text:
## Combinatorial theorems in sparse random sets.(English)Zbl 1351.05204

Summary: We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán’s theorem, Szemerédi’s theorem and Ramsey’s theorem, hold almost surely inside sparse random sets. For instance, we extend Turán’s theorem to the random setting by showing that for every $$\epsilon > 0$$ and every positive integer $$t \geq 3$$ there exists a constant $$C$$ such that, if $$G$$ is a random graph on $$n$$ vertices where each edge is chosen independently with probability at least $$C n^{-2/(t+1)}$$, then, with probability tending to 1 as $$n$$ tends to infinity, every subgraph of $$G$$ with at least $$\left(1 - \frac{1}{t-1} + \epsilon\right) e(G)$$ edges contains a copy of $$K_t$$. This is sharp up to the constant $$C$$. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by M. Schacht [Ann. Math. (2) 184, No. 2, 333–365 (2016; Zbl 1351.05207)] and by E. Friedgut et al. [Random Struct. Algorithms 37, No. 4, 407–436 (2010; Zbl 1228.05284)].

### MSC:

 05C80 Random graphs (graph-theoretic aspects) 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 05C42 Density (toughness, etc.) 05D10 Ramsey theory

### Citations:

Zbl 1228.05284; Zbl 1351.05207
Full Text:

### References:

 [1] J. Balogh, R. Morris, and W. Samotij, ”Independent sets in hypergraphs,” J. Amer. Math. Soc., vol. 28, iss. 3, pp. 669-709, 2015. · Zbl 1310.05154 [2] V. Bergelson and A. Leibman, ”Polynomial extensions of van der Waerden’s and Szemerédi’s theorems,” J. Amer. Math. Soc., vol. 9, iss. 3, pp. 725-753, 1996. · Zbl 0870.11015 [3] B. Bollobás, Extremal Graph Theory, Mineola, NY: Dover Publications, 2004. · Zbl 1099.05044 [4] D. De Caen and Z. Füredi, ”The maximum size of 3-uniform hypergraphs not containing a Fano plane,” J. Combin. Theory Ser. B, vol. 78, iss. 2, pp. 274-276, 2000. · Zbl 1027.05073 [5] D. Conlon, ”Combinatorial theorems relative to a random set,” in Proceedings of the International Congress of Mathematicians 2014, Vol. 4, , pp. 303-328. · Zbl 1373.05175 [6] D. Conlon, W. T. Gowers, W. Samotij, and M. Schacht, ”On the KŁR conjecture in random graphs,” Israel J. Math., vol. 203, iss. 1, pp. 535-580, 2014. · Zbl 1303.05175 [7] P. ErdHos, ”On the number of complete subgraphs contained in certain graphs,” Magyar Tud. Akad. Mat. Kutató Int. Közl., vol. 7, pp. 459-464, 1962. · Zbl 0116.01202 [8] P. ErdHos and M. Simonovits, ”Supersaturated graphs and hypergraphs,” Combinatorica, vol. 3, iss. 2, pp. 181-192, 1983. · Zbl 0529.05027 [9] P. Frankl and V. Rödl, ”Large triangle-free subgraphs in graphs without $$K_4$$,” Graphs Combin., vol. 2, iss. 2, pp. 135-144, 1986. · Zbl 0596.05037 [10] E. Friedgut, ”Sharp thresholds of graph properties, and the $$k$$-SAT problem,” J. Amer. Math. Soc., vol. 12, iss. 4, pp. 1017-1054, 1999. · Zbl 0932.05084 [11] E. Friedgut, V. Rödl, A. Ruciński, and P. Tetali, ”A sharp threshold for random graphs with a monochromatic triangle in every edge coloring,” Mem. Amer. Math. Soc., vol. 179, iss. 845, p. vi, 2006. · Zbl 1087.05052 [12] E. Friedgut, V. Rödl, and M. Schacht, ”Ramsey properties of random discrete structures,” Random Structures Algorithms, vol. 37, iss. 4, pp. 407-436, 2010. · Zbl 1228.05284 [13] Z. Füredi, ”Random Ramsey graphs for the four-cycle,” Discrete Math., vol. 126, iss. 1-3, pp. 407-410, 1994. · Zbl 0792.05128 [14] Z. Füredi and M. Simonovits, ”Triple systems not containing a Fano configuration,” Combin. Probab. Comput., vol. 14, iss. 4, pp. 467-484, 2005. · Zbl 1075.05062 [15] H. Furstenberg and Y. Katznelson, ”An ergodic Szemerédi theorem for commuting transformations,” J. Analyse Math., vol. 34, pp. 275-291 (1979), 1978. · Zbl 0426.28014 [16] S. Gerke, Y. Kohayakawa, V. Rödl, and A. Steger, ”Small subsets inherit sparse $$\epsilon$$-regularity,” J. Combin. Theory Ser. B, vol. 97, iss. 1, pp. 34-56, 2007. · Zbl 1111.05090 [17] S. Gerke, H. J. Prömel, T. Schickinger, A. Steger, and A. Taraz, ”$$K_4$$-free subgraphs of random graphs revisited,” Combinatorica, vol. 27, iss. 3, pp. 329-365, 2007. · Zbl 1136.05067 [18] S. Gerke, T. Schickinger, and A. Steger, ”$$K_5$$-free subgraphs of random graphs,” Random Structures Algorithms, vol. 24, iss. 2, pp. 194-232, 2004. · Zbl 1035.05084 [19] W. T. Gowers, ”A new proof of Szemerédi’s theorem,” Geom. Funct. Anal., vol. 11, iss. 3, pp. 465-588, 2001. · Zbl 1028.11005 [20] W. T. Gowers, ”Quasirandomness, counting and regularity for 3-uniform hypergraphs,” Combin. Probab. Comput., vol. 15, iss. 1-2, pp. 143-184, 2006. · Zbl 1082.05081 [21] W. T. Gowers, ”Hypergraph regularity and the multidimensional Szemerédi theorem,” Ann. of Math., vol. 166, iss. 3, pp. 897-946, 2007. · Zbl 1159.05052 [22] W. T. Gowers, ”Decompositions, approximate structure, transference, and the Hahn-Banach theorem,” Bull. Lond. Math. Soc., vol. 42, iss. 4, pp. 573-606, 2010. · Zbl 1233.05198 [23] R. Graham, V. Rödl, and A. Ruciński, ”On Schur properties of random subsets of integers,” J. Number Theory, vol. 61, iss. 2, pp. 388-408, 1996. · Zbl 0880.05081 [24] B. Green and T. Tao, ”The primes contain arbitrarily long arithmetic progressions,” Ann. of Math., vol. 167, iss. 2, pp. 481-547, 2008. · Zbl 1191.11025 [25] M. Hamel and I. Łaba, ”Arithmetic structures in random sets,” Integers, vol. 8, p. 04, 2008. · Zbl 1195.11037 [26] P. E. Haxell, Y. Kohayakawa, and T. Łuczak, ”Turán’s extremal problem in random graphs: forbidding even cycles,” J. Combin. Theory Ser. B, vol. 64, iss. 2, pp. 273-287, 1995. · Zbl 0828.05056 [27] P. E. Haxell, Y. Kohayakawa, and T. Łuczak, ”Turán’s extremal problem in random graphs: forbidding odd cycles,” Combinatorica, vol. 16, iss. 1, pp. 107-122, 1996. · Zbl 0853.05072 [28] S. Janson, ”Poisson approximation for large deviations,” Random Structures Algorithms, vol. 1, iss. 2, pp. 221-229, 1990. · Zbl 0747.05079 [29] S. Janson, K. Oleszkiewicz, and A. Ruciński, ”Upper tails for subgraph counts in random graphs,” Israel J. Math., vol. 142, pp. 61-92, 2004. · Zbl 1055.05136 [30] S. Janson and A. 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Rödl, ”On $$K^4$$-free subgraphs of random graphs,” Combinatorica, vol. 17, iss. 2, pp. 173-213, 1997. · Zbl 0889.05068 [37] Y. Kohayakawa, V. Rödl, and M. Schacht, ”The Turán theorem for random graphs,” Combin. Probab. Comput., vol. 13, iss. 1, pp. 61-91, 2004. · Zbl 1048.05075 [38] T. Łuczak, ”Randomness and regularity,” in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., 2006, pp. 899-909. · Zbl 1099.05073 [39] T. Łuczak, A. Ruciński, and B. Voigt, ”Ramsey properties of random graphs,” J. Combin. Theory Ser. B, vol. 56, iss. 1, pp. 55-68, 1992. · Zbl 0711.05041 [40] B. Nagle, V. Rödl, and M. Schacht, ”The counting lemma for regular $$k$$-uniform hypergraphs,” Random Structures Algorithms, vol. 28, iss. 2, pp. 113-179, 2006. · Zbl 1093.05045 [41] H. H. Nguyen, ”On two-point configurations in a random set,” Integers, vol. 9, p. a3, 41-45, 2009. · Zbl 1207.11021 [42] R. Rado, ”Note on combinatorial analysis,” Proc. London Math. Soc., vol. 48, pp. 122-160, 1943. · Zbl 0028.33801 [43] F. P. Ramsey, ”On a problem of formal logic,” Proc. London Math. Soc., vol. S2-30, iss. 1, p. 264, 1929. · JFM 55.0032.04 [44] O. Reingold, L. Trevisan, M. Tulsiani, and S. Vadhan, ”Dense subsets of pseudorandom sets,” in Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, Washington DC: IEEE Computer Society, 2008, pp. 76-85. [45] V. Rödl and A. Ruciński, ”Lower bounds on probability thresholds for Ramsey properties,” in Combinatorics, Paul Erd\Hos is Eighty, Budapest: János Bolyai Math. Soc., 1993, vol. 1, pp. 317-346. · Zbl 0790.05078 [46] V. Rödl and A. Ruciński, ”Random graphs with monochromatic triangles in every edge coloring,” Random Structures Algorithms, vol. 5, iss. 2, pp. 253-270, 1994. · Zbl 0790.05079 [47] V. Rödl and A. Ruciński, ”Threshold functions for Ramsey properties,” J. Amer. Math. Soc., vol. 8, iss. 4, pp. 917-942, 1995. · Zbl 0846.05079 [48] V. Rödl and A. Ruciński, ”Rado partition theorem for random subsets of integers,” Proc. London Math. Soc., vol. 74, iss. 3, pp. 481-502, 1997. · Zbl 0880.05080 [49] V. Rödl and A. Ruciński, ”Ramsey properties of random hypergraphs,” J. Combin. Theory Ser. A, vol. 81, iss. 1, pp. 1-33, 1998. · Zbl 0893.05011 [50] V. Rödl, A. Ruciński, and M. Schacht, ”Ramsey properties of random $$k$$-partite, $$k$$-uniform hypergraphs,” SIAM J. Discrete Math., vol. 21, iss. 2, pp. 442-460, 2007. · Zbl 1140.05043 [51] V. Rödl and J. Skokan, ”Regularity lemma for $$k$$-uniform hypergraphs,” Random Structures Algorithms, vol. 25, iss. 1, pp. 1-42, 2004. · Zbl 1046.05042 [52] K. F. Roth, ”On certain sets of integers,” J. London Math. Soc., vol. 28, pp. 104-109, 1953. · Zbl 0050.04002 [53] W. Samotij, ”Stability results for random discrete structures,” Random Structures Algorithms, vol. 44, iss. 3, pp. 269-289, 2014. · Zbl 1290.05131 [54] D. Saxton and A. Thomason, ”Hypergraph containers,” Invent. Math., vol. 201, iss. 3, pp. 925-992, 2015. · Zbl 1320.05085 [55] M. Schacht, ”Extremal results for discrete random structures,” Ann. of Math., vol. 184, pp. 333-365, 2016. · Zbl 1351.05207 [56] I. Schur, ”Über die Kongruenz $$x^m + y^m = z^m \, (\mbox{mod } p)$$,” Jber. Deutsch. Mat. Verein., vol. 25, pp. 114-117, 1916. · JFM 46.0193.02 [57] M. Simonovits, ”A method for solving extremal problems in graph theory, stability problems,” in Theory of Graphs, New York: Academic Press, 1968, pp. 279-319. · Zbl 0164.24604 [58] T. Szabó and V. H. Vu, ”Turán’s theorem in sparse random graphs,” Random Structures Algorithms, vol. 23, iss. 3, pp. 225-234, 2003. · Zbl 1028.05101 [59] E. Szemerédi, ”On sets of integers containing no $$k$$ elements in arithmetic progression,” Acta Arith., vol. 27, pp. 199-245, 1975. · Zbl 0335.10054 [60] E. Szemerédi, ”Regular partitions of graphs,” in Problèmes Combinatoires et Théorie des Graphes, Paris: CNRS, 1978, vol. 260, pp. 399-401. · Zbl 0413.05055 [61] T. Tao, ”A variant of the hypergraph removal lemma,” J. Combin. Theory Ser. A, vol. 113, iss. 7, pp. 1257-1280, 2006. · Zbl 1105.05052 [62] P. Turán, ”Eine Extremalaufgabe aus der Graphentheorie,” Mat. Fiz. Lapok, vol. 48, pp. 436-452, 1941. · Zbl 0026.26903 [63] B. L. van der Waerden, ”Beweis einer Baudetschen Vermutung,” Nieuw. Arch. Wisk., vol. 15, pp. 212-216, 1927. · JFM 53.0073.12 [64] P. Varnavides, ”On certain sets of positive density,” J. London Math. Soc., vol. 34, pp. 358-360, 1959. · Zbl 0088.25702 [65] V. H. Vu, ”A large deviation result on the number of small subgraphs of a random graph,” Combin. Probab. Comput., vol. 10, iss. 1, pp. 79-94, 2001. · Zbl 0982.05092
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### NEET Physics - New - Mechanical Properties of Fluids

#### Question - 1

A capillary tube of radius r is immersed in water and water rises in it to a height h. The mass of the water in the capillary is 5 g. Another capillary tube of radius 2 r is immersed in water. The mass of water that will rise in this tube is_____

• A $20.0 \mathrm{~g}$
• B $2.5 \mathrm{~g}$
• C $5.0 \mathrm{~g}$
• D $10.0 \mathrm{~g}$

#### Question - 2

A small hole of area of cross-section $2 \mathrm{~mm}^{2}$ is present near the bottom of a fully filled open tank of height 2 m. Taking $g=10 \mathrm{~m} / \mathrm{s}^{2}$, the rate of flow of water through the open hole would be nearly____

• A $8.9 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}$
• B $2.23 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}$
• C $6.4 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}$
• D $12.6 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}$

#### Question - 3

A soap bubble, having radius of 1 mm, is blown from a detergent solution having a surface tension of $2.5 \times 10^{-2}$ $\mathrm{N} / \mathrm{m}$. The pressure inside the bubble equals at a point Z below the free surface of water in a container. Taking $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$, density of water $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, the value of $Z_{0}$ is____

• A 10 cm
• B 1 cm
• C 0.5 cm
• D 100 cm

#### Question - 4

Three liquids of densitites $\mathrm{r}_{1}, \mathrm{r}_{2}$ and $\mathrm{r}_{3}$ (with $\mathrm{r}_{1}>\mathrm{r}_{2}>\mathrm{r}_{3}$ ), having the same value of surface tension T, rise to the same height in three identical capillaries. The angle of contact $\mathrm{q}_{1}, \mathrm{q}_{2}$ and $\mathrm{q}_{3}$ obey____

• A $\frac{\pi}{2}>\theta_{1}>\theta_{2}>\theta_{3} \geq 0$
• B $0 \leq \theta_{1}>\theta_{2}>\theta_{3}<\frac{\pi}{2}$
• C $\frac{\pi}{2}<\theta_{1}>\theta_{2}>\theta_{3} \geq \pi$
• D $\pi>\theta_{1}>\theta_{2}>\theta_{3} \geq \frac{\pi}{2}$

#### Question - 5

A wind with speed $40 \mathrm{~m} / \mathrm{s}$ blows parallel to the roof of a house. The area of the roof is $250 \mathrm{~m}^{2}$. Assuming that the pressure inside the house is atmospheric pressure, the force exerted by the wind on the roof and the direction of the force will be $\left(\mathrm{P}_{\text {air }}=1.2 \mathrm{~kg} / \mathrm{m}^{3}\right)$____

• A 4.8×105 N,upwards
• B 2.4×105 N,upwards
• C 2.4×105 N,downwards
• D 4.8×105 N,downwards

#### Question - 6

A certain number of spherical drops of a liquid of radius 'r ' coalesce to form a single drop of radius ' R' and volume ' V . If ' T ' is the surface tension of the liquid, then___

• A energy $=4 V T\left(\frac{1}{r}-\frac{1}{R}\right)$ is released
• B energy $=3 V T\left(\frac{1}{r}+\frac{1}{R}\right)$ is absorbed
• C energy $=3 V T\left(\frac{1}{r}-\frac{1}{R}\right)$ is released
• D energy is nèither released nor absorbed

#### Question - 7

A fluid is in streamline flow across a horizontal pipe of variable area of cross section. For this which of the following statements is correct?

#### Question - 8

The wetability of a surface by a liquid depends primarily on_____

• A surface tension
• B density
• C angle of contact between the surface and the liquid
• D viscosity

#### Question - 9

A rectangular block of mass m and area of cross section A floats in a liquid of density r. If it is given a small vertical displacement from equilibrium it undergoes oscillation. with a time period T. Then_____

• A $T \alpha \frac{1}{\sqrt{A}}$
• B $T \times \frac{1}{\rho}$
• C $T, \alpha \frac{1}{\sqrt{m}}$
• D $T \alpha \sqrt{\mathrm{p}}$

#### Question - 10

The angle ofcontact between pure water and pure glass is_____

• A 0o
• B 45o
• C 90o
• D 135o

format: plain


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# Question #aae31

Jan 5, 2017

$z = 3 + \frac{\log x}{\log \left(x y\right)}$

#### Explanation:

${\left(x y\right)}^{z} = {\left(x y\right)}^{3} x$ now applying $\log$ to both sides

$z \log \left(x y\right) = 3 \log \left(x y\right) + \log x$ and grouping

$\left(z - 3\right) \log \left(x y\right) = \log x$ and finally

$z = 3 + \frac{\log x}{\log \left(x y\right)}$


text:
# jg-you/scm

C++ MCMC sampler for the Simplicial Configuration Model
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# scm

The Simplicial Configuration Model is random null model for simplicial complexes, mathematical objects which can be seen as high-order generalizations of simple graphs (they incorporate multi-node interactions). This repository contains a C++ reference implementation of a Markov chain Monte Carlo (MCMC) sampler for this model, see [Phys. Rev. E, 96, 032312 (2017)] for more information.
A summary of compilation / usage instructions can be found below; but see the tutorial if you are looking for detailed instructions and examples.

## Compilation

The sampler has two mandatory dependencies, namely boost::program_options and cmake (for automated building).

To compile from the terminal, use the following commands:

cmake .;
make

The resulting binaries will be built in bin/. CMakes should also allow Windows users to compile the source easily (not tested---let us know!).

If cmake is not installed, the following manual compilation lines should work (for gcc with C++11 support):

g++ -std=c++11 -o3 -c src/scm/scm.cpp -o src/scm/libscm.a  #compile scm library
g++ -std=c++11 -o3 -L src/scm/ -lboost_program_options -l scm -o bin/mcmc_sampler mcmc_sampler.cpp  #compile the main binaries (mcmc)
g++ -std=c++11 -o3 -L src/scm/ -lboost_program_options -l scm -o bin/rejection_sampler rejection_sampler.cpp  #compile the main binaries (rejection)

## Using the sampler

The sampler randomizes an initial facet list, and samples uniformly from the space of all simplicial complexes with the same degree sequence (degree = number of facet incident on a node) and size sequence (size = number of node in a facet). As such, it requires an initial facet list, either artificial or taken from a real system.
We thus provide not one, but two important binaries: rejection_sampler and mcmc_sampler.

### Rejection sampler

The rejection method is a very inefficient sampler for the SCM, see our paper. However, since there is---so far---no known constructive procedure to generate SCM instances directly from sequences, it can be used to find one instance, which is then plugged into the much more efficient MCMC sampler as an initial condition.

There are two ways to call bin/rejection_sampler.
The first (default) takes a single positional argument, the path to a facet list, and tries to sample from the associated SCM ensemble:

> bin/rejection_sampler datasets/simple_facet_list.txt
# Sample:
0 1
2 3 4
1 2 3

where the output is a facet list.

The second way to call the rejection sampler is in the sequence mode.
This is accomplished by using the flags --degree_seq_file=path-to-degrees.txt and --size_seq_file=path-to-sizes.txt where path-to-degrees.txt and path-to-sizes.txt are paths to files containing the integer sequences (no particular organization required; will use all integers in the file). There is no known simpliciality test yet, so we make no test on the sequences---convergence is not guaranteed and the sampler might run forever.

Here is a simple example, using small sequences, which we know are simplicial:

> echo "2 2 2 1 1" > d.txt
> echo "3 3 2" > s.txt
> bin/rejection_sampler -s s.txt -k d.txt
# Sample:
0 1 4
0 1 2
2 3

Note that we have used the the shorthand flags -k and -s for the sequences, see the full list of option for rejection_sampler below:

Usage:
[Facet list mode] bin/rejection_sampler [--option_1=VAL] ... [--option_n=VAL] path-to-facet-list
[Seq. mode] bin/rejection_sampler [--option_1=VAL] ... -k path-to-degrees.txt -s path-to-sizes.txt
Options:
-d [ --seed ] arg             Seed of the pseudo random number generator
(Mersenne-twister 19937). Seed with time if not
specified.
-c [ --cleansed_input ]       In facet list mode, assume that the input is
already cleansed, i.e., that nodes are labeled
with 0 indexed contiguous integers and that no
facet is included in another.
-k [ --degree_seq_file ] arg  Path to degree sequence file.
-s [ --size_seq_file ] arg    Path to size sequence file.
-v [ --verbose ]              Output log messages.
-h [ --help ]                 Produce help message.

### MCMC sampler

Once we have an initial condition (either by using the rejection sampler or a real system), the MCMC is called with the following commad:

bin/mcmc_sampler -f 10000 -b 2000 -t 200 -d 42 seed_facet_list.txt

Here, -f 10000 specifies that 10000 MCMC move will be applied between each samples (the sampling frequency), -b 2000 specifies that the burn-in time equals 2000 (the number of steps to ignore away before sampling begins), -t 200 sets the number of samples to 200, and -d 42 sets the seeds of the RNG to 42 (it will be seeded with the time by default). seed_facet_list.txt is the path to the initial condition file (notice how it is the only positional argument).

Note: All the above commands have sensible default values and can be omitted.

By default the sampler uses the uniform proposal distribution with L_max = 2 max s, see the paper, but the behavior can be changed. We provide two parameterizable proposal distributions, and it is straightforward to implement additional ones.

The provided distributions are

• exp_prop: Exponential distribution. Draw L with the p.d.f. Pr(l) = exp(lambda * l)/Z where lambda is a parameter, set with --prop_param LAMBDA, and Z is a normalization constant. L is limited to 2,...,L_max.
• pl_prop: Power law distribution. Draw L with the p.d.f. Pr(l) = l ** (-\lambda)/Z. L is limited to 2,...,L_max.
• unif_prop: Uniform distribution [Default]. Draw L with the p.d.f. Pr(l) = 1 /(L_max-2). L is limited to 2,...,L_max.

Note: The sampler can handle arbitrary facet lists as input (lines beginning with # will be ignored). However, it is better if facet lists are cleansed from the get go. By clean we mean that nodes are 0 indexed contiguous integers, and there are no included facet. If the data is already cleansed, use the flag -c to skip the pre-processing cleansing steps. See scm/utilities/ for some lightweight python cleansing tools.

The full list of options for mcmc_sampler:

Usage:
bin/mcmc_sampler [--option_1=VAL] ... [--option_n=VAL] path-to-facet-list
Options:
-b [ --burn_in ] arg                  Burn-in time. Defaults to M log M,
where M is the sum of degrees.
-t [ --sampling_steps ] arg           Number of sampling steps.
-f [ --sampling_frequency ] arg (=10) Number of step between each sample.
Defaults to M log M, where M is the sum
of degrees.
-d [ --seed ] arg                     Seed of the pseudo random number
generator (Mersenne-twister 19937).
Seed with time if not specified.
-l [ --l_max ] arg                    Manually set L_max. The correctness of
the sampler is not guaranteed if L_max
< 2 max s. Defaults to 10% of the sum
of facet sizes.
--exp_prop                            Use exponential proposal distribution.
--pl_prop                             Use power law proposal distribution.
--unif_prop                           Use uniform proposal distribution
[default].
--prop_param arg                      Parameter of the proposal distribution
(only works for the exponential and
power law proposal distributions).
-c [ --cleansed_input ]               Assume that the input is already
cleansed, i.e., that nodes are labeled
with 0 indexed contiguous integers and
that no facet is included in another.
-v [ --verbose ]                      Output log messages.
-h [ --help ]                         Produce this help message.


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# Congestion Game with Varying Price

I molded my problem as the following game (it is a congestion game with varying price):

$N$ players share resources $E$,

$S_i$ is the strategy space of player $i$ which is in $2^E$ (where $2^E$ is the power set of resources).

$P_e^i$ is the price of resource $e \in E$ considering player $i$. The price of resource $e$ is different for different users.

The goal of each player is to select a strategy $S_i$ which minimize its price $\sum_{e\in S_i}P_e^i$ .

My questions are:

1. Does this game have any Nash Equilibrium (NE)? If so under which conditions?
2. If it has any NE, what is a sample algorithm for achieving it?

I searched literature but could not find any appropriate information! Any solution is appreciated!

-

## 1 Answer

I think the answer is yes, there is a Nash equilibrium, and it can be attained just by letting each person in turn change his selection to the best possible alternative.

For more information, read this paper:

Weighted congestion games with separable preferences

by: Igal Milchtaich

http://www.citeulike.org/user/erelsegal-halevi/article/12205199

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# Find the equation of tangent to the curve x = sin 3t, y = cos 2t at t = π/4.

319 views

closed

Find the equation of tangent to the curve x = sin 3t, y = cos 2t at t =$\frac{\pi}{4}.$

+1 vote
by (34.6k points)
selected

Here,

y = cos 2t

∴ $\frac{dy}{dt}$ = - 2sin 2t

Also,

x = sin 3t

∴ $\frac{dx}{dt}$ = 3 cos 3t

Therefore, equation of tangent at t = $\frac{\pi}{4}$ i.e., at $(\frac{1}{\sqrt2},0)$ is given by,


text:
Calculus 10th Edition

Published by Brooks Cole

Chapter 3 - Applications of Differentiation - 3.2 Exercises: 42

Answer

The Mean Value Theorem cannot be applied.

Work Step by Step

$f(x)$ is not continuous over the closed interval. $f(x)$ has a vertical asymptote at $x=0$ and hence the Mean Value Theorem cannot be applied.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Step 2

Introduction

Step 2 is the second tutorial that will walk you through how to compute an inplace real-to-complex (R2C) FFT. There is a subtlety here, in that the result of R2C FFT requires a larger memory than its input. As a result, the user needs to allocate additional memory for the input so that the extra data would fit in the memory. However, that is not all; the user has to use a special padded access pattern in the spatial domain. First an explanation of why there is a memory overhead:

The complex result of a R2C FFT is conjugate symmetric. What this mean is that half of the output data is equal to the conjugate of the other half. So it is really not required to compute those frequencies. This property saves half of the computations and requires half memory. The exact computational cost of computing all the frequencies of a power 2 FFT is $5N\log N$, and using this property it would reduce to $2.5N\log N$.

So where is the memory padding? The devil is in the Nyquist frequency! The output of an $\small N_0\times N_1\times N_2$ R2C FFT would be $\small N_0\times N_1\times (N_2/2+1)$ complex numbers. That means you need to allocate $\small N_0\times N_1\times (N_2/2+1) \times 2\times sizeof(double)$ memory, which is larger than $\small (N_0\times N_1\times N_2)\times sizeof(double)$.

Note that if we had not used the conjugate symmetry property you had to allocate $(N_0\times N_1\times N_2)\times 2\times sizeof(double)$, which is almost twice the above.

AccFFT gives you the right amount of memory to allocate, by calling accfft_local_size_dft_r2c function. So this is already done, however the user must use a special format for initializing the data. The additional allocated memory is stored as if the input matrix was of size $N_0\times N_1\times \bf{(N_2+2)}$. The last dimension is still $N_2$, but when one wants to access elements in memory this padding must be respected.

For example, to access/initialize the (i,j,k) element of the data in the spatial domain (i.e. before doing the R2C transform), one should access $(i\times N_1 + j)\times (N_2+2) +k$.

As you would see in Step 3, there is no such issue for inplace complex-to-complex transforms. That is because, there is no conjugate property for that case!

Code Explanation

The code for this step is very similar to step1, so I would just point out the important differences.

First note that the allocation size is alloc_max which is returned by calling accfft_local_size_dft_r2c function:

This ensures that the data on each process has enough memory to store the output of R2C transform. Moreover, note that the library would create an inplace transform if the second and third arguments of accfft_plan_dft_3d_r2c point to the same memory address (Otherwise an outplace transform would be created).

Finally, note that initialize and check_err functions are a little different than step1. The padded version of $N_2$ is used to read/write data into the spatial array.

Results

You can make and run the code as follows:

On my machine, I get an output like this:

Again, the ERROR is because we were lazy and did not set c_dims. The library corrects it for us and sets it to an appropriate value.

I suggest you run the program with different sizes and different number of MPI processes, and compare the timings with that of step 1.

GPU Code

The GPU version of the code can be found in step2_gpu.cpp file. Note that in this step, I use a GPU kernel to initialize the data instead of first initializing it on CPU and copying it over.


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# What are invariant measures of $E_m \times R_\alpha$ on $S^1 \times S^1$? Are they necessarily product measures?

For $m \in \mathbb{N}$, let $E_m \colon S^1 \to S^1$ be multiplication map $x \mapsto mx$. Also, let $R_\alpha$ be the map $x \mapsto x+\alpha$.

Now, consider $E_m \times R_\alpha \colon S^1 \times S^1 \to \colon S^1 \times S^1$ , and an invariant measure $\mu$ with respect to this map. Then, is $\mu$ necessarily a product measure of the form $\nu \times \text{Leb}$ for some invariant measure $\nu$ with respect to $E_m$?

My intuition says yes, but I cannot quite know how to show this.

Thank you so much!!

No. Define a function $f\colon S^1\to\{0,1\}$ by $f(x)=1$ if $x\in [0,\alpha)$ and 0 otherwise. Now define $\pi\colon S^1\to S^1$ by $\pi(x)=\sum_{n=0}^\infty f(R_\alpha^n x)m^{-(n+1)}$. It is easy to see that $\pi(R_\alpha x)=E_m(\pi(x))$. Now the measure defined on rectangles by $\mu(A\times B)=\lambda(\pi^{-1}(A)\cap B)$ is invariant under the product transformations. It is very far from a product measure: it is supported on the graph of $\pi$.
On the other hand... if you impose the condition that the product measure has Lebesgue measure the first marginal (that is $\mu(A\times S^1)=\lambda(A)$ for all measurable subsets of the circle), then the answer is yes. The property is called disjointness. Two measure preserving transformations are called disjoint if the only invariant measure for the product transformation that projects to each of the two measures under the coordinate projection is the product measure. This property was introduced in a famous paper of Furstenberg of 1967, Disjointness in Ergodic Theory.


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Revision history [back]

The oeis is here: A006982

It is a highly combinatorial job to compute the numbers in the sequence.

Their strategy is to compute first the numbers of the vertically indecomposable, distributive lattices. Which are lattices that cannot be decomposed as a vertical sum of at least two distributive lattices. The vertical sum of $$(D_A;0_A,1_A;\le_A)$$ (placed in our minds vertically below) and $$(D_B;0_A,1_A;\le_A)$$ (placed in our minds vertically above the previous structure) is the lattice on the set $D=D_A\sqcup D_B/1_A\sim 0_B$ with $0=0_A$ and $1=1_B$ and the induced order that makes $D_A\le D_B$, elementwise.

Each distributive lattice appears uniquely as a vertical sum of vertically independent distributive lattices. The generating series for the vertically independent ones is: $V(t) = t^2+t^4+t^6+3t^8+t^9+6t^{10}+2t^{11}+16t^{12}+8t^{13}+42t^{10}+28t^{15}+112t^{16}+\dots$ and form here on the numbers start to grow "also for the human eye in exponential taste". The coefficient in power $t^{49}$ is $V_{49}= 5 045 200 597$. Having this list, we can generate the list for the distribute lattices, we will get $$D = \sum_{k\ge 0}D_kt^k$$ with $D_{49}=908 414 736 485$.

At the level of enumeration, if we know the coefficients in $V$, then we can find the ones in $D$ simply. Let $W=V/t$. Then $$D=t(1+V+V^2+\dots)=\frac t{1-V}$$ with computations in $\mathbb Z[[ t ]]$ . Sage code for this:

R.<t> = PolynomialRing( QQ )
V = ( t^2 + t^4 + t^6 + 3*t^8 + t^9
+ 6*t^10 + 2*t^11 + 16*t^12 + 8*t^13 + 42*t^14 + 28*t^15 + 112*t^16 + 93*t^17 + 311*t^18 + 295*t^19
+ 869*t^20 + 939*t^21 + 2454*t^22 + 2931*t^23 + 7032*t^24 + 9036*t^25 + 20301*t^26 + 27701*t^27 + 58929*t^28 + 84413*t^29
+ O(t^30) )
W = V/t
D = t/(1-W)
for c in D.coefficients():
print c

This gives:

1
1
1
2
3
5
8
15
26
47
82
151
269
494
891
1639
2978
5483
10006
18428
33749
62162
114083
210189
386292
711811
1309475
2413144
4442221

In accordance with A006982.

So it is enough to write code that generates the vertically independent, distributive lattices.

I will try to do this till degree $64$ in the following days... This is then computing $D$ also till degree $64$.


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# Find the pressure put into a box by a hand-squeezed pump

## Homework Statement

A spherical hand-held pump with a radius r is used to pump air into a cube with a length L. The pump and cube are connected together by rubber tubing. Express the pressure stored in the box as a function of the number of times the pump is fully squeezed.
Imagine the situation as that of a sphygomanometer without the cuff and the gauge replaced by a cube box.

## Homework Equations

I used Pcube*Vcube=Ppump*Vpump
Therefore: Pcube = (Ppump*Vpump) / Vcube

## The Attempt at a Solution

I am assuming the cube and pump are at the same temperature and that the pump holds 1 atm pressure when it is not compressed. My expression is the following:
P(x) = ((4/3)(pi)(r^3) * (1 atm) * (x)) / (L^3)
where x is the number of times the pump is squeezed.
I am not confident in my assumption that the initial pump pressure is 1 atm.
Anyone know how to express this situation?
Thanks

#### Attachments

• 65.1 KB Views: 251

The number of moles injected in each pumping is (1 atm)V/RT, where V is the volume of the sphere and T is the air temperature. So, when x pumping take place, the total number of moles in the cube is $n=\frac{(1\ atm)V}{RT}x$. This number of moles, at temperature T, is present in the cube of volume $L^3$. So the pressure in the cube is $$P=\frac{nRT}{L^3}=(1\ atm)\frac{xV}{L^3}$$This agrees with the answer you obtained. But, for this relationship to be correct, the cube needs to be under vacuum (empty) to begin with. If the cube has air in it to begin with, then the result is different.


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Example of a derived series that never stabilizes.

Definition 1. Let $$G$$ be a group, then $$[x,y]:= xyx^ {-1}y^ {-1}$$, for all $$x,y \in G.$$

Definition 2. We define $$G^{(i)}$$ in the following way, $$G' = G^{(1)} = [G,G] = \{\mbox{group generated by all elements in the form }[x,y];\ x,y\in G\},$$ $$G^ {(i+1)} = \left[G^{(i)},G^{(i)}\right], \ i>1.$$

I am trying to find an example of a group $$G$$ such that the series

$$G \supset G' \supset G^{''} \supset \ldots \supset G^{(i)}\supset \dots$$

satisfies $$G^{(i)}\neq G^{(i+1)}$$ for all $$i\in \mathbb{N}$$. I think that $$G = \mathbb{Z}_2 *\mathbb{Z}_2$$ (free product) might work, but I was not able to argue that the required property holds.

Can anyone help me?

• The free group on two elements the property. Any group that contains a free group of rank $2$ or more will also have it, then. – Arturo Magidin May 26 at 18:11
• In fact $Z_2 * Z_2$ is isomorphic to the infinite dihedral group, in which $G''=1$, so this does not work. – Derek Holt May 26 at 19:27
• @MatheusManzatto The group that Arturo suggested is $\mathbb{Z}*\mathbb{Z}$ – verret May 26 at 19:32
• @ArturoMagidin it does not follow that every group containing $F_2$ has the property. Indeed for $SL_2(\mathbb{Q})$ the sequence stabilizes (indeed it is a perfect group), but it contains $SL_2(\mathbb{Z})$ which famously contains a free group of rank $2$ (see e.g. mathoverflow.net/questions/43726/… ) – Max May 26 at 20:51
• @Max: Thanks for the correction. Have a nonabelian free group as a subgroup only guarantees the derived series does not terminate in the identity, not that it does not stabilize. – Arturo Magidin May 26 at 23:18

Arturo Magidin points out in the comments that $$F_2=\mathbb{Z*Z}$$ works. This is true and can for instance be proved as follows : first of all, note that for any free group $$F(S)$$ on generators $$S\neq \emptyset$$, $$[F(S),F(S)]$$ is a strict subgroup of $$F(S)$$. This follows from the fact that there is a surjective morphism $$F(S)\to \mathbb{Z}^{(S)}$$, for instance (you can also see it by looking at reduced words).
Then note that $$[F_2,F_2]$$ is a subgroup of the free group $$F_2$$, and it is nontrivial (because $$F_2$$ is nonabelian), therefore by the Nielsen-Schreier theorem, it is itself free. One moreover easily checks that it is nonabelian. Therefore its commutator will be a strict subgroup, and it will also be free, and nonabelian, and so on by induction. You get by induction that $$G^{(i)}$$ is free, nonabelian and $$G^{(i+1)} \subset G^{(i)}$$.
As you can see, this proof works for any free group $$F(S)= *_{s\in S}\mathbb{Z}$$
• The surjective morphism $F(S) \to \mathbb{Z}^{(S)}$ argument was very clever. Thx. – Matheus Manzatto May 26 at 22:13
• @SantanaAfton : yes there are, but you're going to have to.clarify how much "non free" you want things to be. Indeed if $G$ is abelian, then clearly $F_2\times G$ works. Now less stupidly any group $G$ that has an $i$ with $G^{(i)}$ free non abelian works as well, and you could artificially construct some of those by hand, starting again from a free group. On the top of my head I don't know any "natural" non free group with the property. – Max May 27 at 6:15


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## Swipe om te navigeren naar een ander artikel

01-12-2020 | Research | Uitgave 1/2020 Open Access

# ´Feet are second class citizens: exploring the perceptions of Scottish and Portuguese older adults about feet, falls and exercise- a qualitative study

Tijdschrift:
Journal of Foot and Ankle Research > Uitgave 1/2020
Auteurs:
Monserrat Conde, Gordon J. Hendry, Jim Woodburn, Dawn A. Skelton
Belangrijke opmerkingen

## Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s13047-020-00434-8.

## Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Abbreviations
UK
United Kingdom
PT
Portugal
FG
Focus Group
NHS
National Health Service

## Background

Ageing brings changes to foot health and foot function [ 1], with a greater prevalence of foot problems in later life [ 2]. Hallux valgus, lesser toe deformities, and foot pain become common foot problems and these may increase falls risk [ 3]. Foot care seeks to address symptoms and prevent foot problems, and includes treatments ranging from nail care to foot and ankle exercises [ 4]. Furthermore, management of foot complaints and understanding foot related exercise may reduce the risk of falls in this population group [ 5].
Qualitative evidence shows that self-footcare is perceived as challenging for both healthy [ 4] and condition-specific groups of community-dwelling older adults [ 6]. Knowledge and awareness on general foot health and foot function also appears to vary greatly in this age group [ 4, 6]. Nonetheless, there appears to be a gap in the literature concerning specific perceptions surrounding the link between foot problems and falls risk.
Little is known about older people’s decision-making process regarding exercise for falls prevention, particularly from culturally and linguistically diverse backgrounds [ 7, 8]. Although falls prevention seems to be a low priority for older people in the United Kingdom, as they rarely perceive falls can be prevented with exercise, there are differences in beliefs and attitudes between ethnic groups [ 8, 9]. Jang et al. (2016) found that cultural values and beliefs had a profound and lasting impact on how older adults perceived and defined ageing and health, their outlooks of cultural appropriateness, family obligations and gender roles and the connotations they attributed to exercise and fall prevention strategies [ 7].
Falls prevention interventions need to be culturally suitable, as programme participation is influenced by cultural and motivational, social and environmental values [ 10]. Negative cultural perceptions result in a wide range of implications from hiding falls episodes, to rejecting walking aids or deeming exercise as not age or gender appropriate [ 7, 10]. The meaning of exercise also displays a great cultural variability, having been shown to influence an older people’s engagement in exercise-based falls prevention interventions [ 7]. Therefore, it’s important to consider those factors to confirm evidence-based recommendations and interventions [ 10].
To the best of our knowledge, there is a lack of comparative research concerning older people’s perceptions about falls prevention across European countries [ 11]. It is well known that the effectiveness of falls prevention interventions depends on the uptake and adherence rates [ 12] . Given that these appear to be influenced by the beliefs, perceptions and preferences of older adults [ 7, 13, 14], it seems relevant to ascertain any underlying cultural variation within the European context. Particularly, considering that variations of ageing perceptions, within social contexts, have been already identified on the north-south European axis [ 15].
Furthermore, the available qualitative evidence does not address the views of older adults surrounding foot related falls risk factors and foot care and the role of falls prevention strategies such as exercise. The aim of this study is to explore the views of Portuguese and Scottish community-dwelling older adults, who have experienced at least one fall in the past year, about these topics.

## Methods

### Study design

This qualitative study using focus group discussions was part of a larger programme of research exploring older adults´ views on a tailored home-based lower limb, ankle and feet exercise programme [ 16]. After participants took part in an introductory group session and a one-week trial of the exercise programme; three focus groups were conducted in Scotland and three in Portugal. Further information about the home-based exercise programme, views of participants on undertaking the programme and the outcome measures proposed are reported elsewhere [ 16]. Focus groups were used as this approach offers high ecological validity, and is adequate for groups less experienced in research participation [ 17]. All advantageous features when conducting cross-national research.

### Participants and recruitment

Scottish and Portuguese community-dwelling older adults, aged ≥60 years, who had at least one fall in the previous 12 months were invited to participate. Participants were recruited via a variety of convenience sampling methods through local community-based organisations for older adults [gatekeepers- (e.g. ROAR connections for life-UK, Envelheseres network -PT), snowballing] and also through a newspaper advertisement (UK). Recruitment strategies differed between nations due to contextual factors (e.g. country-specific differences in how gatekeepers manage access to prospective participants) and available resources.

### Eligibility criteria

Participants were required to be able to ambulate and travel independently to venues, able to read and write in English (Scotland) or in Portuguese (Portugal). Participants were excluded if they were diagnosed with central nervous diseases, cognitive impairment, self-reported depression, significant visual impairment or registered blindness, recent lower limb fractures (< 6 months), recent joint replacements (< 1 year), or other medical contraindications that would not allow participants to exercise safely (e.g. acute medical condition) .

### Data collection

A semi-structured interview guide was used [see Additional file  1 Appendix 2, Supplementary Data]. Its structure was inspired by interview guides used in previous research projects [ 18, 19]. Focus group interviews were adaptable to the participants‘ contributions, allowing further questioning [ 20, 21]. One researcher (MC), trained in this approach, led the interviews with support for flipchart note taking by another researcher (UK) or assistant (Portugal-PT). For data validation, participants were asked if notes in flipcharts expressed their views accurately. Focus groups took place in a meeting room either at the University (UK) or at the community centre (PT) and lasted between 45 and 120 min. Focus group discussions were digitally recorded using a sound recorder (Model MICD-UX560 Sony™).

### Qualitative data analysis

All focus group audio recordings were transcribed verbatim and anonymised [ 20, 21]. The anonymised Portuguese transcripts were translated to English, and then proofread by a certified translator. Data was uploaded and managed in NVIVO 11™ software. Qualitative analysis followed the six phases of thematic analysis framework defined by Braun & Clarke [ 20]: 1) Familiarization with the data, 2) Generating codes, 3) Generating themes, 4) Reviewing potential themes, and 5) Defining and naming themes. This qualitative analysis approach was selected as it captures people’s unique views on the researched topics and is also suitable to analyse data derived from focus groups [ 20]. Flipchart and field notes were considered alongside the transcripts. Three researchers independently analysed 3 randomly selected transcripts each in order to enhance validity of the researcher’s coding procedures through collaborative discussion. Minor discrepancies in terms of the procedures were resolved between the team. The researcher then developed the coding frame for all transcripts, which informed her search for initial themes. These were reviewed in agreement with the research team. Illustrative quotes of the theme and subthemes were extracted from the data.

## Results

### Participants

15 Scottish and 24 Portuguese participants agreed to enrol in the study, but focus group analysis only included 10 and 14 participants respectively, due to a variety of reasons [see Additional file 1 appendix 1, supplementary data].
Ten Scottish adults (mean age 76 years; range 68–80 years, 7 women), and fourteen Portuguese adults (mean 66 years; range 64–74 years, 12 women) participated in this study. The three focus groups in Scotland took place, with 3, 3, and 4 participants respectively. In Portugal, the focus groups occurred in three different small size cities, with 2, 6 and 6 participants each.
Socio-demographic characteristics of the participants are presented (Table  1).
Table 1
Characteristics of participants
Characteristics

Portugal
Scotland
Age
Median (P25; P75)
66 (64; 74)
76 (68; 80)
Sex
Male
n (%)
2 (14.3)
3 (30.0)
Female
n (%)
12 (85.7)
7 (70.0)
Occupation
Retired
n
9
10
Worker
n
5
0
Education
Primary school
n
6
0
Secondary school
n
8
4
College diploma
n
0
4
University degree
n
0
2
Number of falls in last 12 months
Median (P25; P75)
2 (1; 3)
2 (1; 4)
Qualitative findings were categorized into one main overarching theme and three subthemes (Fig. 1). Supporting quotes for each theme are presented as (pseudonym, Focus Group number and Country (UK/PT)). Further illustrative quotes for each subtheme can be found in Appendix 3, Supplementary Data [see Additional file 1].

### Evolving awareness about feet, falls and falls prevention

This theme encompasses the views expressed by focus group participants regarding foot health, foot function and falls, both independently and in relation to each other; as well as to their general health and functional status. Both national groups appear to lack awareness and previous knowledge about these topics, particularly the Portuguese participants.
There was a lack of awareness surrounding foot health, foot function, foot related falls risk factors and foot care management. Participants admitted that feet were largely neglected.
“They pamper their hands, they pamper their faces, they pamper everything else (…) But mostly the Feet… they are second class citizens.” (Anthony, FG3_UK)
It was evident through discussions with participants that there was limited understanding between the link between falls risk and foot related issues. This was highlighted while participants were recalling their individual falls events, they had not considered that their feet could have been a contributing factor to their fall.
“Well…I tell you what I thought: Nothing! Not at all about my feet. I’ve thought it was something either in my head or my ear…” (Lisa, FG2_UK)
Poor awareness on these topics was potentially derived from lack of information. Even the Scottish participants who had been enrolled in falls prevention programmes did not recall being given information on foot-specific falls risk factors.
“The feet are there and we use them and we don’t think about them. Therefore this has been quite an eye opener.” (Louise, FG1_UK)
Participants did sometimes elaborate on their reflections about specific foot falls risk factors.
“I think that if the foot is a healthy foot that has support...and even...I mean...even if one loses balance...But can regain balance better than if...Even if the legs are good, but the foot isn’t... One falls more easily. I think!” (Julia, FG3_PT)
A fatalistic perception of falls was conveyed by quite a few Portuguese participants, attributing these events to fate or chance. The belief in a protective external force was also portrayed.
“No! Thank God! I didn’t fall this week! Oh my God! Thank God!” (Mariana, FG1_PT)
Such comments were sometimes accompanied by hands in praying position, or followed by the cross sign.
It was also common for Portuguese participants to depict their falls as unexplainable, as they appeared to not be aware of any main causes that could be triggering such events. This seemed to be closely related to lack of information on the topic, which could be derived from either not recalling or not being given any explanation by health professionals.
“I fall often, but (...) I don’t know…I don’t really know how to explain it… I don’t need to stumble on anything. Nothing. And when I feel it (….) It’s happens whenever.” (Vera, FG3_PT)
“I don’t know…No one has ever told me, and I don’t have knowledge for those things…But I question myself…” (Sofia, FG3_PT)

#### (i) Feet are often forgotten

Neglecting their feet appeared to be rooted in a combination of aesthetic, social and functional expectations. Foot care was described as reliant on other people’s aesthetic judgement of body appearance. Since feet are often covered by footwear, and not on display, they ended up being less prioritized.
“(…) your feet are poor relations in your body, you know? no one sees your feet”. Anthony (FG3, UK)
However, those who perceived their feet as key elements to perform valued activities of daily living (e.g. sports) acknowledged their importance. Interestingly, these participants would portray themselves to be an exception to the norm.
“I’ve always been aware of foot health because I look on it ‘If you’re not on your feet, you’re in your bed!’” (Lisa, FG2_UK)
Across both nations, participants generally seemed unaware of how relevant feet are to human movement unless they had foot problems (pain or deformities). Most undertones around feet tended to be negative, with participants picturing themselves as having ´bad feet.
“(..) all deformed…and the toes are like this, they don’t lift up” (Sofia, FG3_PT)
Feet were mostly either ignored or perceived as a source of discomfort, in both national groups. Foot problems appeared sometimes to be the main motivators for individuals to take any measures regarding foot health and foot function.
“Yes. because they’re there and mine haven’t broken...So I hadn’t anything to think about it” (Louise, FG1_UK)
“No! I didn’t think that because I don’t have other than cramps I don’t have a lot of stuff in my feet” (Ana, FG1_PT)
Regarding participants´ perceptions about foot care, these mainly concerned hygiene with some mention of skin and nail care.
“(…) They’re there…They get washed and they cleaned and that’s it. ((Laughter))” (Louise, FG1_UK)
Care towards feet was more apparent in those with pre-existing problems (e.g. diabetes mellitus). These participants also demonstrated relatively more knowledge of other aspects of foot health (e.g. sensitivity), due to regular health check-ups.
I have been advised about foot health because I am a diabetic.” (Catarina, FG3_PT)
The Portuguese focus groups displayed overall more negative perceptions, with negative internal representation of feet. Sometimes fear of movement and low self-efficacy derived from previous injuries would come across participants´ remarks:
“(..) And when it hurts, I say alright! I am not going to walk more. I’m going to stop it here. The pain is on the foot, but the head thinks.” (Vera, FG3_PT)
Interestingly, there was an initial lack of awareness with participants from both nations on the range and type of movement available within the foot. The limited understanding regarding the functional abilities of the foot as well as recognising the complexity of movement within the foot was evident across cultures.
“I think when you know how you have to exercise your feet you (are) suddenly aware of how similar they are to your hands…Same bones (…) And I am horrified of how poor my feet are.” (Louise, FG2_UK)
“Between tip toes and heel I didn’t…I mean, I didn’t think that it was as important as it really is.” (Catarina, FG3_PT)
In fact, going through the experience of doing the exercises and discussing them was portrayed as enlightening in many ways:
“You know? (…) you take your feet for granted that you never give them a second thought…But this made you think about it” (Sarah, FG3_UK)
“No, I didn’t think about it a lot. No one had explained it to me before. That walking on your tip toes or heels could be so important for balance.” (Leonor, FG3_PT)
This subtheme has emphasized how older adults hold a wide range of views about constructs surrounding feet, including self-awareness regarding movement and function, discomfort, and aesthetic and social features. Negative views were prevalent across national groups, particularly in Portugal.

#### (ii) Need to look at my feet and do the exercises

Participating in the study appeared to influence older people’s views on the value of specific ankle and foot exercise and self-management. Many admitted it had changed their awareness of the functional status of their ankle and foot complex.
“It has raised awareness of the need to look at my feet and do the exercises. (...) I hadn't really thought very much about what we needed to do... And this was a wake-up thing”. (Louise, FG1_UK)
Most focus groups´ members had not been aware that such exercises could be integrated into a falls prevention plan:
“I also didn’t, right, didn’t know that these exercises were beneficial for falls and that.” (Ana, FG2_PT)
Overall, participants were not used to doing specific foot exercises and expressed frustration at times with how challenging moving their toes could be.
“My toes are all crippled (...) they don't lift up. Lowering even less so” (Sofia, FG3_PT)
Foot problems (e.g. toes deformities, arthritis) also posed an additional challenge to some of them.
And I’ve got arthritis…Well it’s not severe…but doing the exercises was actually heavy going for me…on the barefoot ones…” (John, FG2_UK)
After they had tried the exercises, the idea of lower limb, ankle and foot exercises being beneficial to foot movement and feeling more stable, was repeatedly expressed by participants of both countries.
Some participants mentioned immediate improvements from the one-week trial, which seemed to relate with a sense of increased self-efficacy. Many participants appeared more confident of being able to self-manage their functional status, which helped with overcoming any initial difficulties that they had.
“It’s just the fact that my toes didn’t do it, sit down and think about it, lack of use…Therefore get on and do something about it.” (Louise, FG1_UK)
The idea of ´having a problem and doing something about it´ appeared to resonate across focus groups.
Other participants reported improvement of confidence, function or symptoms, such as pain, following doing the specific ankle and foot exercises.
“I mean… even in a week (…) I feel as if I’m stronger.” (Lisa, FG2_UK)
“The foot arch…These [the exercises] helped me…Helped with the pain that I had to go away…” (Catarina, FG3_PT)
Many participants demonstrated recognizing improved motor skills, but also being aware that their maintenance and improvement needs consistency over time.
“Right, but this we need to continue doing it afterwards. I couldn’t lift my big toe either. And now I can…I can.” (Júlia, FG2_PT)
Links between disuse, functional importance of strength training and specificity of certain movement patterns could be traced across focus groups. Balance training was associated with an ability to better perform activities of daily living.
“So, if someone does these exercises, improves their balance, mobility to…To being able to walk better and have a different balance. That’s what I think” (Vera, FG3_PT)
At times, participants clearly expressed believing that these exercises could contribute to maintaining their functional status the longer term, including their ability to walk independently:
“The stronger your legs get, the less chance you’ll need a Zimmer frame!” (Thomas, FG2_UK)
Even the most active participants seemed to identify specific functional gains that could be obtained from the exercises. They recognised that these would be working different muscles groups and attributes, completing other physical activities. Interestingly, these more in-depth reflections were more evident in the Scottish than Portuguese focus groups.
“Although I think I’m like you, I’m quite active and I am out everyday walking. You’re not using the same muscles…You don’t feel you’re using the same muscles as the exercises pinpoint…They pinpoint the muscles being used” (Jean, FG2_UK)
This subtheme renders a growing understanding of the relevance of specific exercises as part of foot care by the participants, awareness of their own foot function; self-perceived value and implications of specific exercises within a falls prevention scope. These were common underlying experiences in both national groups, despite an apparent lower physical literacy by Portuguese participants.

#### (iii) The important role of footwear

Participants spontaneously brought up the role of footwear in their discussions. Footwear preferences were mainly driven by a search for comfort, without any specific guidance from health professionals. Even in the presence of health conditions such as injury or joint deformity.
“No I’m the one who has to check what fits me” (Leonor, FG3_ PT)
Several participants also expressed adjusting their footwear choice over the life course, as result of the ageing process.
“It’s been many years since I’ve only used flat shoes” (Leonor, FG3_PT)
Scottish participants discussed more the idea of what “safe” footwear was in relation to their capabilities.
“Aye. Have any of yous ever fell when ye had high heels on? Shoes with high heels?” (Thomas, FG2_UK)
“I don’t wear high heels now. I wouldn’t trust myself to be honest with high heels…Because I go over my ankle quite a lot… (…)” (Jean, FG2_UK)
Despite exhibiting some concerns around footwear safety, inquiring about wearing loose slippers and being barefoot, participants were slightly surprised that footwear could influence falls.
“I wasn’t aware of the role…The important role of footwear.” (Elaine, FG1_UK)
An increased awareness that footwear could influence normal foot function, potentially increasing falls risk, appeared to arise from participating in this study for both national groups.
“you brought up a point that I find with these… super comfortable [brand of trainers with gel insoles] …But I… If they stick, I do tend to go forward. I’m not lifting my foot, obviously, high enough. Because it’s only since I got these that I’ve noticed that” (Jean, FG2_UK)
Participants offered mixed remarks about being barefoot. Portuguese participants admitted to being barefoot more often and were more likely to wear sandals or flip flops because of the weather. Interestingly, different to the Scottish focus groups, no considerations about footwear preferences and falls episodes were expressed by them.
This subtheme shows how participants perceived footwear as a coadjutant to maintain/improve foot function, its contribution to falls risk; and both their reasoning and preferences when choosing shoes in their daily lives. Contextual differences between nations played a role on the latter.

## Discussion

This is first qualitative study to explore community-dwelling Scottish and Portuguese older adults´ views about feet within a falls prevention scope. It is also the first study to offer insight into Portuguese older adults´ perceptions about falls and exercise-based falls prevention. The main focus of the findings were participants´ perception of their feet and how this links to understanding risk factors associated with falling and managing/minimising this risk potentially through specific foot exercises. It provides key information concerning a lack of awareness towards foot related falls risk factors and self-foot care within a falls prevention scope; including specific foot and ankle exercise in both national groups. Portuguese participants expressed more fatalistic views about falls, with some typifying their falls as unexplainable. This national group was also apparently less aware and informed that exercise could prevent falls.
A lack of awareness of falls related risk factors associated with their feet was predominant across both nations, even among Scottish participants who had previous experience with exercise-based falls prevention programmes. A recent meta-analysis suggests that foot problems are associated with falls in older adults; recommending health screening and specific referral within falls assessment [ 3]. Most participants mentioned having foot problems, which meant that they could potentially benefit from this information to their falls risk.
Portuguese participants presented more fatalistic views about falls; concurring with previous literature about other national groups [ 8, 9, 22]. Fado (destiny or faith) and catholic doctrine are rooted in Portuguese culture. Although religious beliefs can contribute to developing a feeling of acceptance of falls, they can also act as a coping mechanism for older adults [ 8]. Faith-based social networks can also facilitate exercise engagement [ 23]. These could be considered when planning programmes in Portugal.
Some Portuguese participants characterised their falls as unexplainable, similar to older adults with visual impairment [ 24]. They were at a loss as to why they would fall, emphasizing that they had not received any information from health professionals. There are several reasons why participants may perceive that falls are not preventable or that they have not been provided with falls prevention education. Previous research has shown that older adults may not perceive themselves to be at risk of falling [ 9, 25, 26], believe that falls are not a medical issue [ 27] or just assume that they are an inevitable consequence of ageing [ 25]. Believing that they should be more careful to avoid falls [ 8], dismissing non injurious falls [ 27] or not wanting to burden family members [ 7, 22] also appear to influence reporting of falls. Discussing falls may also cause anxiety [ 27], being perceived as signs of functional decline and frailty [ 25]. All these aspects can contribute to older adults not discussing the issue of falls with health professionals [ 9, 2527].
Concomitantly, health professionals also face several barriers to provide guidance to older patients regarding falls prevention and management, ranging from patient influence, to education training, to communication between staff and patients [ 28]. The lack of structured fall prevention guidelines and insufficient training has also been identified as a major barrier to health professionals providing advice on exercise to prevent falls [ 29], which may be relevant in countries such as Portugal where there is not an established nationwide NHS guidance. Additionally, older adults often present competing clinical priorities with urgent medical issues being prioritized over falls in the short time that health professionals have available to provide care [ 30]. Finally, it should also be noted that health messages may sometimes be misunderstood, or forgotten, by patients [ 31, 32]. This might have also been reflected in our findings.
Interactions with health professionals can positively or negatively influence the uptake of falls prevention interventions, with older adults being more likely to participate in an exercise-based intervention if they are referred or encouraged to participate by a health professional [ 27]. Our findings appear to align with these findings, also highlighting that there is an opportunity for clinicians to address foot-related falls risk factors with their older patients. These aspects should be further researched from both the public and health providers´ perspective and contemplated when planning interventions.
Participants admitted that feet are less prioritised than other body parts, agreeing with previous research [ 4]. Feet not being on display was the main reason given for this ´neglect´. In our study, feelings of privacy were not mentioned by participants, unlike Miikkola et al’s findings [ 4]. Appearance and body image are important to older adults, as they signpost personal identity, physical functionality, and social status; they are also influenced by sociocultural stereotypes of the ageing body [ 33]. Older adults may become self-conscious of age-related changes to feet morphology and appearance [ 1], leading into overlooking them further. In fact, negative views of feet were common across focus groups, particularly among Portuguese. Foot problems were managed rather than prevented, a behaviour previously highlighted among other condition-specific and national groups of older adults [ 4, 6, 34]. Foot pain, discomfort and other problems were common complaints, concurring with strong quantitative evidence [ 35]. These were also the main motivators to seeking a medical appointment for foot health [ 4, 34, 36]. Previous research has identified the importance of positive perceptions of healthcare to facilitate older adults´ future foot consultations [ 34, 36]. Our study reflects the same trend in both nations emphasizing the need for education.
Sometimes participants with a history of foot health problems admitted being protective of their feet, which would trigger some fear and avoidance of movement [ 37]. These potential manifestations of lower self-efficacy can perhaps relate to the fear-avoidance model of pain [ 38]. Reinforcing the role of health professionals in educating older adults about foot health and foot function management [ 4, 6, 34], encouraging regular movement [ 39] and patient-provider communication processes [ 40, 41] are important mediating agents in pain-related outcomes. Our findings also suggest that foot related issues could perhaps be further explored within the patient-provider communication processes.
As with Miikkola et al. [ 4], our participants lacked knowledge of proper foot care which can negatively influence their foot health and foot function, and consequently their foot-specific falls risk factors. Only participants who had foot problems, or who were at a higher risk (diabetes mellitus) reported having received health information about foot self-care through their healthcare services [ 6, 34]. However, assumptions about the quality of self-foot care cannot be made, as this was not explored in depth. This study found that specific foot and ankle exercises were not considered as a strategy for foot health, contrasting with other studies [ 4]. Most Scottish and Portuguese participants were unaware of the role of these exercises in falls prevention [ 5], which had not been previously reported. Moreover, Portuguese participants had very little knowledge that exercise could prevent falls. Similar conclusions were found regarding other national and ethnic groups [ 7, 8, 10], but it is new for Portuguese older adults. Differences in the baseline characteristics of participants and healthcare resources between nations may have influenced our findings. Low health literacy and low education levels are predominant among Portuguese older adults [ 42], which may have also contributed to the findings of this study. Our findings also suggest possible lower physical literacy among Portuguese participants [ 43].
Some participants mentioned immediate benefits during the one-week trial (e.g. pain reduction, improvement of muscle strength, increased confidence). This may be a result of placebo effect related to exercise, particularly for those with negative beliefs about their physical abilities [ 44]. Furthermore, such self-perceived improvements may be coincidental. Motor learning may also have contributed to self-efficacy [ 45], as participants reported observing an improvement in their motor performance. An increased self-awareness of foot function was often mentioned, with many reporting an improvement in observed motor function, potentially demonstrating responsiveness to the neuromuscular effects of specific ankle and foot exercises. This was shared across national groups, independent of previous experience with physical activity or exercise-based interventions, and it has not been previously reported in the literature; possibly, this may have been a manifestation of increased physical literacy [ 43].
Focus groups reported that a specific exercise programme could improve the available range of motion, muscle strength, and balance in the longer-term, linking these as key abilities to performing their activities of daily living. The latter was highly valued, conforming with identified enablers of engagement in exercise-based falls interventions [ 46]. Scottish participants appeared to be more aware of the specificity of these ankle and foot exercises, recognising extra functional elements to their regular physical activity routine. Portuguese were less expressive of specific contributions. Differences in physical literacy may explain these differences; the Scottish participants were overall more educated, more physically active, and considerably more experienced with exercise, all characteristics that contribute to greater levels of physical literacy [ 43]. However, Scottish participants also appeared to have increased their physical literacy in relation to foot function as a result of this study.
Preferences of footwear varied between nations due to weather conditions. Overall, participants expressed being originally unaware that footwear could influence falls, albeit being very curious about optimal footwear for foot health. Participants expressed that shoes should be individualised to their needs and comfort, aligning with existing evidence [ 4]. Most reported that they had not received any guidance about optimal footwear, with all admitting never being informed about footwear as a risk factor for falling. Footwear has been considered to possibly impact fall risk [ 47], however, there is no conclusive evidence supporting a causal relationship between specific types of footwear and falls rates in healthy older adults [ 48]. Nonetheless, footwear impacts older adults´ foot health [ 49, 50], including increasing the chances of acquiring foot problems (e.g. hallux valgus) associated with an increased falls risk [ 3].

### Implications for intervention development

Our findings support the emphasis on regular educational components surrounding foot-related falls risk factors as part of specific ankle and foot exercise interventions for older adults. Participants exhibited a generally low physical literacy related to foot function and specific ankle and foot exercises, these should be assessed and targeted by robust evidence-based educational components.
Falls prevention interventions developed for Portuguese older adults would benefit from accommodating the needs of a population with lower health and physical literacies. This may translate into reinforcing general falls educational components as well as providing greater individualized support when introducing exercise-based interventions to older adults.

## Limitations and future research

This study has some limitations. Recruitment varied between nations due to contextual factors. Selection bias may have resulted in recruiting individuals more interested in these topics. It was not possible to interview those who declined to participate or dropped-out. Differences in characteristics of national groups (particularly age, education and current employment), and contextual settings, could have influenced our findings. Nonetheless, we used adequate approaches for each national setting and were able to capture cultural undertones across the data.
Participants had tried a lower limb, foot and ankle exercise programme for a week. Exploring views separately from participation in a programme may better differentiate between baseline and acquired knowledge, in future studies.
Future research should also explore the views of health professionals, involved in falls management, about the role of foot related conditions and their relationship to falls risk, as well as exercise-based interventions to assist reduce the risk of falls in older people. Resulting evidence could inform clinical practice. More qualitative studies exploring Portuguese older adults´ views about falls prevention would enable culturally adequate evidence to inform national public health policies.

## Conclusions

By performing a thematic analysis of focus group interviews, we found that Scottish and Portuguese older adults had little knowledge of foot related risk factors associated with feet, being unaware of the functional status of their feet and of the role of specific lower limb, ankle and feet exercise in foot care and falls management. Feet were described as neglected body parts, with problems being managed rather than prevented. Footwear was not considered for reducing falls risk. Participating in the study contributed to an evolved awareness about these topics. Furthermore, Portuguese participants demonstrated little knowledge about general falls-risk factors and falls prevention strategies, with some displaying a fatalistic outlook, or considering falls to be unexplainable. Such aspects could be considered when developing culturally appropriate interventions.

## Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s13047-020-00434-8.

## Acknowledgements

We wish to thank the participants who took part in this study, and all the organisations that supported recruitment.
We also wish to thank Dr. Madalena Gomes da Silva (Centro Interdisciplinar de Investigação Aplicada à Saúde; Escola Superior de Saúde- Instituto Politécnico de Setúbal), Ms. Ana Casaca (Physiotherapist, CS Ferreira do Zêzere/URAP/ACES Medio Tejo/ARSLVT), Mr. Artur Silva, Ms. Nereide Ribeiro, Dr. Joanne Hurst (GCU).

### Ethics approval and consent to participate

Approval was granted by the School of Health and Life Sciences Research Ethics Committee of Glasgow Caledonian University (HLS/PSWAHP/17/147) and the Portuguese National Data Commission (CNPD n° 1760/ 2018). Participants gave written informed consent.

Not applicable.

### Competing interests

Dr. Gordon Hendry is an Associate Editor UK of the Journal of Foot and Ankle Research. It is journal policy that editors are removed from the peer review and editorial decision-making processes for papers they have coauthored. Professor Dawn A. Skelton is a Director of Later Life Training, a not for profit training organisation delivering exercise training to health and fitness professionals in the UK and Europe. The remaining authors declare no conflicts of interest in relation to this work.

## Publisher’s Note

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

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text:
# Properties

 Label 1805.1 Level 1805 Weight 1 Dimension 42 Nonzero newspaces 5 Newform subspaces 7 Sturm bound 259920 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$7$$ Sturm bound: $$259920$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1805))$$.

Total New Old
Modular forms 2064 1448 616
Cusp forms 48 42 6
Eisenstein series 2016 1406 610

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 42 0 0 0

## Trace form

 $$42 q - q^{4} + q^{5} + 4 q^{6} - q^{9} + O(q^{10})$$ $$42 q - q^{4} + q^{5} + 4 q^{6} - q^{9} + 2 q^{11} + q^{16} - 24 q^{20} - 3 q^{25} - 4 q^{26} - 4 q^{30} - 3 q^{36} - 32 q^{39} - 2 q^{44} + 3 q^{45} - 3 q^{49} + 2 q^{55} + 2 q^{61} + 3 q^{64} + 4 q^{74} - 3 q^{80} + q^{81} + 32 q^{96} - 2 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1805))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1805.1.c $$\chi_{1805}(721, \cdot)$$ None 0 1
1805.1.d $$\chi_{1805}(1804, \cdot)$$ None 0 1
1805.1.f $$\chi_{1805}(362, \cdot)$$ 1805.1.f.a 2 2
1805.1.h $$\chi_{1805}(69, \cdot)$$ 1805.1.h.a 2 2
1805.1.h.b 4
1805.1.j $$\chi_{1805}(791, \cdot)$$ None 0 2
1805.1.m $$\chi_{1805}(68, \cdot)$$ 1805.1.m.a 4 4
1805.1.n $$\chi_{1805}(116, \cdot)$$ None 0 6
1805.1.o $$\chi_{1805}(299, \cdot)$$ 1805.1.o.a 6 6
1805.1.o.b 12
1805.1.r $$\chi_{1805}(28, \cdot)$$ 1805.1.r.a 12 12
1805.1.t $$\chi_{1805}(94, \cdot)$$ None 0 18
1805.1.u $$\chi_{1805}(56, \cdot)$$ None 0 18
1805.1.y $$\chi_{1805}(58, \cdot)$$ None 0 36
1805.1.z $$\chi_{1805}(31, \cdot)$$ None 0 36
1805.1.bb $$\chi_{1805}(84, \cdot)$$ None 0 36
1805.1.bd $$\chi_{1805}(7, \cdot)$$ None 0 72
1805.1.bg $$\chi_{1805}(14, \cdot)$$ None 0 108
1805.1.bh $$\chi_{1805}(21, \cdot)$$ None 0 108
1805.1.bj $$\chi_{1805}(17, \cdot)$$ None 0 216

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1805))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1805)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 2}$$


text:
how to change the box height of an equation

Here is a simple equation $$\int_a^b f(x) \, dx$$ how do I calculate the box height? How do I round to the nearest multiple of 12pt ?

You could use the \settoheight and \settodepth macros, which take two arguments, to measure the height above and depth below the baseline. For Computer Modern fonts and a basic document font size of 10pt, the total heights of the integral formulas work out to be 25.1pt and 14.0pt for display style and text style, respectively.

To force the heights of the formulas to either 24pt (in the display style case) or 12pt (in the text style case), you could use a \resizebox* directive. Note: don't use \resizebox, use \resizebox*.

\documentclass[10pt]{article}
\newlength\bxheight
\newlength\bxdepth
\newlength\bxtot
\newcommand\myint{\int_a^b f(x)\,dx} % store the formula (it gets used a lot)
\usepackage{graphicx} % for \resizebox macro
\usepackage{amsmath}  % for \text macro
\usepackage{booktabs}
\begin{document}

\settoheight{\bxheight}{$\displaystyle\myint$}
\settodepth{\bxdepth}{$\displaystyle\myint$}
\setlength\bxtot{\dimexpr\bxheight+\bxdepth\relax}
$\displaystyle\myint$ has height \the\bxheight, depth \the\bxdepth, and total height \the\bxtot.

\medskip
\settoheight{\bxheight}{$\myint$}
\settodepth{\bxdepth}{$\myint$}
\setlength\bxtot{\dimexpr\bxheight+\bxdepth\relax}
$\myint$ has height \the\bxheight, depth \the\bxdepth, and total height \the\bxtot.

$\begin{array}{@{}lcc@{}} & \text{natural size} & \text{rounded to nearest } \\ & & \text{multiple of 12pt} \\ \midrule \text{display style} & \displaystyle\myint & \resizebox*{!}{24pt}{\displaystyle\myint} \\[3ex] \text{text style} & \myint & \resizebox*{!}{12pt}{\myint} \end{array}$
\end{document}

• I wonder if it is possible to typeset the equation only once in an \hbox and use height of the box \the\ht to measure the height of equation (instead of using \settoheight)? When I try that, I get \hbox height & depth is 0pt. – reportaman Apr 26 '20 at 9:15
• @reportaman - I'm not sure I understand the gist (or purpose) of your comment. For sure, the instruction \newcommand\myint{\int_a^b f(x)\,dx} ends up (eventually...) placing \int_a^b f(x)\,dx in an \hbox. Executing the three instructions \settoheight{\bxheight}{$\myint$} \settodepth{\bxdepth}{$\myint$} \setlength\bxtot{\dimexpr\bxheight+\bxdepth\relax} as a single instruction, say via \def\docalcs{\settoheight{\bxheight}{$\myint$} \settodepth{\bxdepth}{$\myint$} \setlength\bxtot{\dimexpr\bxheight+\bxdepth\relax}}, is straightforward. Please let me know if I missed something. – Mico Apr 26 '20 at 9:25
• I was looking it from the point of efficiency. If a math equation is put in a \hbox first, and tex builds that \hbox (upon executing \setbox) & if we can query its height & depth then the equation need not be "built" three times by tex... I used the term typeset instead of built. – reportaman Apr 26 '20 at 9:32
• @reportaman - I definitely wrote this answer as a one-off project. Economy (terseness?) was not an objective. Feel free to post a new answer that creates the same output but consumes fewer computing cycles. – Mico Apr 26 '20 at 10:29