we define a tail standard deviation as the square root of TV, then the standard deviation always
has the same limit as the hazard rate. We further demonstrate this with an example of Weibull distributions.
Example 1. A random variable $X$ follows a Weibull distribution, written as $X \sim {\rm{Weibull}}\!\left( {a,b} \right)$ , if its PDF is of the form
\begin{align*}f(x) = a{b^{ - a}}{x^{a - 1}}{\rm{exp}}\!\left[ { - {{\left( {x/b} \right)}^a}} \right],\end{align*}
where $x \in \left[ {0,\infty } \right),a \gt 0$ , and $b \gt 0.$ Then the TCE of the Weibull distribution is
\begin{align*}{\rm{TC}}{{\rm{E}}_q}[X] = \frac{b}{{1 - q}}{\rm{\Gamma }}\!\left( {\frac{1}{a} + 1,{{\left( {{x_q}/b} \right)}^a}} \right),\end{align*}
and the TV of the Weibull distribution is
\begin{align*}{\rm{T}}{{\rm{V}}_q}[X] = \frac{{{b^2}}}{{1 - q}}{\rm{\Gamma }}\!\left( {1 + 2{a^{ - 1}},{{\left( {{x_q}/b} \right)}^a}} \right) - {\left( {\frac{b}{{1 - q}}{\rm{\Gamma }}\!\left( {1 +
{a^{ - 1}},{{\left( {{x_q}/b} \right)}^a}} \right)} \right)^2},\end{align*}
where ${\rm{\Gamma }}\!\left( {s,x} \right)$ is the upper incomplete gamma function and is defined as
\begin{align*}{\rm{\Gamma }}\!\left( {s,x} \right) = \mathop \int \nolimits_x^\infty {t^{s - 1}}{e^{ - t}}{\rm{d}}t.\end{align*}
The limit of ${\rm{T}}{{\rm{V}}_q}[X]$ can be 0 ( $a \gt 1$ ), a non-zero constant ${b^2}$ ( $a = 1$ ), or infinity ( $0 \lt a \lt 1$ ). For a given shape parameter $a$ , the limits of TV and the
hazard rate ${^{ - 2}}$ converge to the same value. In fact, it can be verified that the TV and the square of the reciprocal hazard rate are asymptotically equivalent for Weibull distribution,
(11) \mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{{h^{ - 2}}( {{x_q}})}} = 1.
A numerical illustration for such an asymptotic equivalence in the case of Weibull distribution is demonstrated in Figure 2.
The asymptotic equivalence of Equation (11) implies the same convergence rate for ${\rm{T}}{{\rm{V}}_q}[X]$ and ${h^{ - 2}}({{x_q}})$ , which is a stronger result than Corollary 2. In Theorem 1 and
Corollary 2, we have only the same limit tendency, which can be 0, a positive constant, or $\infty $ . But the convergence rate of the two sides can be different. On the other hand, we do not hold
specific assumptions on the distribution in Theorem 1 and Corollary 2. The asymptotic equivalence of equation (11) actually holds more generally for distributions with certain tails. We discuss this
in the next subsection.
4.2. Asymptotic equivalences of Tail Variances for distributions with polynomial and exponential tails
Theorem 1 and Corollary 2 show that the asymptotic of TV is closely related to the hazard rate. Note that Theorem 1 and Corollary 2 do not require additional distribution assumptions on $X$ . As
such, we do not have explicit asymptotic formulas, and we cannot guarantee the asymptotic equivalence of Equation (11). To further investigate the asymptotic equivalence with more explicit results,
we consider two general distribution families defined by the tails: namely, the distributions families with polynomial and exponential tails. The two distribution families cover a large variety of
standard distributions; we refer to [Reference Rojo33] for more details. To aid the proof of our results in this subsection, we introduce the following notations and prerequisite theorems in regular
varying theory. For more details about regular varying, we refer to [Reference Mikosch31] and the references therein.
Let $U\,:\,\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$ be a measurable function. We say $U$ has regular variation of order $\rho $ at infinity if there exists a real number $\rho $
such that for every $t \gt 0,$
\begin{align*}\mathop {{\rm{lim}}}\limits_{x \to \infty } \frac{{U\!\left( {tx} \right)}}{{U(x)}} = {t^\rho }.\end{align*}
Furthermore, if $\rho = 0$ , the function $U$ is said to be slowly varying at infinity. From the definition, we see that $U(x) = {x^\rho }l(x)$ , where $l$ is slowly varying.
Let ${l_1}$ and ${l_2}$ be nonnegative functions such that
\begin{align*}& \frac{{{l_1}(x)}}{{{x^\epsilon }}} \to 0\ {\rm{and}}\ {l_1}(x){x^\epsilon } \to \infty ,{\rm{as}}\ x \to \infty ,{\rm{for\ all}}\ \epsilon \gt 0,\\ & \text{and}\ \frac{{{l_2}(x)}}{{{e
^{{x^\epsilon }}}}} \to 0\ {\rm{and}}\ {l_2}(x){e^{{x^\epsilon }}} \to \infty ,{\rm{as}}\ x \to \infty ,{\rm{for\ all}}\ \epsilon \gt 0. \end{align*}
Let ${L_1}$ be the set for all ${l_1}$ , and let ${L_2}$ be the set for all ${l_2}$ . [Reference Rojo33] showed that ${L_1}$ consists of all normalized slowly varying functions and ${\rm{that}}\
{L_2}$ consists of all ${l_2}(x)$ for which ${l_2}\!\left[ {{{\left( {{\rm{ln}}x} \right)}^{1/\epsilon }}} \right]$ is slowly varying for all $\epsilon \gt 0$ under the assumption that ${\rm{li}}{{\
rm{m}}_{y \to \infty }}\frac{{l_2^{\prime}\left( y \right)}}{{{l_2}\left( y \right){y^{\epsilon - 1}}}}$ exists for all $\epsilon \gt 0$ . Numerous asymptotic studies focus on regularly varying
random variables, which essentially have polynomial tails, yet fail to cover a vast array of exponential-tail random variables with medium or light tails [Reference Arouri, Boubaker and Nguyen3].
Therefore, investigating the tail asymptotic properties of these two distinct classes of random variables is of importance and interest and provides complementary results to relevant literatures.
Equipped with the notations and theorems just presented, we are able to put forward the definitions of the polynomial tail and the exponential tail as follows:
Theorem 2. If $X$ ’s PDF $f$ and its first derivative, $f^{\prime}$ , exist, then
• X has a polynomial tail if and only if one of the following three holds:
(12) \overline{F}(x)\sim {l_1}(x){x^{ - \alpha + 1}},
which is equivalent to
(13) f(x)\sim \left( {\alpha - 1} \right) \cdot {l_1}(x){x^{ - \alpha }},
(14) f^{\prime}(x)\sim - \left( {\alpha - 1} \right)\alpha \cdot {l_1}(x){x^{ - \alpha - 1}},
for some ${l_1} \in {L_1}$ and $\alpha \gt 1$ .
• X has an exponential tail if and only if
(15) \overline{F}(x)\sim {l_2}(x){e^{ - {x^\beta }}},
which is equivalent to
(16) f(x)\sim \beta \cdot {l_2}(x){e^{ - {x^\beta }}}{x^{\beta - 1}},
(17) f^{\prime}(x)\sim - {\beta ^2} \cdot {l_2}(x){e^{ - {x^\beta }}}{x^{2\beta - 2}},
for some ${l_2} \in {L_2}$ and $\beta \gt 0$ .
Well-known examples of distributions with polynomial tails are, for instance, Student-t distributions and Pareto distributions, while gamma distributions and Weibull distributions are typical
examples of exponential-tailed distributions. Furthermore, based on Theorem 2, we have the following corollary on the asymptotics of TCE:
Corollary 3. If a random variable $X$ has a polynomial tail as defined in Equation (12), then
\begin{align*}{\rm{TC}}{{\rm{E}}_q}[X]\sim \frac{{\alpha - 1}}{{\alpha - 2}}{x_q},\alpha \gt 2,\ when\ q \to 1;\end{align*}
if $X$ has an exponential tail as defined in Equation (15), then
\begin{align*}{\rm{TC}}{{\rm{E}}_q}[X]\sim {x_q},when\ q \to 1.\end{align*}
The asymptotics of TCE have been widely discussed in the literature. For example, [Reference Hua and Joe20] derived similar asymptotic results for distributions of regular variation laws, which are
the polynomial-tailed distributions in this paper; see also [Reference Ignatieva and Landsman22, Reference Vanduffel and Yao37]. Further, [Reference McNeil, Frey and Embrechts30] studied the
asymptotic representation of the TCE for the Generalized Pareto distribution. Since this distribution exhibits a polynomial tail (when the shape parameter $\xi \gt 0$ ) and an exponential tail (when
$\xi = 0$ ), Corollary 3 encompasses their results.
Next, we give the asymptotics of TV for the two distribution families.
Theorem 3. If a random variable $X$ has a polynomial tail as defined in Equation (12), then
\begin{align*}{\rm{T}}{{\rm{V}}_q}[X]\sim \frac{{\alpha - 1}}{{{{\left( {\alpha - 2} \right)}^2}\left( {\alpha - 3} \right)}}x_q^2,\alpha \gt 3,\ when\ q \to 1;\end{align*}
if $X$ has an exponential tail as defined in Equation (15), then
(18) {\rm{T}}{{\rm{V}}_q}[X]\sim {\beta ^{ - 2}}x_q^{2 - 2\beta },when\ q \to 1.
Theorem 3 characterizes the tail behavior of TV for distributions with polynomial and exponential tails. It is consistent with Theorem 4.1 in [Reference Ignatieva and Landsman22], which is a special
case of our results. Specifically, Theorem 4.1 in [Reference Ignatieva and Landsman22] pointed out that if $Z$ is a standardized Student-t distribution, then TV equals $\frac{v}{{{{(v - 1)}^2}\left(
{v - 2} \right)}}z_q^2\left( {1 + O\!\left( {\frac{1}{{z_q^2}}} \right)} \right),$ and if $Z$ is the standard normal distribution, then TV equals $\frac{1}{{z_q^2}}\left( {1 - \frac{6}{{z_q^2}} + O\!
\left( {\frac{1}{{z_q^4}}} \right)} \right)$ . For Student-t distributions, it is straightforward to obtain two similar results by setting $\alpha = v + 1$ . For the normal distribution, by
substituting $x = \frac{1}{{\sqrt 2 }}z$ and setting $\beta = 2$ in Equation (16), Theorem 3 gives TV as being asymptotically equivalent to $z_q^{ - 2}$ , which is consistent with the result
mentioned before.
We give an explicit asymptotic formula for TV of exponential-tailed distributions that supplements the work on univariate cases in [Reference Li28], which pointed out only ${\rm{T}}{{\rm{V}}_q}[X]\
sim o\!\left( {x_q^2} \right)$ . [Reference Li28] mentioned that there could be different converging speeds and named normal distributions and exponential distributions as instances of those speeds.
Our contributions are giving a universal asymptotic formula [Equation (18)] for exponential-tailed distributions, including normal distributions and exponential distributions, and giving the
converging speed in the formula. Furthermore, we can establish an asymptotic equivalence between the TV and the reciprocal square of the hazard rate for the distributions with polynomial tails and
exponential tails.
Corollary 4. If a random variable $X$ has a polynomial tail or an exponential tail defined in Equation (12) or Equation (15), respectively, then it holds that
\begin{align*}{\rm{T}}{{\rm{V}}_q}[X]\sim c \cdot {h^{ - 2}}({{x_q}})\end{align*}
as $q \to 1$ , where
\begin{align*}c = \left( {\begin{array}{l@{\quad}l}{\frac{{{{\left( {\alpha - 1} \right)}^3}}}{{{{\left( {\alpha - 2} \right)}^2}\left( {\alpha - 3} \right)}}} & {\alpha \gt 3\ {\rm{if}}\ X\ {\rm{has
\ a\ polynomial\ tail}};}\\[6pt] 1& \qquad {{\rm{if}}\ X\ {\rm{has\ an\ exponential\ tail}}.}\end{array}} \right.\end{align*}
4.3. Tail categorization based on TV
Risks’ tail behaviors are of importance and interest in many fields, such as insurance, finance, and engineering. There is a large number of papers on the classification of risks’ tails or tail
ordering based on various tools and concepts in the literature. In particular, a tail categorization proposed in [Reference Schuster34] is broadly discussed. The authors suggested using the
asymptotics of hazard rates to classify distributions. Note that we have shown the asymptotic equivalence between TV and hazard rate; we can now categorize risks’ tails using the TV in similar ways.
The TV of each category, when $q \to 1,$ can be computed straightforwardly based on formulas in Section 2.
Proposition 3. Let $X$ be an absolutely continuous random variable, and define ${h_0}\,:\!=\, {\rm{li}}{{\rm{m}}_{x \to + \infty }}\frac{1}{{h(x)}}$ . We have the tail variance of $X$ when $q \to 1$
as follows:
Proposition 3 tells us that the hazard rate essentially gives the same information as TV does when $X$ approaches infinity. The Weibull distribution in Example 1 is a typical example illustrating
such categorization. Its tail decays exponentially, and the shape parameter $a$ indicates whether it is long-tailed ( $0 \lt a \lt 1$ ), medium-tailed ( $a = 1$ ), or short-tailed ( $a \gt 1$ ). More
generally, we present a tail categorization for distributions with polynomial tails and exponential tails.
Remark 5. Let $X$ be an absolutely continuous random variable with a polynomial or an exponential tail as defined in Equation (12) or Equation (15), respectively; we then have the classification and
the TV of $X$ when $q \to 1$ as follows:
TV categorizes the probability distributions into three classes according to tails. Despite the simplicity, it provides a brief insight into the tail behaviors of the concerned risks, which can also
help us have a quick judgment on how the risks could be at the upper tails. For instance, in our case study I, we can observe a significant distinct tendency of TV along $q \to 1$ for the two sets of
insurance data, which suggests that they belong to different categories and should have very different tail behaviors. Note that, from a practical application point of view, TV is much easier to
estimate than the hazard rate. We further provide another practical case study based on financial data to illustrate the advantage of TV in tail categorization.
4.4. Case study II: alternative evidence for financial data categorization
Empirical studies often suggest that high-frequency financial data are usually more heavy-tailed than are low-frequency data. For example, the Gaussian models seem to fit yearly data well in many
circumstances, but the high-frequency data appear to be more heavy-tailed; see [Reference Dacorogna, Müller, Pictet and De Vries9, Reference Zhou39]. In addition to these empirical studies,
Proposition 3 uses the asymptotics of TV to provide an alternative way to verify this conclusion.
We collect the data on the exchange rate in different frequencies between British Pound Sterling and the U.S. Dollar. We consider the minutely, daily, weekly, and yearly data and estimate the TVs
along the tails, respectively, using the bootstrap method (the data encompass 8,253 minute-level points from January 4–12, 2024; 5,230 daily points from December 2003 to January 2024; 1,049 weekly
points in the same interval; and 409 monthly points from January 1990 to January 2024, all sourced from the Wind Economic Database) in order to obtain quasi yearly losses. To facilitate comparison,
the losses are converted into annualized losses, and the results are presented in Figure 3. (The standard errors of TV fluctuate around $0.015{\rm{*}}$ TV for minutely data, $0.008{\rm{*}}$ TV for
daily data, $0.016{\rm{*}}$ TV for weekly data, and $0.014{\rm{*}}$ TV for yearly data.) We can easily observe that the TV increases as the frequency of data gets higher. In particular, minutely
data’s TV shows a tendency to increase to infinity, while the yearly data’s TV converges to 0, which indicates that minutely data have heavy tails and that yearly data have light tails. For daily and
weekly data, the asymptotics of TV appear to have constantcy, which implies medium tails. As such, TV gives us a quick view of the tail categorization of the data, allowing unsuitable distributions
to be precluded from the calibrations. Our results also provide alternative evidence to support the empirical studies on the relationship between the frequency and the tail categorization of
financial data.
5. Conclusion
Tail variance is an instrumental risk measure for the analysis of risks at the tails. In this paper, we review extant explicit formulas of TV and TCE and derive more TV and TCE formulas of frequently
used distributions. These formulas can be used directly in relevant studies. When TCE is used for calculating the risk capital, TV can be taken into account for assessing how far the true loss
deviates from the TCE-based risk capital under the event $X \gt {x_q}$ , which provides valuable insights into the capital adequacy problem. We give sharpened distribution-free lower bounds of the
confidence level of using the TCE-based capital adequacy. Such an approach is straightforward to implement and partially mitigates the limitations of measuring tail risk with TCE exclusivly. More
specifically, using TCE solely as the risk measure plausibly does not exploit the tail information sufficiently and might result in under- or overestimation of the risk capital (see Remark 4).
Moreover, because we observed that the asymptotic properties of TV are eventually determined by the risk heaviness, we studied the asymptotics of tail variance. In particular, for two large
probability distribution families with certain tails (exponential tails and polynomial tails), we unified the asymptotic formulas of TCE and TV, which have been widely studied under various specific
distribution assumptions. More importantly, those formulas also give more accurate convergence rates, which previous literature has failed to do generally. Two case studies based on real data are
presented to support our theoretical results. Our result suggests that tail variance is indeed a useful risk measure that has potential significance to and importance in many applications.
Appendix A. Proofs
A.1. Theorems and propositions in Section 3
A.1.1. Proposition 1.
We provide a proof of Proposition 1 that is enlightened by [Reference Huber21].
Assume $U$ is a uniform random variable on $\left[ { - t,t} \right]$ . Then
\begin{align*}\mathbb{P}\!\left( {Y + U \geq t} \right) & = \mathbb{P}\!\left( {Y \geq t - U} \right)\\[4pt] & = \mathop \int \nolimits_{ - t}^t \,\mathbb{P}\!\left( {Y \geq t - u} \right){f_U}\!\
left( u \right){\rm{d}}u\end{align*}
When $0 \le u \le t$ , $\mathbb{P}\!\left( {Y \geq t - u} \right) = \mathbb{P}\!\left( {Y \geq t} \right) + \mathbb{P}\!\left( {t - u \le Y \lt t} \right)$ . When $ - t \le u \lt 0$ , $\mathbb{P}\!\
left( {Y \geq t - u} \right) = \mathbb{P}\!\left( {Y \geq t} \right) - \mathbb{P}\!\left( {t \le Y \le t - u} \right)$ . Subsequently,
\begin{align*}\mathbb{P}\!\left( {Y + U \geq t} \right) & = \mathop \int \nolimits_{ - t}^t \frac{{\mathbb{P}\!\left( {Y \geq t - u} \right)}}{{2t}}{\rm{d}}u\\[4pt] & = \mathop \int \nolimits_{ - t}^
t \frac{{\mathbb{P}\!\left( {Y \geq t} \right)}}{{2t}}{\rm{d}}u + \mathop \int \nolimits_0^t \frac{{\mathbb{P}\!\left( {t - u \le Y \lt t} \right)}}{{2t}}{\rm{d}}u\\[4pt] & - \mathop \int \nolimits_0
^t \frac{{\mathbb{P}\!\left( {t \le Y \lt t + u} \right)}}{{2t}}{\rm{d}}u\\[4pt] & = \mathbb{P}\!\left( {Y \geq t} \right) + \frac{1}{{2t}}\mathop \int \nolimits_0^t \left[ {\mathbb{P}\!\left( {t - u
\le Y \lt t} \right) - \mathbb{P}\!\left( {t \le Y \lt t + u} \right)} \right]{\rm{d}}u.\end{align*}
Given that $Y$ has a non-increasing PDF, we have $\mathbb{P}\!\left( {t - u \le Y \lt t} \right) \geq \mathbb{P}\!\left( {t \le Y \lt t + u} \right)$ . Hence, $\mathbb{P}\!\left( {Y \geq t} \right) \
le \mathbb{P}\!\left( {Y + U \geq t} \right)$ . Next, we show that $\mathbb{P}\!\left( {Y + U \geq t} \right) \le \frac{{\mathbb{E}[Y]}}{{2t}}$ .
\begin{align*}\mathbb{P}\!\left( {Y + U \geq t} \right) & = \mathbb{E}\!\left[ {\textbf{1}\!\left( {Y + U \geq t} \right)} \right]\\ & = \mathbb{E}\!\left[ {\mathbb{E}\!\left[ {\textbf{1}\!\left( {Y
+ U \geq t} \right)|Y} \right]} \right]\\ & = \mathbb{E}\!\left[ {\mathbb{P}\!\left( {U \geq t - Y|Y} \right)} \right],\end{align*}
where $\textbf{1}\!\left( \cdot \right)$ is an indicator function that equals 1 if the argument is true or equals 0 otherwise.
When $t - Y \le - t$ , $Y \geq 2t$ and $\mathbb{P}\!\left( {U \geq t - Y|Y} \right) = 1$ . When $t - Y \geq t$ , $Y \le 0$ and $\mathbb{P}\!\left( {U \geq t - Y|Y} \right) = 0$ . When $0 \le Y \le
2t$ , $\mathbb{P}\!\left( {U \geq t - Y|Y} \right) = \frac{Y}{{2t}}$ . Hence,
\begin{align*}\mathbb{P}\!\left( {Y + U \geq t} \right) = \mathbb{E}\!\left[ {{f_1}\!\left( Y \right)} \right],\end{align*}
where ${f_1}(y) = \textbf{1}\!\left( {y \geq 2t} \right) + \frac{y}{{2t}}\textbf{1}\!\left( {0 \le y \le 2t} \right)$ . Note that if ${f_1}(y) \le \frac{y}{{2t}}\textbf{1}\!\left( {y \geq 0} \
right),$ we have
\begin{align*}\mathbb{P}\!\left( {Y + U \geq t} \right) = \mathbb{E}\!\left[ {{f_1}\!\left( Y \right)} \right] \le \mathbb{E}\!\left[ {\frac{Y}{{2t}}\textbf{1}\!\left( {Y \geq 0} \right)} \right] = \
Consequently, it follows that
\begin{align*}\mathbb{P}\!\left( {Y \geq t} \right) \le \mathbb{P}\!\left( {Y + U \geq t} \right) \le \frac{{{\rm{TC}}{{\rm{E}}_q}[X]}}{{2t}}.\end{align*}
A.2. Theorems and corollaries in Section 4
Before providing the proofs of theorems and corollaries in Section 4, we need to present the well-known theorems in regularly varying theory and a lemma.
Theorem 4. (Karamata’s theorem) Let $g(x)$ be slowly varying and locally bounded in $\left[ {{x_0}, + \infty } \right)$ for some ${x_0} \geq 0$ . Then
• for $\alpha \gt - 1$ ,
\begin{align*}\mathop \int \nolimits_{{x_0}}^x {t^\alpha }g\!\left( t \right){\rm{d}}t \sim {\left( {\alpha + 1} \right)^{ - 1}}{x^{\alpha + 1}}g(x),x \to \infty ;\end{align*}
• for $\alpha \lt - 1$ ,
\begin{align*}\mathop \int \nolimits_x^\infty {t^\alpha }g\!\left( t \right){\rm{d}}t \sim - {\left( {\alpha + 1} \right)^{ - 1}}{x^{\alpha + 1}}g(x),x \to \infty .\end{align*}
Based on Karamata’s theorem, Corollary 5 can be deduced directly. The proof is provided in [Reference Bingham, Goldie and Teugels8].
Corollary 5. If $g$ is slowly varying and $\alpha \lt - 1$ , then $\mathop \int \nolimits_x^\infty {t^\alpha }g\!\left( t \right)dt$ converges and
\begin{align*}\mathop {{\rm{lim}}}\limits_{x \to \infty } \frac{{{x^{\alpha + 1}}g(x)}}{{\mathop \int \nolimits_x^\infty {t^\alpha }g\!\left( t \right){\rm{d}}t}} = - \alpha - 1.\end{align*}
A.2.1. Theorem 1.
First we can rewrite ${\rm{T}}{{\rm{V}}_q}[X]$ in an integral form using $f(x)$ and ${x_q}$ .
(19) {\rm{T}}{{\rm{V}}_q}[X] & = \mathbb{E}\!\left[ {{X^2}\!\left| X \right\gt {x_q}} \right] - {\mathbb{E}^2}\!\left[ {X | X \gt {x_q}} \right]\nonumber\\ & = \frac{{\mathbb{E}\!\left[ {{X^2}{I_{X \
gt {x_q}}}} \right]}}{{1 - q}} - {\left( {\frac{{\mathbb{E}\!\left[ {X{I_{X \gt {x_q}}}} \right]}}{{1 - q}}} \right)^2}\nonumber\\ & = \frac{{\overline{F}({{x_q}})\mathop \int \nolimits_{{x_q}}^{ + \
infty } {x^2}f(x){\rm{d}}x - {{\left( {\mathop \int \nolimits_{{x_q}}^{ + \infty } xf(x){\rm{d}}x} \right)}^2}}}{{{{\left( {\overline{F}({{x_q}})} \right)}^2}}}
Note that $q \to 1$ is equivalent to ${x_q} \to \infty $ . We compute the following:
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} {\rm{T}}{{\rm{V}}_q}[X] & = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } \frac{{ - \mathop \int \nolimits_{{x_q}}^{ + \infty } {x^2}f(x){\rm{d}}
x - x_q^2\overline{F}({{x_q}}) + 2{x_q}\mathop \int \nolimits_{{x_q}}^{ + \infty } xf(x){\rm{d}}x}}{{ - 2\overline{F}({{x_q}})}}\\ & = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } \frac{{ - {x_q}\
overline{F}({{x_q}}) + \mathop \int \nolimits_{{x_q}}^{ + \infty } xf(x){\rm{d}}x}}{{f({{x_q}})}}\\ & = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } \frac{{ - \overline{F}({{x_q}})}}{{f^{\prime}(
A.2.2. Corollary 2.
First note that
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} {\rm{T}}{{\rm{V}}_q}[X] = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } \frac{{ - \overline{F}({{x_q}})}}{{f^{\prime}({{x_q}})}} = \mathop {{\rm
{lim}}}\limits_{{x_q} \to \infty } \frac{{ - \overline{F}({{x_q}})}}{{f({{x_q}})}} \times \frac{{f({{x_q}})}}{{f^{\prime}({{x_q}})}},\end{align*}
and ${\rm{li}}{{\rm{m}}_{{x_q} \to \infty }}\frac{{ - \overline{F}({{x_q}})}}{{f({{x_q}})}} = {\rm{li}}{{\rm{m}}_{{x_q} \to \infty }}\frac{{f({{x_q}})}}{{f^{\prime}({{x_q}})}}$ . Therefore,
(20) \mathop {{\rm{lim}}}\limits_{q \to 1} {\rm{T}}{{\rm{V}}_q}[X] = \mathop {{\rm{lim}}}\limits_{{x_q} \to \infty } {\left( {\frac{{f({{x_q}})}}{{f^{\prime}({{x_q}})}}} \right)^2} = \mathop {{\rm
{lim}}}\limits_{q \to 1} {\left( {\frac{{ - \overline{F}({{x_q}})}}{{f({{x_q}})}}} \right)^2}.
A.2.3. Theorem 2.
We show that the three definitions of the two types of tails are equivalent, respectively. The equivalences between Equations (12) and (13) and between Equations (15) and (16) have been proved by [
Reference Rojo33]. Hence, we give a proof of the rest of the asymptotic equivalences. Moreover, we say that a function $g(x)$ is eventually monotone if there exists ${\rm{an}}\ {x^{\rm{*}}}$ , for $\
forall {x^{\rm{*}}} \lt {x_1} \lt {x_2}$ , and if we have $g\!\left( {{x_1}} \right) \gt g\!\left( {{x_2}} \right)$ .
(i) Suppose Equation (14) holds. Applying Corollary 3 gives us
\begin{align*}\frac{{{x^{\left( { - \alpha - 1} \right) + 1}}{l_1}(x)}}{{\mathop \int \nolimits_x^\infty {t^{\left( { - \alpha - 1} \right)}}{l_1}\!\left( t \right){\rm{d}}t}} \to \alpha .\end
Given that $\mathop \int \nolimits_x^\infty {t^{\left( { - \alpha - 1} \right)}}{l_1}\!\left( t \right){\rm{d}}t$ converges, the derivative of the integral with respect to $x$ gives us $ - f^{\prime}
(x)$ . Hence, we have Equation (13).
(ii) Now we suppose that Equation (13) holds. For any $0 \lt a \lt b \lt \infty ,$ and given that $f$ is nonnegative and eventually non-increasing, we have
\begin{align*}f\!\left( {bx} \right) - f\!\left( {ax} \right) = \mathop \int \nolimits_{ax}^{bx} f^{\prime}\!\left( t \right){\rm{d}}t \le 0.\end{align*}
As $f^{\prime}(x)$ is non-decreasing and non-positive, when $x$ is large enough, it holds that
(21) \frac{{\left( {b - a} \right)xf^{\prime}\!\left( {ax} \right)}}{{{x^{ - \alpha }}{l_1}(x)}} \le \frac{{f\!\left( {bx} \right) - f\!\left( {ax} \right)}}{{{x^{ - \alpha }}{l_1}(x)}} \le \frac{{\
left( {b - a} \right)xf^{\prime}\!\left( {bx} \right)}}{{{x^{ - \alpha }}{l_1}(x)}} \le 0.
We rewrite the mid-term of the inequalities in Equation (21) as
(22) \frac{{f\!\left( {bx} \right)}}{{{{\left( {bx} \right)}^{ - \alpha }}{l_1}\!\left( {bx} \right)}} \cdot \frac{{{b^{ - \alpha }}{l_1}\!\left( {bx} \right)}}{{{l_1}(x)}} - \frac{{f\!\left( {ax} \
right)}}{{{{\left( {ax} \right)}^{ - \alpha }}{l_1}\!\left( {ax} \right)}} \cdot \frac{{{a^{ - \alpha }}{l_1}\!\left( {ax} \right)}}{{{l_1}(x)}},
which approaches $\left( {\alpha - 1} \right)\left( {{b^{ - \alpha }} - {a^{ - \alpha }}} \right)$ as $x \to \infty $ . Subsequently, for the first inequality of Equation (21), we have
\begin{align*}{\overline {{\rm{lim}}} _{x \to \infty }}\frac{{f^{\prime}\!\left( {ax} \right)}}{{{x^{ - \alpha - 1}}{l_1}(x)}} \le \frac{{\left( {\alpha - 1} \right)\left( {{b^{ - \alpha }} - {a^{ -
\alpha }}} \right)}}{{b - a}}.\end{align*}
Next, taking $b$ as 1 gives
\begin{align*}\mathop {{\rm{lim}}}\limits_{b \downarrow 1} {\overline {{\rm{lim}}} _{x \to \infty }}\frac{{f^{\prime}(x)}}{{{x^{ - \alpha - 1}}{l_1}(x)}} \le - \left( {\alpha - 1} \right)\alpha .\end
Similarly, we have
\begin{align*}\mathop {{\rm{lim}}}\limits_{a \uparrow 1} {\overline {{\rm{lim}}} _{x \to \infty }}\frac{{f^{\prime}(x)}}{{{x^{ - \alpha - 1}}{l_1}(x)}} \geq - \left( {\alpha - 1} \right)\alpha ,\end
which concludes the proof of Equation (14)’s implying Equation (13).
(iii) Suppose Equation (15) holds. Substituting $x$ by ${\left( {{\rm{ln}}\,y} \right)^{1/\beta }}$ gives
\begin{align*}{\overline{F}^{\rm{*}}}(y)\,:\!=\, \overline{F}[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}}]\sim {l_2}[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}}]{y^{ - 1}}.\end{align*}
Note that ${l_2}\!\left[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \right]$ is slowly varying. We then define
\begin{align*}{f^{\rm{*}}}(y) = \frac{{{\rm{d}}{F^{\rm{*}}}(y)}}{{{\rm{d}}y}}\ {\rm{and}}\ {f^{*\prime}}(y) = \frac{{{\rm{d}}{f^{\rm{*}}}(y)}}{{{\rm{d}}y}}.\end{align*}
Because ${f^{\rm{*}}}$ and ${f^{*\prime}}$ are eventually monotone, an argument similar to that in (ii) leads to ${f^{\rm{*}}}(y)\sim {l_2}\!\left[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \
right]{y^{ - 2}}$ and ${f^{*\prime}}(y)\sim - {l_2}\!\left[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \right]{y^{ - 3}}$ , respectively. Taking $f$ back, we have Equation (17).
(iv) Suppose Equation (17) holds. Starting with the definition of asymptotic equivalence, for any $\epsilon \gt 0$ there exists ${x_0}$ . For all $x \gt {x_0}$ , $\left| {\frac{{f^{\prime}(x)}}{{ -
{l_2}(x){e^{ - {x^\beta }}}{x^{2\beta - 2}}}} - {\beta ^2}} \right| \lt \epsilon $ . Subsequently, it holds that
\begin{align*}\mathop \int \nolimits_x^\infty f^{\prime}\!\left( t \right){\rm{d}}t \le - \left( {{\beta ^2} + \epsilon } \right)\mathop \int \nolimits_x^\infty {l_2}\!\left( t \right){e^{ - {t^\beta
}}}{t^{2\beta - 2}}{\rm{d}}t.\end{align*}
Substituting $t$ by ${\left( {{\rm{ln}}\,y} \right)^{1/\beta }}$ , the right-hand side of the last inequality is $\epsilon /\beta \mathop \int \nolimits_{{e^{{x^\beta }}}}^\infty {l_2}\!\left[ {{{\
left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \right]{y^{ - 3}}{\rm{d}}y$ , which is asymptotically equivalent to ${e^{ - 2{x^\beta }}}{l_2}\big( {{e^{{x^\beta }}}}\big)$ by Corollary 3. Therefore,
Equation (17) implies Equation (16).
A.2.4. Corollary 3.
To derive the asymptotic relationship between TCE and VaR, we need to calculate ${\rm{li}}{{\rm{m}}_{q \to 1}}\frac{{{\rm{TC}}{{\rm{E}}_q}[X]}}{{{x_q}}}$ .
If a random variable $X$ has a polynomial tail, Karamata’s theorem gives
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{\mathop \int \nolimits_{{x_q}}^\infty xf(x){\rm{d}}x}}{{{{\overline{F}}_X}( {{x_q}}) \cdot {x_q}}} & = \mathop {{\rm{lim}}}\limits_{q \to 1}
\frac{{\mathop \int \nolimits_{{x_q}}^\infty \,\left( {\alpha - 1} \right){x^{ - \alpha + 1}}{l_1}(x){\rm{d}}x}}{{{l_1}( {{x_q}})x_q^{ - \alpha + 2}}}\\ & = \mathop {{\rm{lim}}}\limits_{q \to 1} \
frac{{\alpha - 1}}{{{l_1}({{x_q}})x_q^{ - \alpha + 2}}}\frac{{x_q^{ - \alpha + 2}{l_1}\!\left( {_q} \right)}}{{\alpha - 2}}\\ & = \mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{\alpha - 1}}{{\alpha -
Hence, ${\rm{TC}}{{\rm{E}}_q}[X]\sim \frac{{\alpha - 1}}{{\alpha - 2}}{x_q},\alpha \gt 2,when\ q \to 1.$
If a random variable $X$ has an exponential tail, by substituting $x$ by ${\left( {{\rm{ln}}\,y} \right)^{1/\beta }}$ , we have
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{\mathop \int \nolimits_{{x_q}}^\infty xf(x){\rm{d}}x}}{{{{\overline{F}}_X}( {{x_q}}) \cdot {x_q}}} & = \mathop {{\rm{lim}}}\limits_{q \to 1}
\frac{{\mathop \int \nolimits_{{x_q}}^\infty x \cdot \beta {l_2}(x){x^{\beta - 1}}{e^{ - {x^\beta }}}{\rm{d}}x}}{{{l_2}({{x_q}}){e^{ - x_q^\beta }} \cdot {x_q}}}\\[-2pt] & = \mathop {{\rm{lim}}}\
limits_{q \to 1} \frac{{\mathop \int \nolimits_{{x_q}}^\infty x \cdot \beta {l_2}(x){x^{\beta - 1}}{e^{ - {x^\beta }}}{\rm{d}}x}}{{{l_2}({{x_q}}){e^{ - x_q^\beta }} \cdot {x_q}}}\\[-1pt] & = \mathop
{{\rm{lim}}}\limits_{q \to 1} \frac{{\mathop \int \nolimits_{{y_q}}^\infty {l_2}\!\left[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \right]{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}{y^{ - 2}}{\
rm{d}}y}}{{{l_2}\!\left[ {{{\left( {{\rm{ln}}{y_q}} \right)}^{1/\beta }}} \right]{{\left( {{\rm{ln}}{y_q}} \right)}^{1/\beta }}y_q^{ - 1}}}.\end{align*}
As mentioned above, ${l_2}\!\left[ {{{\left( {{\rm{ln}}\,y} \right)}^{1/\beta }}} \right]$ is asymptotically equivalent to a slowly varying function. Hence, we can apply Karamata’s theorem again, and
the numerator will be asymptotically equivalent to ${l_2}\big[ {{{\left( {{\rm{ln}}{y_q}} \right)}^{1/\beta }}} \big]{\left( {{\rm{ln}}{y_q}} \right)^{1/\beta }}y_q^{ - 1}$ . Subsequently, TCE and
VaR are asymptotically equivalent for a random variable with an exponential tail.
A.2.5. Theorem 3.
To derive the asymptotic relationship between TV and VaR for a polynomial-tailed distribution, we need to calculate ${\rm{li}}{{\rm{m}}_{q \to 1}}\frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{x_q^2}}$ .
Similarly to the proof of Corollary 3, if a random variable $X$ has a polynomial tail, Karamata’s theorem gives
\begin{align*}\mathop \int \nolimits_{{x_q}}^\infty xf(x){\rm{d}}x & \sim \left( {\alpha - 1} \right)\mathop \int \nolimits_{{x_q}}^\infty \,{x^{ - \alpha + 1}}{l_1}(x){\rm{d}}x\\[-2pt] & = \frac{{\
alpha - 1}}{{\alpha - 2}}x{_q^{ - \alpha + 2}}_{1}({{x_q}})\end{align*}
\begin{align*}\mathop \int \nolimits_{{x_q}}^\infty {x^2}f(x){\rm{d}}x &\sim \left( {\alpha - 1} \right)\mathop \int \nolimits_{{x_q}}^\infty {x^{ - \alpha + 2}}{l_1}(x){\rm{d}}x\\[-2pt] & = \frac{{\
alpha - 1}}{{\alpha - 3}}x_q^{ - \alpha + 3}{l_1}({{x_q}}).\end{align*}
\begin{align*}\mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{{\rm{T}}{{\rm{V}}_q}[X]}}{{x_q^2}} & = \mathop {{\rm{lim}}}\limits_{q \to 1} \frac{{\overline{F}({{x_q}})\mathop \int \nolimits_{{x_q}}^{ +
\infty } {x^2}f(x){\rm{d}}x - {{\left( {\mathop \int \nolimits_{{x_q}}^{ + \infty } xf(x){\rm{d}}x} \right)}^2}}}{{{{\left( {\overline{F}({{x_q}})} \right)}^2}x_q^2}}\\[-2pt] & = \mathop {{\rm{lim}}}
\limits_{q \to 1} \frac{{\frac{{\alpha - 1}}{{\alpha - 3}}x_q^{ - 2\alpha + 4}l_1^2( {{x_q}}) - {{\left( {\frac{{\alpha - 1}}{{\alpha - 2}}} \right)}^2}x_q^{ - 2\alpha + 4}l_1^2( {{x_q}})}}{{l_1^2(
{{x_q}})x_q^{ - 2\alpha + 4}}}\\[-1pt] & = \frac{{\alpha - 1}}{{\alpha - 3}} - {\left( {\frac{{\alpha - 1}}{{\alpha - 2}}} \right)^2}\\[-1pt] & = \frac{{\alpha - 1}}{{{{\left( {\alpha - 2} \right)}^
2}\left( {\alpha - 3} \right)}}.\end{align*}
Therefore, we may conclude the proof of the case of polynomial-tailed random variables. For the case of exponential-tailed random variables, the proof is quite complicated and can be shown upon
A.2.6. Corollary 4.
This is a straightforward result of Theorem 3 along with the definition of the hazard rate.
Appendix B. A table of the summary of TV formulas and asymptotics
The authors would like to thank the anonymous reviewers and editor for their many insightful remarks and invaluable comments that helped improve the paper to a significant extent. The initial version
of Proposition 1 and Table B1 were first prepared by Zihao Li (zl122@hw.ac.uk) and were amended by the authors.
Funding information
The authors acknowledge the financial support from the Israel Science Foundation (Grant N1686/17) and the National Science Foundation of China (Grant 12371474). Jing Yao acknowledges support from the
Chair Professorship of the “BAYU Scholar Program.”
Competing interests
There were no competing interests that arose during the preparation or publication process of this article to declare. | {"url":"http://core-cms.prod.aop.cambridge.org/core/journals/advances-in-applied-probability/article/tail-variance-and-confidence-of-using-tail-conditional-expectation-analytical-representation-capital-adequacy-and-asymptotics/2162958B918116321F70C58C8DD1BCCC","timestamp":"2024-11-05T16:30:22Z","content_type":"text/html","content_length":"1049982","record_id":"<urn:uuid:b960a5a2-382e-44c8-866f-fee23f7a2635>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00722.warc.gz"} |
Informal formative assessments – Tools for improving your teaching practice and providing instant feedback
Informal formative assessments – Tools for improving your teaching practice and providing instant feedback
Informal formative assessments are an essential component of my teaching practice. They provide me and my students with instant feedback such that I can adjust my teaching strategies and they can see
where their comprehension and competencies stand in relation to the learning targets. Additionally, these assessments can turn a passive lecture into a more interactive student-driven lesson. They
are perhaps the most effective and important teaching tools in my toolbox.
An informal formative assessment can take many forms, but it is always ungraded, instant, and reflective for both student and teacher. Informal implies that it is low-key, low-stakes, and often
personalized to the student. Formative means it is a diagnostic tool that helps guide learning towards a summative assessment.
Perhaps the worst (i.e. least informative and reflective) informal formative assessment are the phrases “Does that make sense?” and “Any questions?” I learned this lesson the hard way as a beginner
lab instructor. These are phrases I’d heard a lot as a student myself, so I suppose I picked up the habit somewhere along the way. I’d walk students through a diagram of the Krebs cycle while staring
at my PowerPoint, and then I would look up and say some version of the aforementioned questions and wait a few seconds. I would be met with glassy-eyed nods and then I’d launch into the next slide.
Needless to say, it did not make sense and they had lots of questions, but they didn’t have the framework, time, or space to process their own understanding. And I had the confirmation bias I needed
to move on to the next slide. A few weeks later, the students would fail the Krebs cycle on the summative assessment, and I had to ask myself: why in the world didn’t they say something?
The truth is, many students probably felt confident in their knowledge and skills. Or, at least, they didn’t know that they didn’t know. Thus, a good formative assessment (whether formal or informal)
forces students to reflect on their actual knowledge and skills, and provides instant feedback so that both students and teachers can modify their strategies moving forward.
Okay, enough background. Here is a list of common informal formative assessments that you can use in virtually any classroom situation, from small groups to large lectures:
1. 5-finger Temperature Check (before and after) – When starting a new topic, ask students to give a show of fingers to indicate their level of understanding. “How would you rate your understanding
of ______? 1 means you have never heard of the topic before and 5 means you think you already understand the learning goals.” Generally it’s normal to see a lot of 2’s and 3’s. At the end of the
topic/class, ask the same question. Aim for 4’s and 5’s. If you’re not getting there, ask students to share what’s unclear (see #3 below) and use the opportunity to re-teach. This also gives
feedback to the student–did they ask the right questions during class? Did they improve their understanding?
2. Entry Tasks – In every class I teach, I start with an entry task. This sets the tone for learning and gives students a few minutes to “get in the zone.” It also provides me with a little insight
into how students are thinking about a topic. Typically I have a short open-ended question on the board that students can discuss and think about together. For example, if I’m teaching about
evolution, I might ask, “Why don’t birds have teeth?” I typically give students 2 minutes to discuss during class and then we discuss as a class for up to 5 minutes. While this does cut into
class time, I notice that it gets students to put on their analytical hats and really changes the mood from the typical “sit-and-get” lectures they’re probably coming from.
3. Turn & Talk (aka Pair & Share) – If you have ever asked students a question and received blank stares in response, this single strategy will change your whole game. This also works extremely well
when you have one or more students who are dominating the discussion or if you are trying to get participation from quiet students. The technique is super simple: Instead of asking a question and
soliciting responses from individuals, ask students to turn and discuss with their neighbor. Give them a minute or so to formulate a response and then call on a pair of students randomly to share
their thoughts. Through this method, students are not singled out, but rather asked to share their group’s thoughts. It forces every student to participate since they know any one of them could
be called upon while simultaneously lifting the burden and fear of sharing their personal ideas. Turn and talk provides opportunity for reflection and quick corrections through student-driven
discussion. Furthermore, it creates accountability to stay focused and during the rest of the class because no one wants to let their neighbors down.
4. See/Think/Wonder – This is really a type of “thinking routine.” It’s a chart with three columns (see, think, and wonder). Students can fill it out as a small group or whole class, with the
intention of getting students to really think deeply about a topic. Usually you give students about 5 minutes per column, but it could be a quick brainstorm. I’ve used this strategy to get
students to ask new questions, form hypotheses about complex systems, and abstract engineering concepts from biology. For example, imagine you’re starting a group project where students need to
design a new biomimetic product. This could be a way to start the brainstorming of new ideas. Students spend 3 minutes observing (“see”) a phenomena in nature and writing down what they see. Then
they spend 3 more minutes abstracting those observations (“think”) into physical or engineering phenomena. And finally they spend another 3 minutes applying (“wonder”) these new discoveries to a
biomimetic product. As a teacher, you’ll get to see exactly where their understanding lies, how deep it goes, and what their current limitations are.
5. Online quizzes or clickers – While not truly “informal,” online quizzes or clickers can provide instant formative feedback for students and teachers. Sites like www.mentimeter.com and
www.kahoot.com allow instructors to make free quizzes. Rather than having each student take the quiz independently, you can gamify it by putting the students in groups. This has the added benefit
of encouraging debate and discourse.
6. The QHPM – This is another semi-formal formative assessment that I created about 15 years ago. It’s a scientific framework that allows students to practice asking questions, forming hypotheses
and predictions, and generating methodologies to test those predictions. In a typical biology course, I asked students to write one per week. I provided quick feedback (e.g. Are the hypotheses
based on prior knowledge/logic? Are the predictions testable?) and the QHPMs were graded based solely on participation. Learn more here.
If you have questions or want to implement these in your classroom, I would love to work with you. Please let me know if you have questions as well. | {"url":"https://brettkvo.com/informal-formative-assessments-tools-for-improving-your-teaching-practice-and-providing-instant-student-feedback/","timestamp":"2024-11-09T00:27:55Z","content_type":"text/html","content_length":"59253","record_id":"<urn:uuid:a8f33311-a62f-442d-b8ff-98cc818c4155>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00157.warc.gz"} |
Convert binary classification support vector machine (SVM) model to incremental learner
Since R2020b
IncrementalMdl = incrementalLearner(Mdl) returns a binary classification linear model for incremental learning, IncrementalMdl, using the traditionally trained linear SVM model object or SVM model
template object in Mdl.
If you specify a traditionally trained model, then its property values reflect the knowledge gained from Mdl (parameters and hyperparameters of the model). Therefore, IncrementalMdl can predict
labels given new observations, and it is warm, meaning that its predictive performance is tracked.
IncrementalMdl = incrementalLearner(Mdl,Name,Value) uses additional options specified by one or more name-value arguments. Some options require you to train IncrementalMdl before its predictive
performance is tracked. For example, 'MetricsWarmupPeriod',50,'MetricsWindowSize',100 specifies a preliminary incremental training period of 50 observations before performance metrics are tracked,
and specifies processing 100 observations before updating the window performance metrics.
Convert Traditionally Trained Model to Incremental Learner
Train an SVM model by using fitcsvm, and then convert it to an incremental learner.
Load and Preprocess Data
Load the human activity data set.
For details on the data set, enter Description at the command line.
Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).
Train SVM Model
Fit an SVM model to the entire data set. Discard the support vectors (Alpha) from the model so that the software uses the linear coefficients (Beta) for prediction.
TTMdl = fitcsvm(feat,Y);
TTMdl = discardSupportVectors(TTMdl)
TTMdl =
ResponseName: 'Y'
CategoricalPredictors: []
ClassNames: [0 1]
ScoreTransform: 'none'
NumObservations: 24075
Beta: [60x1 double]
Bias: -6.4280
KernelParameters: [1x1 struct]
BoxConstraints: [24075x1 double]
ConvergenceInfo: [1x1 struct]
IsSupportVector: [24075x1 logical]
Solver: 'SMO'
TTMdl is a ClassificationSVM model object representing a traditionally trained SVM model.
Convert Trained Model
Convert the traditionally trained SVM model to a binary classification linear model for incremental learning.
IncrementalMdl = incrementalLearner(TTMdl)
IncrementalMdl =
IsWarm: 1
Metrics: [1x2 table]
ClassNames: [0 1]
ScoreTransform: 'none'
Beta: [60x1 double]
Bias: -6.4280
Learner: 'svm'
IncrementalMdl is an incrementalClassificationLinear model object prepared for incremental learning using SVM.
• The incrementalLearner function Initializes the incremental learner by passing learned coefficients to it, along with other information TTMdl extracted from the training data.
• IncrementalMdl is warm (IsWarm is 1), which means that incremental learning functions can start tracking performance metrics.
• The incrementalLearner function specifies to train the model using the adaptive scale-invariant solver, whereas fitcsvm trained TTMdl using the SMO solver.
Predict Responses
An incremental learner created from converting a traditionally trained model can generate predictions without further processing.
Predict classification scores for all observations using both models.
[~,ttscores] = predict(TTMdl,feat);
[~,ilcores] = predict(IncrementalMdl,feat);
compareScores = norm(ttscores(:,1) - ilcores(:,1))
The difference between the scores generated by the models is 0.
Specify SGD Solver and Standardize Predictor Data
The default solver is the adaptive scale-invariant solver. If you specify this solver, you do not need to tune any parameters for training. However, if you specify either the standard SGD or ASGD
solver instead, you can also specify an estimation period, during which the incremental fitting functions tune the learning rate.
Load the human activity data set.
For details on the data set, enter Description at the command line.
Responses can be one of five classes: Sitting, Standing, Walking, Running, and Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).
Randomly split the data in half: the first half for training a model traditionally, and the second half for incremental learning.
n = numel(Y);
rng(1) % For reproducibility
cvp = cvpartition(n,'Holdout',0.5);
idxtt = training(cvp);
idxil = test(cvp);
% First half of data
Xtt = feat(idxtt,:);
Ytt = Y(idxtt);
% Second half of data
Xil = feat(idxil,:);
Yil = Y(idxil);
Fit an SVM model to the first half of the data. Standardize the predictor data by setting 'Standardize',true.
TTMdl = fitcsvm(Xtt,Ytt,'Standardize',true);
The Mu and Sigma properties of TTMdl contain the predictor data sample means and standard deviations, respectively.
Suppose that the distribution of the predictors is not expected to change in the future. Convert the traditionally trained SVM model to a binary classification linear model for incremental learning.
Specify the standard SGD solver and an estimation period of 2000 observations (the default is 1000 when a learning rate is required).
IncrementalMdl = incrementalLearner(TTMdl,'Solver','sgd','EstimationPeriod',2000);
IncrementalMdl is an incrementalClassificationLinear model object. Because the predictor data of TTMdl is standardized (TTMdl.Mu and TTMdl.Sigma are nonempty), incrementalLearner prepares incremental
learning functions to standardize supplied predictor data by using the previously learned moments (stored in IncrementalMdl.Mu and IncrementalMdl.Sigma).
Fit the incremental model to the second half of the data by using the fit function. At each iteration:
• Simulate a data stream by processing 10 observations at a time.
• Overwrite the previous incremental model with a new one fitted to the incoming observations.
• Store the initial learning rate and ${\beta }_{1}$ to see how the coefficients and rate evolve during training.
% Preallocation
nil = numel(Yil);
numObsPerChunk = 10;
nchunk = floor(nil/numObsPerChunk);
learnrate = [IncrementalMdl.LearnRate; zeros(nchunk,1)];
beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)];
% Incremental fitting
for j = 1:nchunk
ibegin = min(nil,numObsPerChunk*(j-1) + 1);
iend = min(nil,numObsPerChunk*j);
idx = ibegin:iend;
IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx));
beta1(j + 1) = IncrementalMdl.Beta(1);
learnrate(j + 1) = IncrementalMdl.LearnRate;
IncrementalMdl is an incrementalClassificationLinear model object trained on all the data in the stream.
To see how the initial learning rate and ${\beta }_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(2,1);
ylabel('Initial Learning Rate')
The initial learning rate jumps from 0.7 to its autotuned value after the estimation period. During training, the software uses a learning rate that gradually decays from the initial value specified
in the LearnRateSchedule property of IncrementalMdl.
Because fit does not fit the model to the streaming data during the estimation period, ${\beta }_{1}$ is constant for the first 200 iterations (2000 observations). Then, ${\beta }_{1}$ changes during
incremental fitting.
Configure Performance Metric Options
Use a trained SVM model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warm-up period, during which the updateMetricsAndFit function only fits the
model. Specify a metrics window size of 500 observations.
Load the human activity data set.
For details on the data set, enter Description at the command line
Responses can be one of five classes: Sitting, Standing, Walking, Running, and Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).
Because the data set is grouped by activity, shuffle it to reduce bias. Then, randomly split the data in half: the first half for training a model traditionally, and the second half for incremental
n = numel(Y);
rng(1) % For reproducibility
cvp = cvpartition(n,'Holdout',0.5);
idxtt = training(cvp);
idxil = test(cvp);
shuffidx = randperm(n);
X = feat(shuffidx,:);
Y = Y(shuffidx);
% First half of data
Xtt = X(idxtt,:);
Ytt = Y(idxtt);
% Second half of data
Xil = X(idxil,:);
Yil = Y(idxil);
Fit an SVM model to the first half of the data.
TTMdl = fitcsvm(Xtt,Ytt);
Convert the traditionally trained SVM model to a binary classification linear model for incremental learning. Specify the following:
• A performance metrics warm-up period of 2000 observations
• A metrics window size of 500 observations
• Use of classification error and hinge loss to measure the performance of the model
IncrementalMdl = incrementalLearner(TTMdl,'MetricsWarmupPeriod',2000,'MetricsWindowSize',500,...
'Metrics',["classiferror" "hinge"]);
Fit the incremental model to the second half of the data by using the updateMetricsAndFit function. At each iteration:
• Simulate a data stream by processing 20 observations at a time.
• Overwrite the previous incremental model with a new one fitted to the incoming observations.
• Store ${\beta }_{1}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.
% Preallocation
nil = numel(Yil);
numObsPerChunk = 20;
nchunk = ceil(nil/numObsPerChunk);
ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]);
hinge = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]);
beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)];
% Incremental fitting
for j = 1:nchunk
ibegin = min(nil,numObsPerChunk*(j-1) + 1);
iend = min(nil,numObsPerChunk*j);
idx = ibegin:iend;
IncrementalMdl = updateMetricsAndFit(IncrementalMdl,Xil(idx,:),Yil(idx));
ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:};
hinge{j,:} = IncrementalMdl.Metrics{"HingeLoss",:};
beta1(j + 1) = IncrementalMdl.Beta(1);
IncrementalMdl is an incrementalClassificationLinear model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetricsAndFit checks the
performance of the model on the incoming observations, and then fits the model to those observations.
To see how the performance metrics and ${\beta }_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(3,1);
xlim([0 nchunk]);
h = plot(ce.Variables);
xlim([0 nchunk]);
ylabel('Classification Error')
h = plot(hinge.Variables);
xlim([0 nchunk]);
ylabel('Hinge Loss')
The plot suggests that updateMetricsAndFit does the following:
• Fit ${\beta }_{1}$ during all incremental learning iterations.
• Compute the performance metrics after the metrics warm-up period only.
• Compute the cumulative metrics during each iteration.
• Compute the window metrics after processing 500 observations (25 iterations).
Input Arguments
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'Solver','scale-invariant','MetricsWindowSize',100 specifies the adaptive scale-invariant solver for objective optimization, and specifies processing 100 observations before updating the
window performance metrics.
General Options
SGD and ASGD Solver Options
Adaptive Scale-Invariant Solver Options
Performance Metrics Options
Output Arguments
More About
Incremental Learning
Adaptive Scale-Invariant Solver for Incremental Learning
Estimation Period
Standardize Data
Performance Metrics
Version History
Introduced in R2020b | {"url":"https://es.mathworks.com/help/stats/classificationsvm.incrementallearner.html","timestamp":"2024-11-06T07:49:39Z","content_type":"text/html","content_length":"186655","record_id":"<urn:uuid:7e46aad6-5cba-4374-a487-8582b907be8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00140.warc.gz"} |
How to perform point forecasting in STATA - Datapott Analytics
This article explains how to perform point forecasting in STATA, where one can generate forecast values even without performing ARIMA. Therefore, it is useful in any time series data.
Forecasting is an important part of time series analysis. A point forecast is a singular number which represents the estimate of the true but unknown value of a variable at a future date. The data
on Gross Domestic Product (GDP), Gross Fixed Capital Formation (GFC) and Private Capital Consumption (PFC) for the same time period (1997 to 2017) as considered in the previous article is used.
Performing point forecasting in STATA
Step 1: Declare data as time series
The first step is to declare the data to be time series. For this, follow the below steps.
Stata command:
This will arrange the time series data as follows.
Figure 1: Quarterly time series data of variables for point forecasting in STATA
In the figure above, the last observation is 2017. To use point forecasting, extend the data for some more years depending on which forecast values are needed using ‘tsappend’ command.
tsappend, add(3)
tsappend, last (2020Q4) tsfmt(tq)
This command will add time period until 2020.Alternatively, use the following command.
Figure 2: Extended time periods generated for point forecasting in STATA
Observations up to 2017 are ‘in-sample’ observations whereas observations after 2017 are ‘out of the sample’ observations (figure above).
Step 2: Use regression coefficients
Then use the regression coefficients with the following command.
regress gdp gfcf pfce
After the regression, use the ‘predict’ command for point forecasting, so long as the regressors are available. ‘p’ is any new variable representing GDP (since GDP is the dependent variable in the
regression model). Step 3: Use the ‘predict’ command.
predict p
In-sample and out of sample data: The command ‘predict p’ will generate forecast values for in sample observations and out-of-sample observations.
To restrict the forecasting to be in‐sample (for quarterly data), use the following command.
predict p if t<=tq(2017q4)
predict yp if t>tq(2017q4)
For the out of sample (for quarterly data) prediction use the following command. This command will give forecast values of gdp, (here in p variable) in the data set (figure below).
Figure 3: Point predicted values of GDP for point forecasting in STATA
To plot GDP and ‘p’ in graph, use the following command.
tsline gdp p
The figure below will appear.
Data plot of actual and forecasted values of GDP
The figure above represents the actual values and forecast values of GDP (p variable). The figure shows that there is a very slight difference actual GDP and forecast GDP values. It implies that in
this case, point forecasting can effectively generate forecasted values of GDP for any time period in future.
Shortcomings of point forecasting
In point forecasting, the prediction is possible at each specific point of the data series. However, one criticism of point forecasting is that it does not provide the degree of uncertainty that is
associated with the forecast. To take into account variability in forecasting, interval forecasting is applicable as an alternative to point forecasting. | {"url":"https://datapott.com/how-to-perform-point-forecasting-in-stata/","timestamp":"2024-11-04T08:39:49Z","content_type":"text/html","content_length":"110358","record_id":"<urn:uuid:e3ff04bd-2e6c-41c0-ab89-31c6f8db2779>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00609.warc.gz"} |
Q: How does this wiki treats higher dimensions?
A: A character that is stated to be N-dimensional will have their tier in accordance to that dimension. For example, a character that is stated to be 5-dimensional would be inherently rated as Low
1-C, unless this is contradicted in some way, such as the dimensions in question being portrayed as compactified.
Q: Do infinite-sized characters get infinite speed by default?
A: Yes. As seen in this blog, speed is proportional to your size at a certain point. The bigger you are the faster you naturally become. So an increase of infinity would result in infinite speed. In
addition, any kind of movement in such a scale would by definition cover an infinite distance in a finite time.
Q: Why are dimensions treated as higher infinities/uncountable infinities?
A: Spatial dimensions are defined as powers of the Reals (R; an uncountable infinite set). For example, R^0 would be a point, R^1 would be a line, R^2 would be a plane, R^3 would be tridimensional
space, R^4 would be a space-time continuum, etc. The basic construction of a space with n dimensions is a Cartesian Product, A x B, which is basically just taking all of B and attaching a copy of it
to each point of A. In the case of a space with 3 spatial dimensions and 1 temporal, this would be R x R³, which means attaching a copy of R³ to each point of R. Another example of this is how a line
has R amounts of points attached to it. In more simple terms, a space that is R^2 (for example) is to R^1 what R^1 is to R^0. That is, a difference of uncountable infinite size.
Q: What tier/speed is transcending space and time?
A: While the statement can be used for supporting evidence, by its lonesome, "transcending space-time" doesn't grant any AP or Speed Tier without further context. The reason for this is due to this
kind of statement abridging a variety of meanings of which none can be considered the default without being very arbitrary.
Q: Is destroying multiple 2-A structures a better feat than destroying one?
A: No, as said logic doesn't work when it comes to infinite sets. That is to say, a countable infinite structure multipled by countable infinity would always result in the same countable infinity. In
this fashion, we can draw the conclusion that [2-A x 2-A x 2-A x 2-A...] = 2-A.
Q: Then how can a 2-A character become "above baseline"?
A: It is possible for a 2-A character to become above baseline if they are part of scaling chain that comes from physically beating up baseline 2-As. Alternatively, the verse might state that a
multiverse may be harder to destroy than another, such as how it is portrayed in Maou Gakuin no Futekigousha.
Q: Would a character/structure that is stated or visually shown to be bigger than a 2-A structure be classified as Low 1-C?
A: Yes. When it comes to infinite sets, there is no set whose cardinality is in-between countable infinity and uncountable infinity. A 2-A structure is comprised of a countable infinite amount of
universes. For something to be bigger than such thing, it would strictly have to be an uncountable infinite amount, as otherwise it would still result in the same countable infinite cardinality.
Q: Is containing a 2-A structure Low 1-C as well then?
A: By itself, no. It might be counter-intuitive to think that containing something would not result in the container being bigger than it, but when it comes to infinite sets things are quite
different. For example, the set of rationals encompasses the set of integers (which is countable infinite) and the latter encompasses/contains the set of naturals as a subset (also countable
infinite), but the three are still all the same size. This is the reason as to why the verse needs to state or visually shown that the character/structure in question is actually bigger/larger.
What is Orthogonality?
A: Mathematically, and in simple terms, Orthogonality is the property of 2 lines that are perfectly perpendicular to each other, in other words, they intersect each other at Right Angle. In this way,
while normally, a line can be drawn on a 1-D plane or be contained in it, two orthogonal lines must require a 2-Dimensional plane to draw. For example, you cannot draw a right-angle figure in the
one-dimensional plane, you need a 2-D plane (i.e. a paper) to do so. Similarly, one cannot draw a 3rd line perpendicular to both of the previous lines simultaneously on that piece of paper, and would
instead require one to draw a line perpendicular to the 2-D plane itself, extending to 3-D Space. This process can go on infinitely, and each time, you would need to extend the additional Orthogonal
line perpendicular to the previous N-Dimensional plane, thus requiring N+1 Dimensional Space.
Is a universe with more than one Temporal Dimension Tier 1?
A: Yes. As noted above, structures that contain 3 spatial dimensions and 1 temporal (such as a conventional timeline) would be equated to R x R³ in a cartesian product, which means attaching a copy
of R³ (tri-dimensional space) to each point of R (the uncountable set that represent the time axis), resulting in timelines being comprised of uncountable infinite 3-D moments, each of which would be
akin to a static snapshot instance of the universe. As such, a space-time with three spatial dimensions and two temporal dimensions (for example) would be R x R⁴, resulting in a timeline that
contains uncountable infinite 4-D moments, thus Low 1-C 5D(and so on the more temporal dimensions you add).
Is a cosmology that contains universes each with independent time dimensions Tier 1?
A: Being an independent or unique time axis does not necessarily mean said axis is of a higher dimension compared to a conventional timeline. Much like how you can have infinite lines, each with
different directions, in a 2-D plane without the necessity of 3-D space, merely being an independent time axis with its own direction does not imply a higher dimension for each additional time axis.
However, there is indeed the case that even if those Time Axis are independent of each other in the sense of having a different direction, even if not orthogonal, then they can qualify for additional
higher Dimensions, which would be needed for those time axes to extend to, as explained here in details.
In what cases do additional Temporal Axes require Higher Dimensions?
A: Cases that qualify for additional Higher Dimensions (Tier 1) for Temporal Axes are as follows :
• Overarching Timelines/Hypertimelines (A Timeline containing another Timeline, containing another Timeline.... and so on)
• Unsynchronized Temporal (Finite and Infinite Un-synchrony)
• Temporal Acausality
All such cases are explained here in details.
Q: When is the destruction of a universe considered 3-A?
A: Refer to our Universe page. As long as you have the statement that it is a "universe" or "space-time", creating or destroying such structure would inherently be 3-A.
Q: Is moving in a timeless void considered Infinite speed?
A: No. As we know, the speed formula is defined by [S=D/T]. However, in this case, the "T" would simply not exist, which then becomes impossible to tell how much distance you are actually covering in
a quantifiable manner. You could be moving at 10 m/s, 100 m/s, 1000 m/s, etc, and there would be no way to know in such a place. This is different from moving in 0 seconds, since it's a matter of
time not existing as opposed to just being stopped.
Q: Where do Alephs scale in regards to tiering?
A: This is a trick question. Because by itself, an aleph won't scale you anywhere. It's no different than asking "What tier is the number 10?". Alephs only become a proper measurement for scaling
when used to quantify a certain object. For example, an Aleph-0 (countable infinite) amount of universes would be 2-A, while an Aleph-1 (uncountable infinite) amount of universes would be Low 1-C.
However, things get a bit different when it comes to Aleph-2 and beyond, as they would be strictly bigger than all of the spaces mentioned above, by all rigorous notions of size, regardless of what
their elements are. At this stage, the quality of the elements don't matter, only the cardinality of the whole. As such, in cases like this, an Aleph-2 (or higher) amount of anything would be High
Q: Are dimensionless characters 1-A?
A: No. This is because 1-A is all about being beyond dimensionality, and not merely lacking it. After all, just because you lack something does not mean you transcend that something. A character who
"lacks" dimensions wouldn't possess a "quality" that's too great to be categorized within dimensionality as opposed to just being outside of its influence and confines out of a lack of it. In other
words, it's a negative, reductive quality and not necessarily an additive one that would indicate some level of superiority and thus for this reason, it's not considered 1-A. However, it is possible
for a dimensionless character/structure to be 1-A if they have an established relationship of superiority over the nature of infinite or uncountable infinite dimensions, or if their non-dimensional
nature stems from being akin to a background canvas where any amount of dimensions can be inserted/contained.
Q: Is a character who "transcends dimensions" 1-A?
A: No. A character being stated to "transcend dimensions" or something similar wouldn't qualify. The reasoning for this is given how vague it is as a statement, as it doesn't really tell us anything
about how a character would "transcend" them and furthermore, it's vague on the basis that it could be metaphorical, as the term "transcend" is often used to indicate not being bound to something but
in a figurative sense. For example, a common saying is that "love transcends time", which obviously isn't meant to be literal but is meant to be a reference to how love exists everywhere and in every
time period. On top of this, there's also the fact that "transcend dimensions" also isn't pertaining to the concept of dimensionality. For this reason, it wouldn't be the same as a being who's
existence is "alien" to the concept of dimensionality and thus is beyond any extensions of it. Under the premise that "transcend dimensions" is literal, it would at most qualify for a "+ 1 reality
level" above the cosmology as opposed to straight up 1-A like how beings who possess a ontological superiority to the concept of dimensionality.
Q: Is transcending the concept of space and time 1-A?
A: A character who transcends both space and time wouldn't qualify for 1-A under the basis that most writers don't bother making a connection between the concept of space-time and dimensionality. In
fiction, these two (technically three) things tend to be separated and it would be fallacious to assert that inherently being beyond the concepts of space and time would make one 1-A. In the instance
that a writer bothers to qualify that "Space = Dimensions", then a character being able to exist beyond the concept of space and time would then qualify for 1-A as there's explicit evidence that
dimensionality is considered a fundamental part of space and thus, the character in question can be considered "alien" in relation to how many dimensions one could theoretically impose on them.
A: No. Normally, a being or structure that is regarded as apophatic would only scale a single transcendence higher than the verse's established cosmology. The reason for this is that accepting
apophasis as inherently 1-A is largely grounded on NLF, as it is no different than giving 1-A or High 1-A for a character that scales "beyond all concepts". Furthermore, something that cannot be
named or given a definition wouldn't be any tier on its own unless there is context provided that makes it scale above structures of the verse in question. However, it can be 1-A or higher if the
context of its apophasis makes it clear that it is being beyond the concept/category of dimensions.
See Also | {"url":"https://all-fiction-battles.fandom.com/wiki/FAQ","timestamp":"2024-11-05T17:10:03Z","content_type":"text/html","content_length":"225041","record_id":"<urn:uuid:2a02e553-e37d-4ea6-9d08-3c121413daf6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00000.warc.gz"} |
What are the 4 types of derivatives in finance?
What are the 4 sorts of derivatives in finance?
The commonest by-product varieties are futures, forwards, swaps, and choices.
How do derivatives make cash?
One technique for incomes earnings with derivatives is promoting (also referred to as “writing”) choices to gather premium quantities. Choices usually expire nugatory, permitting the choice vendor to
maintain your complete premium quantity.
What’s an actual life instance of derivatives?
Utility of Derivatives in Actual Life To test the temperature variation. To find out the pace or distance lined resembling miles per hour, kilometre per hour and so forth. Derivatives are used to
derive many equations in Physics. Within the research of Seismology like to search out the vary of magnitudes of the earthquake.
How do you commerce in derivatives?
First, do your analysis. Conduct the transaction by way of your buying and selling account. Choose your shares and their contracts based mostly on the quantity you may have in hand, the margin
necessities, the worth of the underlying shares, in addition to the worth of the contracts.
What are the disadvantages of derivatives?
Excessive threat concerned. Because of the vital volatility of the underlying securities costs, high-risk derivatives contracts are topic to a excessive stage of threat. Expensive options.
Time-bound. Complexity. Imaginative parts. Experience is required.
Is buying and selling derivatives the identical as shares?
Shares and derivatives defined For those who commerce shares instantly, you personal the underlying asset. It is potential to commerce shares and shares in each the lengthy and short-term. Buying and
selling derivatives entails speculating on the worth of an asset at a future cut-off date and having the ability to purchase or promote at a beforehand outlined value.
Are ETFs a by-product?
ETFs aren’t derivatives A by-product is a monetary contract whose worth relies on, or derived from, a standard safety. ETFs aren’t derivatives as a result of, like most mutual funds, they sometimes
make investments instantly within the bodily securities of their goal benchmarks.
What are 4 examples of derivatives?
In finance, there are 4 primary sorts of derivatives: ahead contracts, futures, swaps, and choices.
What are 4 makes use of of derivatives?
Producing choice capability. Avoiding fee of taxes. Offering leverage services. Hedging or assuaging dangers within the underlying. Acquiring publicity for the underlying belongings. Switching asset
allocations between the completely different lessons of belongings. Hypothesis and producing earnings.
Why do we’d like derivatives?
Derivatives are used to search out the speed of modifications of a amount with respect to the opposite amount. The equation of tangent and regular line to a curve of a operate might be calculated by
utilizing the derivatives. Spinoff of a operate can be utilized to search out the linear approximation of a operate at a given worth.
What’s a by-product in finance for dummies?
A by-product is a safety with a value that’s dependent upon or derived from a number of underlying belongings. The by-product itself is a contract between two or extra events based mostly upon the
asset or belongings. Its worth is set by fluctuations within the underlying asset.
What are the three 3 methods wherein derivatives can be utilized?
Buyers sometimes use derivatives for 3 causes—to hedge a place, to extend leverage, or to take a position on an asset’s motion.
What are probably the most used monetary derivatives?
5 of the extra fashionable derivatives are choices, single inventory futures, warrants, a contract for distinction, and index return swaps. Choices let buyers hedge threat or speculate by taking over
extra threat.
What are the fundamentals of derivatives?
Spinoff is a product whose worth is derived from the worth of a number of primary variables, known as bases (underlying asset, index, or reference charge), in a contractual method. The underlying
asset might be fairness, foreign exchange, commodity or every other asset.
How will you clarify by-product in layman’s time period?
Definition: A by-product is a contract between two events which derives its worth/value from an underlying asset. The commonest sorts of derivatives are futures, choices, forwards and swaps.
Description: It’s a monetary instrument which derives its worth/value from the underlying belongings.
Does Warren Buffett use derivatives?
When requested about these positions in an interview on the monetary information channel CNBC quickly after this letter was revealed, Buffett acknowledged: “I-well, we have used derivatives for a lot
of, a few years. I do not suppose derivatives are evil, per se, I believe they’re harmful. I’ve all the time mentioned they’re harmful.
Who advantages from derivatives?
Buyers use derivatives to switch funding portfolio money flows, replicate funding technique returns in money markets, and create exposures unavailable to money market contributors.
The right way to calculate derivatives?
Discover f(x + h). Plug f(x + h), f(x), and h into the restrict definition of a by-product. Simplify the distinction quotient. Take the restrict, as h approaches 0, of the simplified distinction
What are 5 examples of derivatives?
Ahead Contracts. Future Contracts. Choices Contracts. Swap Contracts.
How is a improvement financed?
Growth finance is a short-term funding choice, often for between 6-24 months. It’s designed particularly to help with the acquisition prices and construct prices related to a residential or business
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Comprehensive Guide on Histograms
1. Basics of statistics
2. Basics of probability theory
3. Random variables
4. Point estimation
5. Discrete probability distributions
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Comprehensive Guide on Histograms
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What is a histogram?
A histogram is a diagram that illustrates the distribution of a given set of values. For instance, consider the following set of values:
To draw a histogram, we first need to construct a frequency table, which is a table that shows how many times each value appears:
Value Frequency
Note the following:
• the value 2 appears once.
• the value 3 appears twice.
• and so on.
We can then draw our histogram:
Since the y-axis is the frequency, this histogram is called the frequency-histogram. As we shall explore later, there are other types of histograms that have a different y-axis. In a
frequency-histogram, the height of each bar indicates the frequency of the corresponding value. For instance, we can see that the value 6 is the most frequent value whereas the value 7 does not
appear at all.
The above histogram shows the count of each value, but we can also perform binning, which is a process that involves categorizing values into intervals or so-called bins:
Interval Frequency
[2,3] 3
[4,5] 4
[6,7] 4
[8,9] 1
Here, [2,3] is the closed interval from 2 to 3, which means values between 2 and 3 (both ends inclusive) will fall into this interval. The width of each interval in this case is 1, or we can
equivalently say that the bin width or bin size is 1. Let's now plot the frequency-histogram:
Instead of labelling the bars using interval notation like [2,3], we often indicate the mid point of each interval instead:
Note that this does not mean that the value 2.5 appears 3 times. The correct interpretation here is that there are 3 values between the closed interval $[2.5-0.5,\;2.5+0.5]$.
Drawbacks of histograms
Histogram shape is dependent on bin width
One of the biggest drawbacks about the histograms is that the diagram is heavily dependent on the bin width we choose. For instance, consider the following raw frequency histogram without any
Let's perform binning with bin widths 1 and 2 respectively:
Bin width = 1 Bin width = 2
We can see that both histograms now look different from the frequency without binning. The histogram with bin width equal to 1 hints that the distribution of our values is symmetric with the values
equally spread out. In contrast, the histogram with bin width equal to 2 suggests the opposite - the distribution is skewed with the bulk of the values to the left. Even though these histograms are
based on the same data points, they describe the distribution differently because of the difference in their bin width!
Suppose we have a two sets of values with the following distributions:
$\boldsymbol{x}$ $\boldsymbol{x}$ shifted to the right by $0.2$
The distributions are identical because the second set of values (right) was created by adding 0.2 to every value in first set (right). Let's now plot their frequency histogram both with bin with of
Histogram of $\boldsymbol{x}$ Histogram of $\boldsymbol{x}+0.2$
We can see that even though their underlying distribution is identical, their frequency histograms are completely different. This happens because values such as $2.9$ shift to $3.1$, which means all
the $2.9$s will end up in the $[3,4]$ bin instead of the $[2,3]$ bin. Therefore, the histogram may not always be an accurate representation of the underlying distribution.
Relative frequency histograms
Recall that the y-axis of frequency histograms is the frequency of the values. For instance, the frequency histogram of a set of 10 values with bin width 1 may look like follows:
Relative frequency histograms uses the relative frequency as the y-axis instead. Relative frequencies are defined as the frequency of value divided by the total number of values. For instance, the
value 5 occurs twice and the total number of values we have is 10. Therefore, the relative frequency of the value 5 is calculated as:
$$\text{Relative frequency}= \frac{2}{10}=0.2$$
We can interpret this as meaning 20% of the value are 5. We can easily compute the relative frequencies for the other values and plot the relative frequency histogram:
Note the following:
• the shape of the relative frequency histogram will always be the same as the shape of the frequency histogram because all we're doing is dividing each frequency by a fixed number of values.
• the frequencies now add up to one. In this case:
Advantages of relative frequency histograms
You may be wondering why bother computing the relative frequencies when the raw frequencies already describe the distribution. There are several reasons:
• we're often interested in how often a value occurs as a ratio to the total number of values. For instance, knowing that a value appears 20% of the time in our set of values may be more meaningful
than knowing that the value appears 300 times.
• relative frequencies are normalized such that the number of data points do not matter. For instance, suppose we have two samples taken from a population, but one sample is 100 times more data
observations than the other. Suppose the frequency histograms of the two samples are as follows:
Histogram of smaller sample Histogram of larger sample
Notice how the y-axis (frequency) is not in the same scale so we cannot directly compare the two histograms. The relative frequency normalizes the frequency scale such that we will be able to
make more direct comparisons.
Density histograms
Now we wish to normalize the relative frequency histogram such that the areas of the rectangles add up to one. In this case, because the bin width is equal to one, the sum of the rectangles' areas is
equal to one. Let's show this mathematically - let $\mathrm{RF}_i$ be the relative frequency of the $i$-th bin and $h$ be the bin width:
$$\begin{align*} \sum^6_{i=1}(\mathrm{RF}_i)(h)&= (\mathrm{RF}_1)(h)+(\mathrm{RF}_2)(h)+(\mathrm{RF}_3)(h)+(\mathrm{RF}_4)(h)+(\mathrm{RF}_5)(h)\\ &=(0.1)(1)+(0.2)(1)+(0.3)(1)+(0.1)(1)+(0.2)(1)+(0.1)
(1)\\ &=1 \end{align*}$$
When the bin width is not equal to one, then the areas may not add up to one. For instance, consider the case when the bin width is 2:
In this case, the sum of areas of the rectangles is:
$$\begin{align*} \sum^3_{i=1}(\mathrm{RF}_i)(h)&= (\mathrm{RF}_1)(h)+(\mathrm{RF}_2)(h)+(\mathrm{RF}_3)(h)\\ &=(0.3)(2)+(0.4)(2)+(0.3)(2)\\ &=2 \end{align*}$$
Is it a coincidence that the summed area is equal to the bin width 2? The answer is not at all - we can mathematically show that the summed area is always equal to the bin width:
$$\begin{align*} \sum^3_{i=1}(\mathrm{RF}_i)(h)&= (\mathrm{RF}_1)(h)+(\mathrm{RF}_2)(h)+(\mathrm{RF}_3)(h)\\ &=h(\mathrm{RF}_1+\mathrm{RF}_2+\mathrm{RF}_3)\\ &=h(1)\\ &=h \end{align*}$$
Here, note the following:
• we used the fact that the sum of the relative frequencies add up to one.
• the number of bins is 3 but we can easily see that this generalizes to n number of bins!
This means that in order to get the area to equal one, we can divide the relative frequencies by the bin width $h$:
$$\begin{align*} \frac{1}{h}\sum^3_{i=1}(\mathrm{RF}_i)(h)=1 \;\;\;\;\;\;\;\Longleftrightarrow\;\;\;\;\;\;\; \sum^3_{i=1}\Big(\frac{1}{h}\cdot\mathrm{RF}_i\Big)(h)=1 \end{align*}$$
The relative frequency $\mathrm{RF}_i$ divided by the bin width $h$ gives us the so-called density $D_i$ of the bin:
$$D_i=\frac{1}{h}(\mathrm{RF}_i) \;\;\;\;\implies\;\;\;\; \sum^3_{n=1}D_i(h)=1$$
In our case, the bin width is equal 2 so we simply halve the relative frequencies to obtain the densities:
The key difference between relative frequency and densities is that:
• adding up the relative frequencies of every bin will equal one. This is not true for densities.
• adding up the areas of the rectangles will equal one when the y-axis is density. This is not true for relative frequencies.
Drawing histograms using Python
Plotting histograms is extremely easy using Python's matplotlib graphing library.
Raw frequency histogram
Consider the following data points:
xs = [3,4,4,5,5,5,5,6]
To draw a raw frequency histogram, use the bar(~) method:
# Obtain the unique values as well as their counts
unique, counts = np.unique(xs, return_counts=True)
plt.bar(unique, counts)
This generates the following plot:
Binned frequency histogram
To plot binned frequency histograms, use the hist(~) method. By default, matplotlib will use 10 equal-width bins:
xs = [3,4,4,5,5,5,5,6]
plt.ylabel('Frequency', fontsize=13)
We get the following frequency histogram:
Specifying custom bins
We can specify custom bins by specifying the bins parameter in the hist(~) method.
xs = [3,4,4,5,5,5,5,6]
plt.hist(xs, bins=[3,5,6])
Here, we are specifying two bins:
• the first bin is $[3,5)$, which represents the interval between 3 (inclusive) and 5 (exclusive). The bin width is therefore 2.
• the second bin is $[5,6]$, which represents the interval between 5 (inclusive) and 6 (inclusive). The bin width is therefore 1.
This generates the following plot:
Specifying number of equal-sized bins
We can also pass an integer instead of a list to the bins argument of the hist(~) method. In this case, the integer will be parsed as the desired number of equal-sized bins.
xs = [3,4,4,5,5,5,5,6]
plt.hist(xs, bins=2) # Two equal-width bins!
This produces the following plot:
Styling the histogram
Here's some basic styling techniques for histograms:
xs = [3,4,4,5,5,5,5,6]
plt.hist(xs, bins=2, edgecolor='blue', color='yellow', linewidth=2)
This generates the following beautiful plot:
Final remarks
On the surface, histograms may seem simple at first, there's a lot to discuss:
• the effects of bin size of the resulting shape of the histogram.
• strange cases such as when two identically distributions appear to have entirely different histograms.
• other variants such as relative frequency histograms and density histograms.
There's also a whole field of research on how to pick the optimal bin widths, and so histograms are surprisingly deep!
We have demonstrated how we should not rely on a single histogram to make conclusions about the underlying distribution of the our data points. As a general practise, we should:
• plot multiple histograms with different bin widths.
• plot kernel density estimates on top of histograms to visualize the distribution better. I will eventually write a comprehensive guide on kernel density estimation soon, so please join our
newsletteropen_in_new to be notified when I hit publish!
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Random Integer Range Generator | Random Integer Ranges
Random Integer Range Generator
Generate random integer ranges with our online tool. Create randomized sets of integer ranges for testing, simulations, and data generation.
Random Integer Range Generator
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Mathematics Level 9 - Crossword Quiz Net
Find the solutions to the Crossword Quiz Mathematics Level 9. The answers have been split into Cross and Down answers to help you navigate through them easily. Wrong grid? Look for the right one
Cross Answers
1A: Not including the cue ball, how many balls are used in a game of pool
7A: To make as large as possible
8A: Specific input in a function, also known as an independent variable
Down Solutions
3D: Situation involving chance or probability that leads to results called outcomes
4D: What did the zero say to the eight?
Nice Belt
5D: This word means a brief statement or account of the main points of something
Full Grid Answers
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Quadratic Word Problems Worksheet
Quadratic Word Problems Worksheet
Solve quadratic equations by completing the square. Plus each one comes with an answer key.
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Math 2 unit 2 2 quadratic word problems name.
Quadratic word problems worksheet. Writing and evaluating expressions worksheet. Showing top 8 worksheets in the category quadratic word problems. Nature of the roots of a quadratic.
A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km h more than its original speed. Enjoy these free sheets. Fri feb 25
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If it takes 3 hours to complete the total journey what is its original average speed. Unlucky s company can be represented by the equation y 3×2 18x 4 where y is the amount of profit in hundreds of
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Writing and evaluating expressions worksheet. Wed feb 23 2 3 quadratic word problems handout day 1. You can use any of these methods.
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Read each word problem formulate a quadratic equation and solve for the unknown.
Quadratic word problems worksheet with answers. He realizes his company is on the downturn and wishes to sell before he ends up in. Solve quadratic equations by factoring.
Nature of the roots of a quadratic. Find two consecutive integers that have a product of 42. Complete each word problem using techniques learned in previous concepts.
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Variance Metric
A Variance Metric is a dispersion metric that represents the dispersion of a continuous dataset.
• http://en.wikipedia.org/wiki/Variance
□ QUOTE: In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing
how far the numbers lie from the mean (expected value). In particular, the variance is one of the moments of a distribution. In that context, it forms part of a systematic approach to
distinguishing between probability distributions. While other such approaches have been developed, those based on moments are advantageous in terms of mathematical and computational
The variance is a parameter describing in part either the actual probability distribution of an observed population of numbers, or the theoretical probability distribution of a sample (a
not-fully-observed population) of numbers. In the latter case a sample of data from such a distribution can be used to construct an estimate of its variance: in the simplest cases this
estimate can be the sample variance, defined below.
• http://en.wikipedia.org/wiki/Variance#Definition
□ QUOTE: If a random variable X has the expected value (mean) μ = E[X], then the variance of X is given by: :[math]\displaystyle{ \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \
right]. \, }[/math]
That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. This definition encompasses random variables that are
discrete, continuous, or neither (or mixed). It can be expanded as follows: :[math]\displaystyle{ \operatorname{Var}(X) &= \operatorname{E}\left[(X - \mu)^2 \right] \\ &= \operatorname{E}\
left[X^2 - 2\mu X + \mu^2 \right] \\ &= \operatorname{E}\left[X^2 \right] - 2\mu\,\operatorname{E}[X] + \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - 2\mu^2 + \mu^2 \\ &= \operatorname{E}\
left[X^2 \right] - \mu^2 \\ &= \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2. }[/math]
A mnemonic for the above expression is "mean of square minus square of mean".
The variance of random variable X is typically designated as Var(X), [math]\displaystyle{ \scriptstyle\sigma_X^2 }[/math], or simply σ^2 (pronounced “sigma squared").
• (Lord et al., 2005) ⇒ Dominique Lord, Simon P. Washington, and John N. Ivan. (2005). “Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical
fit and theory.” In: Accident Analysis & Prevention, 37(1). doi:10.1016/j.aap.2004.02.004
□ QUOTE: The mean and variance of the binomial distribution are [math]\displaystyle{ E(Z) = Np }[/math] and [math]\displaystyle{ VAR(Z) = Np(1-p) }[/math] respectively. | {"url":"https://www.gabormelli.com/RKB/variance","timestamp":"2024-11-02T12:23:10Z","content_type":"text/html","content_length":"44867","record_id":"<urn:uuid:271488cd-cc8a-4985-844e-b8cd954391d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00342.warc.gz"} |
is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is
called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
The above text is a snippet from Wikipedia: Circle
and as such is available under the Creative Commons Attribution/Share-Alike License.
1. A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.
The set of all points (x, y) such that (x-1)^2+y^2 = r^2 is a circle of radius r around the point (1, 0).
2. A two-dimensional geometric figure, a disk, consisting of the set of all those points of a plane at a distance less than or equal to a fixed distance from another point.
3. Any thin three-dimensional equivalent of the geometric figures.
Put on your dunce-cap and sit down on that circle.
4. A curve that more or less forms part or all of a circle.
move in a circle
5. A specific group of persons.
inner circle
circle of friends
6. A line comprising two semicircles of 30 yards radius centred on the wickets joined by straight lines parallel to the pitch used to enforce field restrictions in a one-day match.
7. A ritual circle that is cast three times deosil and closes three times widdershins either in the air with a wand or literally with stones or other items used for worship.
8. A traffic circle or roundabout.
9. Compass; circuit; enclosure.
10. An instrument of observation, whose graduated limb consists of an entire circle. When fixed to a wall in an observatory, it is called a mural circle; when mounted with a telescope on an axis and
in Y's, in the plane of the meridian, a meridian or transit circle; when involving the principle of reflection, like the sextant, a reflecting circle; and when that of repeating an angle several
times continuously along the graduated limb, a repeating circle.
11. A series ending where it begins, and repeating itself.
12. A form of argument in which two or more unproved statements are used to prove each other; inconclusive reasoning.
13. Indirect form of words; circumlocution.
14. A territorial division or district.
The ten Circles of the Holy Roman Empire were those principalities or provinces which had seats in the German Diet.
1. To travel around along a curved path.
2. To surround.
3. To place or mark a circle around.
Circle the jobs that you are interested in applying for.
4. To travel in circles.
Vultures circled overhead.
The above text is a snippet from Wiktionary: circle
and as such is available under the Creative Commons Attribution/Share-Alike License.
Need help with a clue?
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