2023第十二届认证杯数学中国数学建模国际赛（小美赛）赛题

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2023
Certificate Authority Cup International Mathematical Contest Modeling
http://mcm.tzmcm.cn
Problem A (MCM)
Sunspot Forecasting
Sunspots are phenomena on the Sun’s photosphere that appear as temporary
spots that are darker than the surrounding areas. They are regions of reduced
surface temperature caused by concentrations of magnetic flux that inhibit con
vection. Sunspots appear within active regions, usually in pairs of opposite
magnetic polarity. Their number varies according to the approximately 11-year
solar cycle.
Individual sunspots or groups of sunspots may last anywhere from a few
days to a few months, but eventually decay. Sunspots expand and contract as
they move across the surface of the Sun, with diameters ranging from 16 km (10
mi)[1] to 160,000 km (100,000 mi). Some larger sunspots can be visible from
Earth without a telescope[2]. They may travel at relative speeds, or proper
motions, of a few hundred meters per second when they first emerge.
Solar cycles last typically about eleven years, varying from just under 10
to just over 12 years. The point of highest sunspot activity during a cycle is
known as solar maximum, and the point of lowest activity as solar minimum.
This period is also observed in most other solar activity and is linked to a
variation in the solar magnetic field that changes polarity with this period.
Sunspot numbers also change over long periods. For example, during the pe
riod known as the modern maximum from 1900 to 1958 the solar maxima trend
of sunspot count was upwards; for the following 60 years the trend was mostly
downwards[3]. Overall, the Sun was last as active as the modern maximum over
8,000 years ago[4].
Due to their correlation with other kinds of solar activity, sunspots can be
used to help predict space weather, the state of the ionosphere, and conditions
relevant to short-wave radio propagation or satellite communications. Many
models based on time series analysis, spectral analysis, and neural networks
have been used to predict sunspot activity, but often with poor results. This
may be related to the fact that most prediction models are phenomenology at
the data level. Although we generally know the length of the solar activity cycle,
the cycle is not completely stable, the maximum intensity of the activity varies
1with time, and the time of the peak and the duration of the peak are difficult
to predict accurately.
We need to forecast sunspots, and usually we need the results to be aver
aged out on a monthly basis. You and your team are asked to develop reason
able mathematical models to make as credible a forecast of sunspots as possi
ble. Relevant observational data are publicly available at many observatories as
well as space science research organizations, including the historical number of
sunspots, the area of sunspotsas well as observations of other indicators that may
be relevant. See for example (not limited to) https://www.sidc.be/SILSO/
datafiles/ and http://solarcyclescience.com/activeregions.html
Tasks:
1. Please forecast the start and end of the current and next solar cycle;
2. Please predict the time of onset and duration of solar maximum for the
next solar cycle;
3. Predict the number and area of sunspots in the current and next solar
cycle and explain the reliability of your model in your paper.
References
[1] https://soho.nascom.nasa.gov/explore/lessons/sunspots6_8.html
[2] Mossman, J. E. A comprehensive search for sunspots without the aid of a
telescope, 1981-1982. Royal Astronomical Society, Quarterly Journal, vol.
30: 59-64, 1989.
[3] https://www.sidc.be/html/wolfaml.html
[4] Solanki SK; Usoskin IG; Kromer B; Schssler M; et al. Unusual activity
of the Sun during recent decades compared to the previous 11,000 years.
Nature, 431 (7012): 10841087, 2004.


[size=15.9403px]2023
[size=15.9403px]Certificate Authority Cup International Mathematical Contest Modeling
[size=15.9403px]http://mcm.tzmcm.cn
[size=15.9403px]Problem B (MCM)
[size=15.9403px]Industrial Surface Defect Detection
[size=15.9403px]Surface defects in metal or plastic products not only affect the appearance of the
[size=15.9403px]product, but may also cause serious damage to the performance or durability of
[size=15.9403px]the product. Automated surface-anomaly detection has become an interesting
[size=15.9403px]and promising area of research, with a very high and direct impact on the
[size=15.9403px]application domain of visual inspection[1]. Kolektor Group provided a dataset
[size=15.9403px]of images of defective production items[2], and we would like to use this dataset
[size=15.9403px]as an example to investigate a mathematical model for automatic detection of
[size=15.9403px]product surface defects through photographs.
[size=15.9403px]Domen Tabernik, Matic ˇSuc, and Danijel Skoˇcaj have built a model for
[size=15.9403px]detecting surface defects using deep learning[3], which is claimed to be able to
[size=15.9403px]provide good discrimination even with a small amount of training. However,
[size=15.9403px]our problem at this point is slightly different; first, we want our model to be
[size=15.9403px]deployable on inexpensive handheld devices. Such devices have only very limited
[size=15.9403px]storage space and computational power, so the model is very demanding in terms
[size=15.9403px]of the amount of computation as well as the storage space required. Second,
[size=15.9403px]since this dataset does not encompass all defect patterns, we would like the
[size=15.9403px]model to have relatively good generalization capabilities when other defect types
[size=15.9403px]are encountered as well. You and your team are asked to build easy-to-use
[size=15.9403px]mathematical models to accomplish the following tasks.
[size=15.9403px]Tasks:
[size=15.9403px]1. Determine whether surface defects appear in a photograph, and measure
[size=15.9403px]the amount of computation and storage space required for your model to
[size=15.9403px]do so;
[size=15.9403px]2. Automatically label the locations or areas where surface defects appear,
[size=15.9403px]and measure the amount of computation, storage space, and labeling accuracy
[size=15.9403px]required by your model.
[size=15.9403px]3. Please clarify the generalization capability of your model, i.e. why is your
[size=15.9403px]model still feasible if you encounter defect types that are not exactly the
[size=15.9403px]same as those in the dataset.
References
[1] Domen Tabernik, Matic ˇSuc, and Danijel Skoˇcaj. Automated detection and
segmentation of cracks in concrete surfaces using joined segmentation and
classification deep neural network, Sep 2023.
[2] https://www.vicos.si/resources/kolektorsdd/.
[3] Domen Tabernik, Samo ˇSela, Jure Skvarˇc, and Danijel Skoˇcaj.
Segmentation-based deep-learning approach for surface-defect detection,
Mar 2020.



[size=13.2835px]2023
[size=13.2835px]Certificate Authority Cup International Mathematical Contest Modeling
[size=13.2835px]http://mcm.tzmcm.cn
[size=13.2835px]Problem C (ICM)
[size=13.2835px]Avalanche Prevention
[size=13.2835px]Avalanches are a supremely dangerous phenomenon. Nowadays, we have a good
[size=13.2835px]understanding of how avalanches form. However, we cannot yet predict in detail
[size=13.2835px]exactly why, when and where an avalanche will be triggered[1]. Villages and
[size=13.2835px]roads can be protected from avalanches in a variety of ways. Refraining from
[size=13.2835px]building in vulnerable areas, preventing avalanche formation by planting forestry
[size=13.2835px]or erecting barriers, minimising avalanche impact by means of protective structures
[size=13.2835px]such as snow sheds, and artificially triggering avalanches using explosives
[size=13.2835px]before too much snow has accumulated are just a few of the possibilities[2].
[size=13.2835px]We are now focusing on the use of explosives to trigger artificial small-scale
[size=13.2835px]avalanches. What needs to be determined is the appropriate timing for triggering
[size=13.2835px]the explosions and the relevant parameters. While the use of more explosives
[size=13.2835px]provides better personal safety, it disrupts the normal life of the resident animals
[size=13.2835px]in these areas. When human safety is involved, making the slides safer
[size=13.2835px]by artificially triggering avalanches is far-reaching in that respect. But the Nature
[size=13.2835px]Conservancy does not agree that artificially triggering avalanches over large
[size=13.2835px]areas, especially in ski areas, has an increasingly negative impact on animals.
[size=13.2835px]Moreover, when snow falls on warm ground, it is compressed by strong winds
[size=13.2835px]and becomes hard[3]. The snow is becoming more and more solid since it has
[size=13.2835px]been subjected to widespread heavy snowfall and strong winds, making the success
[size=13.2835px]rate lower and lower. That’s why we need you and your team to build
[size=13.2835px]sound models to study this problem.
[size=13.2835px]Tasks:
[size=13.2835px]1. Find useful and easily measurable parameters to measure the risk of
[size=13.2835px]avalanches occurring.
[size=13.2835px]2. For a slope at risk of avalanches, we need that a simple field survey will
[size=13.2835px]make it possible to determine the proper timing of the use of blasting to
[size=13.2835px]induce small avalanches, the placement of explosives, and the appropriate
[size=13.2835px]blasting power.
[size=13.2835px]Note: In studying the above problem, if the parameters of snowy environment
[size=13.2835px]are involved, please find the required data by yourself. Alternatively, you
[size=13.2835px]may calculate some virtual examples in your paper, but you should to give a
[size=13.2835px]reasonable definition of the required parameters and a realizable, low-cost measurement
[size=13.2835px]method. So that we can implement the measurement according to
[size=13.2835px]your measurement scheme and give the final result.
[size=13.2835px]References
[size=13.2835px][1] https://www.wsl.ch/en/snow-and-ice/
[size=13.2835px][2] https://www.slf.ch/en/avalanches/
[size=13.2835px][3] Louchet, Francois. Snow Avalanches. Oxford University Press. 2021.


2023
Certificate Authority Cup International Mathematical Contest Modeling
http://mcm.tzmcm.cn
Problem D (ICM)
The Twilight Factor of a Telescope
When we use an ordinary optical telescope to observe a distant target in dim
light, the larger the entrance aperture the more light that enters the binoculars.
The larger the magnification of the telescope, the narrower the field of view and
the darker the image appears. But the higher the magnification, the larger the
target appears and the more detail can be observed[1]. We need a comparative
value for the suitability of binoculars when less light is available. Zeiss uses an
empirical formula called the twilight factor, which is defined like this[2]:
TF = √m × d ,
which m is the magnification and d is the lens diameter (in mm).
Twilight Factor is a number used to compare the effectiveness of binoculars
or spotting scopes used in low light. The larger the twilight factor, the more
detail you can see in low light. However, the twilight factor can also be misleading
as shown in the following example: two binoculars, 8 x 56 and 56 x 8 (such
a model does not exist but would be feasible theoretically), have the identical
twilight factor of 21.2. While an 8 x 56 model is ideal during twilight, a 56 x 8
pair would be totally unusable even during the day[3].
We would like to have a more useful metric that expresses the performance of
the telescope in low light and uses only the basic parameters. This would provide
a specification reference for telescope selection. More detailed metrics reflecting
image quality are beyond our discussion, such as contrast, transmission, color
rendition etc.
Tasks:
1. Please consider the visual properties of human eyes under dim light and establish
a reasonable model to propose the algorithm of twilight coefficient
applicable to binoculars for direct observation by human eyes.
2. If the visual receptor is not the human eye but a cmos video recording
device, please consider the sensing characteristics of cmos under dim light
and build a reasonable mathematical model to propose the twilight coefficient
algorithm for lenses applicable to cmos video recording.
Note: In studying the above problem, if the performance parameters of photoreceptors
are involved, please find the required data by yourself. Alternatively,
you may calculate some virtual examples in your paper, but you should to give
a reasonable definition of the required parameters and a realizable, low-cost
measurement method. So that we can implement the measurement according
to your measurement scheme and give the final result.
References
[1] https://www.celestron.com/blogs/knowledgebase/
what-determines-the-brightness-of-the-image-in-my-binoculars
[2] https://www.celestron.com/blogs/knowledgebase/
what-is-twilight-factor-and-how-do-i-calculate-it
[3] https://blogs.zeiss.com/sports-optics/hunting/en/
twilight-factor/