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## Supervised by Susan E. Martonosi Department of Mathematics Harvey Mudd College

Project Description

Operations research (O.R.) is the use of mathematical models to make decisions under uncertainty regarding efficient use of scarce resources. Resources are rarely scarcer than in the developing parts of our world where people struggle for access to clean drinking water, food and medical aid. Only in recent years have humanitarian logistics problems caught the attention of operations researchers, in the context of treatment and prevention of communicable diseases (Operations Research Toolkit on HIV/AIDS, 2012), water delivery and sanitation (Horner, 2008), and disaster relief (De Angelis, Mecoli, Nikoi, & Storchi, 2007), (Duran, Gutierrez, & Keskinocak, 2011), for example. The existing literature combines optimization, simulation, inventory management, stochastic processes, data analytics and other techniques of operations research in creative ways to tackle these challenging and complex problems. The purpose of this proposed project is to develop a mathematical model to address an important problem in the area of humanitarian logistics in Africa.

Because this is a new area of research for the faculty advisor, students will play a key role in the process of researching the existing literature, and from it, defining an open research problem that interests them.   Unlike many traditional research experiences, approximately half of the summer will be spent reviewing the existing literature and defining a problem. From there, students will spend a few weeks working on the mathematical models.

The ideal student for this team would have taken an operations research course (either 186 or 187), have some background in probability and statistics, and be comfortable programming in Matlab. These are not hard requirements, however. The most important requirements are motivation, persistence, and a desire to develop math models to help society.

## Bibliography

De Angelis, V., Mecoli, M., Nikoi, C., & Storchi, G. (2007). Multiperiod Integrated Routing and Scheduling of World Food Programme Cargo Planes in Angola. Computers and Operations Research , 1601-1615.

Duran, S., Gutierrez, M., & Keskinocak, P. (2011). Pre-Positioning of Emergency Items for CARE International. Interfaces , 223-237.

Horner, P. (2008). Africa Puts O.R. on Trial. OR/MS Today , 46.

Operations Research Toolkit on HIV/AIDS. (2012). Retrieved January 3, 2012, from Population Council: http://www.popcouncil.org/Horizons/ORToolkit/index.htm

## Supervised by Stephan Ramon Garcia Department of Mathematics Pomona College

Project Description

We will use \emph{supercharacter theory}, a powerful new algebraic mechanism recently pioneered by Diaconis-Isaacs and André, to study certain exponential sums which arise in number theory (e.g., Gauss, Kloosterman, and Ramanujan sums).  From this new perspective, revisiting even the most elementary groups can yield interesting results.  For instance, many of the fascinating arithmetic properties of \emph{Ramanujan sums}

$\displaystyle c_n(x) = \sum_{ \substack{ j = 1 \\ \gcd(j,n) = 1} }^n \exp\left( \frac{2\pi i jx}{n}\right)$

follow immediately from basic supercharacter theory and elementary linear algebra applied to $\mathbb{Z} / n \mathbb{Z}$.

Students will learn the rudiments of supercharacter theory and begin working on problems right away.  We expect that a careful survey of familiar groups will reveal \emph{supercharacter tables} which contain exponential sums whose properties are of general interest.  Moreover, the powerful machinery of supercharacter theory should automatically yield simple proofs of many difficult algebraic formulas.

Our project revolves around several open-ended questions and features many points of departure.  If time permits, we will also study some deeper foundational issues.  Being a new area, supercharacter theory is fertile ground for student research and there is plenty of low-hanging fruit to be found on this frontier.

Prerequisites. The ideal candidate has a strong background in Abstract Algebra and Linear Algebra.  Some knowledge of Number Theory and familiarity with \emph{Mathematica} (or comparable software) is also desirable.

## Supervised by Rena Levitt Department of Mathematics Pomona College

Project Description

Given a word whose letters are the generators of a group $G$, is there an algorithm to determine if the word represents the identity in $G$? If so, how many steps will it take? In this project we will study a method to answer these questions by building a metric space that encodes the structure of the group.

One of the main ideas of geometric group theory is that you can recover algebraic properties of a group by encoding the structure of the group in a metric space and analyzing the geometric properties of the space. This method has proved fruitful for answering when many interesting questions about groups. One example is Maxwell Dehn's classical word problem. "Let $G$ be a finitely generated group, and $w$ be a word whose letters are generators of $G$. Is there an algorithm to determine if $w$ equals the identity in $G$?''

In this project we will discuss a common method used by geometric group theorist to solve the word problem for a finitely generated group by relating words in the group to paths in a metric space representing the group's structure. Our main focus will be Dehn functions. A Dehn function is a measure of the computational complexity of the solution to the word problem. Given a word $w$ of length $n$ representing the identity, a Dehn function bounds the number of steps needed to reduce $w$ to the identity.  This turns out to be equivalent to studying the relation between perimeter and area in the encoding metric space. Our main source of examples will be spaces built out of squares and equilateral triangles satisfying a combinatorial nonpositive curvature condition that mimics the geometry of the Euclidean and hyperbolic planes.

The ideal students for this project will have taken linear algebra and a course in which they have written rigorous mathematical proofs. Coursework in group theory or topology will also be helpful but not necessary.

Proudly Serving Math Community at the Claremont Colleges Since 2007


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# Directory tex-archive/dviware/dvidvi

Anyone may freely use, modify, and/or distribute this file, without limitation.

Here's a little hack I threw together for those doing pagination
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The dvidvi program converts a dvi file into another dvi file,
with perhaps certain changes.  Invocation is

dvidvi param infile outfile

What are the parameters?  The 'easy' paramaters are the following:

-f n     page n is first page selected
-l n     page n is last page selected
-n n     select at most n pages. Notice that n is the number of pages
selected, independently of the number of pages actually
contained in a sheet
-i { n1..n2 | n1 }[,...]
include pages (ranges allowed). When this option is
used, ONLY the specified pages are selected. However,
we can exclude from these pages with the option -x
-x { n1..n2 | n1 }[,...]
exclude pages (ranges allowed)
-q       work in quiet mode, that is do not print in the screen
messages of how the work is being done.
-r       reverse the order of the pages.

The page numbers for the above options -f -l -i and -x can be specified
in different ways.

1) If a number n is given, it is interpreted as the n'th page from the
begining of the .dvi file. Of course, this number is independent of the
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2) TeX page numbers are those who are actually written in the page;
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A TeX page number can be specified by preceding the number n with the
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3) However, several pages can have the same TeX page number in a .dvi file.
For example, the introductory pages in a book are numbered i, ii, and so on
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In this case, the pages numbered i and 1 in the .dvi file have the same TeX
page number. If you want to select for example the second occurrence of the
page numbered 1, you can specify a page number as (@2)1. Thus @1 is equivalent
to (@1)1. For example, if you specify -f (@2)1 -l(@2)10 you select the pages
between 1 and 10 of the first chapter, not the introductory pages between
i and x.

There is another parameter that tells dvidvi how you want to change page
layout and specifications. This is the -m parameter.

The number preceding the colon is the modulo value.  Everything
will be done in chunks of pages this big.  If there is no colon, than the
default value is assumed to be one.  The last chunk of pages is padded
with as many blank pages as necessary.

Following the colon is a comma-separated list of page numbers.
These page numbers are with respect to the current chunk of pages, and
must lie in the range zero to the modulo value less one.  If a negative
sign precedes the number, then the page is taken from the mirror chunk;
if there are m chunks, then the mirror chunk of chunk n is the chunk
numbered m-n-1.  Put simply, it is the chunk numbered the same, only
from the end.  This can be used to reverse pages.  If no number is
given, the page number defaults to 1.

Following each page number is an optional offset value in
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Each dimension is a decimal number with an optional unit of measure.
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used.  All units are in true dimensions.  Allowable units of measure
are the same that TeX allows:  in, mm, cm, pt, pc, dd, and cc.

Simple enough, eh?  Okay, let's do some simple things.

-m -      Reverses the order of the pages.  This time, both the modulo and
the page number are defaulted.

-m 2:0    Selects the first, third, fifth, etc. pages from the file.  Print
this one after printing the next, taking the paper out of the
feed tray and reinserting it into the paper feed.

-m 2:-1   Selects the second, fourth, etc. pages, and writes them in reverse
order.
-m 4:-1,2(4.25in,0in)
-m 4:-3,0(4.25in,0in)
Useful for printing a little booklet, four pages to a sheet,
double-sided, for stapling in the middle.  Print the first one,
put the stack back into the printer upside down, and print the
second.  The `in' specifications are superfluous.
-m ,(1pt,1)
blurry with this option, you may want to send enemies letters
printed like this.  *Long* letters.
-m 4:0(5.5in,4.25),3(0,4.25)
-m 4:1(0in,4.25),2(5.5,4.25)
Print a four-page card on one sheet.  Print the first, rotate
the paper 180 degrees and feed it again.  (PostScript people
can do funny tricks with PostScript so this isn't necessary.)

Enjoy; this is an early release, so make suggestions, improvements,
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======================================================================

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This CTAN source tree has been automatically extracted from the texk
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It is not possible to use the supplied configure script.
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This dviljk source can only be compiled in a texk build environment.
http://www.tug.org/texlive/svn/.

It all boils down to a Subversion checkout of
svn://tug.org/texlive/trunk/Build/source
This will be the source tree that you can use for compilation. The
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For your convenience, the original configure script from the texk
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## Files

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dvidvi.c 35454 2013-06-28 15:27
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## dvidvi – Convert one DVI file into another

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text:
Article

# Topological characterization of charge quantization

06/1999; 112(6):639-645. DOI: 10.1007/BF03035874

ABSTRACT

Weinberg has written a paper showing how to calculate gauge coupling constants in (4 +N)-dimensional models withN dimensions forming a compact manifold. Each coupling constant is related to the inverse of an appropriate rootmean-square
circumference of the manifold. We extend this work by showing that this charge is quantized, in the sense of a tower of particles
each carrying a charge which is an integer multiple of a basic unit, if and only if Π1(I) =ZZ where Π1 is the first homotopy group,I is the isometry group of the compact manifold, andZZ is the additive group of integers.

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##### Article: Five-dimensional relativity
[Hide abstract]
ABSTRACT: We study the hypothesis where the universe U is a five-dimensional Riemannian manifold, wich satisfies certain global topological conditions. We postulate the existence of a principle of relativity wich treats on equal basis the live dimensions of U; the laws wich satisfy this principle have an approximate description in a 4-dimensional space-time manifold û; this gives the possibility of comparing them with the usual description of experimental laws. Thus, if we extend to the fifth dimension the invariance of general relativity, we obtain classical electrodynamics: the equations of Maxwell, conservation of electricity, electromagnetic forces, etc. Likewise, the five-dimensional extension of the invariance of the wave equations leads one automatically to electromagnetic terms, such as they are actually observed; the electric charge, for instance, is found to be an integral multiple of an elementary charge which depends neither on the mass, nor on the spin. Among the other consequences of the theory, we find gauge invariance, and charge conjugation; the maximum violation of parity in Β-decays; the existence of two neutrinos of opposite chirality.
Il Nuovo Cimento 09/1963; 30(2):565-578. DOI:10.1007/BF02828833
• ##### Article: Charge Quantization, Compactness of the Gauge Group, and Flux Quantization
[Hide abstract]
ABSTRACT: The relationship between charge quantization and the compactness of the gauge group is discussed. Also, remarks are made about charge quantization and the observation of flux quantization in superconductors.
Physical Review D 04/1970; 1(8). DOI:10.1103/PhysRevD.1.2360 · 4.64 Impact Factor


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# Contribution of apical and basal dendrites to orientation encoding in mouse V1 L2/3 pyramidal neurons

## Abstract

Pyramidal neurons integrate synaptic inputs from basal and apical dendrites to generate stimulus-specific responses. It has been proposed that feed-forward inputs to basal dendrites drive a neuron’s stimulus preference, while feedback inputs to apical dendrites sharpen selectivity. However, how a neuron’s dendritic domains relate to its functional selectivity has not been demonstrated experimentally. We performed 2-photon dendritic micro-dissection on layer-2/3 pyramidal neurons in mouse primary visual cortex. We found that removing the apical dendritic tuft did not alter orientation-tuning. Furthermore, orientation-tuning curves were remarkably robust to the removal of basal dendrites: ablation of 2 basal dendrites was needed to cause a small shift in orientation preference, without significantly altering tuning width. Computational modeling corroborated our results and put limits on how orientation preferences among basal dendrites differ in order to reproduce the post-ablation data. In conclusion, neuronal orientation-tuning appears remarkably robust to loss of dendritic input.

## Introduction

Neocortical pyramidal neurons ramify several basal dendritic arbors laterally and one apical dendritic arbor superficially to receive and integrate synaptic inputs1. Each dendritic-tree of a mouse layer 2/3 (L2/3) pyramidal neuron arborizes in a different cortical map sub-region (e.g., retinotopic region) and/or cortical layer, thereby sampling largely non-overlapping axonal inputs coming in from different brain areas2. Inputs are functionally heterogeneous across dendrites and even within individual dendritic branches3,4. Mouse primary visual cortex (V1) L2/3 pyramidal neurons generate action potentials in response to a narrow range of orientations5 despite receiving highly heterogeneous input and poorly tuned subthreshold responses4, making them ideal for studying the relationship between dendritic input and functional selectivity. Indeed, recent evidence suggests that apical tuft dendritic spikes serve to narrow the orientation tuning function, increasing orientation selectivity of area V1 L2/3 pyramidal neurons6. L2/3 pyramidal neurons are also a good model system for studying the relative roles of apical and basal dendrites: while basal dendrites primarily receive feedforward input from L4 and nearby L2/3 neurons7,8, apical dendrites receive cortico-cortical feedback that may refine orientation selectivity9,10 as well as orientation-tuned thalamo-cortical input from layer 14,11,12,13.

It is difficult to firmly establish a causal relationship between computations that occur in dendritic branches and properties at the soma. Calcium imaging does not always have the temporal resolution to disambiguate signals arising in different dendritic processes or the soma, while in vivo dendritic patch clamping, despite its power at dissecting specific hypotheses, is usually limited to assessing single branch contributions to neuronal response. The existence of functional inputs and activity on dendrites does not causally explain the necessity of these inputs for determining the neuron’s final output. Here we employ in vivo two photon microdissection14,15,16,17 to systematically remove individual dendrites from layer 2/3 mouse V1 pyramidal neurons, allowing us to assess the causal relationship between inputs arriving in different dendritic arbors and the computation of orientation selectivity at the soma.

## Results

### Dendrite ablation in vivo

To assess the causal relationship between inputs arriving in different dendrites and orientation-tuning, we systematically removed individual dendrites from L2/3 mouse V1 pyramidal neurons using in vivo 2-photon micro-dissection14,15,17. We visualized both dendritic structure and functional activity of L2/3 pyramidal neurons by stereotactically co-injecting AAV-flex-GCaMP6s and diluted AAV-CaMKII-Cre (40,000 to 120,000×) into L2/3 of mouse V111 (Fig. 1a, methods). Individual dendrites were then removed by performing 2-photon laser point scans on the targeted fluorescent dendritic segment, 10–30 µm away from the soma (Fig. 1b, c and Supplementary Fig. 1)14,15. Dendritic segments expressing GCaMP6s, located 130–200 µm from the dura, could be typically ablated with one or two 200–400 ms point scans of 140–200 mW at 800 nm. Ablated dendrites distal to the lesion acquired a beads-on-the-string appearance and degraded, disappearing within 1–3 h (Fig. 1b, Supplementary Movie 1,2)14,15. Upon ablation, target neurons transiently became brightly fluorescent (Supplementary Fig. 1d), presumably due to the influx of calcium through the instantaneous opening in the membrane, returning to pre-ablation fluorescence levels over the next 5–90 min. Neurons whose fluorescence did not return to their pre-ablation levels within 90 min were more likely to be destroyed, i.e., visually undetectable with debris present, by the next day. Neurons whose apical dendritic arbors were ablated immediately adjacent to the soma were also more likely to disappear, so we restricted ablation to pyramidal neurons with an extended primary apical trunk (≥20–3 µm from soma to the first apical bifurcation), targeting the point immediately prior to the bifurcation. Target neurons had a soma depth between 150 and 250 µm. Neurons that were still present 24 h post-ablation (46% of cells: 18 out of 39 neurons from 26 mice survived after apical dendrite ablation, see “Methods”) exhibited intact morphology in their residual (non-ablated) dendritic segments and normal spontaneous and visual-evoked somatic calcium transients (Fig. 1b–d, Fig. 2a and Supplementary Fig. 1).

We confirmed that targeted dendrites were successfully removed by generating a custom virus co-expressing GCaMP6s with activity-independent stable red fluorophore mRuby (Supplementary Fig. 2). The elimination of the apical tuft following ablation was clearly identified with red fluorescent signal, and complete structural overlap was observed between GCaMP6 and mRuby. Ablation was further confirmed using post-hoc immunostaining with anti-GFP antibodies (Supplementary Fig. 3). Post-hoc immunostaining of unlabeled axons and dendrites adjacent to the ablation site revealed no overt signs of damage (Supplementary Fig. 3e, Supplementary Movie 3) while adjacent GCaMP-labeled processes imaged in vivo near the ablation site were not noticeably affected (Supplementary Fig. 1c), suggesting that ablation effects are spatially restricted within a radius of ~5 µm around the ablation point, in line with previous reports14,15.

### Loss of apical input does not alter orientation preference

L2/3 pyramidal neurons receive orientation-tuned inputs scattered pseudo-randomly on their apical dendritic-trees3,4,11,18, but it is not clear how these contribute to the neuron’s orientation preference. We tested the collective contribution of these inputs to orientation selectivity by ablating the apical tuft from L2/3 pyramidal neurons in fentanyl-dexmedetomidine sedated mice. The ablation of apical dendrite removes ~40% of the total excitatory input received by the neuron19. Figure 1c (and Supplementary Movie 2) shows the reconstructed morphology of one neuron before and after apical tuft ablation. Figure 2a illustrates single-trial dF/F responses of an example neuron to oriented gratings moving in 12 different directions, before as well as 1, 3, and 5 days after apical tuft ablation. Remarkably, removal of the entire apical tuft did not affect the neuron’s preferred orientation (Fig. 2b). The ablated neuron still responded maximally to its pre-ablation preferred orientation, and responses to other orientations were also very similar to pre-ablation (Fig. 2c).

Plotting the absolute-value change in orientation preference before and 5 days after ablation demonstrated that orientation preference did not change significantly following apical tuft ablation (Fig. 2d: blue circles corresponding to 18 ablated neurons all fall close to the diagonal as do black squares corresponding to 17 neighboring control neurons imaged together with ablated neurons). There was no significant difference in orientation preference change between the two groups (Fig. 2d, inset; p = 0.4, t-test; p = 0.95, Mann–Whitney U test). Tuning-width (width at half maximum) and orientation selectivity index (OSI = [Responsepref − Responsenull]/[Responsepref + Responsenull]) were also not significantly affected on average by apical tuft ablation (Fig. 2e, p = 0.81, t-test; p = 0.37, Mann–Whitney U test; Fig. 2f, p = 0.88, t-test; p = 0.47, Mann–Whitney U test). Bootstrap analysis to assess per-neuron shift in orientation preference and tuning width (see “Methods”) also suggests that there is no consistent shift in orientation preference after removal of the apical dendrite beyond that observed in controls (Supplementary Figs. 45, Supplementary Table 1). The narrow distributions of bootstrap-estimated orientation preferences after ablation indicates that the tuning of neurons remains robust even following apical dendrite ablation (Supplementary Fig. 4). No consistent changes in response amplitude or baseline firing rates were observed across neurons a day or more following ablation (Supplementary Fig. 6, see figure legend for statistics). Pre- and post-ablation tuning curves for all recorded neurons are shown in Supplementary Fig. 7. This demonstrates that, under the conditions tested, inputs to the basal dendrites are sufficient for determining the orientation preference and basic orientation tuning function properties of V1 L2/3 pyramidal neurons7.

### Orientation tuning is robust to loss of multiple basal dendrites

We then applied our ablation strategy to the basal dendrites to explore how individual primary basal dendrites contribute to orientation selectivity (Fig. 3). Mouse V1 L2/3 pyramidal neurons typically have 5–8 (median 6) primary basal dendrites19. The point scan was targeted to a randomly-chosen primary basal dendrite, 5–10 µm away from the soma, before secondary branching. Neurons still present 24 h post-basal dendrite ablation survived long-term and were studied. Overall, 36% of functionally studied neurons survived basal dendrite ablation (13 out of 33 neurons survived single basal dendrite ablation and 17 out of 50 neurons survived double basal dendrite ablation, from 34 mice). This was not significantly different from the survival rate after apical dendrite ablation (46%; p = 0.29, χ2 = 1.1, chi-square test).

Similar to what we observed with apical dendrite ablation (Fig. 2), cutting one basal dendrite (Fig. 3a, 8–12% of total dendritic input) did not alter the target neuron’s preferred orientation, nor its tuning-width or OSI (Fig. 3b, Supplementary Fig. 7b). Neurons in which two basal dendrites were removed (Fig. 3c, 16–24% of total input) also demonstrated remarkable tuning curve stability (Fig. 3d, Supplementary Fig. 7b), but did show a small, yet significant, shift of ~12.5 ± 2.7° on average (max: 30°) in orientation preference following ablation (Fig. 3e–f, two-basal ablation vs. control: p = 0.003, ANOVA with Tukey correction for multiple comparisons; p = 0.025, Kruskal–Wallis test with Tukey correction for multiple comparisons. Two-basal-ablation vs. pooled one-basal-ablation + control: p = 9.5 × 10−12, t-test; p = 0.0086, Mann–Whitney U test). We again performed bootstrap analysis to estimate the reliability of orientation preference measurements neuron by neuron. The distribution of the bootstrap estimates of orientation preference before and after ablation was narrow, suggesting that the tuning properties of ablated neurons were measured reliably (Supplementary Figs. 45, Supplementary Table 1). Per-neuron bootstrap analysis also demonstrated that the number of two-basal dendrite ablated neurons that shifted their orientation preference post -ablation was significantly different compared to controls (two-basal: 7/17 neurons significantly shifted, control: 2/35 significantly shifted, p = 0.0015, χ2 = 10.1, chi-square test), while this was not true for apical or one-basal dendrite ablations, which were not significantly different from controls (apical: 3/18 significantly shifted, p = 0.19, χ2 = 1.67; one-basal: 3/13 significantly shifted, p = 0.08, χ2 = 3.1, chi-square test, Supplementary Table 1). Tuning-width and OSI differences did not reach significance (Fig. 3g–h, tuning width: p = 0.19; OSI: p = 0.36, ANOVA with Tukey multiple-comparisons test; See Supplementary Fig. 8 for histogram distribution of changes in tuning width and OSI; See figure legend for Fig. 3 for Kruskal–Wallis test results). This result further emphasizes that V1 L2/3 neurons’ orientation tuning is remarkably robust to loss of dendritic input (Supplementary Fig. 9 and Supplementary Movie 4).

### Modeling tuning robustness of apical dendrite ablation

Our finding that apical dendrite ablation has no effect on the orientation-tuning curve of L2/3 pyramidal neurons suggests a redundant contribution of the apical compared to the basal tree in orientation encoding. Otherwise, ablation should cause a change in neuronal orientation preference. To explore the possible input structures producing this result, we simulated a morphologically detailed L2/3 pyramidal neuron of mouse V1, with validated active and passive properties6,20 as well as literature-based synaptic density and single-synapse orientation-tuning properties (Fig. 4a–c and Supplementary Fig. 10, see Supplementary Data for details of the model neuron)4,19,21. Three main parameters were varied: standard deviation (σapical) of the distribution of single-synapse orientation preferences (prefsyn) in the apical dendritic-tree, standard deviation (σbasal) of the distribution of prefsyn in the basal dendritic-tree, and the difference in the mean orientation preference of these two distributions (∆(µbasal, µapical) = |µbasal–µapical|) (Fig. 4a, Supplementary Fig. 10). We found that to generate tuning curves comparable to experiments (OSI ≥ 0.2, tuning-width ≤ 80°), (σbasal, σapical) cannot be simultaneously larger than 45° (Supplementary Fig. 11a-e). Interestingly, for all Δ(µbasal, µapical) ranging from 0° to 90°, the model neuron’s orientation preference was consistently biased towards that of the basal tree (Fig. 4d), indicating the relative dominance of the basal dendrites.

To further constrain the input parameter space, we modeled the effects of apical dendrite ablation on preferred orientation, OSI and tuning-width (Fig. 4e, Supplementary Fig. 11f). We set σapical at 30° as approximated from prior measurements of single-synapse orientation preference distribution on apical tuft dendrites (Fig. 4e, Supplementary Fig. 11f)11. Figure 4e shows example tuning curves for σapical = σbasal = 30° with varying Δ(µbasal, µapical). For large Δ(µbasal, µapical) (>40°, Fig. 4e right, Supplementary Fig. 11f upper half of heat-maps), the apical dendrite input functions as noise, broadening the orientation-tuning curve, and therefore its removal leads to shifts in orientation preference, narrower tuning-width and higher OSI, contrary to experiment. For intermediate ∆(µbasal, µapical) (20°− 40°) and σbasal ≤ σapical (30°), the basal dendrite dominates, and apical dendrite ablation has no effect (Supplementary Fig. 11f), consistent with experiment. For σbasal > σapical (30°), the apical dendrite input sharpens orientation-tuning, and therefore ablation of the apical dendrite leads to wider tuning width and lower OSI (Fig. 5f right two panels, Supplementary Fig. 11f). For small Δ(µbasal, µapical) (0–10°, Fig. 4e left panel, Supplementary Fig. 11f lower half of heat-maps), the model neuron’s preferred orientation does not change (i.e., shift in orientation preference <10°) following ablation as long as σbasal is <60°.

### Basal dendrite ablation: the shift vs. the drift hypothesis

There are at least two possible explanations for the basal dendrite ablation results in Fig. 3. According to the shift hypothesis, the change in orientation preference following two-basal-dendrite ablation is due to a shift toward apical tree inputs assuming Δ(µbasal, µapical) >0°. Computational analysis, removing basal dendrites from model neurons with different Δ and σ (Supplementary Fig. 11i, j), weighs against the shift hypothesis. Although it was possible to generate model neurons with shifted orientation preference following two-basal-dendrite ablation (i.e., for Δ(µbasal, µapical) = 30–50°, σbasal ≤ 30°, σapical = 15°, mode change = 10°, Supplementary Fig. 11j), in these neurons apical ablation also led to a significant shift in orientation preference (Supplementary Fig. 11h), contrary to experiment. There was no parameter set for which the two-basal dendrite ablation, but not apical dendrite ablation, caused a shift in orientation preference.

The drift hypothesis entails that different basal dendrites are tuned to different preferences: cutting a subset of them will thus cause a drift due to loss of tuned basal inputs. To test this hypothesis, we generated model neurons with differentially tuned basal dendrites. The mean of the distribution of prefsyn in each primary basal dendrite (µb) was set independently. µb was selected from a range of Δμ, where Δμ represents the maximum deviation of one basal dendrite from the mean orientation preference of the soma (here arbitrarily set to 0°, Fig. 5a). Δμ = 0° represents the null hypothesis, i.e., where all basal dendrites sample from the same salt-and-pepper arrangement of input orientation preferences centered at 0°. In this case, any post-ablation changes are due to sampling noise of the finite number of synapses and, as expected, basal dendrite ablation rarely causes changes in orientation preference (Fig. 5b, top row, Fig. 5d far left). The same was true for small disparity values (Δμ = 20°, Fig. 5b second row, Fig. 5d). Intermediate disparity values (Δμ = 30–40°) appear to be necessary to achieve a modest change (mode = 10°, mean = 8°, max = 30° for Δμ = 40°) in orientation preference after two-basal-dendrite cut, while maintaining stability of orientation preference following apical or one-basal-dendrite cut, in agreement with experiment (Fig. 5b, c). Disparities ≥50° failed to generate tuning-curves compatible to the experimental data (OSI ≤ 0.2 and/or tuning-width ≥ 80°). As expected, the neuron’s post-ablation changes in orientation preference strongly correlated with the magnitude of the change in mean orientation preference of the aggregates of the basal dendrites before and after two basal dendrite ablation ($$\widetilde {\Delta \upmu }_{{\mathrm{ablated}}}$$, Supplementary Fig. 12). Figure 5 was generated using a narrow distribution of prefsyn in basal dendrites (σb = σbasal = 15°). σb( = σbasal) of 30° (Supplementary Fig. 13) also induced a shift in orientation preference (max change = 20°, mean = 4.4°) but the fit to experimental data was worse than σb = 15°, indicating the need for sharply tuned input structures into basal dendrites.

In summary, these simulations suggest that to explain our experiment data, a) the difference in mean input tuning between apical and basal arbors, Δ(µbasal, µapical), must be small, b) the distribution of input tuning to the basal dendrites should be similar to or narrower than the distribution of synaptic input to the apical tree (σbasal ≤ σapical = 30°) and c) there needs to be a degree of heterogeneity of tuning (30°–40°) across basal dendrites.

## Discussion

We showed that orientation selectivity in L2/3 pyramidal neurons, as measured by orientation preference, tuning-width and OSI, is robust to a complete loss of the apical tuft. This suggests that inputs to the basal dendritic domains are sufficient to determine the neuron’s orientation selectivity7,22. This is surprising given that apical dendrites receive orientation-tuned input11, with at least four input sources accessible to apical dendrites that could potentially modulate orientation selectivity in V1: (1) feedback cortico-cortical projections from extrastriate cortex thought to refine orientation selectivity9,10,23,24, (2) direction-selective projections from the LGN12, (3) thalamo-cortical projections from lateral posterior thalamus13, and (4) dendrite-targeting inhibitory inputs25,26. Furthermore, several studies provided in vivo evidence of dendritic properties27 in V1 apical dendrites that have the capacity to contribute to the neuron’s orientation selectivity6,28, one arguing that apical tuft input functions to narrow the neuron’s tuning curve increasing selectivity6. However, these studies do not provide a causal connection between dendritic computations and the emergence of orientation selectivity at the soma. Our results suggest that with respect to orientation tuning apical dendritic tuft inputs are dominated by basal dendritic activity, at least under our experimental conditions, favoring a feedforward model for orientation selectivity29,30. Computational modeling confirmed that basal dendritic input is dominant over a wide range of synaptic input parameters. It is interesting to explore in the future what behavioral conditions31,32,33 might change the relative contribution of apical dendritic inputs to orientation encoding in area V1.

Second, we showed that V1 neurons also remained robust to removal of 2 out of 5–8 primary basal dendrites. Specifically, only a small, though significant, change in orientation preference occurred in about 40% of the neurons that underwent two basal-dendrite ablation. The fact that we did not observe a marked loss of selectivity in any of the ablated neurons (Fig. 3) argues strongly against the possibility that there is a dominant, “master”, dendrite17,34, i.e., the situation in which input from a single dendrite dominates a neuron’s properties. Based on our results, we estimate the probability of a master dendrite dominated neuronal output to be p ≈ 0.00009 for neurons with six basal dendrites (p ≈ (5/6)13 × (4/6)17 ≈ 0.00009 corresponding to 13 neurons with single and 17 neurons with double basal dendrite ablation), or p ≈ 0.0013 for neurons with eight basal dendrites (p ≈ (7/8)13 × (6/8)17 ≈ 0.0013).

Still, to explain the small but significant post-ablation change in orientation preference following two-basal dendrite ablation, simulations suggest that orientation tuning is somewhat heterogeneous across basal dendrites, with basal dendrite orientation preference spanning a range of ~30°–40° (Fig. 5). Such heterogeneity may arise from dendrite-specific forms of plasticity potentially mediated by spatially restricted biochemical signaling35,36,37 and/or dendritic spikes6,27,28,38,39,40.

We note that microdissection comes with some potential limitations: Damage to nearby neuronal processes due to ablation, though minimal, cannot be completely avoided; Astrocytes attracted to the ablation site post-ablation15 could impact glutamate levels and modify the activity of the target neuron; Ion influx during the instant opening of the membrane in response to ablation could potentially induce unwanted plasticity on the connections that remain. Nonetheless our immunohistochemical findings showed that the damage around the ablation site is minimal (Supplementary Fig. 3). Furthermore our results show that neurons are active and remain tuned post-ablation, arguing that these processes are less likely to play a significant role.

These results give new insights into the complex structure-function relationship of the pyramidal neuron, the fundamental computational subunit of the neocortex. In particular, the remarkable robustness of orientation preference under dendritic micro-dissection hints at the extraordinary ability of sensory cortical neurons to maintain functional selectivity following input loss. Our approach emphasizes the importance of applying causal manipulations to study the contribution of dendritic arbors to sensory encoding. Dendritic microdissection is a powerful method for probing causal relations between dendritic structure/function and somatic properties that can be applied to several key questions in systems neuroscience research.

## Methods and data

### Animals in experiments

All experimental protocols were approved by The Baylor College of Medicine (BCM) Institutional Review Board and Brigham and Women’s Hospital (BWH) Institution Animal Care and Use Committee. Male and female wild-type (C57BL/6) mice were purchased from the institutional vivarium in BCM and Jackson Labs and bred for experiments.

### Chronic window implantation and sparse labeling

Viral vectors were purchased from or edited/packaged by University of Pennsylvania Vector Core, Addgene and BCM Vector Core. For viral injections and chronic window implantation, 6–10 week old wild-type C57BL/6 mice (both male and female) were anaesthetized with isoflurane (1–1.5%). Baytril (5 mg kg−1), Carprofen (5 mg kg−1) and Dexamethasone (1.5 mg kg−1) were administered subcutaneously. The depth of anesthesia was assessed via monitoring breathing rates. A headpost was implanted on the skull and a 3 mm diameter craniotomy was made over the visual cortex of the left hemisphere. The craniotomy was centered 2.7 mm lateral to the midline and 1.5 mm posterior to the bregma. To sparsely label pyramidal neurons with GCaMP6s, a mix (~90 nl per site, up to three sites) of diluted CamKII-CRE (AAV5 or AAV1, diluted 40,000–120,000×) and flex‑GCaMP6s (AAV5, diluted up to 2×) (U Penn Vector Core, Addgene) or flex-mRuby-GCaMP6s (AAV8, BCM Vector Core) was injected slowly over 5 min per penetration using a Drummond Nanoject. Two to three penetrations ~0.5 mm apart on average were performed per craniotomy. This approach enabled (i) expression of GCaMP6s at sufficient levels in each cell to image the dendritic tree, while (ii) labeled neurons remained sparse. After the viral injection, a round coverslip was fitted to the craniotomy and sealed with vetbond and dental cement. Most chronic windows in our hands remain clear for 2–3 months. Visual stimulation and 2‑photon imaging were performed on week 3–4 following viral injection, at which time GCaMP6s expression was optimal.

### In vivo calcium imaging of sparsely labeled neurons

Three to four weeks after the injection, the GCaMP6s expressing mouse was sedated with Fentanyl (0.5 mg kg−1) and Dexmedetomidine (0.5 mg kg−1)41 for imaging experiments. A stable level of anesthesia was confirmed by stable breathing rate and lack of movement. The right eye of the mouse was aligned to the center of the monitor. Customized light shielding was attached to the headpost to stimulate the mouse’s right eye effectively without producing light artifacts in the two‑photon images. We imaged calcium activity of ablation candidate neurons and their neighbor control neurons in L2/3 of mouse primary visual cortex during visual presentation. Images were acquired at ~9 frames/s using an Ultima IV microscope in spiral scanning mode with a 20×, 0.95 NA, Olympus objective or 25 × 1.0 NA Nikon objective (5–30 mW laser power, 900 nm). GCaMP6s gives excellent signal to noise ratio, corresponds well to the underlying firing rate, and is well suited for measuring tuning functions. Orientation-tuning measurements and structure imaging were repeated before (day 0) and 1, 3, 5 days after ablation.

### In vivo dendrite ablation

Under two-photon scanning, fluorescent dendrites were clear and visible at low laser power (<20 mW, 910 nm). Dendritic arbors of several L2/3 neurons were imaged using a custom Ultima IV 2-photon microscope. First, we screened for clearly orientation-tuned neurons with a clearly defined primary apical dendrite (Primary apical bifurcation > 20 μm away from the soma, soma depth between 150 and 250 µm, average pre-ablation orientation selectivity index for all ablated neurons was 0.79 ± 0.18, max = 0.99, min = 0.38). Note that these criteria likely excludes the L2 neurons commonly studied, e.g., in refs. 3,4,7. Target apical dendrites were severed at least 15 μm away from the soma (Fig. 1c and Supplementray Movie 1, 2). Basal dendrites were severed at least 10 μm away from the soma (Fig. 3a, c and Supplementary Movie 4).

The ablation point was magnified 13–15 times with 1024 resolution at 910 nm, then ablated via repeated 200–400 ms point scans at 150–200 mW power, 800 nm wavelength. Approximate area impacted by single point scan is an ellipsoid of 0.4 μm (x,y) and 1.2 μm (z) diameter according to our point spread function (Supplementary Fig. 1a). Differences in ablation parameters depended on depth of ablation plane, overlying shadow casting vessels, and window clarity. As reported in previous studies (14–17), when a single point on a fluorescing dendrite received several focused laser pulses, the targeted dendrite formed a beads-on-a-string morphology immediately after the ablation and then degraded within a day (Supplementary Fig. 1). Successfully ablated dendritic segments become transiently brightly fluorescent as calcium enters the membrane, then recover its original brightness. Distal to the ablation point, dendritic segments displayed marked beading within ~1 h and degraded completely within a day. Dendritic segments proximal to the ablation point by about 5–10 μm survive indefinitely. Neurons that do not regain baseline fluorescence (~30% of ablated cells) disappear within 24 h. The remaining 70% of ablated neurons recover to baseline fluorescence levels within 2–3 h post-ablation. Of these, ~50% survive. This gives a net 30–40% survival rate for ablated neurons. For functionally mapped neurons, we found similar chance of survival rate both from apical (18 out of 39, from 26 mice) and basal dendrite ablation (13 out of 33 for one basal dendrite ablation and 17 out of 50 two basal dendrite ablation, from 34 mice), χ2 = 1.1, p = 0.29, chi-square test. Surviving neurons did not demonstrate significant regrowth or degradation at the ablation point (at most 3–5 µm regression toward the soma). The remaining dendritic arbor, as well as neighboring neuronal structures, maintained their original structure for at least 14 days following ablation (Fig. 1b, Supplementary Fig. 1b).

We assessed the possibility of injury to the neuron and/or neuropil in four ways. Evidence for minimal post-ablation damage includes: (1) ablated neurons that survived for a day lasted indefinitely after being stressed for a while (as documented by the calcium influx and increase in fluorescence, Supplementary Fig. 1d), (2) no visible change in spontaneous or visual-evoked calcium activity were observed in the days following ablation of either apical or basal dendrites (Figs. 13, Supplementary Figs. 4, 5), (3) no morphological changes were observed in non-ablated structures from the same neuron or processes of nearby non-ablated neurons, even 5 microns from the ablation site (Supplementary Fig. 1), and (4) immuno-labeled neuronal processes (anti-Tuj1) were minimally affected at the ablation site (Supplementary Fig. 3 and Supplementary Movie 3). Other labs using the same method have shown no effect of this ablation on spine morphology or turnover in remnant proximal segments of ablated dendrites (36), and FIB-SEM reconstructions of ablation sites found a ~5 micron lesion with no prominent glial scar 5 days post-ablation12. Taking all of the data together leads us to conclude, as others have (refs. 14,15), that the off-target damage mediated by 2-photon microdissection is minimal.

### Visual stimulation

Visual stimuli were generated with MATLAB (Mathworks Inc.) PsychToolBox and presented on a Dell monitor (77° × 55° of visual angle), at a fixed mean luminance (80 candela/m2), positioned 32 cm in front of the animal. Prior to orientation mapping the animal was adapted for at least 15 min to the mean luminance level. To measure orientation selectivity, grayscale square‑wave gratings (0.04 cycles/degree, 2 cycles/second) moving in one of 12 (30° steps) or 36 (10° steps) directions were presented in pseudorandom order over the full stimulation field. Stimulus presentation lasted 2 s and the inter‑stimulus interval (uniform illumination set at the mean intensity) lasted 3 s. Twenty to thirty repetitions per stimulus direction of motion were collected and analyzed.

### Data analysis

Calcium trace from each soma was selected and separated using custom MATLAB functions, including the ROI selecting function ‘roigui’ from T.W. Chen at Janelia Institute. Baseline fluorescence (F0) for calculating (F-F0)/F0 was the average of the 20% lowest values in a 20 sec (10 s pre, 10 s post) window around each frame. Visual-evoked calcium responses arose ~200 ms after the onset of the stimulus and peaked around the offset of the stimulus. If the s.e.m. (standard deviation of the mean) of peak responses (for the preferred orientation) overlapped with the s.e.m. of null orientation responses on Day 0 (before ablation), we excluded those neurons from analysis. We also excluded from ablation a few neurons that had multiple orientation preferences (their orientation tuning curve showed 2–3 peaks). Naturally, we also did not use such neurons as controls. The per-trial response was calculated as the average of a 1.5 s window centered at the peak of the mean response across all orientations. Tuning curves were calculated by averaging responses across repetitions of each stimulus orientation. From this tuning curve, orientation selectivity index was calculated with (Rpref−Rortho)/(Rpref + Rortho). All orientation-tuning curves were fitted with von Mises function using Circstat tool box42 after baseline subtraction. Preferred orientation and tuning width was calculated from the fitted tuning curve. Calculating preferred orientation and tuning width from the raw mean tuning curve generated similar results. To assess the effect of dendrite ablation on tuning curves, preferred orientation, tuning width and OSI values before ablation were subtracted from the corresponding values obtained 5 days after ablation. We used both t-test and Mann–Whitney U test for pairwise comparisons and ANOVA and Kruskal–Wallis test for multi-group comparisons to assess how tuning values shift in ablated versus control neurons. Where necessary, Tukey test for multiple comparisons was used to compare between groups (e.g., control vs two-basal dendrite ablation) following ANOVA or Kruskal–Wallis test. Absolute-value was used to assess the shift in preferred orientation.

### Bootstrap analysis of orientation tuning reliability

For the bootstrapping analysis in Supplementary Figs. 45, five single-trial responses were randomly sub-selected from the 20–50 trials acquired per orientation to generate simulated tuning curves. The preferred orientation was calculated from the von Mises-fit orientation tuning curve based on the sub-sampled data. This was repeated 1000 times to generate confidence intervals for the estimate of the preferred orientation pre- and post-ablation per neuron. Distributions of the orientation estimates were shifted to have the mean of the preferred orientation estimates before ablation at 0°. A neuron was said to show significant difference in preferred orientation (or in tuning width) pre- versus post- ablation, if 95% of the confidence intervals established by bootstrap did not overlap with each other. The frequency of detecting neurons with significant difference in preferred orientation pre- and post- ablation was assessed using the chi-square test (Supplementary Figs. 4, 5, Supplementary Table 1).

### Immunofluorescence labeling

Mice were killed by isoflurane overdose followed by cervical dislocation and decapitation. Following decapitation, brains were quickly removed and fixed in ice cold 4% PFA (pH 7.2, 1 h) followed by 20% sucrose immersion for 24 h and then 30% sucrose immersion for 24 h. Fixed brains were frozen in OCT medium and sectioned into 50μm coronal sections. For immunostaining, free floating sections were washed in 0.1 M PB and blocked in PBTGS (0.1 M PB with 0.3% Triton X-100 detergent and 5% goat serum) for 1 h. Sections were incubated in primary antibody over night at room temperature. The following antibodies were used: chicken anti-GFP (Abcam) and mouse anti-beta III Tubulin (Santa Cruz). Primary antibodies were then labeled using AlexaFluor-conjugated secondary antibodies (Invitrogen). Sections were visualized with a Zeiss Imager, Z1 fluorescent microscope with an Apotome attachment.

### Biophysical model

The morphologically detailed L2/3 V1 pyramidal neuron model (Fig. 4a) was implemented in the NEURON simulation environment43. Model neurons (available in ModelDB,(accession number 231185) 3) were constrained against experimental electrophysiological and anatomical data6,44,45,46,47,48,49 (see Supplementary information for detailed description). Synapses impinging in the apical and basal dendrites where either spontaneously activated or stimulus driven. Stimulus driven synapses were assigned with preferred orientations and randomly distributed along the dendrites4. When assigning orientation preference to synapses, we varied the standard deviation of the distributions (σbasal, σapical), the difference (Δ) of μapical and μbasal, as well as the μb of each individual basal dendrite (Δμ). Control condition was the one corresponding to σapical = σbasal = 30°, μapical = μbasal = 0°, Δμ = 0°, under which the OSI was 0.78 and the tuning width 43° (Fig. 4b, c). Ablation was simulated by removing the respective dendritic compartments and adjusting excitatory synaptic weights for the change in neuronal excitability. For each condition, we calculated the preferred orientation, tuning width and OSI of the resulting somatic tuning curves.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

Raw datasets associated with Figs. 2 and 3 will be available from the corresponding authors on request. The biophysical model used during the current study is available in the ModelDB repository with the accession number 231185.

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## Acknowledgements

S.S. and J.P. were supported by NEI R01 grant EY-024019 and NINDS R21 grant NS088457. A.P. and P.P. were supported by the European Research Council Starting Grant dEMORY (GA 311435). We thank Xiaolong Jiang for providing high quality Neurolucida reconstructions of L2/3 pyramidal neurons of mouse V1 visual cortex for the biophysical model study. We thank Matt Rasband for providing equipment and reagents for immunostaining experiments. We thank Sangkyun Lee for MATLAB expertise.

## Author information

Authors

### Contributions

J.P. and S.M.S. designed the study, A.P. and P.P. designed the modeling part of the study, J.P. and M.A.M. performed the experiments, J.P. analyzed the data, A.P. performed the simulations, J.P. and R.T.A. wrote the manuscript. J.P., R.T.A., A.P., P.P. and S.M.S. edited the manuscript together.

### Corresponding authors

Correspondence to Jiyoung Park, Panayiota Poirazi or Stelios M. Smirnakis.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Jesper Sjöström, Ikuko Smith and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Park, J., Papoutsi, A., Ash, R.T. et al. Contribution of apical and basal dendrites to orientation encoding in mouse V1 L2/3 pyramidal neurons. Nat Commun 10, 5372 (2019). https://doi.org/10.1038/s41467-019-13029-0

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• DOI: https://doi.org/10.1038/s41467-019-13029-0

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Cognitive Neurodynamics (2021)


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I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m \times 1$ vector. The distance metric is:d(x_1, If a topological space X has a countable basis. Then if we have an open cover of X, can this cover be refined to a countable one February 10 1 If a topological space X has a countable basis. Then if we have an open cover of X, can this cover be refined to a countable one?Yes. Fix \cal B to be a countable basis. Let \cal U be our open cover. For U\in\cal U define V_U=\{V\in\mathca What does Equational Theorem Prover do February 10 1 http://www.cs.unm.edu/~mccune/eqp/What does EQP do? 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To be more specific, I have a dataset of m x n volumes of size 1024 x 1024 x 100, and each volume overlaps for 100 pixels on the xy plane with the adjacent volumes Minimizing the following objective function with matrices February 10 2 Suppose A and B are known matrices, and we are to find matrix X that minimizes the following function,\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$Taking the relevant derivative w.r.t X gives us,$$ X +(X^TAX-B)(A+A^T)X$$I couldn't reduce t If I have an open cover, can I choose a countable sub-cover such that the union of the open sets strictly increasing February 10 2 If I have an open cover, can I choose a countable sub-cover such that the union of the open sets strictly increasing?Actually I want to use this to prove every noetherian space is compact. Thanks!In general you can't: Consider X = [0, 1] (which is Distance between theorems February 10 1 In automated proving one can define the best proof of a theorem as the one which minimizes the length of the proof. Given a set of known statements one could define the difficulty of a theorem as the minimum length among all its proofs deduced from k Can covering be done on two elements February 10 1 The covering rule is:$$B \bullet (B+C) = B$$and$$B+(B \bullet C)=B$$So does it follow from this rule that:$$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet (A\bullet\bar{C}+D\bullet\bar{F}) = B? $$Or does the covering rule minimize this objective function February 10 I have a function to minimize and I don't understand how I should proceed. The function is coming from a publication.Background: In a 2D image, P_1 and P_2 represents 2 patches of colors (RGB) which overlap one on the other (these 2 patches have The distance covered by the Ant is: February 10 A set of concentric circles of integer radii1,2,...N is shown is the fig above. An ant starts at point A_1, goes round the first circle ,returns to A_1 ,moves to A_2, goes round the ... To test convergence of improper integral  \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\, \mathrm dx February 10 2 I have to test convergence of improper integral$$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$I write as \log(x) \leq x . So x\log(x) \leq x^2. So  \frac{x\log(x)}{(1+x^2)^2} \leq \frac{x^2}{(1+x^2)^2} . Now using comparison test Prove that for any integer n, if b^2 divides n, then b divides n. February 10 2 Prove that for any integer n, if b^2 divides n, then b divides n. Trying to figure out this proof. The proof I'm looking at is written as n = any integer, if 25|n \implies 5|n. I've been trying to figure this for days and have been runn y=e^x\sin x; find all points where slope of tangent line equals 0 February 10 1 I have the derivative already. Using the product rule, I got e^{2}\sin x+e^{2}\cos x. I can't figure out how to find all the points without graphing it.We need$$\dfrac{dy}{dx}=0\implies e^x(\sin x+\cos x)=0$$As for finite real x,e^x>0$$$\implies\ What does a variable superscript above a set mean February 10 I'm not entirely sure I've worded this correctly. An example of what I mean is...$$U = \{0,1\}^n$$What is the meaning of the superscript? Convergence of improper integral$\int_{2}^{\infty} \frac{1}{log(t)}dt$February 10 Convergence of improper integral$\int_{2}^{\infty} \frac{1}{log(t)}dt$How do i start? Is there any relation between a group being unimodular and having equivalent uniform structures February 10 1 Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and right Haar measures on it coincide.For locally com Why do characters on a subgroup extend to the whole group February 10 1 As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally compact abelian group$G,$a closed subgroup$H,$a Continuous functions on compact group and uniformity February 10 If$G$is a compact abelian group and$f\in C(G)$. Then$\forall \epsilon >0$,there exists an open neighbourhood$U$of$0\in G$, such that$\forall g\in G , \forall u_1,u_2\in U$, we have$|f(gu_1)-f(gu_2)|<\epsilon$.Is this statement right? how to For what values of$a, n$the number$2^a\cdot 3^n+1$is prime February 10 Respected All. Please forgive me as I am unable to understand if it is an off topic. I am stuck in a situation. Change the title if you find inappropriateWhat I am willing to know is that if there is any website where I can get complete information o Fraction walk through on hold February 10 1 $$0.824 = \frac{n/20\cdot 1}{n/20\cdot 1+(1-n/20)\cdot 0.5}$$ Please answer this question with step by step. 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Types in Nemo

On this page we discuss the type hierarchy in Nemo and a concept known as parents. These details are quite technical and should be skipped or skimmed by new users of Julia/Nemo. Types are almost never dealt with directly when scripting Nemo to do mathematical computations.

In contrast, Nemo developers will certainly want to know how we model mathematical objects and the rings, fields, groups, etc. that they belong to in Nemo.

Introduction

Julia provides two levels of types that we make use of

• abstract types
• concrete types

Concrete types are just like the usual types everyone is familiar with from C or C++.

Abstract types can be thought of as collections of types. They are used when writing generic functions that should work for any type in the given collection.

To write a generic function that accepts any type in a given collection of types, we first create an abstract type. Then we create the individual concrete types that belong to that abstract type. A generic function can then be constructed with a type parameter, T say, similar to a template parameter in C++. The main difference is that we can specify which abstract type our type parameter T must belong to.

We use the symbol <: in Julia to determine that a given type belongs to a given abstract type. For example the built-in Julia type Int64 for 64 bit machine integers belongs to the Julia abstract type Integer. Thus Int <: Integer returns true.

Here is some Julia code illustrating this with a more complex example. We create an abstract type called Shape and two user defined concrete types square and circle belonging to Shape. We then show how to write methods that accept each of the concrete types and then show how to write a generic function for any type T belonging to the abstract type Shape.

Note that in the type definitions of square and circle we specify that those types belong to the abstract type Shape using the <: operator.

abstract Shape

type square <: Shape
width::Int
border_thickness::Int
end

type circle <: Shape
centre::Tuple{Int, Int}
border_thickness::Int
end

function area(s::square)
return s.width^2
end

function area(s::circle)
end

function border_thickness{T <: Shape}(s::T)
return s.border_thickness
end

s = square(3, 1)
c = circle((3, 4), 2, 2)

area(s)
area(c)
border_thickness(s)
border_thickness(c)

The abstract type hierarchy in Nemo

Abstract types in Julia can also belong to one another in a hierarchy. For example, the Nemo.Field abstract type belongs to the Nemo.Ring abstract type. An object representing a field in Nemo has type belonging to Nemo.Field. But because we define the inclusion Nemo.Field <: Nemo.Ring in Nemo, the type of such an object also automatically belongs to Nemo.Ring. This means that any generic function in Nemo which is designed to work with ring objects will certainly also work with field objects.

In Nemo we also distinguish between the elements of a field, say, and the field itself, and similarly for groups and rings and all other kinds of domains in Nemo. For example, we have an object of type GenPolyRing to model a generic polynomial ring, and elements of that polynomial ring would have type GenPoly.

In order to model this distinction between elements and the domains they belong to, Nemo has two main branches in its abstract type hierarchy, as shown in the following diagram. One branch consists of the abstract types for the domains available in Nemo and the other branch is for the abstract types for elements of those domains.

All objects in Nemo, whether they represent rings, fields, groups, sets, etc. on the one hand, or ring elements, field elements, etc. on the other hand, have concrete types that belong to one of the abstract types shown above.

Why types aren't enough

Naively, one may expect that rings in Nemo can be modeled as types and their elements as objects with the given type. But there are various reasons why this is not a good model.

As an example, consider the ring $R = \mathbb{Z}/n\mathbb{Z}$ for a multiprecision integer $n$. If we were to model the ring $R$ as a type, then the type would somehow need to contain the modulus $n$. This is not possible in Julia, and in fact it is not desirable either.

Julia dispatches on type, and each time we call a generic function with different types, a new version of the function is compiled at runtime for performance. But this would be a disaster if we were writing a multimodular algorithm, say. In such an algorithm many rings $\mathbb{Z}/n\mathbb{Z}$ would be needed and every function we use would be recompiled over and over for each different $n$. This would result in a huge delay as the compiler is invoked many times.

For this reason, the modulus $n$ needs to be attached to the elements of the ring, not to type associated with those elements.

But now we have a problem. How do we create new elements of the ring $\mathbb{Z}/n\mathbb{Z}$ given only the type? Suppose all rings $\mathbb{Z}/n\mathbb{Z}$ were represented by the same type Zmod say. How would we create $a = 3 \pmod{7}$? We could not write a = Zmod(3) since the modulus $7$ is not contained in the type Zmod.

We could of course use the notation a = Zmod(3, 7), but this would make implementation of generic algorithms very difficult, as they would need to distinguish the case where constructors take a single argument, such as a = ZZ(7) and cases where they take a modulus, such as a = Zmod(3, 7).

The way we get around this in Nemo is to have special (singleton) objects that act like types, but are really just ordinary Julia objects. These objects, called parent objects can contain extra information, such as the modulus $n$.

In order to create new elements of $\mathbb{Z}/n\mathbb{Z}$ as above, we overload the call operator for the parent object, making it callable. Making a parent object callable is exactly analogous to writing a constructor for a type.

In the following Nemo example, we create the parent object R corresponding to the ring $\mathbb{Z}/7\mathbb{Z}$. We then create a new element a of this ring by calling the parent object R, just as though R were a type with a constructor accepting an Int parameter.

R = ResidueRing(ZZ, 7)
a = R(3)

This example creates the element $a = 3 \pmod{7}$.

The important point is that unlike a type, a parent object such as R can contain additional information that a type cannot contain, such as the modulus $7$ of the ring in this example, or context objects required by C libraries in other examples.

More complex example of parent objects

Here is some Julia/Nemo code which constructs a polynomial ring over the integers, a polynomial in that ring and then does some introspection to illustrate the various relations between the objects and types.

julia> using Nemo

julia> R, x = ZZ["x"]
(Univariate Polynomial Ring in x over Integer Ring,x)

julia> f = x^2 + 3x + 1
x^2+3*x+1

julia> typeof(R)
Nemo.FmpzPolyRing

julia> typeof(f)
Nemo.fmpz_poly

julia> parent(f)
Univariate Polynomial Ring in x over Integer Ring

julia> typeof(R) <: PolyRing
true

julia> typeof(f) <: PolyElem
true

julia> parent(f) == R
true

Concrete types in Nemo

Finally we come to all the concrete types in Nemo.

These are of two main kinds: those for generic constructions (e.g. generic polynomials over an arbitrary ring) and those for specific implementations, usually provided by a C library (e.g. polynomials over the integers, provided by Flint).

Below we give the type of each kind of element available in Nemo. In parentheses we list the types of their corresponding parent objects. Note that these are the types of the element objects and parent objects respectively, not the abstract types to which these types belong, which the reader can easily guess.

For example, fmpz belongs to the abstract type RingElem and FlintIntegerRing belongs to Ring. Similarly Poly{T} belongs to PolyElem whereas PolynomialRing{T} belongs to PolyRing. We also have that fmpz_poly belongs to PolyElem and FmpzPolyRing belongs to PolyRing, and so on.

All the generic types are parameterised by a type T which is the type of the elements of the ring they are defined over.

• Generic

• GenPoly{T} (GenPolyRing{T})
• GenRelSeries{T} (GenRelSeriesRing{T})
• GenRes{T} (GenResRing{T})
• GenFrac{T} (GenFracField{T})
• GenMat{T} (GenMatSpace{T})
• Flint

• fmpz (FlintIntegerRing)

• fmpq (FlintRationalField)
• fq_nmod (FqNmodFiniteField)
• fq (FqFiniteField)
• padic (FlintPadicField)
• fmpz_poly (FmpzPolyRing)
• fmpq_poly (FmpqPolyRing)
• nmod_poly (NmodPolyRing)
• fmpz_mod_poly (FmpzModPolyRing)
• fq_poly (FqPolyRing)
• fq_nmod_poly (FqNmodPolyRing)
• fmpz_rel_series (FmpzRelSeriesRing)
• fmpq_rel_series (FmpqRelSeriesRing)
• fmpz_mod_rel_series (FmpzModRelSeriesRing)
• fq_nmod_rel_series (FqNmodRelSeriesRing)
• fq_rel_series (FqRelSeriesRing)
• fmpz_mat (FmpzMatSpace)
• nmod_mat (NmodMatSpace)
• perm (PermGroup)
• Antic

• nf_elem (AnticNumberField)

• Arb

• arb (ArbField)

• acb (AcbField)


text:
# Properties

 Label 588.4.i.g Level $588$ Weight $4$ Character orbit 588.i Analytic conductor $34.693$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} -9 \zeta_{6} q^{9} + ( 20 - 20 \zeta_{6} ) q^{11} -4 q^{13} -12 q^{15} + ( -24 + 24 \zeta_{6} ) q^{17} -44 \zeta_{6} q^{19} -72 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} -27 q^{27} -38 q^{29} + ( -184 + 184 \zeta_{6} ) q^{31} -60 \zeta_{6} q^{33} + 30 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{39} -216 q^{41} -164 q^{43} + ( -36 + 36 \zeta_{6} ) q^{45} -520 \zeta_{6} q^{47} + 72 \zeta_{6} q^{51} + ( 146 - 146 \zeta_{6} ) q^{53} -80 q^{55} -132 q^{57} + ( -460 + 460 \zeta_{6} ) q^{59} -628 \zeta_{6} q^{61} + 16 \zeta_{6} q^{65} + ( -556 + 556 \zeta_{6} ) q^{67} -216 q^{69} + 592 q^{71} + ( -1024 + 1024 \zeta_{6} ) q^{73} -327 \zeta_{6} q^{75} + 104 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -324 q^{83} + 96 q^{85} + ( -114 + 114 \zeta_{6} ) q^{87} -896 \zeta_{6} q^{89} + 552 \zeta_{6} q^{93} + ( -176 + 176 \zeta_{6} ) q^{95} -920 q^{97} -180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 4q^{5} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 4q^{5} - 9q^{9} + 20q^{11} - 8q^{13} - 24q^{15} - 24q^{17} - 44q^{19} - 72q^{23} + 109q^{25} - 54q^{27} - 76q^{29} - 184q^{31} - 60q^{33} + 30q^{37} - 12q^{39} - 432q^{41} - 328q^{43} - 36q^{45} - 520q^{47} + 72q^{51} + 146q^{53} - 160q^{55} - 264q^{57} - 460q^{59} - 628q^{61} + 16q^{65} - 556q^{67} - 432q^{69} + 1184q^{71} - 1024q^{73} - 327q^{75} + 104q^{79} - 81q^{81} - 648q^{83} + 192q^{85} - 114q^{87} - 896q^{89} + 552q^{93} - 176q^{95} - 1840q^{97} - 360q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 2.59808i 0 −2.00000 3.46410i 0 0 0 −4.50000 7.79423i 0
373.1 0 1.50000 + 2.59808i 0 −2.00000 + 3.46410i 0 0 0 −4.50000 + 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.i.g 2
3.b odd 2 1 1764.4.k.j 2
7.b odd 2 1 588.4.i.b 2
7.c even 3 1 588.4.a.b 1
7.c even 3 1 inner 588.4.i.g 2
7.d odd 6 1 588.4.a.e yes 1
7.d odd 6 1 588.4.i.b 2
21.c even 2 1 1764.4.k.g 2
21.g even 6 1 1764.4.a.i 1
21.g even 6 1 1764.4.k.g 2
21.h odd 6 1 1764.4.a.d 1
21.h odd 6 1 1764.4.k.j 2
28.f even 6 1 2352.4.a.g 1
28.g odd 6 1 2352.4.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 7.c even 3 1
588.4.a.e yes 1 7.d odd 6 1
588.4.i.b 2 7.b odd 2 1
588.4.i.b 2 7.d odd 6 1
588.4.i.g 2 1.a even 1 1 trivial
588.4.i.g 2 7.c even 3 1 inner
1764.4.a.d 1 21.h odd 6 1
1764.4.a.i 1 21.g even 6 1
1764.4.k.g 2 21.c even 2 1
1764.4.k.g 2 21.g even 6 1
1764.4.k.j 2 3.b odd 2 1
1764.4.k.j 2 21.h odd 6 1
2352.4.a.g 1 28.f even 6 1
2352.4.a.be 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} + 16$$ acting on $$S_{4}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$400 - 20 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$576 + 24 T + T^{2}$$
$19$ $$1936 + 44 T + T^{2}$$
$23$ $$5184 + 72 T + T^{2}$$
$29$ $$( 38 + T )^{2}$$
$31$ $$33856 + 184 T + T^{2}$$
$37$ $$900 - 30 T + T^{2}$$
$41$ $$( 216 + T )^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$270400 + 520 T + T^{2}$$
$53$ $$21316 - 146 T + T^{2}$$
$59$ $$211600 + 460 T + T^{2}$$
$61$ $$394384 + 628 T + T^{2}$$
$67$ $$309136 + 556 T + T^{2}$$
$71$ $$( -592 + T )^{2}$$
$73$ $$1048576 + 1024 T + T^{2}$$
$79$ $$10816 - 104 T + T^{2}$$
$83$ $$( 324 + T )^{2}$$
$89$ $$802816 + 896 T + T^{2}$$
$97$ $$( 920 + T )^{2}$$


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Lua states confusion

Old topic!

Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.

2 replies to this topic

#1vicx  Members

Posted 25 March 2012 - 05:48 PM

Hello there.

After digging deeply in previous subjects of that kind I came up to great confusion regarding lua states.
There is certain scheme of my scripting vision, which I derived from lpmuds and which I want to translate to Lua, more or less.
I managed to achieve similar result while using lua_state per entity basis, but what I've found out is a great amount of memory being absorbed when reaching high entity count. That have should be clear to me if I put my built-in functions to every state, forcing it to grow with huge impact!
After few tests I came up to the point that I need one state for all the entities, but a whole bunch of problems appeared due to the shape of my scripting vision:P

Maybe someone could help me solving this issue. Here's some overview to what I achieved with multiple states.
Some pseudo code of LionClassLib and lion.lua

function LionClassLib:CreateLion()
setting default values
end;

function LionClassLib:LionCallback()
print("ROAR");
end;

-- This function is called from c api -> every time a new entity is created
function create_object()
this():CreateLion();
-- this() -> returns current object table (defined through binding objects to uids in inherit.lua, apply only when
trying to fight with one lua state, in case of many states this() will lead to base object, so this part works anyway)
end;

function do_anything()
this():LionCallback();
end;

-----> lion.lua <-----
require "inherit"
require "LionClassLib"

LocalObject = inheritFrom(LionClassLib); -> sets metatables and returns table/object with class def and standard behavior

function LocalObject:CreateLion()
LionClassLib:CreateLion(); -- Setting the parent's constructor
-- additional local stuff if needed
end;

While being on many states the great advantage was that I could call another object creation through C API and it didn't
destroy globals of the current one. Now I can't, but this part I can omit.
The idea behind many states was that I could lual_dofile every script, pass initial callbacks and it ran ok. Then I was able to call events from api the same way.
But at this point I'll have the following problem:

1. Clone Entity 1 with lion.lua
2. Load script and call create_object (this():CreateLion)
3. Clone Entity 2 with monkey.lua
4. Load script and call create_object (this():CreateMonkey)
5. It's obvious it will be ok until next point:P
6. Clone Entity 3 with dofile lion.lua! CRASH: luadofile won't compile the same file, the last global create_object comes from monkey.lua where there is no function CreateLion.

So, I hope the idea is quite clear. I was trying to play with environments with setfenv/getfenv etc but I'm not sure it's the right way. I can try with lua_state per script, then it might work, but I'm not quite sure if something wouldn't suddenly blow up.

If someone could direct me in proper direction I'd be pleased.

#2yckx  Prime Members

Posted 25 March 2012 - 11:48 PM

You should only call luaL_dofile once for each file. It looks like you're using the same function names in each file and then overwriting them each time you load a new lua file. I'd suggest putting them into tables, so in lion.lua you'd create a table called lion, and put all the functions into that table. Same with all the others. This avoids name clashes. You'd just have to switch to the appropriate table before making your function calls.

#3vicx  Members

Posted 28 March 2012 - 12:37 PM

Thanks a lot! After implementation and some little fixes it works great. The only thing I left (and it seems working) is dofile every time a script is initialized. This is due to the fact there is an object creation at inheritFrom call. It registers every new object behind the scene so everything is set up properly when managing further calls from api, and even I managed to init scripts from within running scripts with unlimited level of depth:P How it will bounce off the performance, don't know, but thanks for a good advice.

Problem solved. Might be closed.

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