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## Results (1-50 of 336 matches)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-810-15.14-c2-0-0 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.346 0 0.122368 Modular form 810.3.b.b.809.2 2-810-15.14-c2-0-1 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.373$ $0$ $0.140488$ Modular form 810.3.b.b.809.5
2-810-15.14-c2-0-10 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.244 0 0.573531 Modular form 810.3.b.c.809.17 2-810-15.14-c2-0-11 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.0432$ $0$ $0.627833$ Modular form 810.3.b.c.809.2
2-810-15.14-c2-0-12 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.263 0 0.670765 Modular form 810.3.b.c.809.20 2-810-15.14-c2-0-13 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.244$ $0$ $0.687084$ Modular form 810.3.b.c.809.8
2-810-15.14-c2-0-14 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.304 0 0.689607 Modular form 810.3.b.c.809.22 2-810-15.14-c2-0-15 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.0432$ $0$ $0.714571$ Modular form 810.3.b.c.809.1
2-810-15.14-c2-0-16 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.126 0 0.730751 Modular form 810.3.b.b.809.11 2-810-15.14-c2-0-17 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.162$ $0$ $0.787333$ Modular form 810.3.b.b.809.8
2-810-15.14-c2-0-18 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.214 0 0.792110 Modular form 810.3.b.a.809.8 2-810-15.14-c2-0-19 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.0672$ $0$ $0.793583$ Modular form 810.3.b.a.809.4
2-810-15.14-c2-0-2 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.409 0 0.176367 Modular form 810.3.b.c.809.15 2-810-15.14-c2-0-20 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.000470$ $0$ $0.940374$ Modular form 810.3.b.c.809.11
2-810-15.14-c2-0-21 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.000470 0 0.954430 Modular form 810.3.b.c.809.14 2-810-15.14-c2-0-22 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.0672$ $0$ $0.984711$ Modular form 810.3.b.a.809.3
2-810-15.14-c2-0-23 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.000470 0 0.998923 Modular form 810.3.b.c.809.13 2-810-15.14-c2-0-24 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.153$ $0$ $1.01192$ Modular form 810.3.b.b.809.16
2-810-15.14-c2-0-25 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.000470 0 1.02448 Modular form 810.3.b.c.809.12 2-810-15.14-c2-0-26 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.285$ $0$ $1.09289$ Modular form 810.3.b.a.809.1
2-810-15.14-c2-0-27 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.346 0 1.12215 Modular form 810.3.b.b.809.1 2-810-15.14-c2-0-28 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.0282$ $0$ $1.13372$ Modular form 810.3.b.b.809.14
2-810-15.14-c2-0-29 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.0432 0 1.13502 Modular form 810.3.b.c.809.23 2-810-15.14-c2-0-3 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.263$ $0$ $0.230977$ Modular form 810.3.b.c.809.5
2-810-15.14-c2-0-30 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.0282 0 1.16328 Modular form 810.3.b.b.809.13 2-810-15.14-c2-0-31 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.263$ $0$ $1.20809$ Modular form 810.3.b.c.809.6
2-810-15.14-c2-0-32 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.162 0 1.21316 Modular form 810.3.b.b.809.7 2-810-15.14-c2-0-33 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.244$ $0$ $1.21412$ Modular form 810.3.b.c.809.7
2-810-15.14-c2-0-34 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.304 0 1.27332 Modular form 810.3.b.c.809.4 2-810-15.14-c2-0-35 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.126$ $0$ $1.30524$ Modular form 810.3.b.b.809.12
2-810-15.14-c2-0-36 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.0432 0 1.40740 Modular form 810.3.b.c.809.24 2-810-15.14-c2-0-37 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.373$ $0$ $1.55191$ Modular form 810.3.b.b.809.6
2-810-15.14-c2-0-38 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.153 0 1.58248 Modular form 810.3.b.b.809.15 2-810-15.14-c2-0-39 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.337$ $0$ $1.66861$ Modular form 810.3.b.b.809.9
2-810-15.14-c2-0-4 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.304 0 0.282151 Modular form 810.3.b.c.809.3 2-810-15.14-c2-0-40 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.263$ $0$ $1.73444$ Modular form 810.3.b.c.809.19
2-810-15.14-c2-0-41 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.244 0 1.77088 Modular form 810.3.b.c.809.18 2-810-15.14-c2-0-42 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.409$ $0$ $1.81176$ Modular form 810.3.b.c.809.16
2-810-15.14-c2-0-43 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.471 0 1.84331 Modular form 810.3.b.b.809.3 2-810-15.14-c2-0-44 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.409$ $0$ $1.92781$ Modular form 810.3.b.c.809.9
2-810-15.14-c2-0-45 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.214 0 1.94113 Modular form 810.3.b.a.809.7 2-810-15.14-c2-0-46 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $0.304$ $0$ $1.96614$ Modular form 810.3.b.c.809.21
2-810-15.14-c2-0-47 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 0.432 0 2.26115 Modular form 810.3.b.a.809.5 2-810-15.14-c2-0-5 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.285$ $0$ $0.299229$ Modular form 810.3.b.a.809.2
2-810-15.14-c2-0-6 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.337 0 0.368440 Modular form 810.3.b.b.809.10 2-810-15.14-c2-0-7 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.432$ $0$ $0.407181$ Modular form 810.3.b.a.809.6
2-810-15.14-c2-0-8 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 15.14 $$2.0 2 -0.409 0 0.444387 Modular form 810.3.b.c.809.10 2-810-15.14-c2-0-9 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 15.14$$ $2.0$ $2$ $-0.471$ $0$ $0.487305$ Modular form 810.3.b.b.809.4
2-810-3.2-c2-0-0 $4.69$ $22.0$ $2$ $2 \cdot 3^{4} \cdot 5$ 3.2 $$2.0 2 -0.250 0 0.0213674 Modular form 810.3.d.c.161.1 2-810-3.2-c2-0-1 4.69 22.0 2 2 \cdot 3^{4} \cdot 5 3.2$$ $2.0$ $2$ $0.5$ $0$ $0.125411$ Modular form 810.3.d.b.161.5
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# Lecture 15: Holocene, Mesolithic and Archaic People

$\star$ ya = years ago

### Holocene Hunter-Gatherers, Mesolithic/Archaic People

• Mesolithic (Europe) and Epipaleolithic (Near East): 12,000ya to Neolithic (or later).
• AND Archaic (Americas): 10,000ya to food production.
• We're currently in the Holocene time period, which started 10,000/ 12,000ya. A major division of change relating to the climate.

### Change in Climate

• Widespread change in lifestyles worldwide due to climate change. The glaciers began to melt and the sea levels started rising, drowning out coastal sites. Wetlands and marshes began to appear (which are resource and nutrient rich).

### Changes in Material Culture

• The resource rich environments were full of birds, fish and a range of other diverse food types. These renewing resources are good for hunter gatherer populations. There are no more megafauna left. Instead they hunted reindeer, caribou, deer and other small animals. They also stored nutrient rich food such as nuts.

### Changes in Subsistence and Settlement

• Some groups continued using the archaic lifestyle, well into the period of farming and agriculture, but they stayed hunter gatherers. Others interacted with the European people and altered their ways. Overall, there were increasing specialized adaptations (food, tools used etc) for local groups as they adapted to their environments, making them very diverse and leading to different cultures and ways of doing things, ex. making ornaments. Broadening their diet was good for the growing population.
• Subsistence means methods of food obtainment: started using smaller points and tools to hunt for smaller game during the Mesolithic period. Microliths were blades broken into different shapes and hafted to make sharp effective tools. Bows and arrows developed later, as they were still using spears. There were now more wooded environments so there were a lot of wooden tools used. More berries, plants and seeds were used. They used grinding stones, fish hooks and nets to catch birds and shellfish. Overall, they predicted and planned how they would catch their food instead of just chasing herds.
• Foragers $\rightarrow$ Hunter gatherers that move frequently to get the resources they need, as they're available seasonally. They don't store much food. This is a common lifestyle in the tropical area where there aren't much seasonal fluctuations and the food is rich and diverse. They have specific areas they move to during specific times of the year, so they're not just wandering aimlessly.
• Collectors $\rightarrow$ Hunter gatherers that stay in one or a few places for a longer period of time. Whole groups or subgroups go out to collect resources as they're available. The resources are then shared and stored if possible (ex. dried meat, fish and nuts). This is a good adaptation method in temperate environments where there is a cold vs. warm and dry vs. wet season.
• Most groups do a bit of both.
• Seasonality and scheduling methods $\rightarrow$ There are many different movements for each group. Ex. they know when the berries are ripe in a certain area or when salmon are plentiful in a certain area, so they travel there during that time. This choice of when and where to move effects social organization and settlement patterns.

### Changes in Social Organization and Settlement Patterns

• Solving problems of population increase - There was an increase in the population and mostly in population density.
• Decrease in population by using birth control or killing off old people. For relatively small mobile populations, more of a forager lifestyle.
• Split up groups $\rightarrow$ For relatively small mobile populations, more of a forager lifestyle.
• Expand territory to include more resources $\rightarrow$ For relatively small mobile populations, more of a forager lifestyle.
• Use certain resources more intensively, which is harder but more efficient: for some increasingly immobile larger groups, who have status differences. More collector lifestyle.
• The choice of what to do is constrained by environmental conditions.

#### Continued small scale hunter-gatherer lifestyles

• Ex. European Mesolithic - Star Carr
• A temperate forested seasonal environment, full of marshes. A hunting camp where a small group comes and stays for a season, mostly during the spring and summer. Intensely hunted, focusing on deer and elk. They processed the meat and dried it for storage.
• The group continues to migrate to other locations, using axes to cut down trees for wood needed to build platforms through the marshy areas. This shows community labor and structure. Arrows were made from wood and bone. Spear points were made from bone and antler. Domesticated dogs used for hunting were common.
• Archaic North America - Great Basin/West Plains
• A high plain desert environment. Small groups of people living in caves. Collected nuts seasonally and ground seeds. Used small archaic spear points. Had sandals, animal traps and baskets were made from fibers. They ate fish, rodents, insects, plants, seeds, rabbits and marsh birds. Made beautiful ornaments and other things such as feather robes. Moved around the landscape in a forager lifestyle.

#### Intensify larger scale hunter gatherer collector lifestyles

• Lived a large scale hunter gatherer and collector lifestyle. Focused on wetlands which were nutrient rich. Moved only a few times a year, according to seasonal shifts so they were able to have permanent housing in many areas. Stored food. This was the starting of social differences because they had settlement issues and had to organize people.
• Example $\rightarrow$ Archaic North America: Poverty Point around 3,500 - 2,500ya.
• A hunter gatherer population. There were no domesticated plants. Instead they hunted small game, nuts, seeds, fish, shellfish etc. They had a wide diet.
• Built their settlement on a series of ridges, which were very large and organized and look a lot of time and effort to make. This group had a high population density and a long distance trading network across North America to get special materials.
• There were various polished tools, such as arrowheads for hunting and fishing. This group disappears or becomes less complex before the Europeans come to the land.

#### Example

• Archaic North America - North West Coast: Ozette. A complex hunter gatherer group living in a rich coastal environment at the time of European contact. Had a high population density.
• They live an archaic lifestyle 300-500ya, not agriculturists. They lived permanently for most of the year in a village made of wood, which was inhabited for 2000 years.
• They focused on fishing, using harpoons to catch whales and seals using boats. They cooked meat in baskets and wooden boxes by placing rocks into it and putting the meat pieces on top of the rocks. They stored food such as salmon and nuts.

### Intensify towards agricultural lifestyles

#### Example

• Near East Epipaleolithic - Kebaran-Natufian. Upper Paleolithic era. Didn't have a lot of storable resources, and the environment wasn't rich so they couldn't support big population sizes.
• They focused on seeds and their storage, putting a lot of time and effort into getting resources.
• Mice were the second domesticated animal. Adopted pottery. Are a mobile group.

#### Mesolithic vs. Neolithic axes

• why spend more time making? In the Mesolithic era, stone was chipped via pressure flaking. It was quick to make, easy to repair and very practical. But overtime pieces of wood get caught in the chipped edges and it becomes dull and useless. In the Neolithic era, they spent hours polishing the stone. This made for a faster, cleaner cut, without causing damage to the stone.
• Overall $\rightarrow$ groups of complex hunter gatherers, which had increased population, were more settled and needed particular resources (mostly storable). They leaned toward domestication. There was an overlap of time periods, so they were living nearby and probably in contact with each other.


text:
IN WHICH Ross Rheingans-Yoo, a sometimes-poet and erstwhile student of Computer Science and Math, occasionally writes on things of interest.

# Reading Feed (last update: April 14)

A collection of things that I was happy I read. Views expressed by linked authors are chosen because I think they're interesting, not because I think they're correct, unless indicated otherwise.

### (13)

Blog: Marginal Revolution | The importance of local milieus — "We find suggestive evidence that co-locating with future inventors may impact the probability of becoming an inventor. The most consistent effect is found for place of higher education; some positive effects are also evident from birthplace, whereas no consistent positive effect can be derived from individuals’ high school location."

Blog: Shtetl-Optimized | How to upper-bound the probability of something bad — an algorithmist's guideline.

Blog: The Unit of Caring | Anonymous asked: you have the most hilariously naive politics i've ever seen... — "[in conclusion...] And I think anon is wrong about whether I need to grow a backbone."

Blog:

# What is there to say?

My grandfather was a career scientist at Oak Ridge National Labs for 36 years. He was an international traveler and an international collaborator, advancing human knowledge of materials science as best he knew how -- by sharing what he knew with fellow seekers of truth, regardless of nationality. As a young man, he left a country rent by war to seek an education -- and a home -- and a future in the United States. Here he raised three sons, international travelers and collaborators themselves -- a businessman, a public servant, and a professor of Law.

I can't count the friends I have with friends and colleagues, seeking an education -- seeking a future -- seeking to advance the knowledge of all mankind -- who have had my nation slam our door in their faces this weekend. I feel sick for what my nation has done in my name,

# Re-Thinking Prejudices

I've decided that this post is retroactively part 1 of ? of a recurring series on approaching debates with a mind toward actually changing minds and the world.

[ | ]

There's a statue visible from the window of my office, a poem inscribed near its base:

Not like the brazen giant of Greek fame,
With conquering limbs astride from land to land;
Here at our sea-washed, sunset gates shall stand
A mighty woman with a torch, whose flame
Is the imprisoned lightning, and her name
Mother of Exiles. From her beacon-hand
Glows world-wide welcome; her mild eyes command
The air-bridged harbor that twin cities frame.
“Keep, ancient lands, your storied pomp!” cries she
I lift my lamp beside the golden


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# 5.4 Synthesising plucked string sounds

We have looked in some detail at the vibration behaviour of stringed instrument bodies: but how important are all these details for the sound of the instrument? It is time to turn our models into something we can listen to. For the violin, that will involve understanding what happens when you bow a string, which will take us into the tricky territory of non-linear systems. We will put this off, until chapter ?. But for plucked-string instruments like the guitar and the banjo we can get a long way with linear ideas.

What happens when you pluck a string? The simplest description is that the player pulls the string to one side (with a plectrum, fingernail or the flesh of a finger), then lets go. The string is then left to its own devices, to vibrate freely and to feed some of its energy into the instrument body. When pulled aside, the string will take up a triangular shape as sketched in Fig. 1. The player can choose the plucking point along the string, and the orientation of the pluck. Otherwise, the only thing likely to vary is the sharpness of the corner: a small, sharp plectrum will make a very sharp corner, the flesh of a finger will make a more rounded one.

The very simplest model is to assume a perfectly sharp corner, and an ideal string without damping. We already know the mode shapes and natural frequencies for this ideal string from section 3.1.1. For this simple system there are three ways we can find the string motion following the pluck. They are all useful in their different ways so we will briefly summarise all three.

First, we can say that after the string has been released, the ensuing free motion must consist of a linear combination of the modes, each vibrating at its own natural frequency. This statement is, after all, true of any free motion of any linear system. All we need to do is find the amplitude and phase of each modal contribution. This can be done by expressing the initial triangular shape as a Fourier series: the slightly messy mathematical details are given in the next link.

SEE MORE DETAIL

One virtue of this approach is that it does not rely on the particular assumed shape of the pluck: it could be applied to any other initial conditions, such as a rounded corner from a finger-flesh pluck. The method would be followed in the same way, based on calculating the Fourier series representation of the initial shape. The effect of rounding off the sharp corner would be to reduce the amplitudes of the higher Fourier components. You need short wavelengths, corresponding to high-frequency modes, to represent a sharp corner, but a rounded corner will have a natural cutoff at the wavelength matching the curvature of the corner, above which the Fourier coefficients will become small. The result is a familiar one: the sound of a finger-flesh pluck is less bright than a plectrum pluck because it contains less energy at high frequency.

We can learn something interesting about the sound of plucked strings from the result of the calculation in the previous link. We can’t calculate ‘sound’ as such from this model string with rigidly fixed ends, but we can calculate the waveform of force exerted by the vibrating string on the bridge. This is the driving force for body vibration, and so is closely related to the sound that an instrument would make. The amplitude of the $n$th harmonic of this force signal is proportional to the mode shape at the pluck point, divided by the harmonic number, and the mode shapes are simply sinusoidal as plotted in Fig. 1 of section 3.1. This amplitude is plotted in Fig. 2, against harmonic number $n$ and relative pluck position $a/L$ (where $L$ is the length of the string and $a$ is the plucking distance).

This plot reveals several interesting features. For every plucking position, the fundamental ($n=1$) has the largest amplitude. But the value of that fundamental amplitude, for a fixed plucking force, is higher when plucking far from the bridge, and gets progressively lower as the pluck point gets nearer to the bridge. Every guitarist is familiar with the change in loudness and tone quality that results from plucking closer to the bridge: a synthesised example of the effect will be given at the end of this section.

There is another factor contributing to this change of sound when the plucking point is varied. Figure 2 shows that the variation of amplitude with $n$ is different for each plucking point, leading to the set of curving ridges which dominates the plot. The origin of this effect is easy to understand. Look first at the line of points on the far side of the plot, for plucking at $a/L=0.5$, in other words at the mid-point of the string. The line is a zigzag, with every second amplitude falling to zero. The reason is that whenever $n$ is an even number, the corresponding mode shape has a nodal point in the centre. It is impossible to excite a mode of any system by driving at a nodal point. So a mid-point pluck can only excite the odd harmonics, $n=1,3,5,…$. This argument extends in a simple way to other plucking positions. If you pluck at the 1/3 point, every third harmonic will be missing. At the 1/4 point, every fourth will be missing, and so on. This leads directly to the pattern of curving ridges and valleys.

The second approach to calculating the motion of our ideal string following a pluck is to notice that for this particular problem of a plucking force applied at a single point, we can short-cut some of the algebraic complication of the Fourier series approach by making use of a general formula for the step response of any linear system. We are applying a force at the chosen plucking point. That force has a constant value $F$ until the player lets go, then it then jumps down to zero. The general formula for step response can be used to find the subsequent motion. The details are given in the previous link: reassuringly, it is found to agree with the result obtained by the Fourier series method.

A virtue of this step response method is that it can easily be extended to the case where the string is connected to the instrument body rather than rigidly fixed at both ends. Although Fig. 1 didn’t explicitly show any body motion, the sideways pull by the player will move the bridge a little, as well as moving the string. One way to solve the coupled problem of the string-body motion after the pluck is first to find the coupled modes of this combined system. Once we know those, the step response formula can be applied directly. We will come back to this shortly.

The third approach to finding the pluck response is very different. Rather than using modes, we can take advantage of a general solution to the differential equation governing the motion of an ideal string, due to the 18th-century French mathematician Jean le Rond d’Alembert. This approach uses travelling waves: d’Alembert showed that any possible free motion of a string can be described by a combination of a wave travelling leftwards at speed $c$, and another wave travelling rightwards at the same speed. These waves can have any shape whatsoever, but to satisfy the particular starting conditions of our plucked string we need both waves to have a triangular shape. The next link gives details of d’Alembert’s solution.

SEE MORE DETAIL

Figure 3 shows an animation of the result. At the top, the two separate waves are seen, travelling in opposite directions. You could picture these waves as travelling on a string which extends off to infinity in both directions beyond the actual vibrating string. We are viewing that infinite string through a window matching the length of our actual string, so that waves appear from one side, travel across the window, and vanish off the other side. The lower plot shows the actual string motion, which is the sum of the two waves. Both ends of the string are stationary, as they are supposed to be. If you watch carefully, you will see that the red and blue travelling waves always have equal and opposite magnitudes at the two end of the string, so that when you add them together they always cancel.

Now we have a good idea how an ideal string behaves following a perfect pluck. However, this is not enough for the purpose of making an accurate synthesis of the motion of a real string on a real instrument. Several factors have been omitted in this simple analysis, and we need to put at least some of them back.

First, real strings do not have natural frequencies that are precisely harmonic multiples of the fundamental frequency. There are two reasons. One is to do with coupling to the body of the instrument, which we deal with shortly. The other arises because any real string has some bending stiffness. This problem is easily dealt with, as described in the next link. We simply need to combine the behaviour of two systems we already know about, the string and a bending beam. Then we can use Rayleigh’s principle to get a good approximation for the natural frequencies allowing for bending stiffness.

SEE MORE DETAIL

The result is that the frequency of the $n$th overtone (we must now stop calling them ‘harmonics’) is no longer simply an integer multiple of the fundamental. Instead, the overtones are spaced progressively wider apart than the harmonic series of the ideal string. The stronger the influence of the string’s bending stiffness, the stronger is this inharmonicity effect. It has an important perceptual consequence, which we can guess from the discussion of tuned percussion instruments in chapter 3. The more accurately harmonic the frequencies are, the more precise is the resulting sense of pitch. So, for example, thin harpsichord strings will have lower bending stiffness and so be more accurately harmonic that thicker piano strings, which is one reason why harpsichordists pay so much more attention to tuning and to tempering systems.

The effects of inharmonicity due to bending stiffness can be reduced by using over-wrapped strings. To design a string of given length with a low pitch but which does not become too slack, the string has to be heavy. The simplest way to achieve that is to choose a thicker gauge of string, but this will increase the bending stiffness. Wrapping metal wire round a core of metal or nylon fibres allows a heavier string to be made without the penalty of higher bending stiffness. This is the reason behind the familiar appearance of sets of piano strings or guitar strings: the high strings are solid monofilaments of one material or another, but the lower strings are over-wrapped.

The modified natural frequencies can be inserted directly into a modal-based expression for the pluck response, such as the expression given by the step response method. However, the effect of bending stiffness invalidates the d’Alembert travelling-wave method. The wave speed is no longer a single constant value; instead it varies with frequency. Waves with high frequency, or equivalently with short wavelengths, travel faster than those with lower frequency or longer wavelength. The result is that the shape of the travelling triangular waves in Fig. 3 will change with time, because the different frequency components making up the shape travel at different speeds. The motion immediately after the pluck will still resemble Fig. 3, but it will gradually diverge from it as the effects of bending stiffness build up.

But there is a more important reason that the motion of a real string will rapidly deviate from the simple animation in Fig. 3. In the plot the motion went on for ever, but of course in a real string it will die away as energy is lost by the string. Furthermore, the different modes of the string will decay at different rates. Usually, the high-frequency components decay more rapidly than the lower-frequency ones, so that the motion will look increasingly smooth and rounded as time goes on. Eventually, it will become more or less sinusoidal because only the fundamental mode has significant amplitude. This frequency-dependent decay pattern is an important part of the audio “signature” of different instruments, and of different strings or different notes on a given instrument. Some sound examples to illustrate this will be given in section 5.5.

There are three main mechanisms for energy loss from a vibrating string. Energy loss is additive, so the total loss factor for a given mode of the string is the sum of three separate loss factors associated with the three mechanisms. One we have already seen, in section 5.1.2. Energy is lost through the bridge to the body of the instrument, and the associated loss factor is determined by the product of the string impedance and the real part of the bridge admittance of the body. This source of energy loss will be included automatically, when we couple the string to the body to perform a synthesis.

The other two energy loss mechanisms are associated directly with the string motion itself, regardless of coupling to the body, and they need to be included explicitly in a model of the string dynamics. One source of energy loss is within the material of the string. The cyclical deformation of the string during the vibration will involve some energy loss. For a monofilament string, this loss is linked to the bending stiffness just discussed. The Young’s modulus of the string is in reality a complex number: the real part is the usual elastic modulus, while the imaginary part describes energy loss during cyclical deformation. This complex modulus can be used to derive an expression for the loss factor: see the next link for details.

SEE MORE DETAIL

For a metal string, as on a piano or steel-string guitar, the intrinsic damping of the material is usually very small, but for a nylon or gut string it will be much bigger. This is one important reason for the difference of sound between strings of different types. For a multi-layer wrapped string, additional energy loss can occur as a result of friction between the windings. The energy loss in that case will not be represented very accurately by the complex modulus model: friction is intrinsically a nonlinear process, whereas the complex modulus representation is based on linear theory. Furthermore, on an instrument like the guitar where the strings are in direct contact with the player’s fingers, this loss mechanism is likely to vary with time. Contaminants from the fingers gradually build up between the windings of the string, increasing the energy dissipation rate. This seems to be the main reason for the difference in sound between new and old guitar strings [1].

The third mechanism for energy loss from a vibrating string is to do with the air surrounding the string: as the string moves, the associated air flow over its surface leads to losses from viscosity. A loss factor to account for this can be estimated using a classical result for flow round a circular cylinder: the details are given in the next link.

SEE MORE DETAIL

We now have all the main ingredients for a realistic model of a vibrating string, and we are ready to couple the string to the instrument body and synthesise a version of the sound of plucking the string. One possible method of synthesis has already been mentioned: if we can describe the body motion in terms of its modes, we could compute the coupled modes of the string/body system, then use the general formula for step response to create a pluck sound. This approach is certainly possible, and we will show examples later, but there are some details that are rather fiddly to get right. We will not discuss the method further here: see [2] for more discussion.

There is an alternative synthesis method that is faster and in many ways superior. This involves working in the frequency domain, and using an inverse FFT at the final stage to create the required time-varying transient response. This method is attractive because the string and the body are coupled together at a single point, and there is a very simple method which can be applied to any such problem: if two systems have input admittances $Y_1$ and $Y_2$ and they are then rigidly connected at the points at which these admittances are defined, then the coupled system has an admittance $Y_{coup}$ at that point satisfying

$$\dfrac{1}{Y_{coup}}=\dfrac{1}{Y_1}+\dfrac{1}{Y_2} . \tag{1}$$

This result expresses the fact that at the coupling point, the two subsystems have equal velocities while the total applied force is the sum of the forces applied to the two separate subsystems.

The body is naturally described in terms of its admittance, as already described. By calculating the corresponding admittance at the end of a string, which is derived in the next link, eq. (1) can be applied immediately to give the admittance of the strung instrument at the bridge.

To use this method to derive a pluck response, use can be made of a general reciprocal theorem for linear vibration response. This states that if you apply a force at some point on a system, and observe the motion at another point, then you would get exactly the same response if you observed at the first point, and applied the force at the second point. This reciprocal theorem is often used when making measurements: it allows you to swap your excitation point and your measurement point if that makes the experimental details easier. The proof of this result has actually already been given: it follows directly from eq. (11) of section 2.2.5. This formula for the frequency response function is completely symmetric between the forcing point (labelled $j$) and the measurement point (labelled $k$ ).

We wish to find the body vibration which results from a step function of force applied at a given position on the string. Reciprocally, we can consider applying the force at the bridge, and calculate the resulting motion at the relevant point on the string. To solve this reciprocal problem, we simply have to multiply two frequency response functions together. The first is the coupled admittance $Y_{coup}$ from eq. (1), which gives the velocity at the bridge when a force is applied there. The second is the dimensionless transfer function between a given displacement applied at one end of a string and the corresponding displacement at the point where the pluck is to be applied. This second transfer function is also derived in the next link.

SEE MORE DETAIL

Performing these operations and then applying an inverse FFT, we can create a time history representing the velocity waveform at the bridge following an ideal pluck at the chosen point on the string. There are two final ingredients to be considered, to make the model a little more realistic. First, we might want to allow for a pluck that is somewhat rounded rather than perfectly sharp. We saw earlier that the effect of a rounded pluck is to reduce the high-frequency content of the string motion, and hence of the sound of the instrument. We can represent this effect in a simple way by applying a suitable low-pass filter to the computed frequency response, before doing the inverse FFT.

The final ingredient involves thinking more carefully about the output of the synthesis. Really, we would like to compute the sound radiated by the vibrating instrument body. To do this accurately is a very complicated matter, as we glimpsed in Chapter 4. We will want to come back to the question of representing radiated sound, but for the moment we can try to find a simple way to synthesise something that gives a reasonably convincing impression of “sounding like a guitar” (or whichever instrument we are considering). A pragmatic approach will be used: empirically, the sound of bridge velocity is a little dull, while the sound of bridge acceleration is a bit too bright. In the frequency domain, the difference between these two things is a factor proportional to frequency. It is easy to move between these two extremes by applying a factor of “frequency raised to some power”, choosing a power lying between 0 and 1 based on listening to some sound examples. Examples of the effect of varying this power will be shown in section 5.5, together with many other sound examples computed by this approach.

As a taster for the usefulness of synthesis, we give a few preliminary examples here. The three sounds below were all synthesised using the frequency domain method just described. A brief snatch from an anonymous lute piece called “The English Huntsuppe” is “played” on a classical guitar. Parameter values for a typical set of nylon strings were used, together with the measured bridge admittance of a guitar. All the details of the modelling can be found in reference [1]. The middle example, Sound 2, uses the measured body admittance directly. Sounds 1 and 3 are the result of doing some “virtual woodwork” in the computer to modify the admittance in a controlled manner.

First, do these sound fairly credibly like guitars? Then, how would you describe the difference between the three sounds? Can you guess what has been changed? The answer is that in Sound 1, all the natural frequencies of the body have been reduced by one semitone, whereas in Sound 3 they have been increased by one semitone. Nothing else was changed. If you want to think of a physical change to a guitar that would have roughly the same effect, then you can picture Sound 1 as coming from a guitar with a slightly larger or thinner top plate, and Sound 3 from one with a slightly smaller or thicker top plate. As we showed in section 3.2.4, the area and the thickness of a plate both affect the modal density.

Earlier in this section, we saw in Fig. 2 that the plucking point on the string has a strong influence on the frequency spectrum of an ideal plucked string. The next pair of sound examples illustrate this effect via synthesised fragments of another lute piece, “Queen Elizabeth’s Galliard” by John Dowland. All the model parameters are the same as for Sound 2, except that the plucking point has been varied. In Sound 4, the strings are plucked 150 mm from the bridge, while in Sound 5 they are plucked 20 mm from the bridge. The open string length for this “guitar” was 650 mm, but as in a real instrument the actual vibrating length for each note varies depending on which fret is used to play the note. The difference of sound between these two examples is encouragingly familiar from real playing.

[1] J. Woodhouse, E.K.Y. Manuel, L.A. Smith and C. Fritz. “Perceptual thresholds for acoustical guitar models”. Acta Acustica united with Acustica 98, 475-486, (2012).  DOI 10.3813/AAA.918531

[2] J. Woodhouse “On the synthesis of guitar plucks”.  Acta Acustica united with Acustica 90, 928–944 (2004).


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Next: Partition Function Up: Applications of Statistical Thermodynamics Previous: System with Specified Mean

# Calculation of Mean Values

Consider a system in contact with a heat reservoir, or with a specified mean energy. The systems in the representative ensemble are distributed over their accessible states in accordance with the canonical distribution. Thus, the probability of occurrence of some state with energy is given by

 (7.31)

The mean energy is written

 (7.32)

where the sum is taken over all states of the system, irrespective of their energy. Note that

 (7.33)

where

 (7.34)

It follows that

 (7.35)

The quantity , which is defined as the sum of the Boltzmann factor over all states, irrespective of their energy, is called the partition function. (Incidentally, the partition function is represented by the symbol because in German it is known as the zustandssumme,'' which means the sum over states.'') We have just demonstrated that it is fairly easy to work out the mean energy of a system using its partition function. In fact, as we shall discover, it is straightforward to calculate virtually any piece of statistical information from the partition function.

Let us evaluate the variance of the energy. We know that

 (7.36)

(See Chapter 2.) Now, according to the canonical distribution,

 (7.37)

However,

 (7.38)

Hence,

 (7.39)

We can also write

 (7.40)

where use has been made of Equation (7.35). It follows from Equation (7.36) that

 (7.41)

Thus, the variance of the energy can be worked out from the partition function almost as easily as the mean energy. Because, by definition, a variance can never be negative, it follows that , or, equivalently, . Hence, the mean energy of a system governed by the canonical distribution always increases with increasing temperature.

Suppose that the system is characterized by a single external parameter (such as its volume). The generalization to the case where there are several external parameters is straightforward. Consider a quasi-static change of the external parameter from to . In this process, the energy of the system in state changes by

 (7.42)

The macroscopic work done by the system due to this parameter change is

 (7.43)

In other words, the work done is minus the average change in internal energy of the system, where the average is calculated using the canonical distribution. We can write

 (7.44)

which gives

 (7.45)

We also have the following general expression for the work done by the system

 (7.46)

where

 (7.47)

is the mean generalized force conjugate to . (See Chapter 4.) It follows that

 (7.48)

Suppose that the external parameter is the volume, so . It follows that

 (7.49)

and

 (7.50)

where is the pressure. Because the partition function is a function of and (the energies depend on ), it follows that the previous equation relates the mean pressure, , to (via ) and . In other words, the previous expression is the equation of state. Hence, we can work out the pressure, and even the equation of state, using the partition function.

Next: Partition Function Up: Applications of Statistical Thermodynamics Previous: System with Specified Mean
Richard Fitzpatrick 2016-01-25


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## Introduction

Schizophrenia is one of the most disabling mental disorders as it is associated with cognitive impairment, poor long-term social and occupational outcomes, and affects over 20 million people worldwide1. Recently, major clinical and research efforts have turned towards early intervention, identifying individuals at clinical high-risk (CHR) for psychosis, with the goal of preventing or postponing psychosis onset. Among CHR individuals, rates of transition to a psychotic disorder following clinical presentation are estimated to be 15% after 1 year, increasing to 25% after 3 years2. In this population, additional nonclinical measures are needed to clarify the pathophysiological mechanisms of psychosis3.

One such measure, termed the auditory mismatch negativity (MMN), has been described as a reproducible, reliable biomarker for understanding psychosis4. The auditory MMN is an event-related potential (ERP) involuntarily elicited in response to an infrequent sound stimulus that deviates from a repeated sequence of regular sound stimuli, e.g. in duration or frequency. The MMN serves as an index of surprise during sensory learning and deficits in the MMN amplitude have been shown to correlate with the level of function in chronic4 and first-episode schizophrenia5, and CHR individuals6. However, studies of the auditory mismatch response in CHR patients are variable, with some7,8,9,10,11,12, but not all13,14,15,16 previous studies reporting attenuated MMN amplitudes, and MMN abnormalities were found to be greatest in CHR participants who later convert to psychosis17. Recent pharmacological studies in healthy human controls reported significant MMN attenuation following the administration of ketamine, an N-methyl-d-aspartate (NMDA) antagonist18. Ketamine has previously been employed as an experimental model of psychosis19,20, and taken together, these results suggest that impairment in NMDA receptor function may contribute to MMN attenuation and symptoms of psychosis.

Computationally, the MMN has been interpreted from a predictive coding perspective as a prediction error (PE) signal that arises from a failure to anticipate an incoming stimulus21,22,23,24,25. In predictive coding and related theories of hierarchical Bayesian inference26,27, each level of a cortical hierarchy provides predictions about the state of the level below and evaluates the discrepancy with actual inputs from that level (i.e. PEs) to update predictions. Importantly, this updating process also depends on the relative weight or precision of the error signal (or its inverse, uncertainty), where the key aspect is the relative precision assigned to information from lower (sensory input) compared to higher levels (or top-down prior predictions).

From a hierarchical Bayesian perspective, the auditory MMN can be viewed as representing statistical learning about environmental regularities through sequential precision-weighted PE (pwPE) updates generated at multiple levels of the auditory hierarchy. Disturbances in this hierarchical information processing have been proposed to underlie psychotic symptoms28,29.

In recent years, the application of mathematical models to behavioural, electrophysiological, and neuroimaging data has allowed for the quantification of PE signals and a mechanistic understanding of psychotic symptoms as disruptions in hierarchical Bayesian inference30,31. Under this framework, psychotic symptoms are thought to originate from the misattribution of salience to irrelevant sensory stimuli, driving erroneous model updates and culminating in maladaptive predictions and unfounded beliefs (i.e. delusions)28,32,33,34. In generic hierarchical Bayesian formulations—such as the hierarchical Gaussian filter (HGF)—a ratio of precisions controls the influence of PEs on belief updating. In this theoretical framework, we place a greater emphasis on PEs when lower-level (sensory) inputs are more precise relative to higher-level predictions35,36. Based on this hierarchical Bayesian framework, two possible mechanisms underlying the emergence of psychotic symptoms have been proposed: first, a failure to attenuate lower-level sensory precision renders stimuli persistently surprising and gives rise to overly large learning rates, and thus a misattribution of salience to irrelevant stimuli37,38. Alternatively, a reduction in the precision of higher-level beliefs—i.e. increased informational uncertainty—may render the environment seemingly unpredictable, leading to an over-reliance on, and amplification of, low-level pwPE updates28,31,39. Notably, these explanations may co-exist, as they are not mutually exclusive. Both mechanisms result in aberrantly strong incoming PEs that outweigh prior predictions. These unexplained error signals may lead to a brittle or uncertain model of the world, thus predisposing an individual to the adoption of extraordinary unfounded higher-order beliefs culminating in delusions29,31.

In this paper, we conduct a computational, single-trial analysis of a previously published auditory MMN dataset in help-seeking antipsychotic-naïve CHR individuals and healthy controls (HC)40,41. The aim is to investigate the extent to which the trial-by-trial expression of hierarchically-related pwPEs underlies auditory MMN generation. Consistent with previous studies, we hypothesised that the MMN represents hierarchical PE learning25,42,43. Furthermore, based on predictive coding and hierarchical Bayesian theories of psychosis29, we hypothesise that the attribution of “aberrant salience” to objectively uninformative or neutral events34 is reflected in increased expression of low-level, sensory pwPEs. There has been recent empirical support for this notion in CHR populations where increased expression of sensory pwPEs in dorsolateral prefrontal cortex was associated with greater overall symptom severity in CHR individuals39.

## Results

To identify the EEG correlates of pwPEs, we used a general linear model (GLM) with these computational variables to explain the observed trial-by-trial ERP responses, over channels and peri stimulus time (PST) with respect to tone onset. Significant correlations between model-based pwPE trajectories and single-trial ERPs was demonstrated in both groups with several significant scalp x time clusters. Significant effects are summarised in Figs. 1 and 2 using maximum intensity projections of the significant clusters over left to right scalp locations, retaining the anterior-posterior and PST dimensions in the plots. In both groups, the expression of hierarchical pwPEs ($$\varepsilon _2,\varepsilon _3$$) was not significantly correlated with sex or age.

### Expression of pwPEs in healthy controls

In HC, there were significant correlations between sensory (low-level) pwPEs (ε2) and EEG amplitudes between 100 and 214 ms post-stimulus, corresponding to a negative potential and peaking at 106 ms in temporal central channels (peak, F(1,22) = 133.1; p = 6.1e-7; Fig. 1A). In this cluster, higher low-level pwPE values, or more surprising events, correlated with more negative EEG amplitudes. The timing of this early cluster coincided with the timing of the auditory MMN. Further effects of low-level pwPEs were observed in central channels between 252 and 300 ms post-stimulus corresponding to a positive potential and peaking at 284 ms (peak, F(1,22) = 28.7; p = 0.0207; Fig. 1A). The timing of this effect coincided with the positive-P3a component, with larger low-level pwPEs correlating with an increased central positivity. Finally, a late cluster occurred between 366 and 400 ms post-stimulus, corresponding to a negative potential and peaking at 378 ms in frontal channels (peak, F(1,22) = 24.4; p = 0.0458; Fig. 1A). The timing of this late cluster may continue past our analysis time window of 100 to 400 ms post-stimulus and may be indicative of the reorienting negativity (RON) ERP component44. The RON component typically peaks around 400 ms, signifying attentional reorientation and has been shown to be attenuated in first-episode psychosis and chronic-schizophrenia patients45,46.

Further correlations between single-trial ERPs and volatility (high-level) pwPEs (ε3) in the HC group resulted in multiple significant activation foci. The earliest significant cluster occurred between 100 and 220 ms post-stimulus, peaking at 160 ms in frontocentral channels (peak, F(1,22) = 218.5; p = 7.1e-09; Fig. 1C). A later cluster occurred between 240 and 304 ms post-stimulus, peaking at 256 ms in central channels (peak, F(1,22) = 97.8; p = 5.2e-06; Fig. 1C). Again, the timing of the high-level pwPE effects coincided with the timing of the MMN and P3a components, respectively. Ergo the MMN component, and to a lesser extent the P3a, may capture differences in one’s beliefs about environmental volatility.

### Expression of sensory (low-level) and volatility (high-level) pwPEs in the clinically high-risk group

In the CHR group, there were significant trial-wise correlations between low-level pwPEs (ε2) and EEG amplitudes between 100 and 228 ms post-stimulus, corresponding to a negative potential and peaking at 186 ms in frontal channels (peak, F(1,30) = 275.8; p = 2.4e-12; Fig. 2A). Once again, this cluster coincides with the timing of the auditory MMN.

Three significant activation clusters occurred for trial-wise correlations with high-level pwPEs (ε3). The first cluster appeared between 104 and 224 ms post-stimulus, peaking at 180 ms in central channels (peak, F(1,30) = 258.8; p = 4.4e-12; Fig. 2C), a second cluster occurred between 246 and 314 ms post-stimulus, peaking at 262 ms in central channels (peak, F(1,30) = 56.1; p = 4.9e-05; Fig. 2C), and a third cluster occurred between 398 and 400 ms post-stimulus, peaking at 400 ms in frontocentral channels (peak, F(1,30) = 23.9; p = 0.0212; Fig. 2C). Similar to low-level pwPE expression in HCs, the three significant clusters associated with high-level pwPE expression mirrored the series of negative-positive-negative waveforms reflective of the MMN/P3a/RON component complex. This complex provides a neurophysiological index of involuntary attention controls following a deviant stimuli46, which may be reflected by hierarchical PE updates.

### Group differences in sensory learning

Details of test statistics for HC and CHR groups are given in Tables S2 and S3, respectively. No significant group differences were found in the expression of ε2, the low-level pwPE about stimulus probability, or in the effects of ε3, the high-level pwPE about environmental volatility. Additionally, the expression of the hierarchically-coupled uncertainty trajectories ($$\sigma _1,\sigma _2,\sigma _3$$), unweighted PEs ($$\delta _1,\delta _2$$) and binary tones, did not differ between groups. Note that for $$\sigma _3$$, there were no surviving voxels in either group. The absence of group differences in our study was not surprising. Our dataset was primarily aggregated from Lepock and colleagues’41 who found no group differences in classical MMN amplitudes. See Fig. S1 for average ERPs of the classic MMN response (standard – deviant tones), and pwPE difference waveforms (15% highest − 15% lowest ε2 and ε3 trials, respectively) at electrodes within significant clusters close to the peak effect for both groups.

### Correlations between model parameters and clinical variables

Of the 31 CHR, three CHR converted to psychosis during a 2-year follow-up period. Due to the low number of converters, we could not examine whether differences in parameter expression was related to a subsequent conversion to psychosis. Instead, we tested for correlations between the effects of the hierarchical pwPEs ($$\varepsilon _2,\varepsilon _3$$) and positive prodromal symptom severity (SIPS) at baseline.

We cannot exclude the possibility that the three CHR converters had an outsized effect on correlations between model parameters and positive symptom severity. We therefore additionally report the results of our high-risk correlation analysis after excluding these data sets.

#### Increased expression of low-level sensory prediction errors in the clinically high-risk group

We found a significant positive correlation between ε2 and SIPS positive-symptom subscale total scores between 124 and 146 ms post-stimulus peaking at 140 ms in frontocentral electrodes (peak, F(1,29) = 29.3; p = 0.009; Cohen’s f2 = 0.338; Fig. 3A). This result also held when excluding the three CHR who later converted to psychosis (now CHR = 28; peak, F(1,26) = 26.4; p = 0.020; Cohen’s f2 = 0.340). Simply put, an increased expression of low-level pwPE values were associated with greater SIPS positive subscale total scores at baseline (i.e. greater psychosis symptom severity).

#### Increased expression of informational belief uncertainty in the clinically high-risk group

We additionally tested three possible mechanisms underlying the positive correlation between low-level sensory pwPE updates and positive prodromal symptom severity in the CHR group: aberrantly large sensory PE updates (δ1), aberrantly high precision of incoming PEs (i.e. irreducible uncertainty, σ1), or aberrantly low high-level belief precision (i.e. informational uncertainty, σ2), all of which could lead to an increase in belief updating (see Eq. (1)).

We observed that σ2 showed significant correlations with SIPS positive symptoms in the CHR group between 190 and 212 ms post-stimulus, peaking at 194 ms (peak, F(1,29) = 21.9; p = 0.045; Cohen’s f2 = 0.227; Fig. 3C) in frontal channels and 208 ms (peak, F(1,29) = 22.9; p = 0.037; Cohen’s f2 = 0.242; Fig. 3C) in frontocentral channels. However, this effect did not hold when excluding the three CHR converters (now CHR = 28; peak, F(1,26) = 13.3; p = 0.341; Cohen’s f2 = 0.129). In other words, CHR individuals with greater prodromal positive symptom severity at baseline exhibited increased effects of informational uncertainty. No significant correlations were found between SIPS positive symptoms and sensory PEs (δ1) or irreducible uncertainty (σ1). For completeness, we also investigated the effect of environmental uncertainty (σ3) (c.f.22) but found no significant effects. These results also held when excluding CHR converters. See Supplementary Information for a description of the correlations between informational uncertainty and single-trial ERPs in the HC and CHR group.

## Discussion

In this study, we combined computational modelling with trial-wise analyses of ERP responses to examine atypical PE learning in CHR individuals compared to HCs. Current predictive coding theories of psychosis suggest that positive symptoms, such as hallucinations and delusions, emerge from the misattribution of salience to irrelevant sensory stimuli driving specious model updates and culminating in unsound beliefs28,32,33. Model updating in the auditory hierarchy has been proposed to underlie MMN generation, and our computational modelling approach allows for separation of the MMN component into two hierarchically-coupled PEs: low-level sensory PEs and high-level volatility PEs. In both HC and CHR groups, our model-based analysis revealed significant correlations between single-trial EEG activity and both low- and high-level pwPE trajectories that coincided with the timing of the MMN and P3a response (see Figs. 1, 2). These findings are consistent with previous studies that modelled single-trial MMN responses of visual oddball42 and auditory roving paradigms43, and taken together, these results offer compelling evidence for the role of hierarchical PEs during auditory perceptual inference and implicit, statistical learning.

We did not find any significant differences between the CHR and HC groups in low- or high-level pwPEs. Importantly, however, we found that within the CHR group, prodromal positive symptom severity was positively correlated with larger low-level sensory pwPE updates (Fig. 3A). We further examined the mechanisms underlying the expression of heightened low-level pwPEs and found that positive symptom severity was correlated with the expression of informational uncertainty (i.e. the inverse of belief precision), however this effect did not hold following the removal of CHR converters. Despite this lack of significance, we believe our findings still hold merit. Amongst the CHR group, the three converters displayed higher than average positive symptom severity (SIPS positive scores were 13, 16 and 22, respectively). Hence CHR individuals with the greatest positive symptom burden also had greater expression of informational uncertainty. This suggest that impairments in predictive processing, demonstrated by increased information uncertainty, may represent a core neurophysiological deficit already present in those at highest risk of developing psychosis and may highlight an important predictive mechanism in psychosis progression.

Specifically, these findings suggest that CHR individuals with a greater positive symptom burden exhibit increased uncertainty about their internal model of the causes of sensory inputs, thus rendering the environment seemingly more unpredictable. This may, in turn, further increase learning from incoming PEs, which may be perceived as aberrant salience of the underlying stimuli. Over time, this may lead to adoption of unusual, incorrect higher-order beliefs that may become resistant to disconfirmatory evidence (i.e. delusions), as explanations for these experiences. Our results are consistent with those of Hauke and colleagues47, who using the same computational approach, found increased expression of low-level pwPEs, peaking at 137 ms in central channels, in CHR individuals who later converted to psychosis. Furthermore, these findings are in line with those of Cole and colleagues39 wherein computational modelling was applied to CHR individuals’ behaviour during a probabilistic learning paradigm. Model-based fMRI analyses revealed that larger low-level pwPE effects in the dorsolateral prefrontal cortex correlated with lower global functioning.

Based on this computational approach, we suggest that the MMN is related to both low- and high-level pwPEs. Recently, a similar computational modelling method was used to investigate the effects of ketamine on the MMN43. By using a crossover, placebo-controlled design during an auditory roving paradigm, the authors replicated previous findings showing that ketamine significantly reduced the auditory MMN in HCs43,48, while also showing that ketamine reduced the expression of high-level (volatility) pwPEs43. These results provide further support for the dysconnectivity hypothesis, which proposes that abnormal neurotransmitter modulation of NMDA receptor mediated plasticity may give rise to psychosis symptoms49. Taken together, increased MMN attenuation throughout the illness course, may be due to progressive NMDA receptor dysfunction, and reflected computationally by reductions in the expression of high-level pwPE updating47. In comparison, increased low-level pwPE expression may be reflective of the early stages of psychosis, where individuals attribute increased salience to irrelevant stimuli giving rise to sensory PEs. In other words, our finding of correlations between low-level pwPEs and positive prodromal symptoms in CHR do not necessarily contradict previous findings of MMN reductions in CHR populations7,8,9,10,11,12.

In line with this hypothesis, we reproduced Lepock and colleagues’41 results that found no group differences in classical MMN amplitudes. Divergent findings have been reported with regard to reductions in the MMN amplitude in the CHR population. In a meta-analysis, Erickson, Ruffle and Gold17 found that the overall effect size of MMN amplitudes in CHR participants who converted to psychosis vs. controls was large (0.79) and that CHR participants who did not convert to psychosis had a nonsignificant effect size of 0.17, which was statistically indistinguishable from controls. Thus, within CHR groups, MMN abnormalities may be a sensitive marker of clinical progression. Because MMN impairments are greater among CHR individuals who later convert to psychosis and MMN deficits progressively worsen over illness progression17, the absence of group differences in our study was not surprising given the low number of converters. In addition, an advantage of our study is that we used a more graded measure of symptom severity (the SIPS Positive subscale score) than conversion to psychosis, and this measure could be more sensitive to relationships between disease progression and MMN parameters. For example, due to the low number of converters, we could not examine whether a more pronounced expression of low-level pwPEs was related to a subsequent conversion to psychosis. However, the correlation between positive prodromal symptom severity and low-level pwPE updates suggest that impairments in predictive processing may represent a core neurophysiological deficit already present in those at highest risk of developing psychosis.

Our findings linking pwPE signalling and symptom expression help elucidate the progressive pathophysiology of psychosis and model-based parameters may hold great promise as objective neurophysiological markers for predicting transition to psychosis. Nonetheless, several important limitations remain. In the present study, the observed tone contingencies remained stable over all trials. Hierarchically-related pwPEs are best disentangled when the stream of sensory input (i.e. tones) exhibit changing volatility, thus allowing higher-level volatility PEs to be disentangled from low-level outcome PEs. Given the stability of our task structure, strong correlations were found between low-level and high-level pwPE expression rendering it difficult to isolate the effects on learning about volatility. An additional limitation in our methodology was the assumption of a Bayes-optimal learner for all participants. The MMN paradigm is a passive task and did not require behavioural responses from participants, hence the HGF model could not be fit to behaviour. Consequently and in line with previous methods42,43, we fit an optimal Bayesian learner model to the tone sequence that each participant perceived to obtain belief trajectories reflecting adaptive learning. Additionally, since participants did not have to respond behaviourally to the task, model selection based on behaviour could not be performed. In the future, it would be valuable to apply the same computational modelling approach performed here, to data collected during an active oddball task e.g. ref. 50, and test it against other theoretical models of MMN generation. A third limitation is our modest sample size, which may have limited the power to identify group differences in PE expression. Longitudinal follow-up in larger samples is needed to determine whether model-based parameters are predictive of illness progression, including conversion to psychosis. Finally, given the correlational nature of our study, the results are therefore consistent with a relation between MMN indices of PEs to psychosis-like symptoms, but do not prove that this relation is causal.

In conclusion, our results highlight the role of hierarchically-related pwPEs in auditory MMN generation and suggest that positive prodromal symptom severity is associated with increased outcome-related learning (low-level pwPEs). Furthermore, we validated the use of mechanistic computational models for understanding the emergence of prodromal symptoms in psychosis.

## Methods and materials

### Participants

We included 31 CHR and 23 HC participants aggregated from two previous studies40,41. Note that 41 CHR individuals had originally been recruited but ten participants were excluded from further analysis for a variety of reasons including concurrent disorders (n = 2; epilepsy and alcohol use disorder), electrode failure (n = 1), failure to sufficiently correct for eye-blink artefacts during preprocessing of EEG data (n = 6) and missing SIPS negative data (n = 1). Groups were matched for age and pre-morbid IQ to help ensure that any group differences in MMN measures were likely to be related to the disease process and not to pre-existing intellectual differences51,52,53. CHR participants were help-seeking outpatients referred to the Focus on Youth Psychosis Prevention outpatient clinic at the Centre for Addiction and Mental Health (CAMH) in Toronto. HCs were recruited from the community through advertisements online, in newspapers and on bulletin boards. All participants gave written informed consent and the study was approved by the CAMH Research Ethics Board. CHR individuals met diagnostic criteria for a psychosis-risk syndrome using the Criteria of Psychosis-Risk States based on the Structured Interview for Psychosis-Risk Syndromes (SIPS)54. All CHR participants were of the attenuated psychosis syndrome subtype55 and had no history of current or lifetime DSM-IV-TR Axis I psychotic disorder, or mood disorder with psychotic features56, as determined via the Structured Clinical Interview for DSM-IV-TR57. Participants had no history of DSM-IV substance use or dependence in the last 6 months (except nicotine) and were antipsychotic-naive. At two-year follow-up, three CHR individuals had converted to psychosis (n = 10 CHR dropped out of the study before two-year follow-up). Table 1 shows group demographic, neuropsychological and clinical characteristics.

Electrophysiological (EEG) was recorded during an auditory oddball paradigm based on previous established protocols58,59,60,61,62. Participants were presented with 1830 tones on average binaurally through foam-inserted earphones (Model ER-3C, Etymotic Research, Elk Grove Village, IL) with an interstimulus interval of 500 ms. All tones were 85-dB, 1-kHz and had a rise-fall time of 1-ms. Standard tones (50 ms) and deviant tones (100 ms) were presented in pseudorandom order in 90% and 10% of the trials, respectively. Meanwhile, participants watched a silent cartoon video to divert attention from the tones.

### EEG collection and preprocessing

A 32-electrode EEG cap (Ag/AgCl, actiCAP system, Brain Products) with an actiCHamp amplifier (Brain Products, Gilching, Germany) was used. The electrodes were embedded in the cap at sites approximately equally spaced across the scalp, positioned according to a modified International 10–20 System (Fp1-Fp2-F7-F3-Fz-F4-F8-FC5-FC1-FC2-FC6-T7-C3-Cz-C4-T8-TP9-CP5-CP1-CP2-CP6-TP10-P7-P3-Pz-P4-P8-PO9-O1-Oz-O2-PO10). Eye movement artefacts (blinks and eye movements) were monitored via electrodes on the supraorbital and infraorbital ridges and on the outer canthi of both eyes. Continuous EEG recordings were referenced online to electrode FCz, and continuously digitised at 500 Hz. Electrode impedance was kept below 25 kΩ. EEG data were pre-processed using SPM 12 (http://www.fil.ion.ucl.ac.uk/spm/; version 7771). Continuous EEG was bandpass-filtered using a Butterworth filter between 0.5 and 30 Hz. The eye-blink correction procedure was consistent with previous single-trial computational analyses of EEG data43. Eye-blinks were detected using a threshold routine on the vertical EOG channel and all trials overlapping with eye-blink events were rejected. Corrected data was epoched into 500 ms segments around tone onsets (−100 to 400 ms) and baseline-corrected using a −100 to 0 ms prestimulus baseline. Remaining artefactual trials were rejected if the signal exceeded 100 μV. Channels were rejected and interpolated for sensor-level statistics if more than 20% of trials in a given channel were artefactual.

Bad channels occurred for one HC (O1) and two CHR (channel F4 and channels FP1 and FP2, respectively) participants. The average number of artefact-free trials was 1423 (SD, 173) for HCs and 1359 (SD, 161) for CHR; the number of remaining trials did not differ significantly between groups (see Table S1). ERPs were analyzed by including all artefact-free trials.

### Computational model

To model perceptual inference during the auditory oddball task, we employed the hierarchical Gaussian filter (HGF), a generic hierarchical Bayesian model of learning, previously used for computational analyses of behaviour63,64,65,66,67 and in the context of modelling single-trial MMN responses of visual oddball42 and auditory roving paradigms43. The HGF captures subject-specific approximations to ideal hierarchical Bayesian inference and describes an individual’s learning under uncertainty. Since the MMN paradigm did not require behavioural responses from participants, we simulated the participants’ belief trajectories using Bayes-optimal parameters, which are defined as the parameters that result in the minimal overall surprise in response to a sequence of tone stimuli presented to the participant, in a given session36. In our current application, we assume the agent infers on hidden states in the world (x) from sensory inputs (u), which is reflected through belief updating across the HGF hierarchy (Fig. 4).

#### Inversion of the model: the update equations

The HGF update equations are derived by variational model inversion and provide approximate Bayes-optimal rules for trial-by-trial updating of an agent’s beliefs, given the agent’s particular set of parameter values35. “Belief” refers to a posterior probability distribution as described by its sufficient statistics, i.e. mean μ and variance σ (or its inverse, the precision π) under Gaussian assumptions.

On trial k, an observed trial outcome u indicates whether the tone was a “standard” tone (i.e. 50 ms duration) with u(k) = 0 or a “deviant” (100 ms) with u(k) = 1. The observed outcome u(k) induces a sensory PE or $$\delta _1^{\left( k \right)}$$, which along with its precision π2(k) can be referred to as the sensory pwPE $${\it{\epsilon }}_2^{(k)}$$ and it is used to update $$\mu _2^{\left( k \right)}$$ or the belief about tone transition contingencies. This update, in turn, leads to a volatility PE or $$\delta _2^{\left( k \right)}$$, which along with its precision π3(k) can be referred to as the volatility pwPE or $${\it{\epsilon }}_3^{(k)}$$. This quantity is used to update $$\mu _3^{\left( k \right)}$$ or the belief about environmental volatility. This cascade of hierarchical belief updates can be summarised as follows:

$${\Delta}\mu _i^{\left( k \right)} \propto \frac{{\hat \pi _{i - 1}^{\left( k \right)}}}{{\pi _i^{\left( k \right)}}}\delta _{i - 1}^{\left( k \right)}$$
(1)

where, $$\mu _i^{\left( k \right)}$$ denotes the posterior mean of the belief on each trial k at level i of the hierarchy.

The PE from the level below ($$\delta _{i - 1}^{\left( k \right)}$$) is weighted by a ratio of precisions: the precision of the prediction about the level below (sensory precision) ($$\hat \pi _{i - 1}^{\left( k \right)}$$) and the precision of the belief at the current level in the hierarchy ($$\pi _i^{\left( k \right)}$$). The precision-ratio is updated with each trial and at each hierarchical level serving as a dynamic learning rate and enabling the flexibility required to adapt to evolving outcome probabilities and environmental volatility. See Supplementary Information for further explanation of model parameters.

### Single-trial EEG analysis

To test whether trial-by-trial pwPEs correlated with EEG amplitude fluctuations, single-trial EEG data were converted into three-dimensional volumes with two spatial dimensions (anterior to posterior and left to right directions of the scalp surface) and a temporal dimension (peristimulus time) for all 32 channels and timepoints in the 100–400 ms post-stimulus interval using a voxel size of 4.25 ms × 5.38 ms × 2 ms. Bad channels were linearly interpolated and images were smoothed using a Gaussian kernel (full-width at half maximum: 16 mm × 16 mm) to ensure that assumptions of random field theory were met68,69. The scalp × time 3D images were subjected to statistical analysis using the GLM analogous to fMRI analyses70 and applied to scalp EEG data using the SPM software71. At the single subject level, we pursued single-trial analyses where the computational model of behavioural responses allowed us to quantify learning trajectories, including pwPE trajectories ($${\it{\epsilon }}_2^{(k)},{\it{\epsilon }}_3^{(k)}$$). Additional GLMs were included to unpack the effects of pwPEs: the hierarchically-coupled uncertainty trajectories ($$\sigma _1^{(k)},\sigma _2^{(k)},\sigma _3^{(k)}$$), and the unweighted PEs ($$\delta _1^{\left( k \right)},\delta _2^{\left( k \right)}$$). A final GLM incorporating the tone sequence was included to assess trial-by-trial predictions about categorical change (standard vs. deviant). For each computational quantity, we tested the null hypothesis that the parameter estimates were zero at each sensor and peristimulus time (PST).

A random-effects group analysis was performed across all subjects in each group. At the group level, significant effects were inferred, from thresholded F statistical parametric maps (SPM) that were family wise error (FWE) corrected using random field theory at peak level (p < 0.05) and at the cluster level (p < 0.05) with a cluster-defining threshold of p < 0.00172. Group differences were assessed using a paired t-test. All reported results survived whole-volume FWE correction at the peak-level (p < 0.05). A measure of the effect size, Cohen’s f273, was calculated from the partial eta-squared using the reported F-statistic and degrees of freedom. Finally, we evaluate the impact of sex and age on pwPE expression at the group level by performing an analysis of covariance (ANCOVA).

### Relationship with clinical variables

We investigated the association between pwPE representation during single-trial ERPs and clinical measures of the CHR state. We performed an ANCOVA at the group level testing the association between pwPE effects and SIPS positive symptoms.


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## 2005.8: Functions Preserving Matrix Groups and Iterations for the Matrix Square Root

2005.8: Nicholas J. Higham, D. Steven Mackey, Niloufer Mackey and Françoise Tisseur (2005) Functions Preserving Matrix Groups and Iterations for the Matrix Square Root. SIAM Journal On Matrix Analysis and Applications, 26 (3). pp. 849-877. ISSN 1095-7162

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## Abstract

For any matrix automorphism group $\G$ associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of $A\in\G$ also lie in $\G$; moreover, the polar factors of $A$ lie in $\G$ if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudo-orthogonal groups. This work is concerned with exploiting the structure of $\G$ when computing the polar and matrix sign decompositions of matrices in $\G$. We give sufficient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pad\'e-based iterations of Kenney and Laub satisfy these conditions. The well-known scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a different and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structure-preserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.

Item Type: Article automorphism group, bilinear form, sesquilinear form, scalar product, adjoint, Fr\'echet derivative, stability analysis, perplectic matrix, pseudo-orthogonal matrix, generalized polar decomposition, matrix sign function, matrix $p$th root, matrix square root, structure preservation, matrix iteration, Newton iteration MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2005.8 Nick Higham 24 November 2005


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## Materials Science and Engineering: An Introduction

$a = \frac{4R}{\sqrt{3}}$
For a BCC structure, the corner atoms do not touch one another, but all touch the center BCC atom. Therefore, we can draw a line from one corner of the unit cell to the other, as this is where the atoms are touching. First, we will find the distance from one corner of the unit cell, to the other, in terms of the unit cell edge length, $a$. We begin by finding the corner to corner distance on one face, $b$ of the unit cell, using Pythagorean's theorem. $a^2 + a^2 = b^2$ $b = \sqrt{2}a$ Next we can use Pythagorean's theorem again to now find the distance from one corner to the other of the entire unit cell, denoted as $c$ since we know the distance of one face in terms of $a$. $(\sqrt{2}a)^2 + a^2 = c^2$ $c = \sqrt{3}a$ Since the corner atoms overlap with adjacent unit cells, if we draw a line from one corner of the unit cell to the other, this runs through 1 corner atom, the entire center atom, and then 1 corner atom on the opposite side. This is a total of 4 radii. $4R = c$ $4R = \sqrt{3}a$ $$a = \frac{4R}{\sqrt{3}}$$


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zbMATH — the first resource for mathematics

Jackson, Lloyd K.

Compute Distance To:
 Author ID: jackson.lloyd-k Published as: Jackson, L.; Jackson, L. K.; Jackson, Lloyd; Jackson, Lloyd K. External Links: MGP
 Documents Indexed: 26 Publications since 1952 Biographic References: 2 Publications
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Co-Authors

 13 single-authored 4 Schrader, Keith W. 2 Klaasen, Gene A. 1 Bebernes, Jerrold W. 1 Beckenbach, Edwin Ford 1 Bohn, Elwood 1 Fountain, L. 1 Henderson, Johnny Lee 1 Innes, Joan E. 1 Palamides, Panos K.
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Serials

 8 Journal of Differential Equations 4 Pacific Journal of Mathematics 3 Proceedings of the American Mathematical Society 3 SIAM Journal on Applied Mathematics 1 Advances in Mathematics 1 Duke Mathematical Journal 1 Portugaliae Mathematica 1 Transactions of the American Mathematical Society

Fields

 15 Ordinary differential equations (34-XX)

Citations contained in zbMATH Open

25 Publications have been cited 385 times in 275 Documents Cited by Year
Subfunction and second-order ordinary differential inequalities. Zbl 0197.06401
Jackson, L. K.
1968
Comparison theorems for nonlinear differential equations. Zbl 0149.29701
Jackson, Lloyd K.; Schrader, Keith W.
1967
Existence and uniqueness of solutions of boundary value problems for third order differential equations. Zbl 0256.34018
Jackson, Lloyd K.
1973
Existence and uniqueness of solutions of boundary value problems for third order differential equations. Zbl 0206.37601
1971
Uniqueness of solutions of boundary value problems for ordinary differential equations. Zbl 0237.34030
Jackson, Lloyd K.
1973
Infinite interval boundary value problems for $$y'' = f(x,y)$$. Zbl 0145.33102
Bebernes, J. W.; Jackson, L. K.
1967
Existence and uniqueness of solutions of boundary value problems for Lipschitz equations. Zbl 0407.34018
Jackson, Lloyd K.
1979
A variation of the topological method of Wazewski. Zbl 0231.34007
Jackson, Lloyd K.; Klaasen, Gene
1971
A generalized solution of the boundary value problem for $$y'' = f(x,y,y')$$. Zbl 0112.05602
Fountain, L.; Jackson, L.
1962
Subfunctions and third order differential inequalities. Zbl 0194.40902
1970
An existence theorem for a nonlinear two-point boundary value problem. Zbl 0562.34008
Jackson, L. K.; Palamides, P. K.
1984
Boundary value problems for ordinary differential equations. Zbl 0371.34011
Jackson, L. K.
1977
Existence and uniqueness of solutions of k-point boundary value problems for ordinary differential equations. Zbl 0511.34010
Henderson, Johnny; Jackson, Lloyd
1983
Boundary value problems for Lipschitz equations. Zbl 0599.34018
Jackson, Lloyd K.
1980
A Nagumo condition for ordinary differential equations. Zbl 0336.34005
Jackson, L. K.
1976
Uniqueness of solutions of boundary value problems for ordinary differential equations. Zbl 0211.11501
Jackson, L.; Klaasen, G.
1970
Disconjugacy conditions for linear third-order differential equations. Zbl 0167.37103
Jackson, L. K.
1968
The Liouville theorem for a quasilinear elliptic partial differential equation. Zbl 0119.30704
Bohn, Elwood; Jackson, Lloyd K.
1962
Subfunctions of several variables. Zbl 0050.10105
Beckenbach, E. F.; Jackson, L. K.
1953
On second order differential inequalities. Zbl 0145.12003
1966
Nagumo conditions for ordinary differential equations. Zbl 0315.34043
Innes, Joan E.; Jackson, Lloyd K.
1975
Uniqueness and existence of solutions of boundary value problems for ordinary differential equations. Zbl 0313.34020
Jackson, Lloyd K.
1972
Subfunctions and the Dirichlet problem. Zbl 0098.06803
Jackson, Lloyd K.
1958
On generalized subharmonic functions. Zbl 0067.32703
Jackson, L. K.
1955
A compactness condition for solutions of ordinary differential equations. Zbl 0336.34020
Jackson, L. K.
1976
An existence theorem for a nonlinear two-point boundary value problem. Zbl 0562.34008
Jackson, L. K.; Palamides, P. K.
1984
Existence and uniqueness of solutions of k-point boundary value problems for ordinary differential equations. Zbl 0511.34010
Henderson, Johnny; Jackson, Lloyd
1983
Boundary value problems for Lipschitz equations. Zbl 0599.34018
Jackson, Lloyd K.
1980
Existence and uniqueness of solutions of boundary value problems for Lipschitz equations. Zbl 0407.34018
Jackson, Lloyd K.
1979
Boundary value problems for ordinary differential equations. Zbl 0371.34011
Jackson, L. K.
1977
A Nagumo condition for ordinary differential equations. Zbl 0336.34005
Jackson, L. K.
1976
A compactness condition for solutions of ordinary differential equations. Zbl 0336.34020
Jackson, L. K.
1976
Nagumo conditions for ordinary differential equations. Zbl 0315.34043
Innes, Joan E.; Jackson, Lloyd K.
1975
Existence and uniqueness of solutions of boundary value problems for third order differential equations. Zbl 0256.34018
Jackson, Lloyd K.
1973
Uniqueness of solutions of boundary value problems for ordinary differential equations. Zbl 0237.34030
Jackson, Lloyd K.
1973
Uniqueness and existence of solutions of boundary value problems for ordinary differential equations. Zbl 0313.34020
Jackson, Lloyd K.
1972
Existence and uniqueness of solutions of boundary value problems for third order differential equations. Zbl 0206.37601
1971
A variation of the topological method of Wazewski. Zbl 0231.34007
Jackson, Lloyd K.; Klaasen, Gene
1971
Subfunctions and third order differential inequalities. Zbl 0194.40902
1970
Uniqueness of solutions of boundary value problems for ordinary differential equations. Zbl 0211.11501
Jackson, L.; Klaasen, G.
1970
Subfunction and second-order ordinary differential inequalities. Zbl 0197.06401
Jackson, L. K.
1968
Disconjugacy conditions for linear third-order differential equations. Zbl 0167.37103
Jackson, L. K.
1968
Comparison theorems for nonlinear differential equations. Zbl 0149.29701
Jackson, Lloyd K.; Schrader, Keith W.
1967
Infinite interval boundary value problems for $$y'' = f(x,y)$$. Zbl 0145.33102
Bebernes, J. W.; Jackson, L. K.
1967
On second order differential inequalities. Zbl 0145.12003
1966
A generalized solution of the boundary value problem for $$y'' = f(x,y,y')$$. Zbl 0112.05602
Fountain, L.; Jackson, L.
1962
The Liouville theorem for a quasilinear elliptic partial differential equation. Zbl 0119.30704
Bohn, Elwood; Jackson, Lloyd K.
1962
Subfunctions and the Dirichlet problem. Zbl 0098.06803
Jackson, Lloyd K.
1958
On generalized subharmonic functions. Zbl 0067.32703
Jackson, L. K.
1955
Subfunctions of several variables. Zbl 0050.10105
Beckenbach, E. F.; Jackson, L. K.
1953
all top 5

Cited by 223 Authors

 29 Henderson, Johnny Lee 11 Howes, Fred A. 11 Schrader, Keith W. 10 Jackson, Lloyd K. 10 O’Regan, Donal 10 Palamides, Panos K. 9 Eloe, Paul W. 9 Schmitt, Klaus 7 Agarwal, Ravi P. 7 Erbe, Lynn Harry 7 Umamaheswaram, S. 6 Bebernes, Jerrold W. 6 Ehme, Jeffrey A. 6 Thompson, Harold Bevan 6 Zhao, Weili 5 Gaines, Robert E. 5 Sadyrbaev, Felix Zh. 4 Feng, Yuqiang 4 Fink, Arlington Michael 4 Kelley, Walter G. 4 Klaasen, Gene A. 4 Lepin, A. Ya 4 Li, Yong 3 Andres, Jan 3 Bernfeld, Stephen R. 3 Lakshmikantham, Vangipuram 3 Lanz, Aprillya 3 Sficas, Yiannis G. 3 Staikos, Vasilios A. 3 Venkata Rama, M. 3 Waltman, Paul Elvis 3 Wang, Huaizhong 3 Yao, Qingliu 3 Yermachenko, Inara 3 Yin, William K. C. 2 Ahmad, Bashir 2 Baxley, John V. 2 Butuzov, Valentin Fëdorovich 2 Chandra, Jagdish 2 Clark, Stephen R. 2 Garner, Jackie B. 2 Hankerson, Darrel R. 2 Hartenstine, David A. 2 Heidel, J. W. 2 Kelevedjiev, Petio S. 2 Kosmala, Witold A. J. 2 Leela, Srinivasa G. 2 Mawhin, Jean L. 2 Miao, Shumei 2 Moorti, V. R. G. 2 Myshkis, Anatoliĭ Dmitrievich 2 Nefedov, Nikolaĭ Nikolaevich 2 Peterson, Allan C. 2 Rao, Vadrevu Sree Hari 2 Sȩdziwy, Stanisław 2 Shampine, Lawrence Fred 2 Shreve, Warren E. 2 Staněk, Svatoslav 2 Tisdell, Christopher C. 2 Wilhelmsen, Russell 2 Zhou, Qinde 1 Aktaş, Mustafa Fahri 1 Al-Malki, Nawal 1 Anderson, Douglas Robert 1 Asif, Naseer Ahmad 1 Avery, Richard I. 1 Bange, David W. 1 Barr, Dennis 1 Ben-Israel, Adi 1 Ben-Tal, Aharon 1 Bobisud, Larry E. 1 Bohé, Adriana 1 Boucherif, Abdelkader 1 Butler, Geoffrey J. 1 Cabada, Alberto 1 Çakmak, Devrim 1 Chang, Sung 1 Chang, Xingjian 1 Chow, Shuinee 1 Chua, Seng-Kee 1 Chyan, Chuan Jen 1 Constantin, Adrian 1 Countryman, M. 1 Dahiya, Rajbir S. 1 Das, Kaustuv Mukul 1 Davis, Paul William 1 De Coster, Colette 1 Debnath, Lokenath 1 Dhar, Sougata 1 do Rosário Grossinho, Maria 1 Dobkevich, Maria 1 Elcrat, Alan R. 1 Feng, Zongxian 1 Fraker, Ross 1 Gabor, Grzegorz 1 Gao, Yongxin 1 Gatica, Juan A. 1 George, John H. 1 Gerova, Marta 1 Goecke, David M. ...and 123 more Authors
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Cited in 63 Serials

 59 Journal of Differential Equations 30 Journal of Mathematical Analysis and Applications 25 Nonlinear Analysis. Theory, Methods & Applications 23 Proceedings of the American Mathematical Society 10 Applied Mathematics Letters 8 Archive for Rational Mechanics and Analysis 8 Annali di Matematica Pura ed Applicata. Serie Quarta 8 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 6 Applied Mathematics and Mechanics. (English Edition) 5 Rocky Mountain Journal of Mathematics 5 Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 5 Tohoku Mathematical Journal. Second Series 4 Applicable Analysis 4 Computers & Mathematics with Applications 4 Applied Mathematics and Computation 4 Transactions of the American Mathematical Society 3 Journal of Computational and Applied Mathematics 3 Journal of Optimization Theory and Applications 2 Mathematical Notes 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Archiv der Mathematik 2 Czechoslovak Mathematical Journal 2 Mathematische Zeitschrift 2 Mathematika 2 Numerische Mathematik 2 Rendiconti del Circolo Matemàtico di Palermo. Serie II 2 Studies in Applied Mathematics 2 Zeitschrift für Analysis und ihre Anwendungen 2 Acta Mathematicae Applicatae Sinica. English Series 2 Journal of Difference Equations and Applications 2 Boundary Value Problems 2 Matematický Časopis, Slovenskej Akadémie Vied 1 Communications on Pure and Applied Mathematics 1 Journal d’Analyse Mathématique 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Periodica Mathematica Hungarica 1 Chaos, Solitons and Fractals 1 Advances in Mathematics 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Approximation Theory 1 Journal of Geometry 1 Mathematische Nachrichten 1 Mathematical Systems Theory 1 Monatshefte für Mathematik 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Optimization 1 Journal of Automated Reasoning 1 Aequationes Mathematicae 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Russian Mathematics 1 Georgian Mathematical Journal 1 NoDEA. Nonlinear Differential Equations and Applications 1 Opuscula Mathematica 1 Journal of Applied Analysis 1 Abstract and Applied Analysis 1 Journal of Biological Systems 1 Advanced Nonlinear Studies 1 Communications on Pure and Applied Analysis 1 Journal of Function Spaces and Applications 1 Bulletin of the American Mathematical Society 1 Proyecciones 1 Differential Equations and Applications
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Cited in 20 Fields

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1. ## indicial roots

ok in this problem, im supposed to find the indicial roots and determine whether or not they differ by an integer..

$2x^{2}y''-6xy'+(x^{2}+4)y=0$

i got the roots, $2\pm \sqrt{2}$

and said that they do not differ by an integer...so did i do the problem right?

2. Originally Posted by slapmaxwell1
ok in this problem, im supposed to find the indicial roots and determine whether or not they differ by an integer..

$2x^{2}y''-6xy'+(x^{2}+4)y=0$

i got the roots, $2\pm \sqrt{2}$

and said that they do not differ by an integer...so did i do the problem right?
Yes your indicial roots are correct.