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Let K 1(3) = {k : k ≡ 1 (mod 3)}. For w ∈ K 1(3) , a (v, K 1(3) ∪ {w * })-PBD is a pairwise balanced design on v points with block size from the set K 1(3) in which there is at least one block of size w. In this paper, we investigate the existence problem for (v, K 1(3) ∪ {w * })-PBDs and give a complete solution to this problem. As its applications, we solve completely the embedding problem for directed designs DB(4, 1; u)s. In addition, we also apply our (v, K 1(3) ∪ {w * })-PBDs to do embeddings for near resolvable triple systems and nested Steiner triple systems and give unified and simple new proofs of two known theorems. Some new 4-GDDs are constructed as well.Keywords Pairwise balanced design · Group divisible design · Directed block design · Nested design · Embedding · Subdesign AMS Classifications 05B05 · 51E05

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Subdifferential, Fréchet subdifferential, Proximal subdifferential, Perturbed optimization problem, Well-posedness, 49J52, 46N10, 49K27,