added best Sobolev constants in W^{1,p}(R^n)

data/Analysis/Best_Sobolev_constant_in_W1pRn/generate.sage

@@ -0,0 +1,71 @@
+import yaml
+import os
+from utils.utils import numbers_to_yaml
+from utils.utils import real_interval_to_sage_string
+
+
+path = 'data/Analysis/Best_Sobolev_constant_in_W1pRn'
+
+prec10 = 100 #relative precision in base 10
+
+#p_range = [1..100]
+p_range = [a/b for a,b in cartesian_product([[1..24],[1..24]])
+			if gcd(a,b) == 1 and a >= b]
+q_range = p_range
+
+npq_range = []
+for p in p_range:
+	for q in q_range:
+		if not q > p:
+			continue
+		n = p*q/(q-p)
+		if n not in ZZ:
+			continue
+		if not n > p:
+			continue
+		n = ZZ(n)
+		npq_range.append((n,p,q))
+
+npq_range.sort(key = lambda x: x[1])
+npq_range.sort(key = lambda x: x[0])
+
+n_range = sorted(set(n for n,p,q in npq_range))
+
+print("len(npq_range):", len(npq_range))
+
+RIFprec = RealIntervalField(prec10 * 3.4 * 2)
+RBFprec = RealBallField(RIFprec.prec())
+
+numbers = {}
+for n in n_range:
+	numbers_n = {}
+
+	for p in p_range:
+		if not n > p:
+			continue
+		q = n*p/(n-p)
+		if (n,p,q) not in npq_range:
+			continue
+
+		#print("n,p,q:",n,p,q)
+
+		if p == 1:
+			number = RIFprec(pi)^(1/2) * n / RIFprec(gamma(1+n/2))^(1/n)
+		else:
+			number = RIFprec(pi)^(1/2) * n^(1/p) * ((n-p)/(p-1))^(1-1/p) * \
+						(gamma(n/p)*gamma(n+1-n/p)/gamma(n)/gamma(1+n/2))^(1/n)
+
+		#print("number:",number)
+
+		number_str = real_interval_to_sage_string(
+			RIFprec(number),
+			max_digits = prec10,
+		).replace('?','')
+
+		numbers_n[str(p)] = {str(q): number_str}
+
+	if len(numbers_n) > 0:
+		numbers[str(n)] = numbers_n
+
+filename = os.path.join(path, 'numbers.yaml')
+yaml.dump(numbers, stream = open(filename, 'w'), sort_keys = False)