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# intensity transformation 16-bit image to a 8-bit image

Give an intensity transformation function T for converting a 16-bit image to a 8-bit image, i.e. T takes an integer from {0,1,2,...,65535} and returns an integer from {0,1,2,...,255}.

Can we use $s = T(r) = \left \lfloor \sqrt{r}\right \rfloor$?

• Without further specification, you could just return zero for any input. That is a fine function. It would turn all images black, however. – Ross Millikan Apr 8 '13 at 14:20
• I wildly guess you'll prefer a linear transformation. – leonbloy Apr 8 '13 at 14:25

You can certainly use whatever you'd like. However, this won't produce an even-looking image. Why? Look at a table of what input maps to what output: $$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\ \hline 1 & 1 \\ \hline \vdots & \vdots \\ \hline 4 & 2 \\ \hline \vdots & \vdots \\ \hline 9 & 3 \\ \hline \vdots & \vdots \\ \hline 16 & 4 \\ \hline \vdots & \vdots \\ \hline 25 & 5 \\\hline \end{array}$$

Ok. So that was interesting and trivial. What does this mean? Notice that the input maps $3$ digits to $1$, $5$ digits to $2$, $7$ digits to $3$, etc. This indicates that the image will be more weighted in the lighter/higher colors.

Another way to look at this is to say that the ideal image transformation would have the midpoint be the same on both. That is, if you plug $\frac{2^{16}}{2}$ into the transformation, you'd get back $\frac{2^8}{2}$. The same goes for all other denominators: $\frac{2^{16}}{4} \to \frac{2^8}{4}$, etc.

You may want to look at using the $\log_2(x)$ function... (not sure how this would pan out, but it could work).

While the full story obviously depends on exactly what the samples in the original 16-bit image represent, and how the output 8-bit image is going to be used, in the absence of more specific instructions, the only sane thing is to scale the samples linearly -- which in this case means simply chopping off the eight least significant bits of each sample.

More technically, the sample values in digital images relate to light intensity by power laws of varying exponents and complexity -- but by convention these are almost always specified as a combination of

1. Scale the digital sample values affinely so they fall into the range 0.0 to 1.0.

2. Apply a power function (whose exponent is conventionally know as the "gamma" of the samples), possibly with some special piecewise definitions near 0 to avoid dealing with unbounded derivatives.

3. Multiply the result with the desired "full" light output of the display device.

Thus, if you want to produce a set of 8-bit samples that work more-or-less like the original ones under this procedure -- and if you don't know any better, assume that the same gamma is used in both cases -- you should aim to have the first of these steps approximate the output of the first step applied to the original sample as well as possible. That calls for a linear scaling.

In some cases, the initial affine mapping maps a restricted range of the sample space into $[0,1]$, such as mapping 0 through 16 to 0.0, 224 through 255 to 1.0 and the range between them interpolated linearly. However, if you don't have specific information about this, the best you can do is still to scale linearly.

And if what you have are color samples in a non-RGB color space, the samples will typically be converted to RGB for display using linear algebra on the raw samples -- in that case doing anything but a linear rescaling will disturb the color reconstruction.

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Limited access

How could one determine if the distribution of sample means is approximately normal?

I. Both $np$ and $n (1-p)$ are greater than $10$, where $n$ is the number of trials and $p$ is the proportion of success.

II. The sample is large, say greater than $30$.

III. The parent population is approximately normal.

A

I only.

B

II only.

C

III only.

D

I and II only.

E

II and III only.

Select an assignment template


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# Vizing

## A generalization of Vizing's Theorem? ★★

Author(s): Rosenfeld

\begin{conjecture} Let $H$ be a simple $d$-\Def[uniform]{hypergraph} hypergraph, and assume that every set of $d-1$ points is contained in at most $r$ edges. Then there exists an $r+d-1$-edge-coloring so that any two edges which share $d-1$ vertices have distinct colors. \end{conjecture}

Keywords: edge-coloring; hypergraph; Vizing


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## Shrub facilitation of Quercus ilex and Quercus pubescens regeneration in a wooded pasture in central Sardinia (Italy)

iForest - Biogeosciences and Forestry, Volume 3, Issue 1, Pages 16-22 (2010)
doi: https://doi.org/10.3832/ifor0517-003

Research Articles

In the woodlands of Sardinia, as in many other areas of the Mediterranean region, grazing of domestic animals is still very common, though often in the absence of any sustainable management logic or technique. The present work analyzes the effects of excessive grazing pressure on a wooded pasture in the municipality of Orgosolo (Nuoro province), emphasizing the effects on the natural regeneration of the oak species (Quercus ilex and Quercus pubescens). This study has revealed the positive effects of the interaction between shrubs and seedlings of tree species. Crataegus monogyna proved to be the most efficient shrub species in favouring the establishment and growth of saplings; Rubus ulmifolius is efficient in the establishing phase but somewhat less so in the following stages. The other shrub species (Prunus spinosa, Genista pichi-sermolliana, Stachys glutinosa) play a lesser facilitating role. Despite the fact that seedlings are found in more than half (56%) of the shrub patches, the average age of the seedlings (14±0.6 yrs) and their small average size (height 24±1.4cm) lead to think that the present grazing pressure is incompatible with any concrete chance of success for their natural regeneration.

# Introduction

Pastures with consistent tree cover, composed mostly of evergreen oaks like Quercus ilex, Quercus rotundifolia and Q. suber, are quite widespread in Western Mediterranean countries. These landscapes assume a peculiar physiognomy and are called “dehesa” in Spain and “montado” in Portugal. The same stands are also present in Italy, especially on the island of Sardinia.

The features and functions of these formations are the result of specific climatic conditions, from somewhat mild summer droughts, combined with grazing and related human activities over the centuries to exploit natural resources ([43], [42], [38]). What we are looking at today must therefore be considered a peculiar agro-forestry system ([33], [34]), rather than the remains of a previously closed forest ecosystem.

Today, a general tendency for overgrazing with progressive symptoms of degradation can be observed ([47], [45], [44], [41], [25]). In particular, the open woods of the meso-Mediterranean belt dominated by the Holm oak, maintain their functionality only with low grazing pressure. High livestock densities produce negative effects, especially on the natural regeneration of the tree component ([21]), which is often confined under the protective shadow of shrubs which are not palatable to the animals. These patches then become potential safe sites for seedlings and saplings as recognized by many authors ([12], [11], [15], [49], [31], [35], [3], [52], [54]).

In the majority of the hilly and mountain areas of Sardinia, continuous stocking is the main grazing method ([16]). This is true also for the Pradu woodland, which is the subject of the present study. In recent decades, the pastoral activity in Mediterranean areas has been subjected to important modifications, with increasing livestock density and the shift from transhumance to more stable and intensive forms of breeding. The Pradu woodland has also experienced these transformations, together with a considerable increase in the number of animal species feeding at the same time (cattle and horses have joined the more traditional herds of sheep, goats and swine). As a result, the natural regeneration of the tree component has sharply decreased in density and spatial distribution, failing to overcome the progressive senescence of the existing stand under natural or human-induced stresses ([47], [24]). The conservation of the Pradu woodland is further compromised by the increasing, uncontrolled harvest of large branches, and even entire trees, for fuel ([1]).

This paper aims to quantify the consistence and potential of natural tree regeneration at the Pradu site, by analyzing in detail the characteristics of the shrub components, with respect to their ability to become nurse plants for tree saplings.

# Material and methods

## Study area

The study was conducted in mid-eastern Sardinia in the municipal territory of Orgosolo (Nuoro province) in a part of the Pradu woodland (N 40°10’; E 9°20’) with an area of about 280 ha at an altitude of about 1000m above sea level, with a mostly western exposure.

The climate, according to the bioclimatic classification of Rivas-Martínez & Rivas-Sáenz ([48]), is mid-Mediterranean with an oceanic footprint and an average yearly temperature and rainfall of 12 °C and 1100 mm respectively ([8]).

Paleozoic intrusive rocks dominate the landscape, with outcrops of granite and filladic schists ([39]). Dystric, Typic and Lithic Xerorthens ([55]) are the more common soil types over both intrusive and metamorphic substrates; they are usually shallow, coarse-textured, rich in rock fragments and with quite a low content of organic matter. Typic and Lithic Dystroxerepts have limited diffusion over more gentle slopes and in concavities. Despite the generally good soil permeability, recent gully phenomena may be observed locally.

Vegetation type refers to the climax of Sardinian mountain forests dominated by Holm oak and Downy oak stands ([2]). The tree component of the vegetation consists of a high forest with an unevenly aged structure (the diameter ranges from 40-160 cm) where Holm oak (Quercus ilex) is the dominant species (96%), and Downy oak (Quercus pubescens) (3%) and Montpellier maple (Acer monspessolanum - 1%) the associated species. The average density is 65 trees per hectare, but the spatial distribution is very irregular with scarcely covered (minimum of 11 plants ha-1) and densely covered (121 plants ha-1) areas. Historical record refer that until 150 years ago the forest physiognomy was like overaged coppice with standards of oak. At those time the forest was irregularly and occasionally logged and the grazing was transhumant and periodically suspended. Actually, the woodland is grazed by several domestic animals which, at various trophic levels, play an important role in favouring and shaping the floristic composition of the herbaceous vegetation. The authorizations for grazing was granted by municipality board to farmers of neighbourhood without any control on the number of livestock and grazing system. The entire study area is, in fact, subject to grazing year round; according to official data from the Orgosolo municipality, the average livestock pressure in 2003 was 1.7 LU (adult livestock unit) per hectare, of which more than 70% are sheep, 14% swine, and the rest divided among cattle, goats and equines ([17]). Shading, nitrification, and selection of species less palatable to animals have favoured the presence of thorny species (Genista pichi-sermolliana, G. corsica, Carlina corymbosa, Centaurea calcitrapa, Ononis spinosa, Ptilostemon casabonae) or toxic species (Euphorbia cupani, Paeonia morisii, Digitalis purpurea, Ranunculus spp.). Prominent among the ruderal and the nitrophilus species are Onopordon illyricum, Urtica atrovirens and U. pilulifera and the endemic Rumex suffocatus. The presence of Rubus ulmifolius is quite remarkable, because although browsed by sheep and especially goats, it partially characterises an anomalous undergrowth, together with dogrose (Rosa canina) which rises in big tufts. Other woody species are wild pear (Pyrus spinosa) and hawthorn (Crataegus monogyna), found sporadically throughout the area. Exploitation, practiced regularly for centuries, as testified by the important archaeological remains of the nuraghic era, has selected a number of species of scarce foraging and grazing interest. The species of the genus Aira, Vulpia, Bromus are the most common among the gramineae, whereas among the compositae the genus Crepis, Reichardia and Carlina are those which give a significant contribution to the grass turf. Among the leguminosae, in addition to the above-mentioned Ononis spinosa, two species are of remarkable pabular interest: Lotus alpinus, exclusively in the high mountain areas of the Gennargentu and the Marghine-Goceano region of Sardinia, but most of all Trifolium subterraneum, which protects itself from overgrazing thanks to its self-reseeding character. Present among the labiatae are thyme (Thymus catharinae) and helichrysum (Helichrysum microphyllum), small fragrant suffrutexes usually associated with gorses. Asphodel (Asphodelus microcarpus) is sporadically browsed in its vegetative stage, but completely rejected during flowering; this facilitates its diffusion, which is also helped by its root system, very rich in nourishing supplies. Similar to asphodel, geophytes provided with tubers, bulbs and rhizomes have an advantage, particularly those of the genus Gagea, Scilla, Ornithogalum, and Leucojum, which also produce showy and characteristic flowerings.

The whole area is crossed by paths open to lorry traffic which favours channelled erosion, as they interrupt the already poor grass turf, compact the soil and constitute preferential path for run off concentration. This process has considerably increased in recent years, adding to the actions of the swine, which are present in far greater number with respect to the sustainability of the area, and dig looking for tubers, bulbs and larvae, thus worsening the deterioration of the entire area.

The flora of the Pradu woodland, despite being poor in species suitable for grazing, has an endemic component common to the other areas of the Gennargentu which increases its environmental value.

## Plot sampling

The research area was delimited and georeferenced using regional orthophotographs (1998) on a scale of 1:10 000 in a GIS environment. The total area was divided into a squared 100 m wide grid following the Gauss-Boaga geographic projection; the area of each square was further divided into 25 smaller 20 m wide squares (400 m2), also referred to as the reticule. Next were the stratification of the big squares according to the degree of canopy cover in the GIS environment (3 classes: A <30 %, 31< B <60 % and C >60%), and the random selection of 45 small squares (specimen areas - SA) in proportion to the frequency observed in each cover class.

## Field observations

For each SA the general characteristics such as slope, exposure, and degree of vegetation cover were recorded. The crown height (H), as well as the bigger measurement (L1) and the one perpendicular to it (L2), were taken for each shrub present.

Every seedling of tree species (height < 5 m) was surveyed distinguishing between those growing in the open and those growing among shrubs, and its height (h) was also measured; in the presence of groups of seedlings, the total number (ns) and the height of the tallest saplings were taken.

Finally, in a random sample of shrubs with seedlings, 50 of these were taken, and their height (h), and the collar diameter (d) were measured. Seedling age was estimated by counting the annual rings using a stereomicroscope.

## Data processing

The area occupied by the crown of each shrub (S) was estimated through the ellipse formula (eqn. 1):

$$S = (\pi \cdot L1 \cdot L2) / 4$$

As a whole and for each shrub species, we determined the number of shrubs per hectare (N ha-1), the average occupied area and the average height. For the seedling sample we also calculated the average yearly increase in height (Ih) and diameter (Id).

All data were elaborated using common tests of inferential statistics:

• χ2 test to assess the differences between the shrub species for the presence or absence of seedlings within (in cases of just 1 degree of freedom Yates’s correction was utilized);
• two-way ANOVA (factorial scheme with interaction) for the variables S, H, h (variation sources: shrub species; presence or absence of seedlings);
• one-way ANOVA for the variables S (variation source: tree cover degrees) and h (variation source: shrub species);
• Duncan’s test in all cases where P <0.05;
• analysis of the regression of h, d, Ih, Id versus the age of the seedlings;
• linear correlation between S, H and h;
• Spearman’s correlation between S, H and the number of seedlings (ns).

# Results

## Shrubs

An overall census of 502 shrubs with an average density of 279 (s.e. ± 73.5) per hectare resulted in a field survey of the 45 sample plots; 8 of them (18%) were lacking in both tree seedlings and shrubs. Of these, 6 are concentrated in the upper part of the research area, near some enclosures used as temporary shelters for livestock, and 2 are located in the lower part, in an area periodically subject to waterlogging and presumably unsuitable for woody plants.

In the remaining 37 specimen areas the following shrub species were registered (Tab. 1): Stachys glutinosa (S.g.), Crataegus monogyna (C.m.); Genista pichi-sermolliana (G.s.); Prunus spinosa (P.s.); Rubus ulmifolius (R.u.). The most abundant species are G.s. (56%) and R.u. (29%) followed by C.m. (7%), S.g. (5%) and P.s. (3%). G.s. is exclusive to the cover class A, where the average density of shrubs is higher (443 plants ha-1; s.e. ± 3.8). This density generally decreases with the increase of the degree of tree cover.

Tab. 1 - Mean density of shrubs (N ha-1 ± standard error) for the 45 sample squares.

Species Tree canopy cover classes Overall
<30% 30-60% >60%
Stachys glutinosa 81 (±0.7) 100 (±15.8) 75 (±0.5) 100 (±14.1)
Crataegus monogyna 56 (±0.2) 25 (±0.5) 50 (±2.4) 47 (±4.1)
Genista pichi-sermolliana 996 (±387.2) - - -
Prunus spinosa 46 (±0.2) 38 (±2.6) 25 (±0.5) 42 (±4.2)
Rubus ulmufolius 152 (±1.8) 175 (±19.3) 68 (±12.1) 135 (±34.6)

The average area of the shrub crown, with the exclusion of G.s. which has an area of 2.6 m2, is not significantly differentiated in the various classes of tree cover (F2;218= 1.48) and is on average 1.3 m2 (s.e. ± 0.12).

## Saplings

The total number of registered seedlings comes to 1079, with an average density of 599 (s.e. ± 122) per ha; Holm oak is the most abundant species (92%) with 553 (s.e.±117) seedlings per hectare, whereas Downy oak (8%) is much less present (46 ± 18 seedlings per hectare).

The average age of the seedlings is 14 years (s.e. ± 0.6) ranging from 6 to 25 years; the subdivision of the sampled individuals into age classes (Fig. 1) reveals a normal distribution with a slight asymmetry towards the right side.

Fig. 1 - Distribution of seedlings in age classes (2 years amplitude).

The height of the seedlings (48 cm; s.e.± 18) is not significantly correlated to their age (r=0.2); on the contrary d (1 cm; s.e. ± 0.4) is positively correlated to this feature (r = 0.7, p < 0.01). The mean Ih is 3.9 cm/year (s.e. ± 0.4) and tends to decrease considerably with age (r = -0.68, p < 0.01), the average Id is 0.7 cm/year (s.e. ± 0.04), with no significant correlation to age.

## Interaction between shrubs and saplings (shrub facilitation)

Fifty-five percent of the shrubs were shown to host saplings (Fig. 2, Tab. 2) with significant differences among species (χ2[4] = 59.1, p<0.01).

Fig. 2 - Frequency of shrubs with or without saplings.

Tab. 2 - Comparison among sapling densities inside the different shrub species (n.s.= not significant).

Comparisons d.f. χ2 P
1. All species 4 59.1 <0.01
2. C.m. vs. R.u. 1 1.1 n.s.
3. P.s. vs. G.s. vs. S.g. 2 0.13 n.s.
4. 2 vs. 3 1 58.4 <0.01

C.m. is the species with the highest frequency of patches containing saplings (83%) with values comparable to those of R.u. (76 %; χ2[1] = 1.1); all in all, the other kinds of shrubs have significantly lower values with respect to the previous ones (χ2[1] = 58.4, p < 0.01), with comparable frequencies (χ2[2] = 0.13) though consistently below 45%.

The ANOVA test (Tab. 3) has highlighted significant differences with respect to the average area (S), both among the examined species and regarding the presence or absence of saplings inside.

Tab. 3 - ANOVA test, variable S, variation sources are shrub species (1), the presence or absence of seedlings (2) and the interaction 1x2 (n.s.= not significant).

Effect d.f. F P
1. Shrub species 4 17.35 <0.001
2. Sapling occurrence 1 7.15 <0.01
3. Interaction 1x2 4 1.55 n.s.
Error 492 - -

On the basis of the results of the Duncan test, G.s. has an average area (2.6 m2) considerably larger than S.g. (0.2 m2) and P.s. (0.8 m2), whereas R.u. and C.m. have intermediate values (1.4 and 1.9 m2 respectively); moreover, the shrubs giving shelter to saplings have a larger average area than the others (2.5 and 1.5 m2 respectively).

The ANOVA test (Tab. 4) showed significant differences with respect to the average H among species.

Tab. 4 - ANOVA test, variable H, variation sources are shrub species (1), the presence or absence of seedlings (2) and the interaction 1x2 (n.s.= not significant).

Effect d.f. F P
1. Shrub species 4 15.8 <0.001
2. Sapling occurrence 1 1.45 n.s.
3. Interaction 1x2 4 0.13 n.s.
Error 492 - -

Based on the results of the Duncan test, the average height of C.m. shrubs (1.34 m) is higher than that of all other shrubs, those of P.s. and S.g. are the lowest (0.41 and 0.29 m respectively), whereas R.u. and G.s. have intermediate heights (0.82 and 0.83 m respectively).

Considering all shrubs with saplings inside, the average ns (Fig. 3) varies considerably among the species (F4;270=11.67, p<0.01); from the Duncan test it appears that C.m. has a higher average ns (9), whereas G.s. and S.g. have the lowest (2), and P.s. and R.u. have intermediate values (5).

Fig. 3 - Means of sapling number (ns) and shrub height (h) by species (vertical bars indicate 1 standard error).

The average height of the saplings (24 ± 1.4 cm) varies quite considerably among the shrub species (F4;270= 5.31, p<0.01). In particular, the Duncan test shows that the average h is higher for C.m. (44.8 cm), lower for S.g. and G.s. (13.7 and 18.8 cm respectively), whereas R.u. and P.s. have intermediate values (27.3 and 31 cm).

The number of saplings per shrubs is positively correlated (Tab. 5) to the S and H of the shrubs (r = 0.23 and 0.32 respectively, p < 0.05), but the correlation is very high and significant for R.u. (r = 0.46 and 0.48 respectively, p < 0.01) and for C.m. (r= 0.81 and 0.66 respectively, p < 0.01).

Tab. 5 - Values of the Spearman correlation between H, S and ns. (*) = P < 0.05; (**) = P < 0.01).

Species H S N. of cases
S.g. 0.40 -0.12 10
P.s. -0.04 0.15 29
G.s. 0.06 0.19 119
Ru. 0.48** 0.46** 7
C.m. 0.66** 0.81** 110
Total 0.32* 0.23* 275

The height of the saplings is positively correlated to S and most of all to H (r = 0.28 and 0.6 respectively, p < 0.01). The highest correlation values are those of C.m. and R.u. (Tab. 6).

Tab. 6 - Values of the linear correlation between h, H and S.(*) = P < 0.05; (**) = P < 0.01).

Species h-H h-S N. of cases
S.g. 0.66* 0.47 10
P.s. 0.72 0.04 29
G.s. 0.33* 0.35** 119
R.u. 0.57** 0.58** 7
C.m. 0.81** 0.68** 110
Total 0.60** 0.28** 275

# Discussion

## Shrubs

At present the shrub cover consists of five main species and represents a minor part (12%) of the total ground cover, compared to grass (31%) and bare soil (57%). Over half of the registered shrubs are Genista (56%); they form very dense patches located in areas with large granitic outcrops, thus indicating a preference for relatively scant pedological conditions (more dryness). Rubus ulmifolius is the second most common shrub species (29%), followed by Crataegus monogyna (7%) Prunus spinosa and Stachys glutinosa (3% and 5% respectively), both were distributed quite homogeneously over the examined area. All these shrub species have thorns basically as a result of the selecting action of animal grazing ([57], [58]). It is a known fact that in environments with a Mediterranean climate, excessive grazing pressure initially causes a simplification of biodiversity with the disappearance of the most palatable species and/or the substitution of perennial herbaceous species. The last regression stage is the complete disappearance of vegetation and the beginning of erosive phenomena ([42]).

The density and composition of the shrub cover in the Pradu woodland are clearly connected with tree cover: where the latter is higher, the shrub cover is minimal. This condition is largely ascribable to the ecological needs of the shrub species, all of which are basically heliophilus; the action of animal grazing, however, also influences their distribution, which tends to thicken in the less browsed areas and in the presence of large rock outcrops.

## Seedlings

The specific composition of regeneration is directly related to the species proportion of the tree layer: in fact, Holm oak seedlings are the most common (80%) and are occasionally associated with those of the Downy oak (17%) which is sometimes the only species present inside the shrubs. Under no circumstances has the regeneration of Montpellier maple been observed.

The light microclimate inside small shrub patches is known as a potential selecting factor for the saplings of tree species ([26], [36]). In our case, the higher shade tolerance of Holm oak compared to Downy oak and Montpellier maple could facilitate its survival inside shrubs. Moreover, the morphology of seedling leaves should also add competitive advantages to Holm oak: under conditions of repeated/continuous grazing the leaf tends to become more coriaceous and thorny, expressing a substantial adaptability to grazing ([6]).

The age of the saplings is undoubtedly a very important factor within the Pradu woodland. As in other studies carried out in the Mediterranean region ([15], [47]), our observations have demonstrated that a general, uneven age distribution exists; a shortage of young and very young subjects is reported, however. More specifically, the sampled individuals are rather old and their age class distribution shows some peaks that might be the consequence of years of larger seed production. In such years, in fact, the population of predators leaves a higher number of seeds available for germination ([56], [22], Murphy & Janzen in [5]). The absence of seedlings younger than 6 years of age could be symptomatic of a tendency towards a worsening of the seed quality produced by the plants: according to Cierjacks & Hensen ([21]), increased grazing pressure results in a higher percentage of aborted Holm oak acorns. However, worsening climatic conditions (dryness) should not be ruled out.

According to Smit et al. ([54]) the facilitating action exerted by shrubs on tree regeneration decreases as biotic stress increases: intensive grazing causes severe damage to shrubs, thus leading to heavier grazing and higher seedling mortality. The yearly average height growth of the seedlings (3.9 cm) decreases considerably with the increase in age (r = -0.7, p < 0.01) as a result of livestock browsing; nonetheless, the diameter growth remains constant (0.7 cm/year), thus confirming that oak species tolerate grazing well (Shaw 1974 and Hilton et al. 1987, in [40], [59]).

## Shrubs facilitation

In the Pradu woodland, shrub patches represent the only habitats where processes of natural regeneration of the tree components have been observed: no seedlings have been found in the open. This result confirms that facilitation is especially important in cases of heavy grazing pressure ([7]). However, the fact that no regeneration has been observed in many shrubs could indicate that intensive grazing causes severe damage even to shrubs and reduces their defensive action ([54]).

The interaction between shrubs and tree regeneration occurs in different ways during the various stages of the regeneration process ([26], [53]). Due to their morphological characteristics and to the richness of nourishing supplies, the seeds of oak species are more likely to be consumed ([51], [4]). The same authors highlight the role played by rodents and birds in the dispersal of these seeds: among them, the jay (Garrulus glandarius) appears abundant in the study area (Shenk & Torre in [16]). The dispersal of seeds from the parent plants and their establishment in small repositories thanks to these “secondary dispersal agents” become more important when the number of safe sites is small ([46]). It has been widely confirmed that shrub patches are very important sites both in the dispersal phase ([9], [23], [26], [15], [49]) and in the establishment phase, since seeds and seedlings are protected from predation and trampling ([32], [49], [35], [52], [54]). This role remains consistent even if seed consumption by insects, rodents and birds reaches significant levels ([37], [53], [59]). Moreover, the shrub patches create the most favourable micro environmental conditions for the germination of seedlings ([29], [30]): less soil compaction, higher humidity, less radiation, greater availability of nutrients, and less competition with herbaceous species ([11], [26], [49], [20], [3], [28], [36]).

As observed in research conducted in Holland ([3]), the size of the shrubs has an influence on the seedlings’ chance of survival. Our results indicate that at this stage the height of the shrubs does not affect the establishment of seedlings. On the contrary, the crown area is a decisive factor; the wider the shrubs, the greater the difficulty for animals (especially swine) to feed on acorns and browse seedlings. At Pradu the minimum effective area has proven to be somewhat larger than 2 m2; there are, however, species-related differences, with the cover of Crataegus monogyna and Rubus ulmifolius representing the safest site for the establishment of seedlings. A minor portion of Genista pichi-sermolliana shrubs has seedlings inside, but the seedling density is nevertheless low. This may be due to the scarce massiveness of the crown of this species, with a greater accessibility to animals compared to previous ones; however, it must also be considered that Genista patches concentrate on rock outcrops, which have more adverse edaphic conditions for seed development and are generally located away from the most vigorous seed-bearer trees.

Prunus spinosa and Stachys glutinosa are not very common and generally form small-sized shrubs less effective in facilitating the establishment of seedlings.

In the consolidation stage the survival and growth of the seedlings are mainly influenced by animal browsing ([40], [12], [11], [27], [10], [49], [35], [3], [19], [54]). Competition between shrubs and saplings for light, water and nutrients is also important at this stage, and, according to some authors, results in reduced growth of tree species ([10], [50]).

The average height of saplings at Pradu is considerably greater in shrubs of Crataegus monogyna, Prunus spinosa and Rubus ulmifolius compared to the other species; nevertheless no sampling has reached such a size as to definitely escape browsing action by livestock (roughly estimated within 2 m).

Seedlings have shown the tendency to parallel the height growth of the hosting shrubs; this correlation is particularly marked in the case of Crataegus monogyna. This phenomenon is probably due to the morphological characteristics of hawthorn which, in optimum conditions and unlike Prunus spinosa, is able to grow in height at the same rate of the saplings. Shrubs of Rubus ulmifolius tend to grow enveloping seedlings and follow them in their growth; since young shoots of this plant are susceptible to grazing, the height growth is, nonetheless, conditioned. Studies conducted in California ([59]) have also shown that shrubs of Rubus discolor facilitate the establishment of seedlings of oak species; however, they interfere with their subsequent development by confining their growth.

At Pradu, the seedlings’ growth inside Genista pichi-sermolliana patches, though proportional to the shrubs’ size, is less marked: this could be due both to the poor conditions of the site where these shrubs grow and to the poor ability of this species to protect itself from grazing. Clefts between rocks occasionally proved to be safe sites for the establishment and survival of seedlings of tree species; this phenomenon has already been described for Picea abies seedlings in the timbered grasslands of Switzerland ([52]). Still our observations indicate that saplings remain confined inside the rock clefts with no chance for further growth, both because of punctual animal browsing as soon as the seedlings emerge, and because of the poor soil condition of such microenvironments.

# Conclusions

The effectiveness of the interaction between shrubs and seedlings with respect to tree species regeneration in wooded pastures has been demonstrated at the Pradu site as well. In particular, shrub facilitation has proved to be species-specific ([13], [14], [28]). Crataegus monogyna is the most efficient species in facilitating the establishment and development of tree species seedlings; Rubusulmifolius is efficient in the establishment phase, but somewhat less so in the following stages. The other shrubs species have a weaker facilitating role.

Despite the fact that the density of shrubs hosting Holm oak or Downy oak seedlings is quite considerable, the high average age and the small size of the seedlings lead us to think that the present grazing pressure is incompatible with any concrete chance of consolidation for these saplings, and thus for the natural regeneration of the tree component of this peculiar ecosystem. At present, most of the shrubs and the hosted saplings have not attained a height greater than the browsing line of cattle and equines; this means that their future is still dubious.

At the same time, the local social and economic situation does not allow us to advance any hypothesis about even a temporary suspension of grazing activities; as a result, the survival of these ecosystems seems unlikely, and there are important economic, hydrologic and aesthetic implications. A great cultural heritage, based on the local population’s management of natural resources over the centuries, risks extinction as well. The conservation of these stands must therefore rely on the rationalisation of human activities through the exploitation of a whole range of resources (not least tourist appeal). This approach appears to be the only way to face this threat and effectively solve the problem of tree regeneration at Pradu.

The adjustment of the grazing pressure to sustainable levels is certainly a critical measure. According to recent studies conducted in Q. ithaburensis forests in Israel ([25]), a cattle density of 0.7 head per hectare is already able to cause considerable damage to regeneration. It is obvious that an agreement must be reached with local breeders, perhaps by compensating the reduced yield with a better profitability of dairy and meat products. Another possibility could be the introduction of systems that provide for the integration of natural regeneration by means of planting seedlings either inside natural shrub patches, as positively experimented in Spain for Q. pirenaica ([18], [19]), or inside special enclosures.

# Acknowledgements

This work was conducted with the cooperation of the Orgosolo municipality (Nuoro province). The authors would like to thank Prof. Ignazio Camarda of the University of Sassari, for his assistance with the floristic classification. Special thanks also go to Dr. Tonina Muggianu and Dr. Nicola Sanna for their contribution to field surveys. The experiments comply with the current laws of the country in which they were performed.

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Gómez-Aparicio L, Gómez JM, Zamora R, Boettinger JL (2005a). Canopy vs. Soil effects of shrubs facilitating tree seedlings in mediterranean montane ecosystems. Journal of Vegetation Science 16: 191-198.
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Gómez-Aparicio L, Valladares F, Zamora R, Quero JL (2005b). Response of tree seedlings to the abiotic heterogeneity generated by nurse shrubs: an experimental approach at different scales. Ecography 28: 757-768.
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Gómez-Aparicio L, Gómez JM, Zamora R (2005c). Microhabitats shift rank in suitability for seedling establishment depending on habitat type and climate. Journal of Ecology 93: 1194-1202.
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Gómez JM, Hódar JA, Zamora R, Castro J, García D (2001). Ungulate damage on Scots pines in mediterranean environments: effects of association with shrub. Can. J. Bot. 79 (6): 739-746.
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Herrera J (1995). Acorn predation and seedling production in a low-density population of Cork oak (Quercus suber l.). Forest Ecology and Management 76: 197-201.
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Joffre R, Vacher J, De Los Llanos, Long G (1988). The dehesa: an agrosilvopastoral system of the Mediterranean region whit special reference to the Sierra Morena area of Spain. Agroforestry System 6: 71-96.
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Kuiters AT, Slim PA (2003). Tree colonisation of abandoned arable land after 27 years of horse-grazing: the role of bramble as a facilitator of oak wood regeneration. Forest Ecology and Management 181: 239-251.
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Leiva MJ, Fernández-Alés R (2003). Post-dispersive losses of acorn from mediterranean Savanna-like forests and shrublands. Forest Ecology and Management 176: 265-271.
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Linares AM (2007). Forest planning and traditional knowledge in collective woodlands of Spain: the dehesa system. Forest Ecology and Management 249: 71-79.
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Madrau S (1995). Valutazione della attitudine all’utilizzo forestale dei suoli della Sardegna. L’area del parco del Gennargentu. Annali Accademia Italiana di Scienze Forestali 44: 279-305.
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Munoz DC (2004). Algunos apuntes para la conservaciòn de las dehesas. Escuela Tecnica Superior De Ingeneros Agronomos Y De Montes. Univeridad De Cordoba, Spain.
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#### Authors’ Affiliation

(1)
S Alias
University of Sassari, v. Sa Terra Mala, I-08100 Nuoro (Italy)
(2)
L Bianchi
G Calamini
S Sioni
Department of Environmental Forestry Science and Technology, University of Florence, v. San Bonaventura 16, I-50145 Florence (Italy)
(3)
E Gregori
CRA - ABP Florence, p.zza M. D’Azeglio 30, I-50121 Florence (Italy)

#### Citation

Alias S, Bianchi L, Calamini G, Gregori E, Sioni S (2010). Shrub facilitation of Quercus ilex and Quercus pubescens regeneration in a wooded pasture in central Sardinia (Italy). iForest 3: 16-22. - doi: 10.3832/ifor0517-003

#### Paper history

Accepted: Sep 24, 2009

First online: Jan 22, 2010
Publication Date: Jan 22, 2010
Publication Time: 4.00 months

© SISEF - The Italian Society of Silviculture and Forest Ecology 2010

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text:
# How can I setup a controller in the BGE?

Is it possible to get a DS4 controller to work in the BGE?

I've tried this project from Blend Swap, but only the buttons seem to work, not the axes. Is there any way to fix this?

If it's relevant, I'm using Xubuntu, with the builtin kernel driver, but I've also tried it with the old Ds4drv but it won't work at all.

In fairness, that file from blendswap did not work for me either. However making a controller's axes work is pretty easy.

There are two ways to access a controller in the BGE, that is with the joystick sensor or the joystick class through python. They are both very much the same.

The notable difference is in the way it reports the input from the axes. Using the logic brick you will get a rather arbitrary range between -32767 and 32767 (if this is the horizontal axis,-32767 is left, 0 is no moment, and 32767 is full right.)
Now getting the input through the python class normalizes the values: between -1.0 and 1.0, again 0 is no movement.

For the remainder of the answer I will be using the just the python class, but everything is easily adaptable to the logic brick.

Trouble is there are no event handlers for controllers (buttons, hats, or axes), so we have to make our own.

# The buttons

For the button, using the python class, all you get is a list of the active buttons. They are listed by id 0-12. As an example button 0 is the Square Button on a DS4, and the blue X button on the xbox layout.

To turn the list in to something useful for input you will need a few lines like these:

joystick = bge.logic.joysticks[0]
btns = joystick.activeButtons
if 0 in btns and len(btns) == 1:
doSomething()

The top line gets the controller, the second line gets the button list. Then it is just checking if the button we are targeting (button 0) is in the list. I have the len(btns) == 1 part in there so that it is only triggered by an exclusive button press (there are no other buttons held down).

# The hats

Similarly the active hat is expressed in a list. However there will only ever be one active "direction" in this hat list. There are eight directions on the hat:

0:None
1:Up
2:Right
4:Down
8:Left
3:Up - Right
6:Down - Right
12:Down - Left
9:Up - Left

from the api

Now if there is no active direction (the hat is not being pressed) then the list will be 0, not an empty list.

This is one way you can use the input from the hats:

up = 1
down = 4
left = 8
right = 2
upLeft =9
upRight = 3
downLeft = 12
downRight = 6

hats = joystick.hatValues
if up in hats:
player.applyMovement([0.01, 0, 0], True)
elif left in hats:
player.applyMovement([0, 0.01, 0], True)

# The axes

As mentioned above the axis values are in a list axisValues, which is different for the logic brick and the python class. Also, I do not have a DS controller, I have a "Logitech Dual Action" controller, so the axes may be (probably are) different.

At least for a xbox style controller the axes are in this order: left analog stick X, left analog stick Y, then right analog stick X and right analog stick Y.

So some (overly simplified) code to use this would be:

av = joystick.axisValues
left_analog_x = av[0]
left_analog_y = av[1]
right_analog_x = av[2]
right_analog_y = av[3]

Note the dead zone there, if that is not done the player will always drift, even when there is not input.) Now I have done this on two different computers, and both have had wildly different amounts of initial drift. So what I do is just print(joystick.axisValues) and then add or subtract those values to each value in the axisValues list. It is not an ideal solution, but if the values are small it works.


text:
• Hello!

when I run exe that I create, I have the next error. "No se encuentra el punto de entrada del procedimiento _ZN20QFutureInterfaceBase13setThreadPoolEP11QThreadPool en la biblioteca de vinculos dinamicos Qt5Core.dll".

I have windows 7 with 64 bits. And I use Qt Creator 5.5.
How can I resolve this problem? thanks.

• Are you sure your application is using the right version of Qt5Core.dll?
In debug, it should be Qt5Cored.dll, and in release it must be Qt5Core.dll.

• yes, I saw that qt5cored.dll is in debug and in release qt5core.dll. I included the libraries with my exe program and then I have the error that I mention.

• Hi, perhaps you copied the Qt dlls from the wrong place, instead of copying from C:\Qt\5.5\mingw492_32\bin you copied from C:\Qt\Tools\QtCreator\bin?


text:
# Tangent of angles approaching 90 degrees

Last week a colleague came to me with a puzzle. He asked me to punch in the tangent of 89 degrees into a nearby TI-83 calculator.

$\tan(89^\circ) = 57.28996163$

He asked me what was surprising about this number. I wasn’t surprised. I didn’t have an answer for him, although in retrospect I probably should have. He had to tell me this was the number of degrees in a radian. Oh! So it is.

Even further, he said, try punching in tan(89.9), or tan(89.99), etc.

$\begin{array}{rl} \tan(89^\circ) &= 57.28996163\\ \tan(89.9^\circ)&=572.9572134\\ \tan(89.99^\circ) &= 5729.577893\\ \tan(89.999^\circ) &= 57295.77951\end{array}$

Each one is (about) ten times the previous. (With the TI-83, replacing the last result with each new one, I didn’t see the “about” until later.) This is kinda neat! His question to me was: WHY is this true?

Tangent is a function that accepts an angle and spits out a ratio of lengths. It seems weird that the answer for tan(89) looks like a number of degrees. It is a unitless output, though, and ~57 degrees per radian is also unitless, so I suppose this isn’t much of an issue. The question is, why does it appear that

$\tan(89^\circ) = \cfrac{180^\circ}{\pi \text{ rad}}\text{ ?}$

Ditching the degree measure,

$\tan\left(\cfrac{\pi}{2}-\cfrac{\pi}{180}\right) \stackrel{?}{=} \cfrac{180}{\pi}$

My problem was in trying to first tackle this question visually.

An equivalent image:

I tried to explain this by imagining rolling the circle over the tangent line, wrapping the line around the circle, etc. I didn’t get anywhere.

I also tried considering the fact that there’s nothing particularly special about degree measure, except for the fact that 360 is an easy to divide number. Does this happen with other angle units? For example, what about a unit that was, instead of 1/360 of a circle, a larger 1/100 of a circle? We could instead take the equation above and ask,

$\tan\left(\cfrac{\pi}{2}-\cfrac{\pi}{100}\right) \stackrel{?}{=} \cfrac{100}{\pi},$

Is the tangent of one one-hundredth of a circle short of $\pi/2$ equal to the number of hundredths of a circle in a single radian? It looks to be true!

$\begin{array}{rl} \tan\left(\cfrac{\pi}{2}-\cfrac{\pi}{100}\right)&= 31.82051595\dots\\\\ \cfrac{100}{\pi} &=31.83098862\dots \end{array}$

But this is only approximate. We could extend this to any fractional unit $1/n$ of a circle:

$\tan\left(\cfrac{\pi}{2}-\cfrac{2\pi}{n}\right) \approx \cfrac{n}{\pi}$

Using this different unit, where the approximation is less accurate, I was able to see that the degree version wasn’t exactly true, either. It definitely looks like dividing the circle into a larger number (360, rather than 100) yields a closer approximation:

https://www.desmos.com/calculator/vqbu4zni5p?embed

I was comfortable in concluding now that this wasn’t just a coincidence that relied on degree measure, and could extend this to include using 89.9, 89.99 etc degrees as well. In fact, tacking on .9s to the 1/nths of a circle units works. Just plugging in a bunch of numbers, it looks like

$\tan\left(\cfrac{\pi}{2}-\cfrac{2\pi}{n}\ 10^{-a}\right) \approx \cfrac{n}{\pi}\ 10^a$

works for any n, and also extends to any power a, not just the integers.

https://www.desmos.com/calculator/ctcicykylj?embed

The question remained, why is this true? Now that I saw it’s only an approximation, I realized that I should be going about this algebraically from the start.

A trick, called the small angle approximation, is used in physics often to get rid of pesky sines and tangents when you’d rather just have an expression with the angle inside.

$\begin{array}{rl} \sin x &\approx x\\ \tan x& \approx x \qquad\text{when }x\ll1 \end{array}$

This behavior is clear when the functions are written in their Taylor series form:

$\begin{array}{rl} \sin x &= x - \cfrac{x^3}{6} + \cfrac{x^5}{120} - \cfrac{x^7}{5040} +\dots \\\\ \tan x &= x +\cfrac{x^3}{3} + \cfrac{2x^5}{15} +\cfrac{17x^7}{315}+\dots \end{array}$

When x is real small, all the higher power terms get super small, and the approximation becomes more accurate.

This approximation was my first thought, but there’s a problem: it works for small angles, but my colleague’s puzzle was about angles near 90 degrees. In fact, we can’t even fudge the Taylor series of tangent near here, because there is no Taylor series around 90 degrees. (This is a consequence of the fact that tan(x) blows up to infinity at 90 degrees.)

The problem is solved by noting that working with tangent near 90 degrees is the same as working with another trig function, cotangent, near 0 degrees.

$\tan\left(\cfrac{\pi}{2}-x\right) = \cot(x) = \cfrac{1}{\tan(x)}.$

Setting everything up:

$\begin{array}{rl} \tan\left(\cfrac{\pi}{2}-\cfrac{2\pi}{n}\ 10^{-a}\right) &= \left(\tan\left(\cfrac{2\pi}{n}\ 10^{-a} \right) \right)^{-1} \\ & = \left(\left(\cfrac{2\pi}{n}\ 10^{-a} \right) +\cfrac{1}{3}\left(\cfrac{2\pi}{n}\ 10^{-a} \right)^3 +\dots \right)^{-1}\\ &\approx \left(\cfrac{2\pi}{n}\ 10^{-a} \right)^{-1} \qquad \text{(the approximation)}\\ &= \cfrac{n}{2\pi}\ 10^a \end{array}$

Done! Having the $10^a$ instead of any old number is unneccessary — this works for any multiple. However, integer a‘s makes the trick of having the same digits show up in tan(89), tan(89.9), etc. work.

So, we can show this algebraically. I just wish I had a nice geometric argument.


text:
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Nate has 4 1/3 yards of fabric. He uses 2.5 yards of fabric to make a pillow. How much fabric does he have left?

13/3 - 5/2 = $$\frac{26-15}{6} = 11/6$$


text:
# What type of Euler angles is used in Blender?

My understanding

There are several possibilities for representing a rotation, e.g. direction + angle, which is the most visual but requires two distinct objects, and matrix or quaternion, which are preferred for computation but not visual at all.

Blender uses the previous methods and also an Euler angle representation, similar to a gimbal system. Any transform in a 3D space, excluding pure translations, can be expressed by three angles representing a chain of three rotations around three different axes in a specific order.

There is an infinity of choices for the axes. Usually axes are part of a Cartesian frame XYZ, and either fixed (extrinsic) or rotated (intrinsic). Proper Euler angles use a chain with the same axis for the first and third rotation, Tait-Bryan angles use different axes for the three rotations, but are also often referred to as Euler angles. One commonly used possibility is:

The red axes initially aligned with XYZ have been rotated by φ about Z, then by Φ about ξ, the older X axis now rotated, and finally by Ψ about ζ, the old Z. This Euler mode would be referred to as ZX'Z'' to give a meaning to the triplet (φ, Φ, Ψ).

In Blender these three angles can be grouped into a Vector of floats when a single object is preferred to represent the rotation. Still the angles mean nothing unless we know the chain of axes used (mode).

What the documentation says

From the Advanced section:

The special thing about this is that the axes have a hierarchical relationship between them: one of the axes is at the top of the hierarchy, and has one of the other two axes as its immediate child; at the same time, this child axis is the parent of the remaining axis, the one at the very bottom of the hierarchy.

Which axis is on top, which one in the middle and which at the bottom, depends on the particular Euler gimbal: there are six types of them, as there are six possible combinations: XYZ, XZY, YXZ, YZX, ZXY and ZYX Euler rotation modes. These modes are named using the letters of the axes in order, starting from the axis at the bottom of the hierarchy, and finishing with the one on top.

It doesn't indicate which mode is used.

My experiment

I setup an experiment to try understand which mode Blender uses:

It's just three Cartesian systems, a fixed set of XYZ axes, one set with darker colors randomly oriented using a Transform node and used as a target. One last in lighter colors can be oriented manually using a Rotate Euler (Euler + Object options) connected to a Transform.

I'm able to match the darker system by orienting the mobile system. When matching, the Euler angle values match. But I still have trouble understanding the axes used.

When manipulating the Y input of Rotate Euler, and looking along the world Y axis:

It can be seen that the size of the rotated light green axis changes during the rotation, which indicates the rotation is not about the world Y axis. I tried other possibilities but I was not able to determine which axis is associated with the Y input.

Question

What are the axes and the axis order used in Euler defined rotation?

• Note: I had to select an answer, but actually the three existing answers are interesting to read and provide their own elements. Thanks to the three authors.
– mins
Aug 16 at 9:19
• Fun semirelated math fact: If you use the three Euler angles in some order, applied to global coordinate axes, then you get exactly the same result as if you apply the same angles in the opposite order, but to local axes. Aug 16 at 11:09
• ("Local axes" as in they start out the same as the global axes, but follow the rotating object as it rotates.) Aug 16 at 11:15
• @Arthur: See Davenport angles.
– mins
Aug 16 at 12:53
• Oh, there is even a name for it. Cool, didn't know that. Aug 16 at 12:57

## 3 Answers

In the viewport, you can select the order of Euler component rotations:

But in GN, you are not offered the choice. It's XYZ.

Unless you're offered a 'Local' option for entities with their own axes, such as instances (or rotations, which you can think of as axes themselves), transforms in GN are of the geometry, in its object's space, as previously transformed at object-level. (Just like other modifiers.) So rotations are about the modified object's axes.

Euler rotations are compounded; as in a physical gimbal, each axis becomes the child of the axis before it. Chaining single-axis Eulers in another order does not work. The axes revert to the object's, after a rotation is executed.

If you want to assemble your own compound Euler rotation in GN, setting the axis order, you can do this...

... for example, to get a ZXY compound Euler rotation of 30 in Z, followed by 10 in X, followed by 20 in Y.

Note the rotations are 'Local', and read backwards from their effect.

• My edited answer has some of the information in your answer. However, you seem to actually know this, while I've only drawn empirical conclusions from experimenting while I was writing. I bow to thee. +1 Aug 15 at 19:51
• "rotations are about the modified object's axes", if I understand correctly, the elementary rotations are processed this way: Rotation about local X by the value of the first angle, then about rotated local Y by the second angle, and about twice rotated local Z by the last angle?
– mins
Aug 15 at 19:54
• @InTheAbsenceOfFear .. sorry, didn't mean to tread on your perfectly good answer, I was phrasing mine and checking everything, to make sure it was right, as far as it goes, while you were writing yours. No bow required, +1 too. Aug 15 at 19:55
• You posted your answer while I was editing mine. Consequently, I didn't know of yours until I was done, and you didn't know of my edit. I do in no way feel trodden on. And don't put to much into my "I bow to thee". I just like such phrases. :) Aug 15 at 20:02
• @mins Right! I just deleted a comment repeating exactly what you said. The axes start out aligned to the object's. then each axis is rotated by the previous component of the Euler, in order. Like a mechanical gimbal. Aug 15 at 20:06

As far as I can tell from the source code, Rotate Euler is simply multiplying two matrices after translating the Euler rotations into them.

The translation from Euler rotation to matrix is as follows:

/******************************** XYZ Eulers *********************************/

void eul_to_mat3(float mat[3][3], const float eul[3])
{
double ci, cj, ch, si, sj, sh, cc, cs, sc, ss;

ci = cos(eul[0]);
cj = cos(eul[1]);
ch = cos(eul[2]);
si = sin(eul[0]);
sj = sin(eul[1]);
sh = sin(eul[2]);
cc = ci * ch;
cs = ci * sh;
sc = si * ch;
ss = si * sh;

mat[0][0] = (float)(cj * ch);
mat[1][0] = (float)(sj * sc - cs);
mat[2][0] = (float)(sj * cc + ss);
mat[0][1] = (float)(cj * sh);
mat[1][1] = (float)(sj * ss + cc);
mat[2][1] = (float)(sj * cs - sc);
mat[0][2] = (float)-sj;
mat[1][2] = (float)(cj * si);
mat[2][2] = (float)(cj * ci);
}

Source

So it looks like an XYZ translation.

The axis order for Euler rotations can be set. Click the drop down list Mode in the Transform tab in Object Properties.

I did some testing on this, by setting the Euler rotation from python on one cube and then on another cube I rotated manually (using R followed by angle) on the global axes in the same order as I did in python, and the two cubes lined up. My conclusion is that the axis order refers to the global axes, i.e. Tait-Bryan rotation.

The comments I received made me want to understand this. I'm not sure the following will fully answer the OP's question, but I'm hoping it'll at least be a step in the right direction. A little more testing (after a more thorough reading of the question) gave this result. I did enable displaying the object local axes to make things clearer.

Look at the rotation set in the Object Properties panel, and at the local axis indicators. It seems that the local axes inferred by the rotation in the Object Properties are taken by Geometry Nodes as the axes around which to calculate rotations and along which to do scaling and translation. Look at the rotational difference between the two images, and you'll note that the cylinder is rotated about the object's local axis, which is indicated by an X (there are also lines going to the object origin, but those are obscured by the object's geometry).

• Thanks, it's an interesting addition. +1, but I've two comments. This is valid for the N-Panel UI, I don't see an equivalence to select the Euler mode in Transform GN (so one mode must be used by default). Also ref. you conclusion: I tested changing Y value in my setup and looking from the global Y axis, the rotation was not preserving the distances to the center, hence it was not about the world Y. I may be wrong, just a comment.
– mins
Aug 15 at 18:29
• @mins I guess I didn't read thoroughly enough. I missed that you were doing this in GN. I just did a very quick test, and it looks like Tait-Bryan XYZ to me. If it doesn't preserve the distances to the centre, are you sure the object origin is at the centre of the object? Rotations are done about the object origin, which may very well be offset from the centre. Aug 15 at 18:45
• @mins Like I said, I guess I didn't read thoroughly enough. Apologies for that. Unfortunately I've not bothered to learn GN yet, and I can't really provide any more insight. Just a thought though. I noticed that if geometry is added from within GN, it does seem to behave differently, as if its centre and/or origin is offset. This may well also affect rotations. Aug 15 at 19:06
• @mins Did some further testing, and edited my answer to include a bit more info. Aug 15 at 19:40
• I appreciate your additional info and effort.
– mins
Aug 15 at 19:59


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# 능동소나 linear frequency modulation 신호의 fractional Fourier transform 분석에 기반한 표적의 거리 추정

Hyung, Sungwoong;Park, Myungho;Hwang, Soobok;Bae, Keunsung
형성웅;박명호;황수복;배건성

• Received : 2015.10.08
• Accepted : 2015.12.14
• Published : 2016.01.31
• 36 157

#### Abstract

As a generalized form of the conventional Fourier transform, fractional Fourier transform (FrFT) can analyze a signal at intermediate domain between time and frequency domains with a transform order ${\alpha}$. Especially, FrFT has a number of advantages in the analysis of LFM (Linear Frequency Modulation) signals due to its robustness to noise. In this paper, we have proposed a new method to detect and estimate the distance of the target from the FrFT spectrum of the received echo signal. Experimental results have validated the proposed method, and shown that reliable target distance could be estimated in noise and reverberation environments.

#### Keywords

FrFT;Time-frequency domain;LFM;Active sonar;Target detection

#### References

1. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley, Chichester, New York, USA, 2001), pp. 117-137.
2. C. Capus and K. Brown, "Fractional Fourier transform of the Gaussian and fractional domain signal support," IEE Proc.-Vis. Image Signal Process., 150, 99-106 (2003). https://doi.org/10.1049/ip-vis:20030313
3. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, "Some important fractional transformations for signal processing," Optics Comm., 125, 18-20 (1996). https://doi.org/10.1016/0030-4018(95)00748-2
4. V. Katkovnik, "A new form of the Fourier transform for time-varying frequency estimation," 1995 URSI International Symposium on. IEEE, 3, 179-182 (1995).
5. R. Jacob, A. Unnikrishnan, and T. Thomas, Development of Time-Frequency Techniques for Sonar Applications, (Ph.D. thesis, Cochin University, 2010).
6. J. W. Seok, T. H. Kim, and K. S. Bae, "LFM Signal Separation Using Fractional Fourier Transform" (in Korean), J. Korea Inst. Inf. Commun. Eng., 17, 540-545 (2013). https://doi.org/10.6109/jkiice.2013.17.3.540
7. S. W. Hyung, K. S. Bae, S. B. Hwang, and M. H. Park, "Detection Method of LFM Signal Interval from Noisy Received Signal" (in Korean), 2015 IEIE Summer Conference, 2015, 525-526 (2015).

#### Acknowledgement

Supported by : 국방과학연구소


text:
Pragmatic Algebra
 Pragmatic Algebra (ISBN: 81-88296-02-3) CONTENTS Previous Next Block Designs and Strongly Regular Graphs Constructed from Some Linear and Unitary Groups Dean Crnkovic, Vedrana Mikulic and Sanja Rukavina Conditions for p-Nilpotency of Finite Soluble Groups Gongzhi Luo, Bin Hu and Xiaolan Yi Power Menger Algebras with Union and Their Endomorphisms P. Glubudom and K. Denecke A Note on the Minus and Star Partial Orderings of Nonnegative Definite Matrices and Their Powers Liu Xiaoji On One Question of the Theory of $\omega$-Composition Formations Awni Faez Al-Dababaseh Previous Next


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# nLab Łoś ultraproduct theorem

Contents

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Idea

The Łoś ultraproduct theorem characterizes the first-order theory of an ultraproduct of $\mathcal{L}$-structures (for $\mathcal{L}$ some signature). It generalizes the transfer principle from nonstandard analysis, by replacing the hyperreals $^*\mathbb{R}$ with an ultraproduct and formulas from the standard reals $\mathbb{R}$ with formulas which are true on an “ultrafilter-large” subset of factors of the ultraproduct.

## Definition

The theorem is usually given in this form:

###### Theorem

let $(A_i)_{i \in I}$ be a family of nonempty $\mathcal{L}$-structures, and let $\mathcal{U}$ be an ultrafilter on $I$. Let $^* A$ be the ultraproduct of $A_i$ with respect to $\mathcal{U}$. Since each $A_i$ was nonempty, $^* A$ is the quotient of the product $\prod_{i \in I} A_i$ by the equivalence relation which identifies sequences which coincide on a subset of indices belonging to $\mathcal{U}$. Let $(a_i)_{i \in I}$ be such a sequence, and let $[(a_i)]$ denote its equivalence class. Then for each $\mathcal{L}$-formula $\varphi(x)$,

$^* A \models \varphi([a_i]) \iff \{j \in I | A_j \models \varphi(a_j)\} \in \mathcal{U}.$

The above follows from a more fundamental fact:

###### Proposition

Any functor

$[\mathcal{U}] : \mathbf{Set}^I \to \mathbf{Set}$

which specifies a choice of ultraproduct (and hence comparison maps, since ultraproducts are colimits) is elementary (logical in the terminology of Makkai-Reyes, i.e. a pretopos morphism.)

###### Proof

A functor is a pretopos morphism precisely when it preserves initial and terminal objects, pullbacks, disjoint sums, and quotients by internal equivalence relations.

Initial objects: a product of empty sets is empty, and a colimit of empty sets is empty.

Terminal objects: a product of terminal objects is terminal, and the quotient of a singleton is a singleton.

Pullbacks: a product of pullbacks is a pullback, and finite limits commute with filtered colimits.

Disjoint sums: a product of disjoint sums is a disjoint sum of products, and colimits commute with colimits.

Quotients: a product of quotients $X_i / E_i$ is a quotient of products $\prod_I X_i / \prod_I E_i$, and colimits commute with colimits.

Now we can show the theorem:

###### Proof of theorem

Since the ultraproduct functor is elementary, then the process of taking points $M(X)$ of a definable set $X$ inside models $M$ commutes with taking ultraproducts. In symbols,

$\left(\prod_{i \in I} M_i / \mathcal{U}\right) (X) \simeq \left(\prod_{i \in I} M_i(X) / \mathcal{U}\right).$

Since this is a filtered colimit, a sequence $\overline{x}$ satisfies that its germ $[\overline{x}]$ is in $\prod_{i \in I} M_i / \mathcal{U}(X)$ if and only if there is some $J \in \mathcal{U}$ such that the restriction of $\overline{x}$ to $J$ is in $\prod_{j \in J} M_j(X)$. i.e. if $x_j \in M_j(X)$ for each $j \in J$.

## Examples

An immediate consequence of the Łoś theorem is the transfer principle for the hyperreals.

The compactness theorem also follows quickly from the Łoś theorem, so anything that you build using compactness can be realized a bit more concretely as an ultraproduct.

## Remarks

The proposition should be interpreted as saying that Set simultaneously carries a pretopos structure and an ultracategory structure, and that these two structures commute. Indeed, one can view Set as a dualizing object for a generalized Stone duality between pretoposes and ultracategories; this is the starting point of Makkai duality.

There is an analogous statement for ultraproducts of structures in continuous logic.

## References

• Hisashi Aratake, Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Łoś’s Theorem , arXiv:2012.04317 (2020). (abstract)

• Michael Makkai, Stone duality for first order logic, Advances in Mathematics, 65(2):97–170, 1987.

Last revised on December 9, 2020 at 03:37:33. See the history of this page for a list of all contributions to it.


text:
Volume 37 • Number 4 • 2014
 • Linearization for Systems with Partially Hyperbolic Linear Part Yong-Hui Xia Abstract. To study the linearization problem of dynamic system on measure chains (time scales), the authors in the previous work assumed that linear system $x^{\Delta}=A(t)x$ should possess exponential dichotomy. In this paper, the assumption is weakened and the setting on the whole linear part $x^{\Delta}=A(t)x$ need not to be hyperbolic. We only need assume partially hyperbolic linear part. More specifically, if system $x^{\Delta}=A(t)x$ is rewritten as two subsystems $\left\{\begin{array}{lll} x_1^{\Delta}=A_1(t)x_1, \\ x_2^{\Delta}=A_2(t)x_2 \end{array}\right.$, it requires that the first subsystem $x_1^{\Delta}=A_1(t)x_1$ has exponential dichotomy, while there is no requirement on the other linear subsystem $x_2^{\Delta}=A_2(t)x_2$. That is, the whole linear system $x^{\Delta}=A(t)x$ need not to possess exponential dichotomy. The previous result is improved in this paper. 2010 Mathematics Subject Classification: 34N05, 26E70, 39A21 Full text: PDF


text:
# Special relativity

1. Aug 17, 2015

### thecosmos123456

special relativity says speed can be additive and subtractive ,our motion influences the speed , i did not get this part .how can our motion influence the speed of other object ?and how can speed be additive and subtractive

2. Aug 17, 2015

### Staff: Mentor

What do you mean by this? It's not clear what your question is.

3. Aug 18, 2015

### Mordred

The only thing I can think of, as the question isn't very clear. Is how you measure the speed of an object is influenced by the speed of the observer. Ie if your on a train moving east at 100 km per hour and look at a car moving at the same speed and direction. The car will appear to not be moving.

So the speed is additive and subtractive compared to your momentum and direction.
The speed of the object itself isn't influenced but how you measure the speed relative to your motion is influenced.

However that's just a guess, if your dealing with relativistic apparent velocity then it gets more complicated.

In special relativity, while those speeds will change with respect to change of reference, they are not "additive". In the scenario above, A will observe C's velocity (C's velocity relative to A) to be $$\frac{u+ v}{1+ \frac{uv}{c^2}}$$


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## CryptoDB

#### Publications

Year
Venue
Title
2019
EUROCRYPT
The notion of covert security for secure two-party computation serves as a compromise between the traditional semi-honest and malicious security definitions. Roughly, covert security ensures that cheating behavior is detected by the honest party with reasonable probability (say, 1/2). It provides more realistic guarantees than semi-honest security with significantly less overhead than is required by malicious security.The rationale for covert security is that it dissuades cheating by parties that care about their reputation and do not want to risk being caught. But a much stronger disincentive is obtained if the honest party can generate a publicly verifiable certificate when cheating is detected. While the corresponding notion of publicly verifiable covert (PVC) security has been explored, existing PVC protocols are complex and less efficient than the best covert protocols, and have impractically large certificates.We propose a novel PVC protocol that significantly improves on prior work. Our protocol uses only “off-the-shelf” primitives (in particular, it avoids signed oblivious transfer) and, for deterrence factor 1/2, has only 20–40% overhead compared to state-of-the-art semi-honest protocols. Our protocol also has, for the first time, constant-size certificates of cheating (e.g., 354 bytes long at the 128-bit security level).As our protocol offers strong security guarantees with low overhead, we suggest that it is the best choice for many practical applications of secure two-party computation.
2018
ASIACRYPT
Two-party Secure Function Evaluation (SFE) allows two parties to evaluate a function known to both parties on their private inputs. In some settings, the input of one of the parties is the definition of the computed function, and requires protection as well. The standard solution for SFE of private functions (PF-SFE) is to rely on Universal Circuits (UC), which can be programmed to implement any circuit of size $s$ . Recent UC optimizations report the cost of UC for $s$ -gate Boolean circuits is $\approx 5 s \log s$ .Instead, we consider garbling that allows evaluating one of a given set $\mathcal {S}$ of circuits. We show how to evaluate one of the circuits in $\mathcal {S}$ at the communication cost comparable to that of evaluating the largest circuit in $\mathcal {S}$ . In other words, we show how to omit generating and sending inactive GC branches. Our main insight is that a garbled circuit is just a collection of garbled tables, and as such can be reused to emulate the throw-away computation of an inactive execution branch without revealing to the Evaluator whether it evaluates active or inactive branch.This cannot be proven within the standard BHR garbled circuits framework because the function description is inseparable from the garbling by definition. We carefully extend BHR in a general way, introducing topology-decoupling circuit garbling. We preserve all existing constructions and proofs of the BHR framework, while allowing this and other future constructions which may treat garbled tables separately from function description.Our construction is presented in the semi-honest model.
2017
EUROCRYPT
2017
ASIACRYPT
2015
EPRINT
2015
EPRINT
2015
EPRINT
2015
TCC
2015
ASIACRYPT
2015
ASIACRYPT
2014
CRYPTO
2014
CRYPTO
2014
EPRINT
2013
CRYPTO
2010
TCC
2010
EPRINT
Two-party Secure Function Evaluation (SFE) allows mutually distrusting parties to (jointly) correctly compute a function on their private input data, without revealing the inputs. SFE, properly designed, guarantees to satisfy the most stringent security requirements, even for interactive computation. Two-party SFE can benefit almost any client-server interaction where privacy is required, such as privacy-preserving credit checking, medical classification, or face recognition. Today, SFE is a subject of immense amount of research in a variety of directions, and is not easy to navigate. In this paper, we systematize some of the vast research knowledge on \emph{practically} efficient SFE. It turns out that the most efficient SFE protocols are obtained by combining several basic techniques, such as garbled circuits and computation under homomorphic encryption. As an important practical contribution, we present a framework in which these techniques can be viewed as building blocks with well-defined interfaces. These components can be easily combined to establish a complete efficient solution. Further, our approach naturally lends itself to automated protocol generation (compilation). We believe, today, this approach is the best candidate for implementation and deployment.
2010
EPRINT
The power of side-channel leakage attacks on cryptographic implementations is evident. Today's practical defenses are typically attack-specific countermeasures against certain classes of side-channel attacks. The demand for a more general solution has given rise to the recent theoretical research that aims to build provably leakage-resilient cryptography. This direction is, however, very new and still largely lacks practitioners' evaluation with regard to both efficiency and practical security. A recent approach, One-Time Programs (OTPs), proposes using Yao's Garbled Circuit (GC) and very simple tamper-proof hardware to securely implement oblivious transfer, to guarantee leakage resilience. Our main contributions are (i) a generic architecture for using GC/OTP modularly, and (ii) hardware implementation and efficiency analysis of GC/OTP evaluation. We implemented two FPGA-based prototypes: a system-on-a-programmable-chip with access to hardware crypto accelerator (suitable for smartcards and future smartphones), and a stand-alone hardware implementation (suitable for ASIC design). We chose AES as a representative complex function for implementation and measurements. As a result of this work, we are able to understand, evaluate and improve the practicality of employing GC/OTP as a leakage-resistance approach. Last, but not least, we believe that our work contributes to bringing together the results of both theoretical and practical communities.
2010
CHES
2010
EPRINT
In this work, we consider Authentication and Key Agreement (AKA), a popular client-server Key Exchange (KE) protocol, commonly used in wireless standards (e.g., UMTS), and widely considered for new applications. We discuss natural potential usage scenarios for AKA, attract attention to subtle vulnerabilities, propose a simple and efficient AKA enhancement, and provide its formal proof of security. The vulnerabilities arise due to the fact that AKA is not a secure KE in the standard cryptographic sense, since Client C does not contribute randomness to the session key. We argue that AKA remains secure in current deployments where C is an entity controlled by a single tamper-resistant User Identity Module (UIM). However, we also show that AKA is insecure if several Client's devices/UIMs share his identity and key. We show practical applicability and efficiency benefits of such multi-UIM scenarios. As our main contribution, we adapt AKA for this setting, with only the minimal changes, while adhering to AKA design goals, and preserving its advantages and features. Our protocol involves one extra PRFG evaluation and no extra messages. We formally prove security of the resulting protocol. We discuss how our security improvement allows simplification of some of AKA security heuristics, which may make our protocol more efficient and robust than AKA even for the current deployment scenarios.
2008
EPRINT
Abstract: We study the problem of Key Exchange (KE), where authentication is two-factor and based on both electronically stored long keys and human-supplied credentials (passwords or biometrics). The latter credential has low entropy and may be adversarily mistyped. Our main contribution is the first formal treatment of mistyping in this setting. Ensuring security in presence of mistyping is subtle. We show mistyping-related limitations of previous KE definitions and constructions. We concentrate on the practical two-factor authenticated KE setting where servers exchange keys with clients, who use short passwords (memorized) and long cryptographic keys (stored on a card). Our work is thus a natural generalization of Halevi-Krawczyk and Kolesnikov-Rackoff. We discuss the challenges that arise due to mistyping. We propose the first KE definitions in this setting, and formally discuss their guarantees. We present efficient KE protocols and prove their security.
2006
TCC
2006
EPRINT
We propose a new model for key exchange (KE) based on a combination of different types of keys. In our setting, servers exchange keys with clients, who memorize short passwords and carry (stealable) storage cards containing long (cryptographic) keys. Our setting is a generalization of that of Halevi and Krawczyk \cite{HaleviKr99} (HK), where clients have a password and the public key of the server. We point out a subtle flaw in the protocols of HK and demonstrate a practical attack on them, resulting in a full password compromise. We give a definition of security of KE in our (and thus also in the HK) setting and discuss many related subtleties. We define and discuss protection against denial of access (DoA) attacks, which is not possible in any of the previous KE models that use passwords. Finally, we give a very simple and efficient protocol satisfying all our requirements.
2005
ASIACRYPT
2004
ASIACRYPT

Crypto 2019
Eurocrypt 2017
PKC 2014


text:
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# Predicate Logic(谓词逻辑)

### Predicate谓词

describe the property of the subject term (in a sentence)

• A predicate is a function from a domain of individuals to {𝐓,𝐅}
• 𝒏-ary  predicate𝒏元谓词: a predicate on 𝑛 individuals𝐼: “is an integer” // unary𝐺: “is greater than” //binary
• Predicate constant谓词常项: a concrete predicate // 𝐼,𝐺
• Predicate variable谓词变项: a symbol that represents any predicate

### Individual个体词

the object you are considering (in a sentence)

“√(1+2√(1+3√(1+⋯)) )  is an integer”
“𝑒^𝜋 is greater than 𝜋^𝑒”

• Individual Constant个体常项: √(1+2√(1+3√(1+⋯)) ) ,  𝑒^𝜋, 𝜋^𝑒
• Individual Variable个体变项: 𝑥,𝑦,𝑧
• Domain个体域: the set of all individuals in consideration

### Propositional function命题函数

**$𝑃(𝑥_1,…,𝑥_𝑛)$, where 𝑃 is an 𝑛-ary predicate **

𝑃(𝑥,𝑦):“𝑥 is greater than 𝑦”
𝑃(𝑥,𝑦) gives a proposition when we assign values to 𝑥,𝑦
𝑃(𝑒^𝜋,𝜋^𝑒) is a proposition (a true proposition)
𝑃(𝑥,𝑦) is not a proposition

EXAMPLE:

 𝑝:“Alice’s father is a doctor”; 𝑞:“Bob’s father is a doctor”

• Individuals: Alice’s father, Bob’s father; Predicate 𝐷: “is a doctor”
• 𝑝=𝐷(Alice′ s father), 𝑞=𝐷(Bob′ s father)

### Function of Individuals:  a map on the domain of individuals

• 𝑓(𝑥)=𝑥’s father
• 𝑝=𝐷(𝑓(Alice));𝑞=𝐷(𝑓(Bob))

### Universal Quantifier全称量词

#### DEFINITION:

Let $P(x)$ be a propositional function. The universal quantification全称量化 of $P(x)$ is “$P(x)$ for all x in the domain”.

notation: $∀x P(x)$; read as “for all $x$ $P(x)$” or “for every $x$ $P(x)$”

“∀” is called the universal quantifier全称量词

“∀x P(x)” is true iff P(x) is true for every x in the domain

“∀x P(x)” is false iff there is an x_0 in the domain such that $P(x_0)$ is false

Counterexample反例: an x_0 such that P(x_0) is false

### Existential Quantifier

#### DEFINITION:

Let P(x) be a propositional function. The existential

    quantification存在量化 of P(x) is “there is an x in the domain

    such that P(x)”

• notation: ∃x P(x); read as “for some x P(x)” or “there is an x s.t. P(x)”
• “∃” is called the existential quantifier****存在量词
• “∃x P(x)” is true iff there is an x in the domain such that P(x) is true
• “∃x P(x)” is false iff P(x) is false for every x in the domain

#### EXAMPLE:

$P(x):“x^2-x+1=0”$

•“$∃x P(x)$”  is false when D=R and is true when D=C

REMARK: If the domain is empty, then "∃x P(x)" is false for any P.

REMARK: if not stated, the individual can be anything.

### Binding Variables and Scope 绑定范围和变量

#### DEFINITION:

An individual variable x is bound约束的 if a quantifier (∀, ∃) is used on x; otherwise, x is said to be free自由的.

• ∃x(x+y=1)
• x is bound and y is free
• scope辖域 of a quantifier: the part of a formula to which a quantifier is used
• the scope of ∃x in ∃x(x+y=1) is (x+y=1)

#### Predicate Logic谓词逻辑:

the area of logic that deals with predicates and quantifiers (a.k.a. predicate calculus)处理谓词和量词的逻辑区域（又称谓词演算)

• predicate logic is an extension of propositional logic

## Well-Formed Formulas

### Elements that may appear in Well-Formed Formulas合式公式

• Propositional constants: T,F, p,q,r,…（命题常数）
• Propositional variables: p,q,r,…  （命题变量）
• Logical Connectives: ¬,∧,∨,→, ↔（逻辑连接词）
• Parenthesis: (, )（插入语）
• Individual constants: a,b,c,…(个体常项)
• Individual variables: x,y,z,…（个体变项）
• Predicate constants: P,Q,R,…（命题常项）
• Predicate variables: P,Q,R,…（命题变项）
• Quantifiers: ∀, ∃（量词）
• Functions of individuals: f,g,…（个体函数）

### DEFINIITON: well-formed formulas合式公式/formulas

1. propositional constants, propositional variables, and propositional functions without connectives are WFFs（无连接词的命题常数，命题变量和命题函数是WFF）
2. If A is a WFF, then ¬A is also a WFF（如果A是WFF，则¬A也是WFF）
3. If A,B are WFFs and there is no individual variable x which is bound in one of A,B but free in the other, then (A∧B), (A∨B), (A→B), (A↔B) are WFFs.（如果A，B是WFF，并且没有单个变量x绑定到A，B中的一个而在另一个中自由，则（A∧B），（A∨B），（A→B），（A↔ B）是WFF。）
4. If A is a WFF with a free individual variable x, then ∀x A, ∃x A are WFFs.（如果A是具有自由个体变量x的WFF，则∀xA，∃xA是WFF。）
5. WFFs can be constructed with.（WFF由1到4构造而成）

#### Precedence

∀, ∃ have higher precedence than ¬, ∧, ∨, → , ↔

### From Natural Language to WFFs

#### The Method of Translation

1. Introduce symbols to represent propositional constants, propositional variables, individual constants, individual variables, predicate constants, predicate variables, functions of individuals
2. Construct WFFs with 1-4 such that WFFs reflect the real meaning of the natural language


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# Rearranging after solving first-order nonlinear differential equation

My friend posed to me an interesting physics problem that reduces to the differential equation $$\frac{dy}{dx} = \frac{a}{y} + b$$ where $a$ and $b$ are known, nonzero constants. After reading up on how to solve differential equations like this, I got it down to $$\frac{by - a\space \ln (a + by)}{b^2} = x + C$$ and this is where I hit a dead-end. Wolfram|Alpha tells me that $$y = \dfrac{a\left(-W\left(-\dfrac{e^{\dfrac{(x+C)b^2}{a}-1}}{a}\right)\right)-a}{b}$$ where $W(z)$ is the Lambert W-function, but this is incredibly complex and I have no idea how it got to that point.

How one would get to this solution, and is there possibly a simpler way of expressing this?

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I think you got the sign of the logarithm wrong, and you should take the absolute value of its argument; but neither of those will help make the solution any nicer. –  joriki Nov 3 '11 at 22:57
Thanks, fixed. W|A refuses to give an answer altogether if I use absolute values, though, so I'm not even sure how that would even be reflected in the solution. –  algorithmshark Nov 3 '11 at 23:21

$$\frac{by - a \ln (a + by)}{b^2} = x + C$$

and do a few rearrangements:

$$\frac{b}{a}y - \ln (a + by) = \frac{b^2}{a}(x + C)$$

and complicate things a little:

\begin{align*}1+\frac{b}{a}y - \ln\left(1 + \frac{b}{a}y\right) &= 1 + \frac{b^2}{a}(x + C)+\ln\,a\\\frac{\exp\left(1+\frac{b}{a}y\right)}{1+\frac{b}{a}y}&=\exp\left(1 + \frac{b^2}{a}(x + C)+\ln\,a\right)\\\left(-1-\frac{b}{a}y\right)\exp\left(-1-\frac{b}{a}y\right)&=-\frac1{a}\exp\left(- \frac{b^2}{a}(x + C)-1\right)\end{align*}

which is the cue for Lambert to enter the fray:

$$-1-\frac{b}{a}y=W\left(-\frac1{a}\exp\left(- \frac{b^2}{a}(x + C)-1\right)\right)$$

and finally

$$y=-\frac{a}{b}\left(1+W\left(-\frac1{a}\exp\left(- \frac{b^2}{a}(x + C)-1\right)\right)\right)$$

Which branch of the Lambert function will be needed will depend on your application.

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# Breaking laws

1. Jun 15, 2010

### binbots

How come we say perpetual motion is not possible according to the laws of thermodynamics. Yet our universe is expanding faster and faster. How can we say nothing is faster than the speed on light but the universe expanding is? Keep it simple please.

2. Jun 15, 2010

### Galap

The universe is not expanding faster than C. if it were, light from distant galaxies would never reach us because the space between them would grow faster than the light could cross it.

As for perpetual motion, It is not the same as a force. We do not know why the universe is expanding faster and faster, but it seems to be due to some accumulating acceleration. We do not say an orbiting satellite is perpetual motion, yet it perpetually moves. Be careful with the definitions.

3. Jun 15, 2010

### LostConjugate

I think what bin is asking is that if it is expanding at any rate, wouldn't it eventually exceed the speed of light far enough away. The answer is no, relative to any observer in the universe nothing is moving faster than the speed of light.

Speed is distance over time, but take the time of any relative object and you get

$$t' = \frac{t}{1-\frac{v^2}{c^2}}$$

So

$$\frac{d}{t'}$$

will never exceed c where c is the speed of light.

4. Jun 15, 2010

### AUK 1138

Back in one of my astro classes my professor tried to explain that what's happening is that the objects aren't 'moving' away faster than the speed of light, but instead, the distance between the objects is getting longer! kind of like drawing two dots on a balloon and filling it up, they aren't moving with respect to the balloon, but are becoming farther apart due to its expanding nature. now, obviously this balloon example is wholly crude, but the point is that it's not that the objects are moving, but instead, the way we perceive distances have changed and are changing with respect to time. i hope this helps.


text:
1. ## Residual Analysis

I'm wondering if someone can give me a brief explanation of what residual analysis is and what it's used for. I know it can be used to test for normality, and for detecting outliers, but I'm not sure how. My book doesn't really help.

Any help would be greatly appreciated.

2. Are you using a statistical program when analyzing the residuals?

3. Yes. I'm using SAS.

I forgot to mention that it's multivariate data, instead of univariate.

4. I haven't used that, but for instance, with eViews, I can go to residual table, and it will show which values lie outside two standard deviations. Look for something like that.

5. When the residuals are graphed, if one residual is particularly farther away from the x-axis than others, then we can conclude that the corresponding y-value is most likely an outlier. Also, the residuals are used in finding the Root Mean Square Error (RMSE), and this determines how good a model is relative to a data set.

$\sum_{n=2}^{n} \frac{(z_i-x_i)^2}{n-2}$

Where z represents the model, x represents the data set. Notice that the upper part of the fraction is actually just the residuals squared. Sorry about the summation symbol, Latex was being uncooperative.

Where Y is an (nxm) matrix and B is an [(r+1)xn) matrix. Y can also be writen as $Y = [Y_1 | Y_2]$ where Y1 and Y2 are both n x 1 vectors.