Medical Gross Anatomy
Anatomy Tables - Arteries of the Abdomen

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Artery Source Branches Supply to Notes
anterior cecal ileocolic a. occasionally the appendicular a. cecum anterior cecal a. supplies the ileocecal junction
anterior inferior pancreaticoduodenal inferior pancreaticoduodenal a. pancreatic brs., duodenal brs. lower duodenum and head of the pancreas anterior inferior pancreaticoduodenal a. anastomoses with the anterior superior pancreaticoduodenal a. to form the anterior pancreatic arcade
anterior inferior pancreaticoduodenal inferior pancreaticoduodenal a. pancreatic brs., duodenal brs. lower duodendum and head of the pancreas anterior inferior pancreaticoduodenal a. anastomoses with the anterior superior pancreaticoduodenal a. to form the anterior pancreatic arcade
anterior superior pancreaticoduodenal gastroduodenal a. pancreatic brs., duodenal brs. upper duodenum and head of the pancreas anterior superior pancreaticoduodenal a. anastomoses with the anterior inferior pancreaticoduodenal a. to form the anterior pancreatic arcade
aorta, abdominal the continuation of the descending thoracic aorta inferior phrenic aa. (2), celiac trunk, middle suprarenal aa. (2), lumbar aa. (4 pairs), superior mesenteric a., renal aa. (2), testicular/ovarian aa. (2), inferior mesenteric a., median sacral a., common iliac aa. (2) abdominal wall; gastrointestinal tract; body below the level of the respiratory diaphragm abdominal aorta passes posterior to the diaphragm (aortic hiatus) at the level of the T12 vertebral body
appendicular posterior cecal, anterior cecal or ileocolic no named branches vermiform appendix despite its variable origin, appendicular artery is nearly constant in its course posterior to the terminal part of the ileum
artery of the ductus deferens umbilical a.; may be a branch of the superior or inferior vesical a. no named branches ductus deferens, seminal vesical; possibly supplies the ureter artery of the ductus deferens is also known as: deferential a.
caudal pancreatic splenic a. numerous small aa. that supply the pancreas tail of the pancreas caudal pancreatic a. enters the tail of the pancreas near the hilum of the spleen; it may arise from the left gastro-omental a.
cecal, anterior ileocolic a. may give off the appendicular a. anterior surface of the cecum anterior cecal a. supplies the ileocecal junction
cecal, posterior ileocolic a. may give off the appendicular a. posterior surface of the cecum posterior cecal a. supplies the ileocecal junction
celiac trunk abdominal aorta at the level of the T12-L1 intervertebral disc left gastric a., splenic a., common hepatic a. stomach, lower esophagus, liver, upper duodenum, pancreas, spleen celiac trunk supplies the foregut derivatives
circumflex iliac, deep external iliac a. unnamed muscular branches iliacus muscle and the lower abdominal wall deep circumflex iliac a. courses along the iliac crest on the inner surface of the abdominal wall
circumflex iliac, superficial femoral a. unnamed muscular branches superficial fascia of lower abdomen and thigh superficial circumflex iliac a. crosses the upper thigh parallel to the inguinal ligament
colic, left inferior mesenteric a. ascending br., descending br. descending colon left colic a. anastomoses with the middle colic a and the sigmoid a.to form part of the marginal a.
colic, middle superior mesenteric right br., left br. transverse colon middle colic a. anastomoses with the right colic a. and the left colic a. to form part of the marginal a.
colic, right superior mesenteric (or br. of) ascending br., descending br. ascending colon right colic a. anastomoses with the ileocolic a. and the middle colic a. to form part of the marginal a.
common hepatic celiac trunk gastroduodenal a., proper hepatic a. liver, upper part of the duodenum, part of the pancreas, part of the stomach common hepatic a. is one of three brs. of the celiac trunk (see also: left gastric a., splenic a.)
common iliac abdominal aorta external iliac a., internal iliac a. pelvis, lower limb abdominal aorta bifurcates at the level of the L4 vertebral body to form the right and left common iliac aa.; the common iliac artery bifurcates anterior to the sacroiliac articulation into its terminal brs. (external iliac a. and internal iliac a.)
cremasteric inferior epigastric a. no named branches cremaster m., coverings of the spermatic cord cremasteric a. is one of the contents of the spermatic cord; it anastomoses with the testicular artery distally
cystic right hepatic a. (or superior mesenteric a., proper hepatic a., left hepatic a., gastroduodenal a.) no named branches gall bladder cystic a. arises from the right hepatic a. in ~72% of cases; other possible origins are noted at left and are clinically relevant during gall bladder surgery
dorsal pancreatic splenic a. right br., left br. neck of the pancreas right br.of the dorsal pancreatic a. anastomoses with the anterior superior pancreaticoduodenal a. in the prepancreatic arcade
ductus deferens, artery of umbilical a.; may be a branch of the superior or inferior vesical a. no named branches ductus deferens, seminal vesical; possibly supplies the ureter artery of the ductus deferens is also known as: deferential a.
epigastric, inferior external iliac a. cremasteric a. lower rectus abdominis m., pyramidalis m., lower abdominal wall inferior epigastric a. anastomoses with the superior epigastric a. within the rectus abdominis m.
epigastric, superficial femoral a. cutaneous brs. superficial fascia and skin of the lower abdominal wall superficial epigastric a. is one of three superficial arteries that arise from the femoral a. (see also: superficial circumflex iliac a. and superficial external pudendal a.)
epigastric, superior internal thoracic a. no named branches upper rectus abdominis m., upper abdominal wall superior epigastric a. is the direct continuation of the internal thoracic a.; it s anastomoses with the inferior epigastric a. within the rectus abdominis m.
esophageal descending thoracic aorta; left gastric a. no named branches lower 2/3rds of the thoracic esophagus there are usually two (sometimes 3) esophageal brs. off of the aorta; the left gastric a. gives esophageal brs. that ascend to supply the abdominal esophagus and lowest part of the thoracic esophagus; anastomoses occur between the esophageal brs. of the aorta and left gastric a.
gastric, left celiac a. esophageal brs. superior part of the stomach near the lesser curvature left gastric a. anastomoses with the right gastric a. on the lesser curvature of the stomach; it also anastomoses with the esophageal brs. of the thoracic aorta
gastric, right proper hepatic a. no named branches inferior part of the stomach near the lesser curvature right gastric a. anastomoses with the left gastric a. on the lesser curvature of the stomach
gastric, short splenic a. no named branches fundus and upper part of the stomach near the greater curvature short gastric aa. reach the stomach by passing through the gastrosplenic ligament; they are usually 4-5 in number
gastro-omental, left splenic a. gastric brs., omental brs. mid-part of the stomach near the greater curvature; greater omentum left gastro-omental a. anastomoses with the right gastro-omental a. near the greater curvature of the stomach; also known as: left gastroepiploic a.
gastro-omental, right gastroduodenal a. gastric brs., omental brs. lower part of the pyloric region of the stomach, right part of greater curvature of the stomach, greater omentum right gastro-omental a anastomoses with the left gastro-omental a. near the greater curvature of the stomach; also known as: right gastroepiploic a.
gastroduodenal common hepatic a. supraduodenal aa., retroduodenal aa., posterior superior pancreaticoduodenal a., anterior superior pancreaticoduodenal a., right gastro-omental a. upper duodenum, upper part of the head of the pancreas; greater curvature of the stomach on the right gastroduodenal a. supplies the lowest portion of the foregut and its derivatives
great anterior radicular the spinal br. of the lower posterior intercostal a., subcostal a., or upper lumbar a., usually on left no named branches lower spinal cord great anterior radicular a. anastomoses with the anterior spinal a. at lower thoracic or upper lumbar spinal cord levels
hepatic, common celiac trunk gastroduodenal a., proper hepatic a. liver, upper parts of the duodenum, upper part of the pancreas, right side of the stomach common hepatic a. supplies some of the foregut derivatives
hepatic, left proper hepatic a. segmental aa. left lobe of the liver, quadrate lobe of the liver, part of the caudate lobe of the liver left hepatic a. supplies the liver parenchyma and stroma; an aberrant left hepatic a. may arise from the left gastric a.
hepatic, proper common hepatic a. right gastric, right & left hepatic liver, lesser curvature of the stomach proper hepatic a. supplies the foregut derivatives associated with the liver bud
hepatic, right proper hepatic a. cystic a., segmental aa. right lobe of the liver, part of the caudate lobe of the liver right hepatic a. supplies the liver parenchyma and stroma
ileocolic superior mesenteric a. colic br., anterior cecal br., posterior cecal br., appendicular a., ileal br. cecum, appendix, terminal portion of the ileum colic br. of the ileocolic a. participates in the formation of the marginal a.
iliac, common abdominal aorta external iliac a., internal iliac a. pelvis, lower limb abdominal aorta bifurcates at the level of the L4 vertebral body to form the right and left common iliac aa.; the common iliac artery bifurcates anterior to the sacroiliac articulation into its terminal brs. (external iliac a. and internal iliac a.)
iliac, external common iliac a. inferior epigastric a., deep circumflex iliac a., femoral a. lower limb external iliac a. is continuous with the femoral a., the name change occurs at the inguinal ligament; the common iliac artery bifurcates anterior to the sacroiliac articulation
iliac, internal common iliac a. anterior division gives rise to the : umbilical a., obturator a., uterine a., vaginal a., inferior vesical a., middle rectal a., internal pudendal a., inferior gluteal a.; posterior division gives rise to the: iliolumbar a., lateral sacral a., superior gluteal a. pelvic viscera, gluteal region, hip, medial thigh common iliac artery bifurcates anterior to the sacroiliac articulation to form the internal iliac a. and the external iliac a.
iliolumbar internal iliac a., posterior division iliac br., lumbar br. iliacus m., psoas major m., quadratus lumborum m. lumbar br. of the iliolumbar a. sends a small spinal br. into the vertebral canal
inferior epigastric external iliac a. cremasteric a. lower rectus abdominis m., pyramidalis m., lower abdominal wall inferior epigastric m. anastomoses with the superior epigastric m. within the rectus abdominis m.
inferior mesenteric abdominal aorta at the level of the L3 vertebral body left colic a., sigmoid aa.(2-3), superior rectal a. splenic flexure, descending colon, sigmoid colon, superior part of rectum branches of the inferior mesenteric a anastomose in the marginal artery
inferior pancreatic dorsal pancreatic, left br. no named branches lower part of the body of the pancreas inferior pancreatic a. may be imbedded in the substance of the gland
inferior pancreaticoduodenal superior mesenteric a. anterior inferior pancreaticoduodenal a., posterior inferior pancreaticoduodenal a. lower duodenum, head of the pancreas inferior pancreaticoduodenal a. supplies both the anterior and posterior surfaces of the lower duodenum and lower part of the head of the pancreas
inferior phrenic abdominal aorta superior suprarenal aa. diaphragm, suprarenal gland inferior phrenic a. is the first abdominal branch of the aorta; it may arise from the celiac trunk
inferior rectal internal pudendal a. no named branches anus, ischioanal fossa inferior rectal a. anastomoses with the middle rectal a. and the superior rectal a.
inferior suprarenal renal a. numerous unnamed branches inferior aspect of the suprarenal gland inferior suprarenal a. may anastomose with the superior suprarenal a.
intercostal, posterior highest intercostal (upper 2 intercostal spaces), descending thoracic aorta (3rd-11th intercostal spaces) posterior br., spinal br., anterior br., collateral br., lateral cutaneous br. intercostal muscles, spinal cord and vertebral column, deep back muscles, skin and superficial fascia overlying the intercostal spaces posterior intercostal aa. supply the lateral and posterior portions of the intercostal space; anterior intercostal aa. supply the anterior portions of the intercostal spaces
internal iliac common iliac a. anterior division gives rise to the : umbilical a., obturator a., uterine a., vaginal a., inferior vesical a., middle rectal a., internal pudendal a., inferior gluteal a.; posterior division gives rise to the: iliolumbar a., lateral sacral a., superior gluteal a. pelvic viscera, gluteal region, hip, medial thigh common iliac artery bifurcates anterior to the sacroiliac articulation to form the internal iliac a. and the external iliac a.
intestinal superior mesenteric a. arterial arches jejunum, ileum intestinal aa. are 12-15 in number and are found in the mesentery
lateral sacral internal iliac a., posterior division spinal brs. sacrum, sacral nerve rootlets, meninges, adjacent muscles there are usually 2 lateral sacral aa. on each side, a superior one and an inferior one
left colic inferior mesenteric a. ascending br., descending br. descending colon left colic a. anastomoses with the middle colic a and the sigmoid a.to form part of the marginal a.
left gastric celiac a. esophageal brs. superior part of the stomach near the lesser curvature left gastric a. anastomoses with the right gastric a. on the lesser curvature of the stomach; it also anastomoses with the esophageal brs. of the thoracic aorta
left gastro-omental splenic a. gastric brs., omental brs. mid-part of the stomach near the greater curvature; greater omentum left gastro-omental a. anastomoses with the right gastro-omental a. near the greater curvature of the stomach; also known as: left gastroepiploic a.
left hepatic proper hepatic a. segmental aa. left lobe of the liver, quadrate lobe of the liver, part of the caudate lobe of the liver left hepatic a. supplies the liver parenchyma and stroma; an aberrant left hepatic a. may arise from the left gastric a.
lumbar abdominal aorta anterior br., posterior br., spinal br. psoas major m., psoas minor m., quadratus lumborum m., spinal cord and vertebral column, deep back muscles the 4 lumbar arteries on each side arise from the posterior surface of the aorta at the level of vertebrae L1-L4; they course posterior to the psoas major m.
marginal formed by anastomoses of branches of the ileocolic a., right colic a., middle colic a., left colic a., sigmoid a. colic brs. colon an important anastomosis for the large intestine
median sacral abdominal aorta 5th lumbar aa. sacrum median sacral a. appears to be the continuation of the abdominal aorta in the median plane, although it is much smaller in size
mesenteric, inferior abdominal aorta at the level of the L3 vertebral body left colic a., sigmoid aa.(2-3), superior rectal a. splenic flexure, descending colon, sigmoid colon, superior part of rectum branches of the inferior mesenteric a anastomose in the marginal artery
mesenteric, superior abdominal aorta at the level of the lower 1/3 of the L1 vertebral body inferior pancreaticoduodenal a., middle colic a., jejunal a., ileal a., right colic a., ileocolic a. inferior part of the head of the pancreas, distal duodenum, jejunum, ileum, cecum, appendix, ascending colon, transverse colon superior mesenteric a. supplies the midgut derivatives; brs. of the superior mesenteric a. participate in formation of the marginal artery
middle colic superior mesenteric right br., left br. transverse colon middle colic a. anastomoses with the right colic a. and the left colic a. to form part of the marginal a.
middle rectal internal iliac, anterior division no named branches middle portion of the rectum middle rectal a. anastomoses with the inferior rectal a and the superior rectal a.
middle suprarenal abdominal aorta numerous unnamed branches medial side of the suprarenal gland middle suprarenal a. anastomoses with the superior suprarenal a. and the inferior suprarenal a.
musculophrenic internal thoracic a. anterior intercostal aa. anterior diaphragm, anterior aspects of intercostal spaces 7-10 or 11 musculophrenic a. supplies muscles that develop in the septum transversum
pancreatic, caudal splenic a. numerous small aa. that supply the pancreas tail of the pancreas caudal pancreatic a. enters the tail of the pancreas near the hilum of the spleen; it may arise from the left gastro-omental a.
pancreatic, dorsal splenic a. right br., left br. neck of the pancreas right br.of the dorsal pancreatic a. anastomoses with the anterior superior pancreaticoduodenal a. in the prepancreatic arcade
pancreatic, inferior dorsal pancreatic, left br. no named branches lower part of the body of the pancreas inferior pancreatic a. may be imbedded in the substance of the gland
pancreatic, superior splenic a. no named branches superior part of the body of the pancreas superior pancreatic aa. are multiple small brs. arising from the splenic a.
pancreatica magna splenic a. numerous unnamed brs. within the pancreas body of the pancreas pancreatic magna is the largest pancreatic branch to arise from the splenic a.; it enters the pancreas at the junction of its middle 1/3 and left 1/3
pancreaticoduodenal, anterior inferior inferior pancreaticoduodenal a. pancreatic brs., duodenal brs. lower duodenum and head of the pancreas anterior inferior pancreaticoduodenal a. anastomoses with the anterior superior pancreaticoduodenal a. to form the anterior pancreatic arcade
pancreaticoduodenal, anterior superior gastroduodenal a. pancreatic brs., duodenal brs. upper duodenum and head of the pancreas anterior superior pancreaticoduodenal a. anastomoses with the anterior inferior pancreaticoduodenal a. to form the anterior pancreatic arcade
pancreaticoduodenal, inferior superior mesenteric a. anterior inferior pancreaticoduodenal a., posterior inferior pancreaticoduodenal a. lower duodenum, head of the pancreas inferior pancreaticoduodenal a. supplies both the anterior and posterior surfaces of the lower duodenum and lower part of the head of the pancreas
pancreaticoduodenal, posterior inferior inferior pancreaticoduodenal a. pancreatic brs., duodenal brs. lower duodenum and head of the pancreas posterior inferior pancreaticoduodenal a. anastomoses with the posterior superior pancreaticoduodenal a. to form the posterior pancreatic arcade
pancreaticoduodenal, posterior superior gastroduodenal a. pancreatic brs., duodenal brs. upper duodenum and head of the pancreas posterior superior pancreaticoduodenal a. anastomoses with the posterior inferior pancreaticoduodenal a. to form the posterior pancreatic arcade
phrenic, inferior abdominal aorta superior suprarenal aa. diaphragm, suprarenal gland inferior phrenic a. is the first abdominal branch of the aorta; it may arise from the celiac trunk
posterior cecal ileocolic a. may give off the appendicular a. posterior surface of the cecum posterior cecal a. supplies the ileocecal junction
posterior inferior pancreaticoduodenal inferior pancreaticoduodenal a. pancreatic brs., duodenal brs. lower duodenum and head of the pancreas posterior inferior pancreaticoduodenal a. anastomoses with the posterior superior pancreaticoduodenal a. to form the posterior pancreatic arcade
posterior intercostal highest intercostal (upper 2 intercostal spaces), descending thoracic aorta (3rd-11th intercostal spaces) posterior br., spinal br., anterior br., collateral br., lateral cutaneous br. intercostal muscles, spinal cord and vertebral column, deep back muscles, skin and superficial fascia overlying the intercostal spaces posterior intercostal aa. supply the lateral and posterior portions of the intercostal space; anterior intercostal aa. supply the anterior portions of the intercostal spaces
posterior superior pancreaticoduodenal gastroduodenal a. pancreatic brs., duodenal brs. upper duodenum and head of the pancreas posterior superior pancreaticoduodenal a. anastomoses with the posterior inferior pancreaticoduodenal a. to form the posterior pancreatic arcade
radicular, great anterior the spinal br. of the lower posterior intercostal a., subcostal a., or upper lumbar a., usually on left no named branches lower spinal cord great anterior radicular a. anastomoses with the anterior spinal a. at lower thoracic or upper lumbar spinal cord levels
rectal, inferior internal pudendal a. no named branches anus, ischioanal fossa inferior rectal a. anastomoses with the middle rectal a. and the superior rectal a.
rectal, middle internal iliac, anterior division no named branches middle portion of the rectum middle rectal a. anastomoses with the inferior rectal a and the superior rectal a.
rectal, superior inferior mesenteric a. two unnamed branches superior part of the rectum superior rectal a. is the continuation of the inferior mesenteric a. after the sigmoid brs. are given off; it anastomoses with the middle rectal a. and the inferior rectal a.
renal abdominal aorta at the level of the upper border of the L2 vertebra inferior suprarenal a., apical segmental a., anterior br: superior anterior segmental a., middle anterior segmental a., inferior segmental a.; posterior br.: posterior segmental a. kidney, upper ureter, suprarenal gland the right renal a. is longer than the left renal artery; the right renal artery passes posterior to the inferior vena cava
retroduodenal gastroduodenal a. multiple unnamed brs. posterior portion of the 1st part of the duodenum retroduodenal aa. arise as multiple small brs.
right colic superior mesenteric (or br. of) ascending br., descending br. ascending colon right colic a. anastomoses with the ileocolic a. and the middle colic a. to form part of the marginal a.
right gastric proper hepatic a. no named branches inferior part of the stomach near the lesser curvature right gastric a. anastomoses with the left gastric a. on the lesser curvature of the stomach
right gastro-omental gastroduodenal a. gastric brs., omental brs. lower part of the pyloric region of the stomach, right part of greater curvature of the stomach, greater omentum right gastro-omental a anastomoses with the left gastro-omental a. near the greater curvature of the stomach; also known as: right gastroepiploic a.
right hepatic proper hepatic a. cystic a., segmental aa. right lobe of the liver, part of the caudate lobe of the liver right hepatic a. supplies the liver parenchyma and stroma
sacral, lateral internal iliac a., posterior division spinal brs. sacrum, sacral nerve rootlets, meninges, adjacent muscles there are usually 2 lateral sacral aa. on each side, a superior one and an inferior one
sacral, median abdominal aorta 5th lumbar aa. sacrum median sacral a. appears to be the continuation of the abdominal aorta in the median plane, although it is much smaller in size
short gastric splenic a. no named branches fundus and upper part of the stomach near the greater curvature short gastric aa. reach the stomach by passing through the gastrosplenic ligament; they are usually 4-5 in number
sigmoid inferior mesenteric a. ascending br., descending br. sigmoid colon sigmoid aa. are 2-3 in number; they anastomose with the left colic a. to help form the marginal a.
splenic celiac trunk dorsal pancreatic a., pancreatic brs., pancreatica magna a., caudal pancreatic a., short gastric aa., left gastro-omental a., splenic brs. pancreas, spleen, greater curvature of the stomach splenic a. courses through the substance of the pancreas near its superior border; splenic artery is very tortuous
subcostal descending thoracic aorta spinal br., collateral br., lateral cutaneous br. vertebrae, spinal cord; muscles, skin & fascia of the upper abdominal wall subcostal a. is equivalent to a posterior intercostal a., but is named subcostal because it courses inferior to the 12th rib
superficial epigastric femoral a. cutaneous brs. superficial fascia and skin of the lower abdominal wall superficial epigastric a. is one of three superficial arteries that arise from the femoral a. (see also: superficial circumflex iliac a. and superficial external pudendal a.)
superior mesenteric abdominal aorta at the level of the lower 1/3 of the L1 vertebral body inferior pancreaticoduodenal a., middle colic a., jejunal a., ileal a., right colic a., ileocolic a. inferior part of the head of the pancreas, distal duodenum, jejunum, ileum, cecum, appendix, ascending colon, transverse colon superior mesenteric a. supplies the midgut derivatives; brs. of the superior mesenteric a. participate in formation of the marginal artery
superior pancreatic splenic a. no named branches superior part of the body of the pancreas superior pancreatic aa. are multiple small brs. arising from the splenic a.
superior rectal inferior mesenteric a. two unnamed branches superior part of the rectum superior rectal a. is the continuation of the inferior mesenteric a. after the sigmoid brs. are given off; it anastomoses with the middle rectal a. and the inferior rectal a.
superior suprarenal inferior phrenic a. multiple small aa. superior aspect of the suprarenal gland superior suprarenal aa. are numerous small vessels that enter the suprarenal gland independently
supraduodenal gastroduodenal a. no named branches superior portion of the 1st part of the duodenum supraduodenal a. supplies branches to both the anterior and posterior surfaces of the duodenum
suprarenal, inferior renal a. numerous unnamed branches inferior aspect of the suprarenal gland inferior suprarenal a. may anastomose with the superior suprarenal a.
suprarenal, middle abdominal aorta numerous unnamed branches medial side of the suprarenal gland middle suprarenal a. anastomoses with the superior suprarenal a. and the inferior suprarenal a.
suprarenal, superior inferior phrenic a. multiple small aa. superior aspect of the suprarenal gland superior suprarenal aa. are numerous small vessels that enter the suprarenal gland independently
trunk, celiac abdominal aorta at the level of the T12-L1 intervertebral disc left gastric a., splenic a., common hepatic a. stomach, lower esophagus, liver, upper duodenum, pancreas, spleen celiac trunk supplies the foregut derivatives
umbilical internal iliac a., anterior division superior vesical aa., a. of the ductus deferens superior part of the bladder; ductus deferens distal to the branches described at left, the lumen of the umbilical a. becomes obliterated after birth and the remnant of the vessel becomes the medial umbilical ligament
vesical, superior umbilical a. no named branches superior aspect of the bladder superior vesical aa.arise from the umbilical a. proximal to where its lumen becomes obliterated

The material presented in these tables is contained in the book:
MedCharts Anatomy by Thomas R. Gest & Jaye Schlesinger
Published by ILOC, Inc., New York
Copyright © 1995, unauthorized use prohibited.
The excellent editorial assistance of
Dr. Pat Tank, UAMS
is gratefully acknowledged.