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The Drosophila compound eye is formed by selective recruitment of undifferentiated cells into clusters called ommatidia during late larval and early pupal development. Ommatidia at the edge of the eye, which often lack the full complement of photoreceptors and support cells, undergo apoptosis during mid-pupation. We have found that this cell death is triggered by the secreted glycoprotein Wingless, which activates its own expression in peripheral ommatidia via a positive feedback loop. Wingless signaling elevates the expression of the pro-apoptotic factors head involution defective,grim and reaper, which are required for ommatidial elimination. We estimate that approximately 6-8% of the total photoreceptor pool in each eye is removed by this mechanism. In addition, we show that the retinal apoptosis previously reported in apc1 mutants occurs at the same time as the peripheral ommatidial cell death and also depends on head involution defective, grim and reaper. We consider the implications of these findings for eye development and function in Drosophila and other organisms.

The appropriate removal of excess cells is an important aspect of patterning and morphogenesis during development. Cell removal is achieved by localized induction of programmed cell death (PCD), or apoptosis. Classic examples include the sculpting of digits in the vertebrate limb bud(Zuzarte-Luis and Hurle, 2002)and the removal of excess neurons in the developing central nervous system(Oppenheim, 1991). Identifying which signaling pathways influence this life-or-death choice and the mechanisms by which they interact with the cellular apoptotic machinery remains a major goal of developmental biology.

Elimination of excess cells is essential for the formation of the compound eye of Drosophila. The fly eye consists of approximately 750 repeated units called ommatidia, which consist of eight photoreceptor neurons (R cells), four cone cells (which secrete the lens) and two primary pigment cells(Wolff and Ready, 1993). Differentiation of these cells is triggered by the passage of the morphogenetic furrow (MF). During late larval/early pupal stages, as R cells differentiate in regularly spaced clusters behind the MF, undifferentiated cells furthest from these clusters undergo PCD(Baker and Yu, 2001). One day after the start of pupation, after all the cells of each ommatidium have been specified and organized into an ordered array, an additional round of apoptosis reduces the number of interommatidial cells. The survivors of this trimming become secondary and tertiary pigment cells(Brachmann and Cagan, 2003). In both these cases, the epidermal growth factor receptor (EGFR)/Ras signaling pathway is thought to provide a survival signal(Baker and Yu, 2001; Miller and Cagan, 1998; Yu et al., 2002). The primary target of this survival signal is the proapoptotic factor Head involution defective (Hid) (Yu et al.,2002). Ras signaling inhibits Hid activity post-translationally(Bergmann et al., 1998) and downregulates hid transcript levels(Kurada and White, 1998).

Hid belongs to a group of proteins that promote PCD by inhibiting the activity and/or stability of inhibitor of apoptosis proteins (IAPs)(Martin, 2002; Shi, 2002). These IAP inhibitors share limited sequence similarity, most notably a short RHG domain at their N-termini that binds to IAPs(Shi, 2002). In flies, the founding members of this group were hid, reaper (rpr) and grim (Chen et al.,1996; Grether et al.,1995; White et al.,1994). More recently, an additional family member called sickle was identified in flies(Christich et al., 2002; Srinivasula et al., 2002; Wing et al., 2002), and SMAC/Diablo (Du et al., 2000; Verhagen et al., 2000) and HtrA2/Omi (Suzuki et al.,2001) have been identified in mammals.

In addition to the apoptotic events mentioned above, there is a later occurrence of PCD in the fly eye that is less well understood. A ring of apoptotic cells is found at the periphery of the eye at mid-pupation(Wolff and Ready, 1991). These dying cells at the edge of the ommatidial field contain photoreceptors, cone cells and primary pigment cells (Hay et al., 1994; Wolff and Ready,1991).

We have found that Wg expression coincides with the mid-pupal ring of apoptosis. Wg signaling at the eye's perimeter activates Wg expression in peripheral ommatidia. Wg signaling also induces the expression of hid,rpr and grim, and these genes are required for the elimination of approximately 80-100 perimeter ommatidia/eye. The ommatidia destined for PCD are often incomplete, probably resulting from an insufficient number of precursor cells at the edge of the eye for recruitment into clusters.

Under normal conditions, Wg signaling is tightly correlated with the stability of cytosolic Armadillo (Arm), the fly β-catenin homolog(Peifer et al., 1994; van Leeuwen et al., 1994). apc1 is a fly homolog of the adenomatous polyposis coli (APC) tumor suppressor gene, which encodes a negative regulator of Arm/β-catenin stability (Polakis, 2000). Mutation of apc1 results in arm-dependent mid-pupal apoptosis of all photoreceptors (Ahmed et al., 1998). We found that this PCD occurred at the same time as the peripheral ommatidial cell death and that it is also dependent on hid,rpr and grim. Mutations in the APC gene in humans cause retinal lesions in which photoreceptor neurons degenerate(Traboulsi et al., 1990). This and other results will be discussed in regard to the possible role Wnt induction of apoptosis may play in vertebrate eye development and disease.

### Drosophila genetics

Fly stocks were maintained on standard medium at 25°C unless otherwise indicated. Pupal developmental ages are expressed as hours after pupal formation (APF) with white pre-pupae defined as 0 hours APF. The P[GMR-Gal4], P[UAS-wg] stock was described previously(Cadigan et al., 2002). The UAS-lacZ, UAS-p35, UAS-wgts and wgIL lines were from the Bloomington Stock Center (Indiana University) and UAS-rasV12 was from Mike Simon. hidP, hidX14, Df(3L)H99,Df(3L)X25 and Df(3L)XR38 were from Kristin White. The apc1S76 and apc1Q8 alleles(Ahmed et al., 1998) were from Yashi Ahmed and Eric Wieschaus. The fz9131, fz2M2-8 P[FRT, w+]2Achromosome was generously provided by Rob Howes and Roel Nusse. The P[FRT,hsneo]82B, pygo10 chromosome was from David Parker. Clones of fz, fz2 and pygo were generated in the eye using Eye-FlpT12(Newsome et al., 2000),provided by Barry Dickson. Clones were marked by armlacZ as previously described (Cadigan et al.,2002; Parker et al.,2002) or with Ubi-GFPnls(Davis et al., 1995).

### Histology and scanning electron microscopy

Immunostainings were performed as described(Cadigan and Nusse, 1996). Rat anti-Elav (1:100) and mouse anti-Cut (1:75) were from the University of Iowa Hybridoma Bank. Affinity purified mouse anti-Pros (1:5000) was from Richard Carthew. Affinity purified rabbit anti-Wg (1:50) was prepared as described(Bhanot et al., 1996). Guinea pig anti-Sens (1:500) was kindly provided by Hugo Bellen and rabbit anti-Bar(1:50) by K. Saigo. Mouse anti-lacZ was purchased from Sigma (St Louis, MO). Cy3- and Cy5-conjugated secondary antibodies were from Jackson Immunochemicals(West Grove, PA) and Alexa Flour 488-conjugated secondaries were from Molecular Probes (Eugene, OR). TUNEL staining was performed as described(Wang et al., 1999) using the Cell Death Detection Kit (Fluorescein) from Roche Diagnostics (Indianapolis,IN). All fluorescent pictures were obtained with a Zeiss Axiophot coupled to a Zeiss LSM510 confocal apparatus.

In-situ hybridizations were performed as previously described(Cadigan et al., 1998). Probes were made by PCR on genomic DNA with the following oligos:5′CTTGCAATTTCACTGGCGGCGATGTGT3′ and 5′GTAATACGACTCACTATAGGGCGGCCAAGTGAAGCTCTGTGGTTTCTTC3′ for hid, 5′CGTTCGTTTTCCCGCCAAATGAGTCAG3′ and 5′GTAATACGACTCACTATAGGGCGCTCGTTCCTCCTCATGTGTCCATAC3′ for grim, 5′GACACCAGAACAAAGTGAACGAACTCG3′ and 5′GTAATACGACTCACTATAGGGCGTGTTGTGGCTCTGTGTCCTTGACTGCA3′ for rpr. Oligos on the reverse strands each has a T7 RNA polymerase site. Antisense dioxygenin probes were synthesized using the Ambion T7 Megascript kit with the Roche DIG RNA labeling mix.

Samples for scanning electron microscopy (SEM) were prepared as previously described (Cadigan and Nusse,1996). All images were processed as Adobe Photoshop files.

### Wg activates apoptosis of ommatidia at the periphery of the eye

Fig. 1A shows that Wg was expressed in a ring around the midpupal eye, i.e. 42 hours after pupal formation (APF), coincident with the peripheral apoptosis previously reported(Fig. 1B)(Hay et al., 1994; Wolff and Ready, 1991). The highest level of Wg expression was found at the very edge of the eye, outside the field of Elav-positive R cells (Fig. 1E-H). Apoptotic nuclei were found in a region of lower Wg expression (Fig. 1E,F) and some of them were still slightly positive for Elav(Fig. 1G), indicating dying R cells.

Fig. 1.

Wg signaling eliminates R cell clusters from the edge of the eye through apoptosis. All micrographs are of 42 hours APF eyes. (A-D) Wild-type eye stained for Wg protein (blue), TUNEL (green) and Elav (red). Wg staining is coincident with the apoptotic cells at the edge of the eye. (E-H) Close-up of the eye perimeter in (A-D). Most of the Wg protein is adjacent to the apoptotic cells, with lower levels in the dying cells, some of which are still Elav-positive (arrows). The edge of this eye is slightly curved, making some of the TUNEL-positive nuclei appear to be present in interior Elav-positive photoreceptors in the optical stack (arrowheads). (I-L)GMR-Gal4::UAS-p35 eyes stained as in wild type. Apoptosis is not observed (J) but a large accumulation of Wg protein is observed (I) coincident with photoreceptors (L; arrow indicates one example). (M-P) Clone mutant for the Wg receptors fz and fz2. Clonal marker (LacZ) is shown in blue, TUNEL in green and Elav in red. No apoptosis is observed inside the fz, fz2 clone. (Q-T) Clones mutant for the Wg signaling component pygo stained in red for Wg (Q,R) or Elav (S,T). Clonal marker (LacZ)is shown in green. Wg expression is still present in the pygo clone. Extra R cell clusters in a pygo clone are apparent (arrows) while decaying clusters are seen in the adjacent pygo+ tissue(arrowheads).

Fig. 1.

Wg signaling eliminates R cell clusters from the edge of the eye through apoptosis. All micrographs are of 42 hours APF eyes. (A-D) Wild-type eye stained for Wg protein (blue), TUNEL (green) and Elav (red). Wg staining is coincident with the apoptotic cells at the edge of the eye. (E-H) Close-up of the eye perimeter in (A-D). Most of the Wg protein is adjacent to the apoptotic cells, with lower levels in the dying cells, some of which are still Elav-positive (arrows). The edge of this eye is slightly curved, making some of the TUNEL-positive nuclei appear to be present in interior Elav-positive photoreceptors in the optical stack (arrowheads). (I-L)GMR-Gal4::UAS-p35 eyes stained as in wild type. Apoptosis is not observed (J) but a large accumulation of Wg protein is observed (I) coincident with photoreceptors (L; arrow indicates one example). (M-P) Clone mutant for the Wg receptors fz and fz2. Clonal marker (LacZ) is shown in blue, TUNEL in green and Elav in red. No apoptosis is observed inside the fz, fz2 clone. (Q-T) Clones mutant for the Wg signaling component pygo stained in red for Wg (Q,R) or Elav (S,T). Clonal marker (LacZ)is shown in green. Wg expression is still present in the pygo clone. Extra R cell clusters in a pygo clone are apparent (arrows) while decaying clusters are seen in the adjacent pygo+ tissue(arrowheads).

To test whether Wg signaling was required for this perimeter PCD, we induced clones mutant for wg, the Wg receptors frizzled(fz) and frizzled 2 (fz2), or the nuclear factor pygopus (pygo). Simultaneous removal of fz, fz2abolishes Wg signaling in all readouts examined(Bhanot et al., 1999; Chen and Struhl, 1999), and the same is true for pygo(Belenkaya et al., 2002; Kramps et al., 2002; Parker et al., 2002; Thompson et al., 2002). Apoptosis was not observed at the edge of the eye in fz, fz2 clones(Fig. 1N) or in pygoor wg clones (data not shown). Wg was still expressed inside the pygo and fz, fz2 clones(Fig. 1Q,R and data not shown),indicating that Wg signaling was not required for the cells at the very edge to express Wg. In fact, Wg protein was found at a greater distance into the eye in pygo clones, which could be similar to the increased spread of Wg in the wing in the absence of Wg signaling(Cadigan et al., 1998).

Inside the mutant clones, extra R cell clusters were apparent(Fig. 1S,T), while apoptotic Elav-positive cells outside the clone were observed(Fig. 1S). Many extra photoreceptor clusters were also seen at the perimeter when the caspase inhibitor p35 was expressed throughout the eye(GMRGal4::UAS-p35) (Fig. 1K), consistent with earlier reports(Hay et al., 1994). Thus Wg removed ommatidia at the edge of midpupal eyes by activation of apoptosis.

### Wg induces its own expression in ommatidia destined to die

When PCD was blocked by expression of p35(Fig. 1J), a dramatic increase in Wg expression was observed (Fig. 1I), which corresponded to the perimeter ommatidia(Fig. 1L). In wild-type eyes at an earlier time (24 hours APF), Wg was expressed uniformly at the edge(Fig. 2A). By 28 hours APF,most of the perimeter of each eye displayed Wg expression in clusters(Fig. 2B), which corresponded to R cells (data not shown, but see Fig. 2F,G). This was more apparent at 32 hours APF(Fig. 2C), and Wg expression in the ommatidia started to fade by 36 hours APF(Fig. 2D). At 24 and 28 hours APF, almost all the apoptotic cells were in the interior(Fig. 2A,B), corresponding to the elimination of inter-ommatidial cells(Hay et al., 1994; Miller and Cagan, 1998). TUNEL-positive nuclei at the edge of the eye were apparent at 32 hours APF,and their number increased by 36 hours APF(Fig. 2D), although they were still far from the numbers seen at 42 hours APF. These data indicate that Wg expression in the peripheral ommatidia preceded PCD. If apoptosis was blocked,elevated Wg expression persisted in the ommatidia that were to be eliminated(Fig. 1I).

Fig. 2.

Accumulation of Wg in ommatidial clusters precedes PCD and depends on Wg signaling. (AD) Wild-type eyes showing time course of Wg protein expression(red) and TUNEL (green) at 24 hours (A), 28 hours (B), 32 hours (C) and 36 hours APF (D). At 24 hours APF, Wg is expressed in a uniform ring at the eye's edge. Starting at 28 hours and peaking at 32 hours APF, Wg expression is found in clusters (arrows). These clusters are fading at 36 hours APF. TUNEL-positive nuclei in the eye interior at 24 and 28 hours APF correspond to dying inter-ommatidial cells. TUNEL-positive nuclei begin to appear at the edge at 32 hours APF and increase by 36 hours APF in the Wg-positive clusters.(E-H) 32 hours APF eye containing a clone of pygo stained for Wg(red), Elav (blue) and clonal marker LacZ (green). Wg expression overlaps with Elav outside the clone (arrows) but not in cells lacking pygo. Note that some of the Wg clusters contain Elav-clusters in a different focal plane(arrowheads), suggesting that they are already undergoing apoptosis. (I,J) pygo clone at 48 hours APF stained for TUNEL (green) and clonal marker lacZ (red). No TUNEL-positive nuclei are observed in the clone. (K,L) pygo clone in 54 hours APF pupal eye stained for Elav (green) and clonal marker lacZ (red) showing two perimeter ommatidia inside the clone(arrow) that have not undergone PCD.

Fig. 2.

Accumulation of Wg in ommatidial clusters precedes PCD and depends on Wg signaling. (AD) Wild-type eyes showing time course of Wg protein expression(red) and TUNEL (green) at 24 hours (A), 28 hours (B), 32 hours (C) and 36 hours APF (D). At 24 hours APF, Wg is expressed in a uniform ring at the eye's edge. Starting at 28 hours and peaking at 32 hours APF, Wg expression is found in clusters (arrows). These clusters are fading at 36 hours APF. TUNEL-positive nuclei in the eye interior at 24 and 28 hours APF correspond to dying inter-ommatidial cells. TUNEL-positive nuclei begin to appear at the edge at 32 hours APF and increase by 36 hours APF in the Wg-positive clusters.(E-H) 32 hours APF eye containing a clone of pygo stained for Wg(red), Elav (blue) and clonal marker LacZ (green). Wg expression overlaps with Elav outside the clone (arrows) but not in cells lacking pygo. Note that some of the Wg clusters contain Elav-clusters in a different focal plane(arrowheads), suggesting that they are already undergoing apoptosis. (I,J) pygo clone at 48 hours APF stained for TUNEL (green) and clonal marker lacZ (red). No TUNEL-positive nuclei are observed in the clone. (K,L) pygo clone in 54 hours APF pupal eye stained for Elav (green) and clonal marker lacZ (red) showing two perimeter ommatidia inside the clone(arrow) that have not undergone PCD.

This induction of Wg expression in the perimeter ommatidia was dependent on Wg signaling. In pygo clones, the expression of Wg in ommatidia did not occur (Fig. 2F), although the expression in non-ommatidial cells at the very edge of the eye was still present. Similar results were observed in fz, fz2 clones (data not shown). This Wg pattern inside the clones was very similar to the wild-type pattern at 24 hours APF (Fig. 2A).

An alternative explanation for the lack of Wg-positive ommatidia in the pygo clones at 30 hours APF is that these mutant cells were developmentally delayed. In such a scenario, Wg expression would occur later and eventually cause PCD. However, Wg was not found in perimeter ommatidia of pygo clones at 36 hours APF (data not shown) or 42 hours APF(Fig. 1R and data not shown). Consistent with this, no sign of apoptosis was observed in mutant clones at 48 hours APF (Fig. 2J) or 54 hours APF (data not shown), long after control ommatidia had undergone PCD. Perimeter ommatidia were still found in the pygo clones at 54 hours(Fig. 2L) and likely persisted until adulthood (data not shown and Fig. 5H). There was no detectable developmental lag in cells unresponsive to Wg. Rather, these cells were unable to initiate autoregulation of Wg expression or Wg-dependent PCD.

Fig. 5.

Ommatidia destined to die are often incomplete. (A,D) Wild-type eye at 5 hours APF stained for Bar (red) and Elav (green). Bar is a marker for R1 and R6. Several ommatidia at the edge contained only one positive Bar cell(arrows), while smaller Elav-positive clusters contained none (arrowheads in D). (B,E) GMR-Gal4::UAS-p35 eye at 32 hours APF stained for Elav(green), Prospero (Pros; red) and Wg (blue). Pros is a marker for R7 and Wg for ommatidia that will later undergo PCD. About 20% of the Wg-positive ommatidia do not contain an R7 (data not shown). Most of the Wg-positive ommatidia containing an R7 lack some R cells (arrowheads,) but occasionally they appear to have a full complement (arrow). (C,F)GMR-Gal4::UAS-p35 eye at 32 hours APF stained for Cut (red) and Wg(green). Cut is a marker for cone cells. Some Wg-positive ommatidia had four cone cells of normal or near normal size (arrows, compared with more interior ommatidia), but most four cone cell clusters appeared smaller than normal(arrowheads). Often fewer than four cone cells were observed per ommatidia(asterisk). (G) SEM of adult control eye showing the periphery of the ventral eye. (H) SEM of an adult eye containing a large fz, fz2 clone covering the ventral portion of the eye showing three smaller ommatidia never seen in controls (arrows).

Fig. 5.

Ommatidia destined to die are often incomplete. (A,D) Wild-type eye at 5 hours APF stained for Bar (red) and Elav (green). Bar is a marker for R1 and R6. Several ommatidia at the edge contained only one positive Bar cell(arrows), while smaller Elav-positive clusters contained none (arrowheads in D). (B,E) GMR-Gal4::UAS-p35 eye at 32 hours APF stained for Elav(green), Prospero (Pros; red) and Wg (blue). Pros is a marker for R7 and Wg for ommatidia that will later undergo PCD. About 20% of the Wg-positive ommatidia do not contain an R7 (data not shown). Most of the Wg-positive ommatidia containing an R7 lack some R cells (arrowheads,) but occasionally they appear to have a full complement (arrow). (C,F)GMR-Gal4::UAS-p35 eye at 32 hours APF stained for Cut (red) and Wg(green). Cut is a marker for cone cells. Some Wg-positive ommatidia had four cone cells of normal or near normal size (arrows, compared with more interior ommatidia), but most four cone cell clusters appeared smaller than normal(arrowheads). Often fewer than four cone cells were observed per ommatidia(asterisk). (G) SEM of adult control eye showing the periphery of the ventral eye. (H) SEM of an adult eye containing a large fz, fz2 clone covering the ventral portion of the eye showing three smaller ommatidia never seen in controls (arrows).

The induction of Wg expression in the ommatidia destined for elimination suggested that Wg is required for PCD at this time. To confirm this, we attempted to use the conditional mutant wgIL, which encodes a temperature-sensitive protein(Baker, 1988). However, even at 16.5°C, where wgIL is wild-type for most Wg activities(Baker, 1988; Cadigan and Nusse, 1996), very few Wg-positive ommatidia or subsequent apoptosis was observed (see Fig. S1),precluding the use of this allele to determine the critical phase for Wg activity in regulating PCD. As an alternative, the wgtsgene was expressed under the control of the GMR promoter(Fig. 3). When reared at the restrictive temperature (25°C), GMR-wgts eyes had a normal pattern of apoptosis (Fig. 3D). If shifted to 16.5°C at 24 hours APF and cultured until an equivalent stage, considerable R cell apoptosis was observed(Fig. 3A,B). Less PCD was evident in shifts at 30 hours APF (Fig. 3C) and shifts at 36 hours APF resulted in almost no ectopic apoptosis (data not shown). These data demonstrate that Wg expression at 24 hours APF can induce R cell apoptosis, consistent with the expression pattern of endogenous Wg.

Fig. 3.

Ectopic Wg expressed at 24 hours APF can induce R cell PCD. GMR-Gal4::UAS-wgts eyes were stained for TUNEL (green) and Elav (red). Animals were reared at 25°C (where wgts is inactive) and shifted to 16.5°C at 24 hours APF (A,B), 30 hours (C) or not shifted (D). After the shift to the lower temperature (activating Wg activity), animals were cultured until the equivalent of 42 hours APF(development occurs 2.7× slower at the lower temperature). A shift down at 24 hours APF results in considerable apoptosis in the interior R cells(A,B) while less TUNEL-positive cells are observed when shifted at 30 hours APF. No ectopic PCD was observed in eyes cultured continuously at 25°C(D).

Fig. 3.

Ectopic Wg expressed at 24 hours APF can induce R cell PCD. GMR-Gal4::UAS-wgts eyes were stained for TUNEL (green) and Elav (red). Animals were reared at 25°C (where wgts is inactive) and shifted to 16.5°C at 24 hours APF (A,B), 30 hours (C) or not shifted (D). After the shift to the lower temperature (activating Wg activity), animals were cultured until the equivalent of 42 hours APF(development occurs 2.7× slower at the lower temperature). A shift down at 24 hours APF results in considerable apoptosis in the interior R cells(A,B) while less TUNEL-positive cells are observed when shifted at 30 hours APF. No ectopic PCD was observed in eyes cultured continuously at 25°C(D).

### Wg activates hid, grim and rpr expression at the edge of the eye and these genes are required for perimeter apoptosis

The pro-apoptotic gene hid is required for the apoptosis of undifferentiated and inter-ommatidial cells during larval and early pupal development (Kurada and White,1998; Yu et al.,2002). Therefore, we explored the role of hid in mid-pupal ommatidial PCD. The hidP allele is a transposon insertion allele in the hid ORF(Grether et al., 1995). In hidP homozygotes, apoptosis was still observed at the perimeter (Fig. 4B), although it was distinct from wild type in that high Wg expression was still found in the apoptotic ommatidia and the Elav staining in the perimeter clusters still appeared normal (see Fig. S2). This suggests that the rate of ommatidial apoptosis is significantly reduced in hid mutants.

Fig. 4.

Requirement of hid, grim and rpr for perimeter apoptosis. All eyes are 42 hours APF except where noted. (A,B) hidPmutant clone stained for TUNEL (green) and clonal marker LacZ (red). No detectable reduction in apoptosis is observed. (C,D) Df(3L)X25deficiency clone stained as for (A,B). In one clone there is little reduction in TUNEL (arrow), while a strong reduction is observed in the other(arrowhead). (E-H) Df(3L)H99 clone stained for clonal marker LacZ(blue), TUNEL (green) and Elav (red). Very little TUNEL is observed inside the clone (F) and extra R cell clusters are apparent (arrows in G). (I,J) Df(3L)H99 clone at 30 hours APF stained for Wg (red) and clonal marker lacZ (green). Wg-positive ommatidia are present inside the clone. (K,L) Df(3L)H99 clone at 54 hours APF stained for Elav (red) and clonal marker GFP (green). Perimeter ommatidia are still observed inside the clone at this late time. (M-O) GMR-Gal4::UAS-p35 eyes stained for hid(M), grim (N) or rpr (O) transcripts. Elevated expression of all three genes is observed at the edge of the eye. (P-R)GMR-Gal4::UAS-GPI-fz2 eyes stained for hid (P), grim (Q) or rpr (R). No elevated expression of these genes was observed at the edge of the eye.

Fig. 4.

Requirement of hid, grim and rpr for perimeter apoptosis. All eyes are 42 hours APF except where noted. (A,B) hidPmutant clone stained for TUNEL (green) and clonal marker LacZ (red). No detectable reduction in apoptosis is observed. (C,D) Df(3L)X25deficiency clone stained as for (A,B). In one clone there is little reduction in TUNEL (arrow), while a strong reduction is observed in the other(arrowhead). (E-H) Df(3L)H99 clone stained for clonal marker LacZ(blue), TUNEL (green) and Elav (red). Very little TUNEL is observed inside the clone (F) and extra R cell clusters are apparent (arrows in G). (I,J) Df(3L)H99 clone at 30 hours APF stained for Wg (red) and clonal marker lacZ (green). Wg-positive ommatidia are present inside the clone. (K,L) Df(3L)H99 clone at 54 hours APF stained for Elav (red) and clonal marker GFP (green). Perimeter ommatidia are still observed inside the clone at this late time. (M-O) GMR-Gal4::UAS-p35 eyes stained for hid(M), grim (N) or rpr (O) transcripts. Elevated expression of all three genes is observed at the edge of the eye. (P-R)GMR-Gal4::UAS-GPI-fz2 eyes stained for hid (P), grim (Q) or rpr (R). No elevated expression of these genes was observed at the edge of the eye.

Three other genes in the fly genome, rpr, grim and sickle, are known to encode proteins with limited sequence similarity to Hid (Shi, 2002; Christich et al., 2002; Srinivasula et al., 2002; Wing et al., 2002). These proteins promote apoptosis by inhibiting the activity and/or stability of IAPs(Martin, 2002, Shi, 2002; Srinivasula et al., 2002). Therefore, we explored whether functional redundancy could account for the partial inhibition of Wg-dependent apoptosis in hid mutants.

The H99 deficiency completely removes hid, grim and rpr but not sickle(White et al., 1994; Christich et al., 2002; Wing et al., 2002). The X25 deletion removes hid and grim while the XR38 deficiency removes rpr and sickle(Peterson et al., 2002). Clones of X25 showed a strong reduction of apoptosis 60% of the time(arrowhead in Fig. 4C,D), with the reminder having a much smaller reduction in TUNEL (arrow in Fig. 4C,D). H99/+ heterozygotes displayed a phenotype at the eye's perimeter similar to that of hid mutants, while H99/XR38 eyes displayed slightly less TUNEL signal than in H99/+ (data not shown). In H99 homozygous clones,there was almost no apoptosis (Fig. 4F) and extra R cell clusters were observed(Fig. 4G; see arrows). No evidence for a developmental lag was observed in H99 clones, with Wg-positive ommatidia appearing at the same time as controls(Fig. 4J) and perimeter R cells persisting until 54 hours APF with no sign of PCD(Fig. 4L and data not shown).

The above data suggest that all three pro-apoptotic factors in the H99 interval are required for the wg-dependent PCD at the edge of the eye. hid, grim and rpr transcripts were all elevated at the periphery in GMR-Gal4::UAS-p35 eyes(Fig. 4M-O). The expression pattern was clustered in a similar pattern to that found for Wg expression. The perimeter expression of all three genes was completely abolished in GMR-Gal4::UAS-GPI-fz2 eyes (Fig. 4P-R). GPI-Fz2 is known to be an efficient inhibitor of Wg signaling (Cadigan et al.,1998) and blocked peripheral apoptosis as completely as p35 (data not shown). These results strongly suggest that hid, grim and rpr are expressed in the Wg-positive peripheral ommatidia and that Wg signaling is required for this expression.

### Perimeter ommatidia are often incomplete

Ommatidia are formed by the sequential recruitment of cells into clusters. The process starts with the specification of evenly spaced R8 neurons, which then recruit four additional neurons (R2-R5). After a wave of mitosis, R1 and R6 are added and then specification of R7 completes the eight R cell complement. Four cone cells and two primary pigment cells are then added(Wolff and Ready, 1993).

Staining with the general neuronal marker Elav revealed that many ommatidia at the edge of the eye were incomplete(Fig. 5B,D and data not shown). We examined the composition of the peripheral ommatidia using cell-specific markers. Senseless (Sens) is a marker for R8(Frankfort et al., 2001), Bar for R1/R6 (Higashijima et al.,1992), Prospero for R7(Kauffmann et al., 1996) and Cut for cone cells (Blochlinger et al.,1993). Wild-type or GMR-Gal4::UAS-p35 eyes were stained for these proteins at 5 or 30-32 hours APF, before perimeter apoptosis occurred, with similar results. This indicates that PCD did not influence the perimeter ommatidia composition before 32 hours APF. All the ommatidia containing at least two Elav-positive cells had one R8 and almost all single Elav-positive cells were Sens-positive (data not shown). Some peripheral ommatidia were missing either R1 or R6(Fig. 5A,D) and many had fewer than four cone cells (e.g. asterisk in Fig. 5C). We conclude that many of the ommatidia destined to die are incomplete, perhaps from lack of cells at the edge of the eye field for recruitment into the developing clusters. Interestingly, most (>80%)perimeter ommatidia had an R7 cell, even if they had fewer than the normal complement of R cells (arrowheads in Fig. 5B).

The incomplete ommatidia were clearly eliminated, since they were not found at 42 hours APF (data not shown). To determine whether complete ommatidia are also eliminated, we used Wg expression at 32 hours APF as a marker for ommatidia that will undergo PCD. Many Wg-positive ommatidia had an R7 cell(Fig. 5B,E) and four cone cells(Fig. 5C,F). Occasionally,Wg-positive ommatidia appeared to have the normal number of Elav-positive R cells (arrow in Fig. 5B,E). The Wg-positive cone cells clusters were often smaller than normal(Fig. 5C,F; arrowheads),although occasionally we observed some of near normal size (arrows in Fig. 5C,F). These cells are capable of secreting lens, as evidenced by the presence of surface ommatidia of reduced size at the edge of adult eyes containing large clones of pygo or fz, fz2 (Fig. 5H and data not shown). The clone shown is at the ventralmost portion of the eye, where inhibition of Wg signaling did not result in expansion of the eye field at the expense of head anlage (data not shown), as is observed when Wg signaling is blocked dorsally(Heslip et al., 1997). Small extra ommatidia are also observed at the edge of adult GMR-Gal4::UAS-p35 eyes (data not shown)(Hay et al., 1994).

### The R cell apoptosis seen in apc1 mutants depends on hid, grim and rpr

Mutants in the apc1 gene, which encodes a negative regulator of Wg signaling, cause R cells to undergo PCD during mid-pupation(Ahmed et al., 1998). In addition, expression of an activated form of Arm results in a similar phenotype (Ahmed et al., 1998; Freeman and Bienz, 2001). We found that R cell apoptosis in apc1 mutants occurred at the same time as the perimeter cell death in wild type, with TUNEL-positive cells arising around 35 hours APF (data not shown) and peaking at 42 hours APF(Fig. 6E). All Elav-positive cells were eliminated by 54 hours APF (data not shown). Adult apc1mutants survive to adulthood and have slightly smaller eyes (K. Cadigan,unpublished observations) with no photoreceptors(Ahmed et al., 1998).

Fig. 6.

The R cell apoptosis in apc1 mutants is absent in H99clones. All micrographs are of the interior of eyes at 42 hours APF. (A-C)Wild-type eye stained for Elav (red) and TUNEL (green). No PCD is apparent.(D-F) apc1S76/apc1Q8 mutant eye stained as above. Many of the Elav-positive cells are TUNEL-positive. (G-I)Large Df(3L)H99 mutant clone in an apc1S76/apc1Q8 mutant background stained as above. No cells outside the clone are visible in the field shown. There is a complete block of apoptosis and the Elav pattern is identical to wild type.

Fig. 6.

The R cell apoptosis in apc1 mutants is absent in H99clones. All micrographs are of the interior of eyes at 42 hours APF. (A-C)Wild-type eye stained for Elav (red) and TUNEL (green). No PCD is apparent.(D-F) apc1S76/apc1Q8 mutant eye stained as above. Many of the Elav-positive cells are TUNEL-positive. (G-I)Large Df(3L)H99 mutant clone in an apc1S76/apc1Q8 mutant background stained as above. No cells outside the clone are visible in the field shown. There is a complete block of apoptosis and the Elav pattern is identical to wild type.

To test whether apoptosis in apc1 mutants is similar to that which occurs at the perimeter, we examined whether hid, grim and rpr were required for this PCD. We found a slight reduction in the amount of TUNEL signal in hid, apc1 mutant eyes and the signal was greatly reduced in a H99/+ background (data not shown). Apoptosis was completely abolished in H99 mutant clones(Fig. 6G-I). As in the case of the perimeter apoptosis, the PCD found in apc1 mutants appeared to depend on hid, grim and rpr.

Interestingly, we found no elevation of Wg expression in the interior of apc1 mutant eyes, even when PCD was completely blocked in H99 mutant clones (data not shown). Thus, the elevated Wg signal in the absence of apc1 was not sufficient to activate the Wg autoregulatory loop.

### Wg signaling induces apoptosis through activation of hid, grim and rpr expression

Elevated Wg signaling has previously been reported to induce apoptosis in the photoreceptor neurons of the mid-pupal eye(Ahmed et al., 1998; Freeman and Bienz, 2001). However, the physiological relevance and mechanism of this Wg signaling-dependent PCD were not clear. In this report, we demonstrated that endogenous Wg induced apoptosis at the edge of mid-pupal eyes. Wg signaling induced the expression of the apoptotic activators hid, grim and rpr (Fig. 4M-R). Using a combination of different deficiencies in the hid/grim/rpr region, we demonstrated that all three genes were required for Wg-dependent perimeter ommatidia PCD(Fig. 4A-F). This is distinct from the apoptosis that occurs earlier in eye development, where removal of hid is sufficient to prevent PCD(Kurada and White, 1998; Yu et al., 2002). Coordinated regulation of these three genes has not previously been reported, and it will be interesting to determine whether the activation by Wg signaling is direct or occurs through an intermediary.

### Mid-pupal apoptosis at the eye's perimeter depends on Wg signaling

During mid-pupation (36-44 hours APF), ommatidia at the edge of the eye are eliminated through PCD that depends on Wg signaling. Our results indicate the following order of events, summarized in Fig. 7. The ring of Wg expression bordering the eye anlage is initially established around 6 hours APF (Cadigan et al., 2002)(data not shown). At 24 hours APF, after eversion of the pupal eye, Wg is found in cells at the edge of the eye that are distinct from the perimeter ommatidia (Fig. 2F-H and data not shown). Several hours later, between 28-32 hours APF, Wg expression is found in some of these ommatida (Fig. 2B,C). The establishment of this pattern depends on the ommatidial cells being able to transduce the Wg signal(Fig. 2E-G). Wg signaling then activates the expression of hid, grim and rpr and apoptosis ensues. Under normal conditions, Wg expression is largely faded from apoptotic ommatidia by 36 hours APF (Fig. 2D). When apoptosis is blocked, the positive feedback loop of Wg expression is unchecked, and Wg and the pro-apoptotic factors can accumulate to high levels (Fig. 1I; Fig. 4M-O).

Fig. 7.

Cartoon of Wg-induced PCD at the eye's perimeter. Non-ommatidial cells at the edge of the eye are depicted as oblongs and ommatidial cells as circles. A red outline indicates Wg expression and purple represents hid, grimand rpr expression. At 24 hours APF, Wg is found only in the edge cells. Between 28-32 hours APF, Wg expression is induced in the perimeter ommatidia, which are often lacking the full complement of cells. Wg signaling is necessary but not sufficient for this induction of ommatidial Wg expression. By 36 hours APF, Wg expression in the ommatidia is fading and the expression of the pro-apoptotic genes hid, grim and rpr are triggering apoptosis, causing the elimination of these ommatidia. See text for further details.

Fig. 7.

Cartoon of Wg-induced PCD at the eye's perimeter. Non-ommatidial cells at the edge of the eye are depicted as oblongs and ommatidial cells as circles. A red outline indicates Wg expression and purple represents hid, grimand rpr expression. At 24 hours APF, Wg is found only in the edge cells. Between 28-32 hours APF, Wg expression is induced in the perimeter ommatidia, which are often lacking the full complement of cells. Wg signaling is necessary but not sufficient for this induction of ommatidial Wg expression. By 36 hours APF, Wg expression in the ommatidia is fading and the expression of the pro-apoptotic genes hid, grim and rpr are triggering apoptosis, causing the elimination of these ommatidia. See text for further details.

Our data cannot distinguish whether Wg from the edge cells or the induced Wg in the perimeter ommatidia is required for activating PCD, since the clones we examined removed wg activity from both cell types. However, the expression of Wg in the ommatidia precedes the appearance of TUNEL-positive nuclei (Fig. 2B,D) and the level of Wg expression is at similar levels as found in the edge cells(Fig. 2C). Therefore, we favor a model in which Wg expression in the ommatidia is necessary for apoptosis to occur.

Incomplete ommatidia are not found at the edge of wild-type eyes (data not shown) (Hay et al., 1994). Therefore, the partial ommatidia observed during early/mid-pupation are destined to die. These ommatidia are also marked by Wg expression at 30-32 hours APF (Fig. 2C,F). The smallest of these clusters has just one cell, usually expressing a marker for the R8 cell type (data not shown). All ommatidia containing more than one R cell contain an R8, but many are missing either R1 or R6(Fig. 5A,D). Around 20% of the ommatidia destined to die were lacking an R7 cell (data not shown). There are also Wg-positive clusters containing 0-3 cone cells, instead of the normal four (Fig. 5C,F and data not shown). We suspect that the clusters at the perimeter of the eye field simply run out of surrounding cells to recruit.

While most of the eliminated ommatidia appear to have been incomplete, we believe that some did possess the full complement of neurons and support cells. Some Wg-positive ommatidia appeared to contain eight R cells(Fig. 5B,E), and ones containing four cones cells were not uncommon(Fig. 5C,F). Many also possessed two primary pigment cells (data not shown). Almost all the Wg-positive ommatidia that possessed four cone cells appeared significantly smaller than normal (arrowheads in Fig. 5C,F and data not shown). These correspond to the stunted ommatidia' previously reported (Hay et al., 1994; Wolff and Ready,1991). These ommatidia probably give rise to the smaller ommatidia observed at the edge of adult eyes containing fz, fz2 or pygo clones (Fig. 5Hand data not shown).

How many ommatidia are eliminated by Wg signaling during development? There are approximately 80-100 Wg-positive ommatidia in GMR-Gal4::UAS-lacZor GMR-Gal4::UAS-p35 eyes at 32 hours APF. These estimates do not include the 1-2 cell clusters that are difficult to distinguish. Assuming an average of five R cells/Wg cluster, 400 to 500 R cells would be eliminated per eye. There are about 6000 photoreceptors in the adult eye. This calculation suggests that between 6.2 and 7.7% of the pupal photoreceptors are removed by Wg signaling during mid-pupation.

Besides the elimination of ommatidia described in this study, Wg signaling at the perimeter of the eye has also been shown to specify the pigment rim epithelia (Tomlinson, 2003)and to induce Homothorax expression, which is necessary for specification of the dorsal rim inner photoreceptors, which are specialized for detection of polarized light (Wernet et al.,2003). Thus, Wg signaling has multiple roles in refining the perimeter of the developing eye.

### The function of ommatidia removal

What purpose does the elimination of these perimeter ommatidia achieve? The answer probably lies in the connections between the six outer R cells of each ommatidium and their post-synaptic targets in the laminal layer of the optic ganglia. The lamina is organized into units called cartridges, which underlie each ommatidium and form synapses with the outer R cells. Because of the precise arrangement of the photoreceptors in the ommatidia and the curvature of the eye, a single line of sight is perceived by six different outer photoreceptors (R1-R6) residing in six different ommatidia. These R cells do not innervate the underlying cartridge; rather, each cell forms a synapse with a distinct adjacent cartridge. In this way, visual excitation in the curved surface of the retina is transformed into a smooth topographic map in the optic ganglia (for reviews see Meinertzhagen and Hanson,1993; Clandinin and Zipursky,2002).

Because there is not a one-to-one relationship between ommatidia and lamina cartridges, a problem arises at the edge of the eye, where there are not enough adjacent cartridges for perimeter ommatidia to form synapses with. The development of the optic ganglia is stimulated by the projection of axons from the overlying ommatidia during larval and early pupal development(Meinertzhagen and Hanson,1993). The peripheral-most ommatidia are thought to induce underlying cartridges before they are eliminated. Thus, there are extra laminal targets at the eye's edge for the remaining perimeter outer R cells to innervate (Meinertzhagen and Hanson,1993). Removal of ommatidia by wg-dependent PCD should minimize the number of incorrect connections that would compromise the fly's peripheral vision.

### What is the signal triggering the Wg autoactivation circuit?

The cells at the edge of the eye express Wg from 6 hours APF(Cadigan et al., 2002). However, Wg from these edge cells does not activate Wg in the adjacent ommatidia until 26-32 hours APF. Wg signaling alone is not sufficient to trigger Wg autoactivation, since elevated Wg expression is not observed in apc1 mutant eyes (data not shown), even though the pathway is activated enough to induce R cell apoptosis(Fig. 6E)(Ahmed et al., 1998). There must be some other signal(s) that triggers the wg self-activation in the perimeter ommatidia.

The edge cells expressing Wg could also express another signal that would allow the Wg signal to activate Wg expression in the ommatidia at the appropriate time. This simply moves the problem back one step; i.e. what then triggers the expression/activation of this signal? An alternative is the presence of a signal that counteracts Wg signaling until 24-28 hours APF. One candidate is Ras signaling, which can block the activity of Wg in the eye(Freeman and Bienz, 2001; Hazelett et al., 1998). It has been previously suggested that elevated Ras signaling during larval and early pupal stages prevents overexpression of a stablized form of Arm from inducing PCD until mid-pupation (Freeman and Bienz,2001). However, we found that overexpression of an activated form of ras did not block the perimeter apoptosis (H. Lin and K. Cadigan,unpublished). Therefore, we think it unlikely that a decrease in the Ras pathway is the signal allowing Wg to initiate the apoptotic cascade.

As discussed above, some eliminated ommatidia appear to have the full complement of cell types, so incompleteness does not appear to be the signal. However, the ommatidia destined to die are almost always smaller than normal(Fig. 5C,F and data not shown). Small size combined with Wg from the edge cells could initiate Wg expression in the ommatidia.

Another candidate for the trigger is the failure of synapse formation between the R cell neurons and their targets in the optic ganglia. Projection of the R cell axons into the ganglia is complete by early pupation, but the formation of synapses between the outer R cells and the lamina neurons does not occur until 24-38 hours APF (Clandinin and Zipursky, 2002). Perhaps the R cells from perimeter ommatidia cannot find enough post-synaptic targets because they are at the edge of the field. The absence of a retrograde signal from neurons in the optic ganglia could lead to the accumulation of Wg in these ommatidia.

### Implications for eye development in other organisms and human disease

The expression pattern of Wg in the eyes of other arthropods suggests that it may function in a similar fashion as in Drosophila melanogaster. The flour beetle Tribolium castaneum is a primitive holometabolous insect species, in which the adult eye forms at the end of larval development from lateral head ectoderm known as optic placodes. During early pupation, the beetle ortholog of wg is expressed in a ring around the optic placodes (Friedrich and Benzer,2000). In the crustacean Mysidium columbiae (mysid),retinal morphogenesis occurs in a posterior-to-anterior direction as in Drosophila (Duman-Scheel et al.,2002). As this occurs, the mysid ortholog of wg is expressed in a half ring around the posterior of the developing eye(Duman-Scheel et al., 2002). It will be interesting to determine whether this ring becomes complete as retinal differentiation proceeds and whether apoptosis occurs at the edge of the eye in beetles and mysids.

The finding that Pax6 orthologs in flies, mice and humans are necessary for eye development has suggested a common origin for eyes(Halder et al., 1995). However, vertebrate and insect eyes have completely different morphologies, as well as distinct embryonic origins. This has led to the idea that a commonprimitive unit' consisting of light-sensing photoreceptors has been recruited for organogenesis several times during animal evolution(Gehring and Ikeo, 1999). However, many interesting parallels exist in the development of insect and vertebrate eyes, perhaps because of utilization of genetic circuits that were developed before the divergence of these species(Pichaud et al., 2001).

The periphery of the vertebrate eye contains the ciliary marginal zone(CMZ), which contains undifferentiated retinal progenitor cells, which can differentiate to add neurons and glia to the periphery of the eye(Reh and Levine, 1998). In the developing chick eye, Wnt2b is expressed in the CMZ. Blocking Wnt signaling causes premature differentiation of neuronal progenitors(Kubo et al., 2003). Whether Wnt signaling acts in part through inducing apoptosis in the CMZ remains to be examined.

A possible connection between Wnt signaling and PCD in insect and vertebrate eyes is also suggested by similarity in APC mutations. In flies, apc1 mutants have retinal apoptosis(Ahmed et al., 1998)(Fig. 6). Some humans heterozygous for mutations in the APC gene also display retinal lesions, termed congenital hypertrophy of the retinal pigment epithelium(CHRPE). While an increase in the thickness of the retinal pigment epithelium is the most consistent feature of the condition in humans(Buettner, 1975; Traboulsi et al., 1990) and in mice carrying a similar mutation (Marcus et al., 2000), photoreceptor degeneration is also observed in several lesions. Because these lesions are probably due to somatic inactivation of the wild-type APC allele, they are variable and difficult to study systematically. A more direct comparison with the fly apc1 mutants is needed, perhaps with conditional knockouts of the mouse APC gene.

We have shown that in the fly eye, Wg induces apoptosis by activating the expression of hid, grim and rpr. These proteins then inactivate the caspase inhibitor IAP through an RHG domain at their N-termini(Shi, 2002). hid,grim and rpr have no obvious orthologs in vertebrates, although Smac/Diablo and HtrA2/Omi possess an RHG at their N-termini and inhibit IAP activity (Du et al., 2000; Verhagen et al., 2000; Suzuki et al., 2001). If Wnt signaling induces PCD in the vertebrate eye, does this occur through activation of Smac and/or HtrA2 expression? A similar parallel has been found in p53-induced apoptosis, where rpr is activated in flies(Brodsky et al., 2000) and HtrA2 in human cell culture (Jin et al.,2003). The involvement of Wnt signaling in the human eye though the induction of IAP inhibitors clearly deserves further consideration.

Supplemental data available online

We thank all the researchers mentioned above who provided fly stocks and antibodies, with special thanks to R. Howes and R. Nusse for providing the fz, fz2 double mutants prior to publication. Special thanks to Chris Sutter for performing the clonal analysis with wg. Thanks to Bruce Donahoe and Chris Edwards of the Microscopy and Image Analysis Laboratory for help with confocal microscopy. Thanks also to Steve Clark for assistance with Scanning Electron Microscopy. We are grateful to C. Desplan for discussion and advice and also want to thank D. Parker, J. Guan, Tim Blauwkamp and other members of the Cadigan lab for helpful discussions. Supported by the University of Michigan Cancer Center Predoctoral Training Grant to H.V.L. and NIH grant RO1 GM59846 to K.M.C.

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text:
# [XeTeX] bopomofo ruby characters

Jonathan Kew jonathan_kew at sil.org
Mon Apr 7 22:01:29 CEST 2008

On 7 Apr 2008, at 6:53 pm, Ján Zahornadský wrote:

> Hello everyone!
>
> First, thanks for being such a great community!
>
> Now I'm in a problem of attaching a bopomofo pronunciation hints into
> Chinese characters. I've found Wang fonts, which have these as a
> part of
> a glyph, but due to multiple readings of the characters it's necessary
> to switch between these fonts, often not having idea which fonts
> carries
> which reading.
>
> The second approach I tried is to write a command to attach those
> symbols to a character. As I'm pretty unskilled in (Xe)LaTeX, my
> code so
> far looks like this (for 綽/ㄔㄨㄛˋ character):
>
> \fontsize{36}{36}\ming\fbox{綽\kern 8pt\raise 28pt\hbox{%
> \fontsize{12}{12}%
> \fontspec[RawFeature={vertical:}]{Ming(for ISO10646)}%
> \rotatebox{270}{\raise 1pt\hbox{\makebox[32pt]{\hfil ㄔㄨㄛ
> {\kern-16pt\
> lower-7pt\hbox{ˋ}}\hfil}}}%
> }\kern -4pt}

I don't have the same fonts as you, but here's a version of an \annot
macro that seems to work OK for me; you can just change font names in
the test doc to match your setup. A couple of points to note here:

- I've used "em" rather than absolute "pt" units in the \annot macro
so that it will scale to whatever text size you're using. You can of
course adjust the various dimensions to fine-tune the results, but it
looks pretty reasonable to my (ignorant) eye.

- It's very inefficient to use \fontspec{....} repeatedly for a
typeface that you use frequently; better to use \newfontfamily to
declare it ahead of time.

- The use of \addfontfeature here is also inefficient, but was
simpler than using \newfontfamily to declare a vertical variant of
each font, so I took the easy way out.

For more Chinese support, Google for the zhspacing package. I've also
seen a reference to something called xeCJK, but don't know anything
about it (the pages I saw were in Chinese, which I don't understand
beyond a handful of words).

HTH,

JK

% - - - - - - - - - - - - - - - -

\documentclass[10pt]{report}
\usepackage{fontspec}
\usepackage{xltxtra}
\usepackage{xunicode}
\setmainfont[Mapping=tex-text]{Gentium}

\newfontfamily\song{STSong}
\newfontfamily\kai{STKaiti}
\newfontfamily\hei{STHeiti}

\newcommand\annot[3]{#1%
\raise.85em\hbox{%
\rotatebox{270}%
{\raise.2em\hbox{\makebox[1em]{\fontsize{0.3em}{0.3em}%
\addfontfeature{Vertical=RotatedGlyphs}#2%
\kern-1.1em\raise.7em\hbox{#3}}}}\kern-.1em}}

\begin{document}

\fontsize{36}{36}

\song
\fbox{\annot{綽}{ㄔㄨㄛ}{ˋ}}

\kai
\fbox{\annot{綽}{ㄔㄨㄛ}{ˋ}}

\hei
\fbox{\annot{綽}{ㄔㄨㄛ}{ˋ}}

\end{document}

% - - - - - - - - - - - - - - - -

More information about the XeTeX mailing list

format: plain


text:
×

# Astronomical Unit

Top
FAQ

## Define Astronomical Unit

View Notes

How to define astronomical unit? Astronomers use astronomical units – or AU – to represent the distance of the solar system. We can say that one astronomical unit (AU) outlines the mean distance between the Earth and our Sun for general reference. An AU is nearly 93 million miles (150 million km). It’s approximately eight light minutes. The definition of AU also means distances in astronomical units to pre-eminent solar system objects. The astronomical unit is applied primarily for measuring distances around other stars or within the Solar System.

It is also an essential element in the definition of another unit of astronomical length, the parsec. The astronomical unit is quite instrumental in formulating and understanding the distance between stellar objects and is crucial in the calculations and computation involving astronomical problems. One astronomical unit is equal to 92955807 miles.

### Astronomical Unit Definition

According to the prevailing astronomical convention, 1 astronomical unit is equal to 149,597,870.7 kilometres (or 92,955,807 miles). As the earth orbits the sun with a varying orbital distance, we need to consider the average distance, therefore, one astronomical unit is the average distance between the Earth and the Sun. In other words, the Earth and the Sun mid-distance varies in a single year. The varying distance between the earth and the sun is maximum at the aphelion ( 152,100,000 km or 94,500,000 miles or 1.016 AUs) and the minimum at the perihelion (147,095,000 km or 91,401,000 miles or 0,983 AUs).

1 astronomical unit value = 149597870700 metres (exactly)

≈ 92955807 miles

≈ 499.00478384 light-seconds

≈ 4.8481368×10−6 parsecs

≈ 1.5812507×10−5 light-years

The speed of light can be represented in terms of astronomical units. As we know  299792458 m/s is the speed of light which is equal to precisely 299792458 × 86400 ÷ 149597870700 or about 173.144632674240 AU/d, some 60 parts per trillion less than the 2009 estimate.

### Development of Astronomical Unit

The earliest documented example of astronomers calculating the distance between the Earth and the Sun dates back to Classical Antiquity. In the 3rd century BCE, Greek mathematician Aristarchus of Samos imputed that the distance was estimated to be between 18 and 20 times the distance between the Earth and the Moon.

According to the oldest Chinese mathematical writing, Zhoubi Suanjing the 1st century BCE treatise also estimates the distance between the Earth and Sun. According to the anonymous exposition, the distance is calculated by conducting geometric measures of the length of noontime shadows formed by objects aligned at specific distances. However, the predictions were based on the belief that the Earth was flat.

By the 19th century, ascertainments of the speed of light and the constant deviation of light resulted in the first direct measurement of the Earth-Sun distance in kilometres. By 1903, for the first time, the term “astronomical unit” emerged. Developments in precision have always been a key to developing astronomical understanding. Throughout the twentieth century, measurements became more precise and ever more reliant on accurate observation of the effects described by Einstein’s theory of relativity and upon the analytical tools it used.

### Astronomical Unit Modern Usage

The astronomical unit finds great applicability in the measurement of the stellar distance of extraterrestrial objects. It can be used to calculate the heliocentric distance of an asteroid or measure the distance of a planet’s orbiting moon. In the solar system, it finds usage in the development of mathematical and numerical methods for computational purposes. Also, it can be used in general to measure the distance between planetary systems and understand the extent of gaseous clouds around planets. However, an astronomical unit is an inept way of measuring distance when it comes to interstellar objects. For interstellar purposes, it is best to use measures such as light-years and parsec. While fabricating a numerical model of the Solar System, the astronomical unit proffers a relevant scale that minimizes floating-point calculations errors.


text:
• May 16, 2022, 02:58:21 AM
• Welcome, Guest

Pages: 1 2 [3] 4   Go Down

0 Members and 1 Guest are viewing this topic.

max_thehitman

• SAS~Area51
• Modder
• member
• Offline
• Posts: 8977
• Beer...Girls...IL2+Mods!
Re: Demoiselle
« Reply #24 on: May 29, 2017, 07:52:31 AM »

Very cool little aircraft  Thank you. Great fun to fly!
Those early pilot pioneers had lots of fun with these airplanes.

Here is something you can do to make this 3d-model look a little bit better in your game.
You will have to do this to your personal game, so I am not posting here all the files.

Go open up your folder .... \3do\Plane\Demoiselle(Multi1)
Now open each file with NOTEPAD and look to see if you see the words... [LightParams] written on it.
When you see this word, change the number values on it to this (see below).

[LightParams]
Ambient 1.0
Diffuse 1.0
Specular 0.2
SpecularPow 2

Shine 0.0

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Do this to ALL files,

except the two files named... Pilot1 and Rigging
For these 2 files write the following number values....

[LightParams]
Ambient 1.0
Diffuse 1.0
Specular 0.0
SpecularPow 0

Shine 0.0

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This will make your 3d-model look a little less shiny and less bright and have a more "natural look".

Some of the texture skins used for this airplane need their colors to be changed to have less bright colors
and less saturation of colors, so to make the wood parts and canvas parts have a more natural look.

But this will need for someone to make a new skin-template or use Photoshop to do some color channel tweaks.

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Edlor

• Modder
• member
• Offline
• Posts: 859
Re: Demoiselle
« Reply #25 on: May 29, 2017, 08:42:18 PM »

Thanks all.

Glad you like it and that screenshot should be framed and hung in a museum Alfie. I really like that.

As to the Gunbus I've got too many projects on the go to take on another one at this time. If one doesn't show up by the time I get caught up I'd be happy to bring it to life then.

I'm happy to hear you got it working Uzin and I'll be looking forward to your Irkutsk map once you're done your testing.

Thanks Max and thanks for the tips on adjusting the lightparams values. They'll definitely come in handy for future projects.

Cheers and thanks again,

Edlor
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stiv_069

• Skinner
• member
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• Posts: 461
Re: Demoiselle
« Reply #26 on: June 01, 2017, 10:06:23 AM »

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Alfie Noakes

• Art Director
• member
• Offline
• Posts: 2560
Re: Demoiselle
« Reply #27 on: June 02, 2017, 12:22:58 AM »

Wow....excellent skin for the Demoiselle  stiv_69
Superb detailing......is it completed ?.......are you planning to release it ?

Cheers

Alfie
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stiv_069

• Skinner
• member
• Offline
• Posts: 461
Re: Demoiselle
« Reply #28 on: June 02, 2017, 12:35:20 AM »

Wow....excellent skin for the Demoiselle  stiv_69
Superb detailing......is it completed ?.......are you planning to release it ?

Cheers

Alfie
I'm no. But I know where you can get it.
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UberDemon

• UberDemon/danzigzag
• Modder
• member
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• Posts: 758
Re: Demoiselle
« Reply #29 on: June 05, 2017, 09:09:09 PM »

Muito legal!

Vi uma copia to Demoiselle em Paris.  Vai tentar o 14 bis?

[Very nice.  I saw a copy of the Demoseille in Paris.  Are you going to try making the 14 bis? (Also from Santos Dummont)]
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Edlor

• Modder
• member
• Offline
• Posts: 859
Re: Demoiselle
« Reply #30 on: June 06, 2017, 07:20:50 AM »

Thanks UberDemon.

Code: [Select]
I saw a copy of the Demoseille in Paris.  Are you going to try making the 14 bis? (Also from Santos Dummont)]
That would have been cool to see and no plans for the 14 Bis at this time.

Cheers,

Edlor
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David Prosser

• member
• Offline
• Posts: 3538
Re: Demoiselle
« Reply #31 on: June 07, 2017, 02:43:45 AM »

Every time I see a clip of the 14 bis flying, I think. No way. It's facing the wrong way. Not going to happen. Then, it takes off. Simply amazing.

fighter1976

• member
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• Posts: 86
Re: Demoiselle
« Reply #32 on: June 15, 2017, 12:44:21 PM »

Hi! Have problem: when mission starts, the airplane shakes violently and explode, even with throttle at idle
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Edlor

• Modder
• member
• Offline
• Posts: 859
Re: Demoiselle
« Reply #33 on: June 15, 2017, 11:43:36 PM »

Hi,

Could you provide more info such as which game version you're using and a log file.
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fighter1976

• member
• Offline
• Posts: 86
Re: Demoiselle
« Reply #34 on: June 16, 2017, 12:22:44 AM »

I'm running DBW 1.71
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fighter1976

• member
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• Posts: 86
Re: Demoiselle
« Reply #35 on: June 16, 2017, 12:23:53 AM »

About the log, I don't know how to do it( or find it).Sorry
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text:
# Does there always exist a monochromatic solution to ma+mb = nc+nd when m,n are coprime and N is coloured using 4 colours?

Let $$m \ge 2 ,n \ge 2$$ be positive integers which are coprime (that means that the greatest common divisor of $$m,n$$ is $$1$$). Is it possible to paint the set $$\mathbf{N}$$ of all natural numbers using $$4$$ colours such that there are no positive integers $$a,b,c,d$$ of the same colour satisfying $$ma+mb = nc+nd?$$

The answer is YES for the specific case when $$m=3, n=2.$$ For instance if $$h \in \mathbf{N}$$ is of the form $$h = 5^m (5k+i)$$ for $$i \in \{1,2,3,4 \}$$ then if we paint $$h$$ using the $$i$$ th color then by chasing the coefficient of $$i$$ it is easy to see that we do not have a monochromatic solution to the equation $$3a+3b=2c+2d.$$

In fact if one goes through the proof for this particular case then it seems to imply a much weaker relation where $$4$$ gets replaced by a larger number of colors.

• Why 4 colors specifically? Do you know the answer when you use fewer or more than 4 colors? – Wojowu Feb 22 at 18:39
• This might be relevant sfu.ca/~vjungic/RamseyNotes/sec_RadoThm.html; proposition 4.4.11 – Vlad Matei Feb 23 at 6:53
• @VladMatei The last proposition there (Rado's Theorem) implies the claim even with any finite number of colors: take $(c_1,c_2,c_3,c_4)$ in the paper equal to the $(m,m,-n,-n)$ in the question. Even though the question seems more suitable for math.stackexchange.com you probably should make this into an answer. – Yaakov Baruch Feb 23 at 7:36
• I doubt that. For the case when $m=3,n=2$ we do have a coloring. For instance if $n = 5^m (5k + i)$ for $i \in \{1,2,3,4 \}$ then we paint the natural number $n$ with the $i$ th color. It is easy to see that for this coloring we do not have a monochromatic solution to $3a+3b=2c+2d.$ – Aditya Guha Roy Feb 23 at 8:02
• My comment was meant to show that for sufficiently many colors it is easy. It does not provide an answer for $4$ colors. – Vlad Matei Feb 23 at 8:31


text:
Search by Topic

Resources tagged with Working systematically similar to Domino Magic Rectangle:

Filter by: Content type:
Stage:
Challenge level:

There are 339 results

Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically

Seasonal Twin Sudokus

Stage: 3 and 4 Challenge Level:

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

Corresponding Sudokus

Stage: 3, 4 and 5

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

Intersection Sums Sudoku

Stage: 2, 3 and 4 Challenge Level:

A Sudoku with clues given as sums of entries.

Twin Corresponding Sudoku III

Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

Twin Corresponding Sudokus II

Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

Twin Corresponding Sudoku

Stage: 3, 4 and 5 Challenge Level:

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

Instant Insanity

Stage: 3, 4 and 5 Challenge Level:

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Wallpaper Sudoku

Stage: 3 and 4 Challenge Level:

A Sudoku that uses transformations as supporting clues.

Integrated Sums Sudoku

Stage: 3 and 4 Challenge Level:

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

Twinkle Twinkle

Stage: 2 and 3 Challenge Level:

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Rectangle Outline Sudoku

Stage: 3 and 4 Challenge Level:

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

Intersection Sudoku 2

Stage: 3 and 4 Challenge Level:

A Sudoku with a twist.

Diagonal Sums Sudoku

Stage: 2, 3 and 4 Challenge Level:

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

Crossing the Bridge

Stage: 3 Challenge Level:

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

Sandwiches

Stage: 2, 3, 4 and 5 Challenge Level:

Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.

First Connect Three

Stage: 2 and 3 Challenge Level:

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

Cayley

Stage: 3 Challenge Level:

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Colour in the Square

Stage: 2, 3 and 4 Challenge Level:

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Diagonal Product Sudoku

Stage: 3 and 4 Challenge Level:

Given the products of diagonally opposite cells - can you complete this Sudoku?

You Owe Me Five Farthings, Say the Bells of St Martin's

Stage: 3 Challenge Level:

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Ratio Sudoku 1

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios.

Teddy Town

Stage: 1, 2 and 3 Challenge Level:

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Ratio Sudoku 3

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios or fractions.

Intersection Sudoku 1

Stage: 3 and 4 Challenge Level:

A Sudoku with a twist.

Colour Islands Sudoku 2

Stage: 3, 4 and 5 Challenge Level:

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

Bochap Sudoku

Stage: 3 and 4 Challenge Level:

This Sudoku combines all four arithmetic operations.

Stage: 3 and 4 Challenge Level:

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Ratio Sudoku 2

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios.

Creating Cubes

Stage: 2 and 3 Challenge Level:

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Games Related to Nim

Stage: 1, 2, 3 and 4

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Tetrahedra Tester

Stage: 3 Challenge Level:

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Tea Cups

Stage: 2 and 3 Challenge Level:

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

When Will You Pay Me? Say the Bells of Old Bailey

Stage: 3 Challenge Level:

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Stage: 1 and 2 Challenge Level:

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Triangles to Tetrahedra

Stage: 3 Challenge Level:

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

The Third Dimension

Stage: 1 and 2 Challenge Level:

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Colour Islands Sudoku

Stage: 3 Challenge Level:

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

Star Product Sudoku

Stage: 3 and 4 Challenge Level:

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

Stage: 3 Challenge Level:

A few extra challenges set by some young NRICH members.

Pole Star Sudoku 2

Stage: 3 and 4 Challenge Level:

This Sudoku, based on differences. Using the one clue number can you find the solution?

Fault-free Rectangles

Stage: 2 Challenge Level:

Find out what a "fault-free" rectangle is and try to make some of your own.

One to Fifteen

Stage: 2 Challenge Level:

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Reach 100

Stage: 2 and 3 Challenge Level:

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Code Breaker

Stage: 2 Challenge Level:

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Shape Times Shape

Stage: 2 Challenge Level:

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

Difference

Stage: 2 Challenge Level:

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Multiples Sudoku

Stage: 3 Challenge Level:

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

Magic Potting Sheds

Stage: 3 Challenge Level:

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Four Triangles Puzzle

Stage: 1 and 2 Challenge Level:

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?


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# Learning Certified Control using Contraction Metric

Abstract: In this paper, we solve the problem of finding a certified control policy that drives a robot from any given initial state and under any bounded disturbance to the desired reference trajectory, with guarantees on the convergence or bounds on the tracking error. Such a controller is crucial in safe motion planning. We leverage the advanced theory in Control Contraction Metric and design a learning framework based on neural networks to co-synthesize the contraction metric and the controller for control-affine systems. We further provide methods to validate the convergence and bounded error guarantees. We demonstrate the performance of our method using a suite of challenging robotic models, including models with learned dynamics as neural networks. We compare our approach with leading methods using sum-of-squares programming, reinforcement learning, and model predictive control. Results show that our methods indeed can handle a broader class of systems with less tracking error and faster execution speed. Code is available at https://github.com/sundw2014/C3M. The paper is available at https://arxiv.org/abs/2011.12569.

## Problem definition

Formally, given a control-affine system $\dot{x} = f(x) + B(x)u$, we want to find a feedback controller $u(\cdot,\cdot,\cdot)$, so that for any reference $(x^*(t), u^*(t))$ solving the ODE, the closed-loop system perturbed by $d$ $$\dot{x} = f(x(t)) + B(x(t))u(x(t),x^*(t),u^*(t)) + d(t)$$ satisfies that the tracking error $|x(t) – x^*(t)|$ is upper bounded, for all $|d(t)| \leq \epsilon$ and all initial condition $x(0) \in \mathcal{X}$.

## Contraction analysis

Contraction analysis can be viewed as a differential version of Lyapunov’s theory. It analyzes the incremental stability by considering the evolution of the distance between two neighboring trajectories of the system. Lyapunov’s theory considers whether the trajectories will finally converge to a point (equilibrium), while contraction theory considers whether all the trajectories converge to a common trajectory. Control contraction metric (CCM) theory extends the analysis to cases with control input, which enables tracking controller synthesis.

Consider a system $\dot{x} = f(x) + B(x) u$, its virtual displacement $\delta_x$ between any pair of arbitrarily close neighboring trajectories evolves as $\dot{\delta}_x = A(x, u) \delta_x + B(x) \delta_u$, where $A(x,u) := \frac{\partial f}{\partial x} + \sum_{i=1}^{m}u^i\frac{\partial b_i}{\partial x}$.

We say $M : \mathcal{X} \mapsto \mathbb{S}_n^{\geq 0}$ is a CCM if there exists a controller $u(x,x^*,u^*)$ s.t. $\forall x, x^*, u^* \in \mathcal{X} \times \mathcal{X} \times \mathcal{U}$, $\forall \delta_x \in \mathcal{T}_{x}{\mathcal{X}}$, $$\frac{d}{dt}\left(\delta_x^\intercal M(x) \delta_x\right) \leq -\lambda \delta_x^\intercal M(x) \delta_x,$$ which implies $\|\delta_x\|_M := \sqrt{\delta_x^\intercal M(x) \delta_x}$ converges to zero exponentially at rate $\lambda$. Such a closed-loop system is referred to be contracting (under metric $M$), which implies good tracking performance.

$$\frac{d}{dt}\left(\delta_x^\intercal M(x) \delta_x\right) \leq -\lambda \delta_x^\intercal M(x) \delta_x,\, \forall \delta_x,\, \forall x,x^*,u^*,$$ $$\dot{\delta}_x = A(x, u) \delta_x + B(x) \delta_u,$$ is equivalent to $\forall x,x^*,u^*$, $$\dot{M} + \mathtt{sym}\left(M(A+BK)\right) + 2 \lambda M \preceq 0,~~~~~(*)$$ where $K := \frac{\partial u}{\partial x}$ and $\mathtt{sym}\left(A\right) = A + A^\intercal$.

## Learning a controller with a controller

The loss function is designed as follows.

$$\begin{multline*} \mathcal{L}_{M,u} (\theta_M, \theta_u) := \\\mathop{\mathbb{E}}_{x,x^*,u^* \sim \mathtt{Unif}}\left[L_{NSD}\left(\dot{M} + \mathtt{sym}\left(M(A+BK)\right) + 2 \lambda M\right)\right], \end{multline*}$$ where $L_{NSD} : \mathbb{R}^{n \times n} \mapsto \mathbb{R}_{\geq}$ and $L_{NSD}(A) = 0$ iff. matrix $A$ is negative semi-definite.

Obviously, by assuming the LHS of $(*)$ is continuous, $$\mathcal{L}_{M,u} = 0~~~\Longleftrightarrow~~~~~(*)\text{ holds } \forall x,x^*,u^* \in \mathcal{X} \times \mathcal{X} \times \mathcal{U}.$$

## Experimental results

The above figures show the tracking error of the proposed method and several others. The proposed method outperforms all the others. It is worth mentioning that $\texttt{Quadrotor}$ is a $9$-dimensional system, and part of the dynamics of $\texttt{Neural Lander}$ is represented as an NN.


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How to use modular polynomials in Sage?

I want to perform some calculations using modular polynomials, and I saw that Sage already offers some databases for this. Most notably, I came across this. But, I couldn't find any documentation or tutorial as how this database is actually used, or what it actually includes. I can do something like this, but it doesn't give much information:

db = ClassicalModularPolynomialDatabase()
print db

Is there any tutorial or documentation as how to use modular polynomials in Sage?