Add files using upload-large-folder tool

.gitattributes

@@ -108,3 +108,18 @@ knowledge_base/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf filter=lfs diff=lfs
 knowledge_base/BdE4T4oBgHgl3EQf5Q5A/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 knowledge_base/BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf filter=lfs diff=lfs merge=lfs -text
 knowledge_base/KdFLT4oBgHgl3EQfLC8K/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+VNAyT4oBgHgl3EQf8vqZ/content/2301.00862v1.pdf filter=lfs diff=lfs merge=lfs -text
+B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf filter=lfs diff=lfs merge=lfs -text
+BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf filter=lfs diff=lfs merge=lfs -text
+ytE1T4oBgHgl3EQf4AXn/content/2301.03497v1.pdf filter=lfs diff=lfs merge=lfs -text
+zNFLT4oBgHgl3EQfni-P/content/2301.12128v1.pdf filter=lfs diff=lfs merge=lfs -text
+jdE0T4oBgHgl3EQf7QI8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf filter=lfs diff=lfs merge=lfs -text
+Q9FKT4oBgHgl3EQfiC5d/content/2301.11840v1.pdf filter=lfs diff=lfs merge=lfs -text
+pNE1T4oBgHgl3EQf2AWA/content/2301.03474v1.pdf filter=lfs diff=lfs merge=lfs -text
+KdFLT4oBgHgl3EQfLC8K/content/2301.12010v1.pdf filter=lfs diff=lfs merge=lfs -text
+pNE1T4oBgHgl3EQf2AWA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf filter=lfs diff=lfs merge=lfs -text
+CNFAT4oBgHgl3EQfsx5b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+VNAyT4oBgHgl3EQf8vqZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf filter=lfs diff=lfs merge=lfs -text

0dE5T4oBgHgl3EQfOg4z/content/tmp_files/2301.05497v1.pdf.txt

@@ -0,0 +1,450 @@
+New universal partizan rulesets
+Koki Suetsugu
+January 2023
+Abstract
+Universal partizan ruleset is a ruleset in which every game value of
+partizan games can be appear as a position. So far, generalized konane
+and turning tiles have been proved to be universal partizan rulesets.
+In this paper, we introduce two rulesets go on lattice and beyond the
+door and prove that they are universal partizan rulesets by using game
+tree preserving reduction.
+1
+Introduction
+Combinatorial game theory (CGT) studies two-player perfect information games
+with no chance moves. We say a game is under normal play convention if the
+player who moves last is the winner and a game is partizan game if the options
+for both players can be different in some positions. Here, we introduce some
+definitions and theorems of CGT for later discussion. For more details of CGT,
+see [1, 3].
+In this theory, the two players are called Left and Right. Since the term
+“game” is polysemous, we refer to each position as a game. The description of
+what moves are allowed for a given position is called the ruleset.
+A game is defined by Left and Right options recursively.
+Definition 1.
+• {|} is a game, which is called 0.
+• For games GL
+1 , GL
+2 , . . . , GL
+n, GR
+1 , GR
+2 , . . . , and GR
+m, G = {GL
+1 , GL
+2 , . . . , GL
+n |
+GR
+1 , GR
+2 , . . . , GR
+m} is also a game. GL
+1 , GL
+2 , . . . , GL
+n are called left options
+of G and GR
+1 , GR
+2 , . . . , GR
+m are called right options of G.
+Let G be the set of all games.
+In terms of the player who has a winning strategy, G is separated into four
+sets. Let L, R, N, and P be the set of positions in which Left, Right, the Next
+player, and the Previous player have winning strategies, respectively.
+The sets are called outcomes of the games. Every position belongs to exactly
+one of the four outcomes. For a game G, let o(G) be the outcome of G. We
+define the partial order of outcomes as L > P > R, L > N > R.
+1
+arXiv:2301.05497v1  [math.CO]  13 Jan 2023
+
+The disjunctive sum of games is an important concept in Combinatorial
+Game Theory. For games G and H, a position in which a player makes a move
+for one or the other on their turn is called a disjunctive sum of G and H, or
+G + H. More precisely, it is as follows:
+Definition 2. If the game trees of G and H are isomorphic, then we say these
+games are isomorphic or G ∼= H.
+Definition 3. For games G ∼= {GL
+1 , GL
+2 . . . GL
+n | GR
+1 , GR
+2 , . . . , GR
+m} and H ∼=
+{HL
+1 , HL
+2 . . . , HL
+n′ | HR
+1 , HR
+2 , . . . , HR
+m′}, G + H ∼= {G + HL
+1 , G + HL
+2 , . . . , G +
+HL
+n′, GL
+1 + H, GL
+2 + H, . . . , GL
+n + H | G + HR
+1 , G + HR
+2 , . . . , G + HR
+m′, GR
+1 +
+H, GR
+2 + H, . . . , GR
+m + H}.
+We also define equality, inequality and negative of games.
+Definition 4. If for any X, o(G + X) is the same as o(H + X), then we say
+G = H.
+Definition 5. If o(G+H) ≥ o(H +X) holds for any X, then we say G ≥ H. On
+the other hand, if o(G + H) ≤ o(H + X) holds for any X, then we say G ≤ H.
+We also say G ≷ H if G ̸≥ H and G ̸≤ H.
+Definition 6. For a game G ∼= {GL
+1 , GL
+2 , . . . , GL
+n | GR
+1 , GR
+2 , . . . , GR
+m}, let −G ∼=
+{−GR
+1 , −GR
+2 , −GR
+m | −GL
+1 , −GL
+2 , . . . , −GL
+n}.
+G + (−H) is denoted by G − H.
+It is known that (G, +, =) is an abelian group and (G, ≥, =) is a partial
+order.
+The question arises here, will there be a ruleset in which for any game there
+is a position equal to the game? If the games appearing in each ruleset are
+restricted, then perhaps we should think in a narrower framework.
+In fact,
+however, it is known that for every game, a position equal to the game appears
+in some rulesets.
+1.1
+Universal partizan ruleset
+Definition 7. A ruleset is universal partizan ruleset if every value in G is equal
+to a position of the ruleset.
+Early results showed that generalized konane and turning tiles are
+universal partizan ruleset ([2, 4]). In this study, we will use the latter ruleset.
+Definition 8. The ruleset of turning tiles is as follows:
+• Square tiles are laid out. The front side is red or blue, and the back side
+is black.
+• Some pieces are on tiles.
+• Each player (Left, whose color is bLue and Right, whose color is Red),
+in his/her turn, take a piece and move the piece straight on the tiles of
+his/her color.
+2
+
+• Tiles on which the piece pass over are turned over.
+• The player who moves last is the winner.
+Turning tiles is proved to be universal partizan ruleset even if the number
+of pieces is restricted to be only one.
+To distinguish this ruleset from the ruleset defined below, we will also refer
+to it as blue-red turning tiles.
+For games that use two colors, red and blue, corresponding to two players,
+we often consider a variant that adds green, which can be used by both players.
+For example, in blue-red-green hackenbush, Right can remove red or green
+edges and Left can remove blue or green edges. From this point of view, we
+consider a varant of turning tiles.
+Definition 9. The ruleset of blue-red-green turning tiles is as follows:
+• Square tiles are laid out. The front side is red, blue, or green, and the back
+side is black.
+• Some pieces are on tiles.
+• Each player (Left, whose color is bLue and Right, whose color is Red),
+in his/her turn, take a piece and move the piece straight on the tiles of
+his/her color or of green.
+• Tiles on which the piece pass over are turned over.
+• The player who moves last is the winner.
+Figure 1: Positions in blue-red turning tiles and blue-red-green turn-
+ing tiles
+Figure 1 is an example of positions in blue-red turning tiles and blue-
+red-green turning tiles.
+3
+
+L
+LL
+RＲ
+LL
+RR
+L
+R
+LLLG
+R
+GG
+RR-
+G
+L
+RR
+R
+R
+G
+R
+G
+R
+L
+L
+R
+L
+R
+R
+R
+R
+RR
+R
+R
+R
+RRR
+R
+R
+RR
+R
+G
+G
+RR
+R
+RR
+L
+7
+RR
+R
+一
+R
+G
+RR
+G
+R
+R
+P
+L
+R
+G
+G
+L
+R
+R
+LL
+R
+R
+R
+R
+L
+G
+R
+G
+R
+GObviously, blue-red-green turning tiles is also a universal partizan
+ruleset because every position in blue-red turning tiles can be appear in
+blue-red-green turning tiles.
+As we have seen here, if two rulesets have an inclusion relation in terms of
+the sets of positions, it can be used for proving universality of the rulesets.
+Theorem 1. Let Γ and ∆ be rulesets and assume that Γ be a universal partizan
+ruleset. If for every position g ∈ Γ, there is at least one position in ∆ whose
+game value is the same as g, then ∆ is also a universal partizan ruleset.
+Proof. This is trivial from the definition of universal partizan ruleset.
+Corollary 1. Let Γ and ∆ be rulesets and assume that Γ be a universal partizan
+ruleset. If for every position g ∈ Γ, there is at least one position in ∆ whose
+game tree is the same as g, then ∆ is also a universal partizan ruleset.
+If a ruleset is proved to be universal partizan ruleset by using Corollary 1,
+we say that it is proved by game tree preserving reduction.
+In the next section, we introduce two rulesets and prove that they are univer-
+sal partizan ruleset by game tree preserving reduction. In Secton 3, we conclude
+this study.
+2
+New universal partizan rulesets
+2.1
+Go on lattice
+Definition 10. The rule of go on lattice is as follows:
+• There is a lattice graph. There are pieces on some nodes. The edges are
+colored red, blue, or dotted.
+• A player, in his/her turn, chooses a piece and moves it straight on edges
+colored his/her color.
+• After a piece passed a node, every piece cannot get on or pass the node.
+• If a player moves a piece to a node adjacent to a dotted edge, then the edge
+changes to solid edge colored by the opponent’s color.
+• The player who moves last is the winner.
+Figure 2: Play of go on lattice
+4
+
+17
+MFigure 2 is a play of go on lattice. We use double line for red edges for
+monochrome printing.
+Theorem 2. Go on lattice is a universal partizan ruleset.
+Proof. Let f be a function from a position in turning tiles to a position in
+go on lattice as follows:
+Let G be a position in turning tiles. In f(G) there are as many nodes
+as tiles in G. The tiles in G and the nodes in f(G) are arranged exactly the
+same. For each piece on a tile in G, there is a corresponding piece on the node
+corresponds to the tile. For any adjacent tiles A and B in G, let A′ and B′ are
+corresponding nodes in f(G). If the color of A, and B are the same, then edge
+between A′ and B′ is solid and the same color as A and B. If there is a piece
+on A or B, then the edge between A′ and B′ is solid and the color is the same
+as the other tile. Finally, if the color of A and B are different and no piece is
+on each tile, then the edge between A′ and B′ is dotted line.
+●
+B
+B
+B
+B
+B
+R
+R
+R
+R
+●
+B
+R
+R
+B
+B
+B
+Figure 3: Corresponding positions in turning tiles and go on lattice
+.
+Figure 3 shows this corresponding. Here, the game tree of G and f(G) are
+isomorphic. We prove that every move in one game has a corresponding move
+in the other game.
+Assume that in G Left can move a piece on tile A0 to
+tile An through tiles A1, A2, . . . , An−1. Then, A1, A2, . . . , An are blue tiles. Let
+A′
+0, A′
+1, . . . , A′
+n be the corresponding nodes in f(G). Let (A′, B′) be the edge be-
+tween A′ and B′. Then, from the definition of f, all of (A′
+1, A′
+2), (A′
+2, A′
+3), . . . , (A′
+n−1, A′
+n)
+are blue edge. In addition, if A0 was a blue tile before turning, then (A′
+0, A′
+1)
+is a blue edge, and if A0 was a red tile, (A′
+0, A′
+1) had been a dot edge and
+after Right moved the piece to A′
+0, it changed to a blue edge. Therefore for
+both case, (A′
+0, A′
+1) is a blue edge and Left can move a piece from A′
+0 to A′
+n
+through A′
+1, A′
+2, . . . , A′
+n−1. Conversely, assume that in f(G), Left can move a
+piece from A′
+0 to A′
+n through A′
+1, A′
+2, . . . , A′
+n−1. Then, in G, all corresponding
+tiles A1, A2, . . . , An are blue tiles. Therefore, Left can move a piece from A0 to
+5
+
+Figure 4: Play of beyond the door.
+An through A1, A2, . . . , An−1 in the corresponding position in turning tiles.
+Similar proof holds for Right’s moves.
+Thus, from Corollary 1, go on lattice is a universal partizan ruleset.
+2.2
+Beyond the door
+Definition 11. The rule of beyond the door is as follows:
+• Square rooms are arranged in a grid pattern. There are doors between the
+rooms. The front and back of the doors are painted red or blue. There are
+pieces in several rooms.
+• A player, in his/her turn, chooses a piece and moves it in a straight line.
+When a piece moves beyond the door, the color of the piece’s side of the
+door must be the player’s color.
+• After a piece passed a room, every piece can not enter the room.
+• The player who moves last is the winner.
+Figure 4 shows a play of beyond the door. The red sides are masked for
+monochrome printing.
+Theorem 3. Beyond the door is a universal partizan ruleset.
+Proof. Let f ′ be a function from a position in turning tiles to a position in
+beyond the door as follows:
+Let G be a position in turning tiles. In f ′(G) there are as many rooms
+as tiles in G and the tiles in G and the rooms in f ′(G) are arranged exactly the
+same. For each piece on a tile in G, there is a corresponding piece in the room
+corresponding to the tile. For any adjacent tiles A and B in G, let A′ and B′
+are corresponding rooms in f ′(G). The color of the door between A′ and B′ is
+the same as the color of A on the B′ side, and the same as the color of B on
+the A′ side.
+Figure 5 shows this corresponding. Here, the game tree of G and f ′(G) are
+isomorphic. We prove that every move in one game has a corresponding move
+in the other game. Assume that in G Left can move a piece on tile A0 to tile
+An through tiles A1, A2, . . . , An−1. Then, A1, A2, . . . , An are blue tiles. Let
+A′
+0, A′
+1, . . . , A′
+n be the corresponding rooms in f ′(G). Let A′ → B′ be the color
+6
+
+Figure 5: Corresponding positions in turning tiles and beyond the door
+.
+Figure 6: f and f ′ have no inverse functions.
+of the door between A′ and B′ on the A′ side. Then, from the definition of f ′,
+all of A′
+0 → A′
+1, A′
+1 → A′
+2, . . . , A′
+n−1 → A′
+n are blue. Therefore, Left can move
+a piece from A′
+0 to A′
+n through A′
+1, A′
+2, . . . , A′
+n−1. Conversely, assume that in
+f(G), Left can move a piece from A′
+0 to A′
+n through A′
+1, A′
+2, . . . , A′
+n−1. Then,
+in G, all corresponding tiles A1, A2, . . . , An are blue tiles. Therefore, Left can
+move a piece from A0 to An through A′
+1, A′
+2, . . . , A′
+n−1 in the corresponding
+position in turning tiles. Similar proof holds for Right’s moves.
+Thus, from Corollary 1, beyond the door is a universal partizan ruleset.
+Note that f and f ′ have no inverse functions. For instance, Fig. 6 shows
+positions in go on lattice and beyond the door. No position in turning
+tiles is mapped to these positions by f and f ′ because depending on the order
+of moves, both Left and Right may move pieces to the same node or the same
+room in these positions.
+This is somewhat interesting.
+That is, even though in some ways these
+rulesets are more complex than turning tiles, considering what kind of values
+7
+
+LLLL
+L
+RRR
+R
+R
+Rcan appear in the rulesets, all of them are the same.
+3
+Conclusion
+In this paper, we proved go on lattice and beyond the door are universal
+partizan rulesets by using game-tree preserving reduction. The method of re-
+duction has been used primarily for proving complexity of problems. Since this
+study shows that reduction is also effective in the proof of universality of a game,
+we can expect that the knowledge accumulated in the study of computational
+complexity will be utilized in the study of combinatorial game theory, and we
+can expect further development of combinatorial game theory.
+References
+[1] M. H. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in play: An Iintro-
+duction to combinatorial game theory, A K Peters, Ltd. / CRC Press(2007).
+[2] A. Carvalho, C. P. Santos: A nontrivial surjective map onto the short
+Conway group, Games of No Chance 5 (U. Larsson, Ed.), MSRI Book
+Series 70, Cambridge University Press, pp. 271–284(2019).
+[3] A. N. Siegel, Combinatorial Game Theory, American Mathematical Soci-
+ety(2013).
+[4] K.
+Suetsugu,
+Discovering
+a
+new
+universal
+partizan
+ruleset,
+arXiv:2201.06069 [math.CO](2022).
+8
+

0dE5T4oBgHgl3EQfOg4z/content/tmp_files/load_file.txt

@@ -0,0 +1,288 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf,len=287
+page_content='New universal partizan rulesets  Koki Suetsugu  January 2023  Abstract  Universal partizan ruleset is a ruleset in which every game value of  partizan games can be appear as a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' So far, generalized konane  and turning tiles have been proved to be universal partizan rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  In this paper, we introduce two rulesets go on lattice and beyond the  door and prove that they are universal partizan rulesets by using game  tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  1  Introduction  Combinatorial game theory (CGT) studies two-player perfect information games  with no chance moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' We say a game is under normal play convention if the  player who moves last is the winner and a game is partizan game if the options  for both players can be different in some positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Here, we introduce some  definitions and theorems of CGT for later discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For more details of CGT,  see [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  In this theory, the two players are called Left and Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Since the term  “game” is polysemous, we refer to each position as a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The description of  what moves are allowed for a given position is called the ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  A game is defined by Left and Right options recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  {|} is a game, which is called 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  For games GL  1 , GL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GL  n, GR  1 , GR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , and GR  m, G = {GL  1 , GL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GL  n |  GR  1 , GR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GR  m} is also a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' GL  1 , GL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GL  n are called left options  of G and GR  1 , GR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GR  m are called right options of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Let G be the set of all games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  In terms of the player who has a winning strategy, G is separated into four  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let L, R, N, and P be the set of positions in which Left, Right, the Next  player, and the Previous player have winning strategies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The sets are called outcomes of the games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Every position belongs to exactly  one of the four outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For a game G, let o(G) be the outcome of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' We  define the partial order of outcomes as L > P > R, L > N > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='05497v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='CO]  13 Jan 2023  The disjunctive sum of games is an important concept in Combinatorial  Game Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For games G and H, a position in which a player makes a move  for one or the other on their turn is called a disjunctive sum of G and H, or  G + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' More precisely, it is as follows:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If the game trees of G and H are isomorphic, then we say these  games are isomorphic or G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For games G ∼= {GL  1 , GL  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' GL  n | GR  1 , GR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GR  m} and H ∼=  {HL  1 , HL  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , HL  n′ | HR  1 , HR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , HR  m′}, G + H ∼= {G + HL  1 , G + HL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , G +  HL  n′, GL  1 + H, GL  2 + H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GL  n + H | G + HR  1 , G + HR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , G + HR  m′, GR  1 +  H, GR  2 + H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GR  m + H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  We also define equality, inequality and negative of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If for any X, o(G + X) is the same as o(H + X), then we say  G = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If o(G+H) ≥ o(H +X) holds for any X, then we say G ≥ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' On  the other hand, if o(G + H) ≤ o(H + X) holds for any X, then we say G ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  We also say G ≷ H if G ̸≥ H and G ̸≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For a game G ∼= {GL  1 , GL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GL  n | GR  1 , GR  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , GR  m}, let −G ∼=  {−GR  1 , −GR  2 , −GR  m | −GL  1 , −GL  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , −GL  n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  G + (−H) is denoted by G − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  It is known that (G, +, =) is an abelian group and (G, ≥, =) is a partial  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The question arises here, will there be a ruleset in which for any game there  is a position equal to the game?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If the games appearing in each ruleset are  restricted, then perhaps we should think in a narrower framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  In fact,  however, it is known that for every game, a position equal to the game appears  in some rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='1  Universal partizan ruleset  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' A ruleset is universal partizan ruleset if every value in G is equal  to a position of the ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Early results showed that generalized konane and turning tiles are  universal partizan ruleset ([2, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' In this study, we will use the latter ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The ruleset of turning tiles is as follows:  Square tiles are laid out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The front side is red or blue, and the back side  is black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Some pieces are on tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Each player (Left, whose color is bLue and Right, whose color is Red),  in his/her turn, take a piece and move the piece straight on the tiles of  his/her color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  2  Tiles on which the piece pass over are turned over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Turning tiles is proved to be universal partizan ruleset even if the number  of pieces is restricted to be only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  To distinguish this ruleset from the ruleset defined below, we will also refer  to it as blue-red turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  For games that use two colors, red and blue, corresponding to two players,  we often consider a variant that adds green, which can be used by both players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  For example, in blue-red-green hackenbush, Right can remove red or green  edges and Left can remove blue or green edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' From this point of view, we  consider a varant of turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The ruleset of blue-red-green turning tiles is as follows:  Square tiles are laid out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The front side is red, blue, or green, and the back  side is black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Some pieces are on tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Each player (Left, whose color is bLue and Right, whose color is Red),  in his/her turn, take a piece and move the piece straight on the tiles of  his/her color or of green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Tiles on which the piece pass over are turned over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 1: Positions in blue-red turning tiles and blue-red-green turn-  ing tiles  Figure 1 is an example of positions in blue-red turning tiles and blue-  red-green turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  3  L  LL  RＲ  LL  RR  L  R  LLLG  R  GG  RR-  G  L  RR  R  R  G  R  G  R  L  L  R  L  R  R  R  R  RR  R  R  R  RRR  R  R  RR  R  G  G  RR  R  RR  L  7  RR  R  一  R  G  RR  G  R  R  P  L  R  G  G  L  R  R  LL  R  R  R  R  L  G  R  G  R  GObviously, blue-red-green turning tiles is also a universal partizan  ruleset because every position in blue-red turning tiles can be appear in  blue-red-green turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  As we have seen here, if two rulesets have an inclusion relation in terms of  the sets of positions, it can be used for proving universality of the rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let Γ and ∆ be rulesets and assume that Γ be a universal partizan  ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If for every position g ∈ Γ, there is at least one position in ∆ whose  game value is the same as g, then ∆ is also a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' This is trivial from the definition of universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let Γ and ∆ be rulesets and assume that Γ be a universal partizan  ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If for every position g ∈ Γ, there is at least one position in ∆ whose  game tree is the same as g, then ∆ is also a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  If a ruleset is proved to be universal partizan ruleset by using Corollary 1,  we say that it is proved by game tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  In the next section, we introduce two rulesets and prove that they are univer-  sal partizan ruleset by game tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' In Secton 3, we conclude  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  2  New universal partizan rulesets  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='1  Go on lattice  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The rule of go on lattice is as follows:  There is a lattice graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' There are pieces on some nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The edges are  colored red, blue, or dotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  A player, in his/her turn, chooses a piece and moves it straight on edges  colored his/her color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  After a piece passed a node, every piece cannot get on or pass the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  If a player moves a piece to a node adjacent to a dotted edge, then the edge  changes to solid edge colored by the opponent’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 2: Play of go on lattice  4  17  MFigure 2 is a play of go on lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' We use double line for red edges for  monochrome printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Go on lattice is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let f be a function from a position in turning tiles to a position in  go on lattice as follows:  Let G be a position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' In f(G) there are as many nodes  as tiles in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The tiles in G and the nodes in f(G) are arranged exactly the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For each piece on a tile in G, there is a corresponding piece on the node  corresponds to the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For any adjacent tiles A and B in G, let A′ and B′ are  corresponding nodes in f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If the color of A, and B are the same, then edge  between A′ and B′ is solid and the same color as A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' If there is a piece  on A or B, then the edge between A′ and B′ is solid and the color is the same  as the other tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Finally, if the color of A and B are different and no piece is  on each tile, then the edge between A′ and B′ is dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' B  B  B  B  B  R  R  R  R B  R  R  B  B  B  Figure 3: Corresponding positions in turning tiles and go on lattice  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 3 shows this corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Here, the game tree of G and f(G) are  isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' We prove that every move in one game has a corresponding move  in the other game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Assume that in G Left can move a piece on tile A0 to  tile An through tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let  A′  0, A′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n be the corresponding nodes in f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let (A′, B′) be the edge be-  tween A′ and B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then, from the definition of f, all of (A′  1, A′  2), (A′  2, A′  3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , (A′  n−1, A′  n)  are blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' In addition, if A0 was a blue tile before turning, then (A′  0, A′  1)  is a blue edge, and if A0 was a red tile, (A′  0, A′  1) had been a dot edge and  after Right moved the piece to A′  0, it changed to a blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Therefore for  both case, (A′  0, A′  1) is a blue edge and Left can move a piece from A′  0 to A′  n  through A′  1, A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Conversely, assume that in f(G), Left can move a  piece from A′  0 to A′  n through A′  1, A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then, in G, all corresponding  tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Therefore, Left can move a piece from A0 to  5  Figure 4: Play of beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  An through A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An−1 in the corresponding position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Similar proof holds for Right’s moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Thus, from Corollary 1, go on lattice is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='2  Beyond the door  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The rule of beyond the door is as follows:  Square rooms are arranged in a grid pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' There are doors between the  rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The front and back of the doors are painted red or blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' There are  pieces in several rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  A player, in his/her turn, chooses a piece and moves it in a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  When a piece moves beyond the door, the color of the piece’s side of the  door must be the player’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  After a piece passed a room, every piece can not enter the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  The player who moves last is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 4 shows a play of beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The red sides are masked for  monochrome printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Beyond the door is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let f ′ be a function from a position in turning tiles to a position in  beyond the door as follows:  Let G be a position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' In f ′(G) there are as many rooms  as tiles in G and the tiles in G and the rooms in f ′(G) are arranged exactly the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For each piece on a tile in G, there is a corresponding piece in the room  corresponding to the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For any adjacent tiles A and B in G, let A′ and B′  are corresponding rooms in f ′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The color of the door between A′ and B′ is  the same as the color of A on the B′ side, and the same as the color of B on  the A′ side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 5 shows this corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Here, the game tree of G and f ′(G) are  isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' We prove that every move in one game has a corresponding move  in the other game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Assume that in G Left can move a piece on tile A0 to tile  An through tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let  A′  0, A′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n be the corresponding rooms in f ′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Let A′ → B′ be the color  6  Figure 5: Corresponding positions in turning tiles and beyond the door  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Figure 6: f and f ′ have no inverse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  of the door between A′ and B′ on the A′ side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then, from the definition of f ′,  all of A′  0 → A′  1, A′  1 → A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1 → A′  n are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Therefore, Left can move  a piece from A′  0 to A′  n through A′  1, A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Conversely, assume that in  f(G), Left can move a piece from A′  0 to A′  n through A′  1, A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Then,  in G, all corresponding tiles A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , An are blue tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Therefore, Left can  move a piece from A0 to An through A′  1, A′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' , A′  n−1 in the corresponding  position in turning tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Similar proof holds for Right’s moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Thus, from Corollary 1, beyond the door is a universal partizan ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Note that f and f ′ have no inverse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' 6 shows  positions in go on lattice and beyond the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' No position in turning  tiles is mapped to these positions by f and f ′ because depending on the order  of moves, both Left and Right may move pieces to the same node or the same  room in these positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  This is somewhat interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  That is, even though in some ways these  rulesets are more complex than turning tiles, considering what kind of values  7  LLLL  L  RRR  R  R  Rcan appear in the rulesets, all of them are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  3  Conclusion  In this paper, we proved go on lattice and beyond the door are universal  partizan rulesets by using game-tree preserving reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' The method of re-  duction has been used primarily for proving complexity of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Since this  study shows that reduction is also effective in the proof of universality of a game,  we can expect that the knowledge accumulated in the study of computational  complexity will be utilized in the study of combinatorial game theory, and we  can expect further development of combinatorial game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Albert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Nowakowski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Wolfe, Lessons in play: An Iintro-  duction to combinatorial game theory, A K Peters, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' / CRC Press(2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Carvalho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Santos: A nontrivial surjective map onto the short  Conway group, Games of No Chance 5 (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Larsson, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' ), MSRI Book  Series 70, Cambridge University Press, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' 271–284(2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content=' Siegel, Combinatorial Game Theory, American Mathematical Soci-  ety(2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  Suetsugu,  Discovering  a  new  universal  partizan  ruleset,  arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='06069 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='CO](2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE5T4oBgHgl3EQfOg4z/content/2301.05497v1.pdf'}

0dE5T4oBgHgl3EQfOg4z/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:72e8c58554f43fd8477be187000c7f95a647b3c92aca61c26f9f96eb287a00d3
+size 40690

0tAzT4oBgHgl3EQfDPqY/content/2301.00973v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c13daab503f1b67bffffdb7c8a8381e73d08fb05ab7c1cd94ef385b4acbc2a33
+size 5629212

0tAzT4oBgHgl3EQfDPqY/content/tmp_files/2301.00973v1.pdf.txt

@@ -0,0 +1,1482 @@
+GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
+1
+Detecting Severity of Diabetic Retinopathy from
+Fundus Images using Ensembled Transformers
+Chandranath Adak, Senior Member, IEEE, Tejas Karkera, Soumi Chattopadhyay, Member, IEEE, and
+Muhammad Saqib
+Abstract— Diabetic Retinopathy (DR) is considered one
+of the primary concerns due to its effect on vision loss
+among most people with diabetes globally. The severity of
+DR is mostly comprehended manually by ophthalmologists
+from fundus photography-based retina images. This paper
+deals with an automated understanding of the severity
+stages of DR. In the literature, researchers have focused on
+this automation using traditional machine learning-based
+algorithms and convolutional architectures. However, the
+past works hardly focused on essential parts of the retinal
+image to improve the model performance. In this paper,
+we adopt transformer-based learning models to capture the
+crucial features of retinal images to understand DR sever-
+ity better. We work with ensembling image transformers,
+where we adopt four models, namely ViT (Vision Trans-
+former), BEiT (Bidirectional Encoder representation for im-
+age Transformer), CaiT (Class-Attention in Image Trans-
+formers), and DeiT (Data efficient image Transformers), to
+infer the degree of DR severity from fundus photographs.
+For experiments, we used the publicly available APTOS-
+2019 blindness detection dataset, where the performances
+of the transformer-based models were quite encouraging.
+Index Terms— Blindness Detection, Diabetic Retinopa-
+thy, Deep learning, Transformers.
+I. INTRODUCTION
+D
+IABETES Mellitus, also known as diabetes, is a disorder
+where the patient experiences increased blood sugar
+levels over a long period. Diabetic Retinopathy (DR) is a mi-
+crovascular complication of diabetes where the retina’s blood
+vessels get damaged, which can lead to poor vision and even
+blindness if untreated [1], [2]. Studies estimated that by twenty
+years after diabetes onset, about 99% (or 60%) of patients
+having type-I (or type-II) diabetes might have DR [1]. With
+a worldwide presence of DR patients of about 126.6 million
+in 2010, the current estimate is roughly around 191 million
+by 2030 [3], [4]. However, about 56% of new DR cases can
+be reduced by timely treatment and monitoring of the severity
+[5]. The ophthalmologist analyzes fundus images for lesion-
+based symptoms like microaneurysms, hard/ soft exudates, and
+hemorrhages to understand the severity stages of DR [1], [2].
+The positive DR is divided into the following stages [5]: (1)
+mild: the earliest stage that can contain microaneurysms, (2)
+C. Adak is with Dept. of CSE, IIT Patna, India-801106, T. Karkera is
+with Atharva College of Engineering, Mumbai, India-400095, S. Chat-
+topadhyay is with the Dept. of CSE, IIIT Guwahati, India-781015, and
+M. Saqib is with Data61, CSIRO, Australia-2122.
+Corresponding author: C. Adak (e-mail: [email protected])
+negative
+mild
+moderate
+severe
+proliferative
+Fig. 1. Fundus images with DR severity stages from APTOS-2019 [7].
+moderate: here, the blood vessels lose the ability to blood
+transportation, (3) severe: here, blockages in blood vessels
+can occur and gives a signal to grow new blood vessels, (4)
+proliferative: the advanced stage where new blood vessels start
+growing. Fig. 1 shows some fundus images representing the
+DR severity stages. Manual examining fundus images for DR
+severity stage grading may bring inconsistencies due to a high
+number of patients, less number of well-trained clinicians, long
+diagnosing time, unclear lesions, etc. Moreover, there may be
+disagreement among ophthalmologists in choosing the correct
+severity grade [6]. Therefore, computer-aided techniques have
+come into the scenario for better diagnosis and broadening the
+prospects of early-stage detection [2].
+Automated DR severity stage detection from fundus pho-
+tographs has been performed for the last two and half decades.
+Earlier, some image processing tools were used [8], [9], but
+the machine learning-based DR became popular in the early
+2000s. The machine learning-based techniques mostly relied
+on hand-engineered features that were carefully extracted from
+the fundus images and then fed to a classifier, e.g., Random
+Forest (RF) [10], KNN (K-Nearest Neighbors) [11], SVM
+(Support Vector Machine) [12], and ANN (Artificial Neural
+Network) [13]. Although SVM and ANN-based models were
+admired in the DR community, the hand-engineered feature-
+based machine learning models require efficient prior feature
+extraction, which may introduce errors for complex fundus
+images [1], [2]. On the other hand, deep learning-based models
+extract features automatically through convolution operations
+[14], [15]. Besides, from 2012, deep learning architectures
+rose to prominence in the computer vision community, which
+also influenced the DR severity analysis from fundus images
+[1]. The past deep learning-based techniques mostly employed
+CNN (Convolutional Neural Network) [1], [16]. However,
+the ability to give attention to certain regions/features and
+fade the remaining portions hardly exists in classical CNNs.
+For this reason, some contemporary methods incorporated
+attention mechanism [17], [18]. Although multiple research
+arXiv:2301.00973v1  [cs.CV]  3 Jan 2023
+
+LOGO2
+GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
+works are present in the literature [1], [2] and efforts were
+made to detect the existence of DR in the initial stages
+of its development, still there is a room for improving the
+performance by incorporating higher degrees of automated
+feature extraction using better deep learning models.
+In this paper, we employ the transformer model for leverag-
+ing its MSA (Multi-head Self-Attention) [19] to focus on the
+DR revealing region of the fundus image for understanding
+the severity. Moreover, the transformer model has shown high
+performance in recent days [19], [20]. Initially, we adopted
+ViT (Visual Transformer) [19] for detecting DR severity due to
+its outperformance on image classification tasks. ViT divides
+the input image into a sequence of patches and applies
+global attention [19]. Moreover, since standard ViT requires
+hefty amounts of data, we also adopted some other image
+transformer models, such as CaiT (Class-attention in image
+Transformers) [21], DeiT (Data-efficient image Transformer)
+[22], and BEiT (Bidirectional Encoder representation for im-
+age Transformer) [23]. CaiT is a modified version of ViT and
+employs specific class-attention [21]. DeiT uses knowledge
+distillation, which transfers the knowledge from one network
+to another and builds a teacher-student hierarchical network
+[22]. BEiT is inspired by BERT (Bidirectional Encoder Rep-
+resentations from Transformers) [24] to implement masking
+of image patches and to model the same for pre-training the
+ViT [23]. For experiments, we used the publicly available
+APTOS-2019 blindness detection dataset [7], where the in-
+dividual image transformers did not perform well. Therefore,
+we ensembled the image transformers to seek better predictive
+performance. The ensembled image transformer obtained quite
+encouraging results for DR severity stage detection. This is
+one of the earliest attempts to adopt and ensemble image
+transformers for DR severity stage detection, which is the main
+contribution of this paper.
+The rest of the paper is organized as follows. § II discusses
+the relevant literature about DR and § III presents the proposed
+methodology. Then § IV analyzes and discusses the experi-
+mental results. Finally, § V concludes this paper.
+II. RELATED WORK
+This section briefly presents the literature on DR severity
+detection from fundus images. The modern grading of DR
+severity stages can be traced in the report by ETDRS research
+group [25]. In the past, some image processing-based (e.g.,
+wavelet transform [8], radon transform [9]) strategies were
+published. For the last two decades, machine learning and
+deep learning-based approaches have shown dominance. We
+broadly categorize the related works into (a) hand-engineered
+feature-based models [11], [26], [27], and (b) deep feature-
+based models [2], which are discussed below.
+A. Hand-engineered Feature-based Models
+The hand-engineered feature-based models mostly em-
+ployed RF [26], KNN [28], SVM [27], ANN [29] for detecting
+DR severity stages. Acharya et al. [26] employed a decision
+tree with discrete wavelet/cosine transform-based features ex-
+tracted from retinal images. Casanova et al. [10] introduced RF
+for DR severity stage classification. In [30], RF was also used
+to assess DR risk. KNN classifier was employed in [11] to
+detect drusen, exudates, and cotton-wool spots for diagnosing
+DR. Tang et al. [28] used KNN for retinal hemorrhage detec-
+tion from fundus photographs. In [27], retinal changes due to
+DR was detected by using SVM. Akram et al. [12] used SVM
+and GMM (Gaussian Mixture Model) with enhanced features
+such as shape, intensity, and statistics of the affected region
+to identify microaneurysms for early detection of DR. ANN
+was employed in [13] to classify lesions for detecting DR
+severity. Osareh et al. [31] employed FCM (Fuzzy C-Means)-
+based segmentation and GA (Genetic algorithm)-based feature
+selection with ANN to detect exudates in DR. In [29], PSO
+(Particle Swarm Optimization) was used for feature selection,
+followed by ANN-based DR severity classification.
+B. Deep Feature-based Models
+The past deep architectures mostly used CNN for tackling
+DR severity. For example, Yu et al. [16] used CNN for
+detecting exudates in DR, Chudzik et al. [32] worked on
+microaneurysm detection using CNN with transfer learning
+and layer freezing, Gargeya and Leng [33] employed CNN-
+based deep residual learning to identify fundus images with
+DR. In [4], CNN was also used to identify DR severity stages
+and some related eye diseases, e.g., glaucoma and AMD (Age-
+related Macular Degeneration). In [34], some classical CNN
+architectures (e.g., AlexNet, VGG Net, GoogLeNet, ResNet)
+were employed for DR severity stage detection. Wang et al.
+[17] proposed Zoom-in-Net that combined CNN, attention
+mechanism, and a greedy algorithm to zoom in the region
+of interest for handling DR. A modified DenseNet169 ar-
+chitecture in conjunction with the attention mechanism was
+used in [18] to extract refined features for DR severity
+grading. In [35], a modified Xception architecture was em-
+ployed for DR classification. TAN (Texture Attention Net-
+work) was proposed in [36] by leveraging style (texture
+features) and content (semantic and contextual features) re-
+calibration mechanism. Tymchenko et al. [5] ensembled three
+CNN architectures (EfficientNet-B4 [37], EfficientNet-B5, and
+SE- ResNeXt50 [38]) for DR severity detection. Very recently,
+a few transformer-based models have come out, e.g., CoT-
+XNet [39] that combined contextual transformer and Xception
+architecture, SSiT [40] that employed self-supervised image
+transformers guided by saliency maps.
+III. METHODOLOGY
+This section first formalizes the problem statement, which
+is then followed by the proposal of solution architecture.
+A. Problem Formulation
+In this work, we are given an image I captured by the
+fundus photography, which is input to the architecture. The
+task is to predict the severity stage of diabetic retinopathy (DR)
+among negative, mild, moderate, severe, and proliferative,
+from I. We formulate the task as a multi-class classification
+problem [15]. Here, from I, features are extracted and fed to
+
+ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
+3
+a classifier to predict the DR severity class labels Á, where
+Á = {0, 1, 2, 3, 4} corresponds to {negative, mild, moderate,
+severe, proliferative}, respectively.
+B. Solution Architecture
+For detecting the severity stage of DR from a fundus
+photograph, we adopt image transformers, i.e., ViT (Vision
+Transformer) [19], BEiT (Bidirectional Encoder representation
+for image Transformer) [23], CaiT (Class-attention in image
+Transformers) [21], and DeiT (Data efficient image Trans-
+formers) [22], and ensemble them. However, we preprocess
+raw fundus images before feeding them into the transformers,
+which we discuss first.
+1) Preprocessing: The performance of deep learning mod-
+els is susceptible to the quality and quantity of data being
+passed to the model. Raw data as input can barely account for
+the best achievable performance of the model due to possible
+pre-existing noise and inconsistency in the images. Therefore,
+a definite flow of preprocessing is essential to train the model
+better [15].
+We now discuss various preprocessing and augmentation
+techniques [15], [41] applied to the raw fundus photographs
+for better learning. In a dataset, the fundus images may be of
+various sizes; therefore, we resize the image I into 256 × 256
+sized image Iz. We perform data augmentations on training set
+(DBtr), where we use centre cropping with central_fraction =
+0.5, horizontal/vertical flip, random rotations within a range
+of [0o, 45o], random brightness-change with max_delta =
+0.95, random contrast-change in the interval [0.1, 0.9]. We
+also apply CLAHE (Contrast Limited Adaptive Histogram
+Equalization) [42] on 30% samples of DBtr, which ensures
+over-amplification of contrast in a smaller region instead of
+the entire image.
+2) Transformer Networks: Deep learning models in com-
+puter vision tasks have long been dominated by CNN (Convo-
+lutional Neural Network) to extract high-level feature maps by
+passing the image through a series of convolution operations
+before feeding into the MLP (Multi-Layer Perceptron) for clas-
+sification [43]. In recent days, transformer models have shown
+a substantial rise in the NLP (Natural Language Processing)
+domain due to its higher performances [20]. In a similar quest
+to leverage high-level performance through transformers, it
+has been introduced in image classification and some other
+computer vision-oriented tasks [19]. Moreover, the transformer
+model has lesser image-specific inductive bias than CNN [19].
+To identify the severity stages of DR from fundus images,
+here we efficiently adopt and ensemble some image transform-
+ers, e.g., ViT [19], BEiT [23], CaiT [21], and DeiT [22].
+Before focusing on our ensembled transformer model, we
+discuss the adaptation of individual image transformers for
+our task, and start with ViT.
+a) Vision Transformer (ViT): The ViT model adopts the idea
+of text-based transformer models [44], where the idea is to take
+the input image as a series of image patches instead of textual
+words, and then extract features to feed it into an MLP [19].
+The pictorial representation of ViT is presented in Fig. 2.
+Here, the input image Iz is converted into a sequence of
+Fig. 2. Workflow of ViT.
+Fig. 3. Internal view of a transformer encoder (TE).
+flattened patches xi
+p (for i = 1, 2, . . . , np), each with size
+wp × wp × cp, where cp denotes the number of channels
+of Iz. Here, cp = 3, since Iz is an RGB fundus image. In
+our task, Iz is of size 256 × 256, and empirically, we choose
+wp = 64, which results np = ( 256
+64 )2 = 16. Each patch xi
+p is
+flattened further and mapped to a D-dimensional latent vector
+(i.e., patch embedding z0) through transformer layers using a
+trainable linear projection, as below.
+z0 = [xclass ; x1
+p E ; x2
+p E ; . . . ; xnp
+p E] + Epos
+(1)
+where,
+E
+is
+the
+patch
+embedding
+projection,
+E
+∈
+Rwp×wp×C×D; Epos is the position embeddings added to
+patch embeddings to preserve the positional information of
+patches, Epos ∈ R(np+1)×D; xclass = z0
+0 is a learnable
+embedding [24].
+After mapping patch images to the embedding space with
+positional information, we add a sequence of transformer
+encoders [19], [45]. The internal view of a transformer encoder
+can be seen in Fig. 3, which includes two blocks As and Fn.
+The As and Fn contain MSA (Multi-head Self-Attention) [19]
+and MLP [15] modules, respectively. LN (Layer Normaliza-
+tion) [46] and residual connection [15] are employed before
+and after each of these modules, respectively. This is shown
+in equation 2 with general semantics. Here, the MLP module
+comprises two layers having 4D and D neurons with GELU
+(Gaussian Error Linear Unit) non-linear activation function
+
+Patch + Position
+Embedding
+0
+MLP Head
+Softmax
+*
+Linear Projection of Flattened Patches
+1
+Transformer Encoder (TE)
+2
+3
+dn
+np
+pLx
+LN
+MSA
+LN
+MLP
+zi
+Zt4
+GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
+similar to [19].
+z′
+l = MSA(LN(zl−1)) + zl−1;
+zl = MLP(LN(z′
+l)) + z′
+l; l = 1, 2, . . . , L
+(2)
+where, L is the total number of transformer blocks. The core
+component of the transformer encoder is MSA with h heads,
+where each head includes SA (Scaled dot-product Attention)
+[19], [45]. Each head i ∈ {1, 2, ..., h} of MSA calculates a
+tuple comprising query, key, and value [19], i.e., (Qi, Ki, V i)
+as follows.
+Qi = XW i
+Q ; Ki = XW i
+K ; V i = XW i
+V
+(3)
+where, X is the input embedding, and WQ, WK, WV are the
+weight matrices used in the linear transformation. The tuple
+(Q, K, V ) is fed to SA that computes the attention required
+to pay to the input image patches, as below.
+SA(Q, K, V ) = ψ
+�QKT
+√Dh
+�
+V
+(4)
+where, ψ is softmax function, and Dh = D/h. The outcomes
+of SAs across all heads are concatenated in MSA, as follows.
+MSA(Q, K, V ) = [SA1 ; SA2 ; . . . ; SAh]WL
+(5)
+where WL is a weight matrix.
+After multiple transformer encoder blocks, the <class>
+token [24] enriches with the contextual information. The state
+of the learnable embedding at the outcome of the Transformer
+encoder (z0
+L) acts as the image representation y [19].
+y = LN(z0
+L)
+(6)
+Now, as shown in Fig. 2, we add an MLP head containing
+a hidden layer with 128 neurons. To capture the non-linearity,
+we use Mish [47] here. In the output layer, we keep five
+neurons with softmax activation function to obtain probability
+distribution s(j) in order to classify a fundus photograph into
+the abovementioned five severity stages of DR.
+b) Data efficient image Transformers (DeiT): For a lower
+amount of training data, ViT does not generalize well. In
+this scenario, DeiT can perform reasonably well and uses
+lower memory [22]. DeiT adopts the ViT-specific strategy and
+merges with the teacher-student scheme through knowledge
+distillation [48]. The crux of DeiT is the knowledge distillation
+mechanism, which is basically the knowledge transfer from
+one model (teacher) to another (student) [22]. Here, we use
+EfficientNet-B5 [37] as a teacher model that is trained apriori.
+The student model uses a transformer, which learns from the
+outcome of the teacher model through attention depending
+on a distillation token [22]. In this work, we employ hard-
+label distillation [22], where the hard decision of the teacher
+is considered as a true label, i.e., yt = argmaxcZt(c). The
+hard-label distillation objective is defined as follows.
+Lhard
+global = 0.5 LCE(ψ(Zs), y) + 0.5 LCE(ψ(Zs), yt)
+(7)
+where, LCE is the cross-entropy loss on ground-truth labels
+y, ψ is the softmax function, Zs and Zt are the student and
+teacher models’ logits, respectively. Using label smoothing,
+Fig. 4. The distillation procedure of DeiT.
+hard labels can be converted into soft ones [22].
+In Fig. 4, we present the distillation procedure of DeiT.
+Here, we add the <distillation> token to the transformer,
+which interacts with the <class> and <patch> tokens through
+transformer encoders. The transformer encoder used here is
+similar to the ViT’s one, which includes As and Fn blocks as
+shown in Fig. 3. The objective of the <distillation> token is to
+reproduce the teacher’s predicted label instead of the ground-
+truth label. The <distillation> and <class> tokens are learned
+by back-propagation [15].
+A linear classifier is used in DeiT instead of the MLP head
+of ViT [19], [22] to work efficiently with limited computa-
+tional resources.
+c) Class-attention in image Transformers (CaiT): CaiT
+usually performs better than ViT and DeiT with lesser FLOPs
+and learning parameters [15], when we need to increase the
+depth of the transformer [21]. CaiT is basically an upgraded
+version of ViT, which leverages layers with specific class-
+attention and LayerScale [21]. In Fig. 5, we show the workflow
+of CaiT.
+LayerScale aids CaiT to work at larger depths, where we
+separately multiply a diagonal matrix Mλ on the outputs of
+As and Fn blocks.
+z′
+l = Mλ(λl
+1, . . . , λl
+D) × MSA(LN(zl−1)) + zl−1;
+zl = Mλ(λ′l
+1, . . . , λ′l
+D) × MLP(LN(z′
+l)) + z′
+l
+(8)
+where, λl
+i and λ′l
+i are learning parameters, and other symbols
+denote the same as the above-mentioned ViT.
+In CaiT, the transformer layers dealing with self-attention
+Fig. 5. Workflow of CaiT.
+
+Self-attention
+lass-attentionO
+口ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
+5
+among patches are separated from class-attention layers that
+are introduced to dedicatedly extract the content of the patches
+into a vector, which can be sent to a linear classifier [21]. The
+<class> token is inserted in the latter stage, so that the initial
+layers can perform the self-attention among patches devotedly.
+In the class-attention stage, we alternatively use multi-head
+class-attention (Ac) [21] and Fn, as shown in Fig. 5, and
+update only the class embedding.
+d) Bidirectional Encoder representation for image Trans-
+former (BEiT): BEiT is a self-supervised model having its
+root in the BERT (Bidirectional Encoder Representations from
+Transformers) [23], [24]. In Fig. 6, we present the workflow
+of the pre-training of BEiT.
+The input image Iz is split into patches xi
+p and flattened
+into vectors, similar to the early-mentioned ViT. In BEiT, a
+backbone transformer is engaged, for which we use ViT [19].
+On the other hand, Iz is represented as a sequence of visual
+tokens vt = [vt1, vt2, . . . , vtnp] obtained by a discrete VAE
+(Variational Auto-Encoder) [49]. For visual token learning, we
+employ a tokenizer Tφ(vt | x) to map image pixels x to tokens
+vt, and decoder Dθ(x | vt) for reconstructing input image
+pixels x from vt [23].
+Here, a MIM (Masked Image Modeling) [23] task is per-
+formed to pre-train the image transformers, where some image
+patches are randomly masked, and the corresponding visual
+tokens are then predicted. The masked patches are replaced
+with a learnable embedding e[M]. We feed the corrupted image
+patches xM = {xi
+p : i /∈ M} �{e[M] : i ∈ M} to the
+transformer encoder. Here, M is the set of indices of masked
+positions.
+The encoded representation hL
+i
+is the hidden vector of
+the last transformer layer L for ith patch. For each masked
+Fig. 6. Workflow of BEiT pre-training.
+position, a softmax classifier ψ is used to predict the respective
+visual token, i.e., pMIM(vt′ | xM) = ψ(WMhL
+i + bM); where,
+WM and bM contain learning parameters for linear transfor-
+mation. The pre-training objective of BEiT is to maximize the
+log-likelihood of the correct token vti given xM, as below:
+max
+�
+x∈ DBtr
+EM
+� �
+i∈M
+log pMIM
+�
+vti | xM�
+�
+where, DBtr is the training dataset. The BEiT pre-training
+can be perceived as VAE training [23], [49], where we follow
+two stages, i.e., stage-1: minimizing loss for visual token
+reconstruction, stage-2: modeling masked image, i.e., learning
+prior pMIM by keeping Tφ and Dθ fixed. It can be written as
+follows:
+�
+(xi,xM
+i
+)
+∈ DBtr
+�
+�
+�
+�Evti∼Tφ(vt|xi) [log Dθ(xi|vti)]
+�
+��
+�
+stage-1
++ log pMIM
+�
+ˆ
+vti|xM
+i
+�
+�
+��
+�
+stage-2
+�
+�
+�
+�
+where, ˆ
+vti = argmaxvt Tφ(vt | xi).
+3) Ensembled Transformers: The abovementioned four im-
+age transformers, i.e., ViT [19], DeiT [22], CaiT [21], and
+BEiT [23] are pre-trained on the training set DBtr. We now
+ensemble the transformers for predicting the severity stages
+from fundus images of the test set DBt, since ensembling
+multiple learning algorithms can achieve better performance
+than the constituent algorithms alone [50]. The pictorial rep-
+resentation of ensembled transformers is presented in Fig. 7.
+For an image sample from DBt, we obtain the softmax
+probability distribution s(j) : {P j
+1 , P j
+2 , . . . , P j
+nc} over jth
+transformer [15], for j = 1, 2, . . . , nT ; where, nc is the total
+number of classes (severity stages), and nT is count of the
+employed image transformers. Here, �nc
+i=1 P j
+i = 1, nc = 5
+(refer to § III-A), and nT = 4 since we use four separately
+trained distinct image transformers, as mentioned earlier.
+We obtain the severity stages/ class_labels Á|wm and Á|mv
+separately using two combination methods weighted mean and
+majority voting [50], respectively.
+Á|wm = argmaxi P µ
+i ; for i = 1, 2, . . . , nc ;
+P µ
+i =
+�nT
+j=1 αjP j
+i
+�nT
+j=1 αj
+(9)
+Fig. 7. Ensembled transformers.
+
+Masked
+Image
+Patches
+Original Image
+latten
+0
+*
+Tokenizer
+L
+1
+BEiT Encoder
+MIM Head
+2
+[M]
+h2
+3
+Unused at
+Pre-training
+Decoder
+Patch + Position
+Embedding
+Reconstructed ImageViT
+s(1)
+ative
+s(2)
+DeiT
+Mild
+Moderate
+Severe
+CaiT
+s(3)
+Proliferative
+c
+(4)
+BEiT6
+GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
+In this task, we choose �nT
+j=1 αj = 1.
+Á|mv = mode
+�
+argmaxi(P 1
+i ), argmaxi(P 2
+i ), . . . , argmaxi(P nT
+i
+)
+�
+= mode
+�
+argmaxi(s(1)), argmaxi(s(2)), . . . , argmaxi(s(nT ))
+�
+;
+for i = 1, 2, . . . , nc
+(10)
+In this task, we use cross-entropy as the loss function [41]
+in the employed image transformers. The AdamW optimizer
+is used here due to its weight decay regularization effect
+for tackling overfitting [51]. The training details with hyper-
+parameter tuning are mentioned in Section IV-B.
+IV. EXPERIMENTS AND DISCUSSIONS
+In this section, we present the employed database, followed
+by experimental results with discussions.
+A. Database Employed
+For our computational experiments, we used the publicly
+available training samples of Kaggle APTOS (Asia Pacific
+Tele-Ophthalmology Society) 2019 Blindness Detection dataset
+[7], i.e., APTOS-2019. This database (DB) contains fundus
+image samples of five severity stages of DR, i.e., negative,
+mild, moderate, severe, and proliferative. Fig. 1 shows some
+sample images from this dataset. In DB, a total of 3662 fundus
+images are available, which we divide into training (DBtr) and
+testing (DBt) datasets with a ratio of 7 : 3. As a matter of fact,
+DBtr and DBt sets are disjoint. The sample counts of different
+severity stages/ class_labels (Á) for DBtr and DBt are shown
+in Fig. 8 individually. Here, 49.3% samples are of negative
+DR (Á= 0). Among positive classes, most samples are from
+the moderate stage (Á= 2). From this figure, it can be seen
+DB is imbalanced due to containing a different number of
+samples corresponding to various severity stages. Therefore,
+we augmented the data during the training of our model as
+mentioned in § III-B.1. The data augmentation also helped in
+reducing the overfitting issue [15].
+Fig. 8. Count of samples in APTOS-2019 [7].
+B. Experimental Results
+This section discusses the performed experiments, analyzes
+the model outcome, and compares them with major state-of-
+the-art methods. We begin with discussing the experimental
+settings.
+1) Experiment Settings: We performed the experiments on
+the TensorFlow-2 framework having Python 3.7.13 over a
+machine with the following configurations: Intel(R) Xeon(R)
+CPU @ 2.00GHz with 52 GB RAM and Tesla T4 16 GB
+GPU. All the results shown here were obtained from DBt.
+The hyper-parameters of the framework were tuned and
+fixed during training with respect to the performance over
+some samples of DBt employed for hyper-parameter tuning.
+For all the image transformers used here (i.e., ViT, DeiT, CaiT,
+and BEiT), we empirically set the following hyper-parameters:
+transformer_layers (L) = 12, embedding_dimension (D) =
+384, num_heads (h) = 6. The following hyper-parameters
+were selected for AdamW [51]: initial_learning_rate = 10−3;
+exponential decay rates for 1st and 2nd moment estimates, i.e.,
+β1 = 0.9, β2 = 0.999; zero-denominator removal parameter
+(ε) = 10−8; and weight_decay = 10−3/4. For model training,
+the mini-batch size was fixed to 32.
+2) Model Performance: In Table I, we present the per-
+formance of our ensembled image transformer (EiT) using
+the combination schemes weighted mean (wm) and majority
+voting (mv), where we obtain 94.63% and 91.26% accuracy
+from EiTwm and EiTmv, respectively. We also ensembled
+multiple combinations of our employed transformers, and
+present their performances in this table. Here, the wm scheme
+performed better than mv. As evident from this table, ensem-
+bling various types of transformers improved the performance.
+Among single transformers (for nT = 1), CaiT performed
+the best. For nT = 2 and nT = 3, “BEiT + CaiT” and
+“DeiT + BEiT + CaiT” performed better than other respective
+combinations. Overall, EiTwm attained the best accuracy here.
+TABLE I
+PERFORMANCE OVER VARIOUS ENSEMBLING OF TRANSFORMERS
+nT
+Ensembled Transformers
+Accuracy (%)
+Weighted
+Majority
+mean
+voting
+1
+ViT
+82.21
+DeiT
+85.65
+BEiT
+86.74
+CaiT
+86.91
+2
+ViT + DeiT
+87.03
+86.55
+ViT + BEiT
+87.48
+87.03
+ViT + CaiT
+87.77
+87.21
+DeiT + BEiT
+88.18
+87.69
+DeiT + CaiT
+88.86
+87.93
+BEiT + CaiT
+89.28
+88.12
+3
+ViT + DeiT + BEiT
+90.53
+88.87
+ViT + DeiT + CaiT
+91.39
+89.56
+ViT + BEiT + CaiT
+92.14
+90.28
+DeiT + BEiT + CaiT
+93.46
+90.91
+4
+ViT + DeiT + BEiT + CaiT
+94.63
+91.26
+( EiT )
+In Fig. 10 of Appendix I, we present the coarse localization
+maps generated by Grad-CAM [52] from the employed indi-
+vidual image transformers to highlight the crucial regions for
+understanding the severity stages.
+a) Various Evaluation Metrics: Besides the accuracy, in
+Table II, we present the performance of EiT with respect to
+some other evaluation metrics, e.g., kappa score, precision,
+recall, F1 score, specificity, balanced accuracy [53]. Here,
+Cohen’s quadratic weighted kappa measures the agreement
+
+2000
+1800
+ DBtr
+ DBt
+1600
+541
+1400
+1200
+ sampl
+1000
+800
+300
+#
+600
+1264
+400
+111
+699
+200
+88
+259
+135
+207
+0
+0
+1
+2
+3
+4
+severity stage/ class
+label (c)ADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
+7
+between human-assigned scores (i.e., DR severity stages)
+and the EiT-predicted scores. Precision analyzes the true
+positive samples among the total positive predictions. Recall
+or sensitivity finds the true positive rate. Similarly, specificity
+computes the true negative rate. F1 score is the harmonic mean
+of precision and recall. Since the employed DB is imbalanced,
+we also compute the balanced accuracy, which is the arithmetic
+mean of sensitivity and specificity. In this table, we can see
+that for both EiTwm and EiTmv, the kappa scores are greater
+than 0.81, which comprehends the “almost perfect agreement”
+between the human rater and EiT [53]. Here, macro means
+the arithmetic mean of all per class precision/ recall/ F1 score.
+TABLE II
+PERFORMANCE OF EiT OVER VARIOUS EVALUATION METRICS
+Metric
+Weighted mean
+Majority voting
+(EiTwm)
+(EiTmv)
+Accuracy (%)
+94.63
+91.26
+Kappa score
+0.92
+0.87
+Macro Precision (%)
+90.55
+84.65
+Macro Recall (%)
+92.88
+88.81
+Macro F1-score (%)
+91.67
+86.55
+Macro Specificity (%)
+98.62
+97.74
+Balanced Accuracy (%)
+95.75
+93.27
+b) Individual Class Performance: Table III presents the
+individual performance of EiTwm and EiTmv for detecting
+every severity stage of DR. From this table, we can see our
+models produced the best precision and recall for negative DR
+(Á= 0), and the lowest for severe DR (Á= 3).
+TABLE III
+PERFORMANCE OF EiT ON EVERY DR SEVERITY STAGE
+class_label (Á)
+0
+1
+2
+3
+4
+EiTwm
+Precision (%)
+98.48
+86.67
+95.00
+83.61
+89.01
+Recall (%)
+95.75
+93.69
+95.00
+87.93
+92.05
+F1-score (%)
+97.09
+90.04
+95.00
+85.71
+90.50
+Specificity (%)
+98.56
+98.38
+98.12
+99.04
+99.01
+EiTmv
+Precision (%)
+96.74
+79.67
+94.14
+70.59
+82.11
+Recall (%)
+93.35
+88.29
+91.00
+82.76
+88.64
+F1-score (%)
+95.01
+83.76
+92.54
+76.19
+85.25
+Specificity (%)
+96.95
+97.47
+97.87
+98.08
+98.32
+In each row, the best result is marked bold, second-best is italic, and lowest is underlined.
+3) Comparison: In Table IV, we present a comparative
+analysis with some major contemporary deep learning archi-
+tectures, e.g., ResNet50 [54], InceptionV3 [55], MobileNetV2
+[56], Xception [57], DenseNet169 (Farag et al. [18]), Efficient-
+Net [37], and SE-ResNeXt50 [38]. We have also compared
+with recently published transformer-based models, i.e., CoT-
+XNet [39], and SSiT [40]. Comparison with some major
+related works [5], [35], [36] can also be seen in this table.
+Our EiTwm outperformed the major state-of-the-art methods
+with respect to accuracy, balanced accuracy, sensitivity, and
+specificity. Our EiTmv also performed quite well in terms of
+balanced accuracy.
+4) Impact of Hyper-parameters:
+We
+tuned
+the
+hyper-
+parameters and observed their impact on the experiment.
+a) MSA Head Count: We analyzed the performance im-
+pact of the number of heads (h) of MSA (Multi-head Self-
+Attention) in the transformer encoder and present in Fig. 9.
+As evident from this figure, the performance (accuracy) of
+TABLE IV
+COMPARATIVE STUDY
+Method
+Accuracy
+Sensitivity
+Specificity
+Balanced
+(%)
+(%)
+(%)
+Accuracy (%)
+ResNet50 [54]
+74.64
+56.52
+85.71
+71.12
+InceptionV3 [55]
+78.72
+63.64
+85.37
+74.51
+MobileNetV2 [56]
+79.01
+76.47
+84.62
+80.55
+Xception [57]
+79.59
+82.35
+86.32
+84.34
+Farag et al. [18]
+82.00
+-
+-
+-
+Kassani et al. [35]
+83.09
+88.24
+87.00
+87.62
+TAN [36]
+85.10
+90.30
+92.00
+-
+EfficientNet-B4 [37]
+90.30
+81.20
+97.60
+89.40
+EfficientNet-B5 [37]
+90.70
+80.70
+97.70
+89.20
+SE-ResNeXt50 [38]
+92.40
+87.10
+98.20
+92.65
+Tymchenko et al. [5]
+92.90
+86.00
+98.30
+92.15
+CoT-XNet [39]
+84.18
+-
+95.74
+-
+SSiT [40]
+92.97
+-
+-
+-
+EiTmv [ours]
+91.26
+88.81
+97.74
+93.28
+EiTwm [ours]
+94.63
+92.88
+98.62
+95.75
+In each column, the best result is marked bold, and the second-best is underlined.
+Fig. 9. Impact of number of heads (h) in MSA on model performance.
+both EiTmv and EiTwm increased with the increment of h
+till h = 6, and started decreasing thereafter.
+b) Weights αj of EiTwm: We tuned the weights αj (refer
+to Eqn. 9) to see its impact on the performance of EiTwm.
+We obtained the best accuracy of 94.63% from EiTwm for
+α1 = α2 = 0.1, and α3 = α4 = 0.4. The performance of
+EiTwm during tuning of αj’s is shown in Table V.
+In Table VI, we also present the tuned αj’s that aided in
+obtaining the best performing ensembled transformers of Table
+I.
+5) Ablation Study: We here present the performed ablation
+study by ablating individual transformers. Our EiT is actually
+an ensembling of four different image transformers, i.e., ViT,
+DeiT, CaiT, and BEiT. We ablated each transformer and
+observed performance degradation than EiT. For example,
+considering the weighted mean scheme, when we ablated CaiT
+from EiT, the accuracy dropped by 4.1%. Similarly, ablating
+BEiT and CaiT deteriorated the accuracy by 7.6%. For our
+task, the best individual transformer (CaiT) attained 7.72%
+lower accuracy than EiTwm. More examples can be observed
+in Table I.
+6) Pre-training with Other Datasets: We checked the perfor-
+mance of our EiT model by pre-training with some other
+dataset. We took 1200 images of MESSIDOR [58] with
+adjudicated grades by [59] (say, DBM). From IDRiD [60],
+we also used “Disease Grading” dataset containing 516 images
+(say, DBI). Here, we made four training set setups from DBM,
+by taking 25%, 50%, 75%, and 100% of samples of DBM.
+
+96
+94.63
+EiTmv
+94
+92.34
+91.92
+92
+91
+90.28
+90
+89.43
+88.59
+87.67
+88
+87
+86
+84
+82
+80
+6
+8
+108
+GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022
+TABLE V
+PERFORMANCE OF EiTwm BY TUNING WEIGHTS αj
+α1
+α2
+α3
+α4
+Accuracy (%)
+0.25
+0.25
+0.25
+0.25
+89.53
+0.85
+0.05
+0.05
+0.05
+82.29
+0.05
+0.85
+0.05
+0.05
+85.78
+0.05
+0.05
+0.85
+0.05
+86.92
+0.05
+0.05
+0.05
+0.85
+87.05
+0.7
+0.1
+0.1
+0.1
+82.35
+0.1
+0.7
+0.1
+0.1
+85.91
+0.1
+0.1
+0.7
+0.1
+87.04
+0.1
+0.1
+0.1
+0.7
+87.20
+0.5
+0.167
+0.167
+0.166
+82.88
+0.166
+0.5
+0.167
+0.167
+86.35
+0.167
+0.166
+0.5
+0.167
+87.62
+0.167
+0.167
+0.166
+0.5
+87.74
+0.3
+0.3
+0.2
+0.2
+88.16
+0.3
+0.2
+0.3
+0.2
+89.58
+0.3
+0.2
+0.2
+0.3
+90.27
+0.2
+0.3
+0.3
+0.2
+90.85
+0.2
+0.3
+0.2
+0.3
+91.67
+0.2
+0.2
+0.3
+0.3
+92.72
+0.4
+0.4
+0.1
+0.1
+91.18
+0.4
+0.1
+0.4
+0.1
+91.49
+0.4
+0.1
+0.1
+0.4
+92.15
+0.1
+0.4
+0.4
+0.1
+92.84
+0.1
+0.4
+0.1
+0.4
+93.47
+0.1
+0.1
+0.4
+0.4
+94.63
+TABLE VI
+TUNED WEIGHTS αj FOR TRANSFORMERS ENSEMBLED WITH
+WEIGHTED MEAN
+Transformerswm
+α1
+α2
+α3
+α4
+ViT + DeiT
+0.25
+0.75
+-
+-
+ViT + BEiT
+0.4
+0.6
+-
+-
+ViT + CaiT
+0.4
+0.6
+-
+-
+DeiT + BEiT
+0.4
+0.6
+-
+-
+DeiT + CaiT
+0.3
+0.7
+-
+-
+BEiT + CaiT
+0.5
+0.5
+-
+-
+ViT + DeiT + BEiT
+0.2
+0.3
+0.5
+-
+ViT + DeiT + CaiT
+0.2
+0.3
+0.5
+-
+ViT + BEiT + CaiT
+0.2
+0.4
+0.4
+-
+DeiT + BEiT + CaiT
+0.3
+0.3
+0.4
+-
+ViT + DeiT + BEiT + CaiT
+0.1
+0.1
+0.4
+0.4
+Similarly, four training setups were generated from DBI. As
+mentioned in § IV-A, we divided APTOS-2019 database (DB)
+in training (DBtr) and test (DBt) sets with a ratio of 7 : 3. In
+Table VII, we present the performance of EiT on DBt, while
+pre-training with DBM and DBI, and training with DBtr.
+It can be observed that the performance of EiT improved
+slightly when pre-trained with more data from other datasets.
+TABLE VII
+ACCURACY (%) OF EiT WITH PRE-TRAINING
+Pre-training data
+25%
+50%
+75%
+100%
+EiTwm
+DBM
+94.71
+94.78
+94.83
+94.88
+DBI
+94.65
+94.67
+94.7
+94.79
+DBM + DBI
+94.73
+94.85
+94.98
+95.13
+N.A.
+94.63
+EiTmv
+DBM
+91.35
+91.48
+91.56
+91.61
+DBI
+91.27
+91.32
+91.34
+91.35
+DBM + DBI
+91.42
+91.6
+91.68
+91.75
+N.A.
+91.26
+N.A.: without pre-training data
+V. CONCLUSION
+In this paper, we tackle the problem of automated severity
+stage detection of DR from fundus images. For this purpose,
+we propose two ensembled image transformers, EiTwm and
+EiTmv, by using weighted mean and majority voting combi-
+nation schemes, respectively. We here adopt four transformer
+models, i.e., ViT, DeiT, CaiT, and BEiT. For experimentation,
+we employed the publicly available APTOS-2019 blindness
+detection dataset, on which EiTwm and EiTmv attained
+accuracies of 94.63% and 91.26%, respectively. Although
+the employed dataset was imbalanced, our models performed
+quite well. Our EiTwm outperformed the major state-of-the-
+art techniques. We also performed an ablation study and
+observed the importance of the ensembling over the individual
+transformers.
+In the future, we will endeavor to improve the model perfor-
+mance with some imbalanced learning techniques. Currently,
+our model does not perform any lesion segmentation, which
+we will also attempt to explore some implicit characteristics
+of fundus images due to DR.
+APPENDIX I
+QUALITATIVE VISUALIZATION
+As mentioned in § IV-B.2, we present the Grad-CAM maps
+of the employed individual image transformers in Fig. 10.
+negative
+mild
+moderate
+severe
+proliferative
+Fig. 10.
+Fundus images (1st row) with Grad-CAM maps for ViT, DeiT,
+BEiT, CaiT as shown in 2nd, 3rd, 4th, 5th rows, respectively.
+REFERENCES
+[1] S. Stolte and R. Fang, “A survey on medical image analysis in diabetic
+retinopathy,” Medical image analysis, vol. 64, p. 101742, 2020.
+[2] N. Asiri et al., “Deep learning based computer-aided diagnosis systems
+for diabetic retinopathy: A survey,” Artificial intelligence in medicine,
+vol. 99, p. 101701, 2019.
+[3] Y. Zheng et al., “The worldwide epidemic of diabetic retinopathy,”
+Indian journal of ophthalmology, vol. 60, no. 5, p. 428, 2012.
+[4] D. S. W. Ting et al., “Development and validation of a deep learning
+system for diabetic retinopathy and related eye diseases using retinal
+images from multiethnic populations with diabetes,” JAMA, vol. 318,
+no. 22, pp. 2211–2223, 2017.
+[5] B. Tymchenko et al., “Deep learning approach to diabetic retinopathy
+detection,” in ICPRAM, 2020, pp. 501–509.
+
+SADAK et al.: DETECTING SEVERITY OF DIABETIC RETINOPATHY FROM FUNDUS IMAGES USING ENSEMBLED TRANSFORMERS
+9
+[6] J. Krause et al., “Grader variability and the importance of reference stan-
+dards for evaluating machine learning models for diabetic retinopathy,”
+Ophthalmology, vol. 125, no. 8, pp. 1264–1272, 2018.
+[7] APTOS 2019 Blindness Detection. [Online]. Available: https://www.
+kaggle.com/competitions/aptos2019-blindness-detection
+[8] G. Quellec et al., “Optimal wavelet transform for the detection of
+microaneurysms in retina photographs,” IEEE transactions on medical
+imaging, vol. 27, no. 9, pp. 1230–1241, 2008.
+[9] L. Giancardo et al., “Microaneurysm detection with radon transform-
+based classification on retina images,” in EMBS, 2011, pp. 5939–5942.
+[10] R. Casanova et al., “Application of random forests methods to diabetic
+retinopathy classification analyses,” PLOS one, vol. 9, no. 6, p. e98587,
+2014.
+[11] M. Niemeijer et al., “Automated detection and differentiation of drusen,
+exudates, and cotton-wool spots in digital color fundus photographs for
+diabetic retinopathy diagnosis,” Investigative ophthalmology & visual
+science, vol. 48, no. 5, pp. 2260–2267, 2007.
+[12] M. U. Akram et al., “Identification and classification of microaneurysms
+for early detection of diabetic retinopathy,” Pattern recognition, vol. 46,
+no. 1, pp. 107–116, 2013.
+[13] D. Usher et al., “Automated detection of diabetic retinopathy in digital
+retinal images: a tool for diabetic retinopathy screening,” Diabetic
+Medicine, vol. 21, no. 1, pp. 84–90, 2004.
+[14] J. D. Bodapati et al., “Deep convolution feature aggregation: an appli-
+cation to diabetic retinopathy severity level prediction,” Signal, Image
+and Video Processing, vol. 15, no. 5, pp. 923–930, 2021.
+[15] A. Zhang, Z. C. Lipton, M. Li, and A. J. Smola, “Dive into deep
+learning,” arXiv:2106.11342, 2021.
+[16] S. Yu et al., “Exudate detection for diabetic retinopathy with convolu-
+tional neural networks,” in EMBC, 2017, pp. 1744–1747.
+[17] Z. Wang et al., “Zoom-in-net: Deep mining lesions for diabetic retinopa-
+thy detection,” in MICCAI, 2017, pp. 267–275.
+[18] M. M. Farag et al., “Automatic severity classification of diabetic
+retinopathy based on densenet and convolutional block attention mod-
+ule,” IEEE Access, vol. 10, pp. 38 299–38 308, 2022.
+[19] A. Dosovitskiy et al., “An image is worth 16x16 words: Transformers
+for image recognition at scale,” arXiv:2010.11929, 2020.
+[20] A. M. Bra¸soveanu and R. Andonie, “Visualizing transformers for nlp:
+a brief survey,” in Int. Conf. on Information Visualisation (IV).
+IEEE,
+2020, pp. 270–279.
+[21] H. Touvron et al., “Going deeper with image transformers,” in ICCV,
+2021, pp. 32–42.
+[22] ——, “Training data-efficient image transformers & distillation through
+attention,” in ICML, 2021, pp. 10 347–10 357.
+[23] H. Bao, L. Dong, and F. Wei, “BEiT: BERT Pre-Training of Image
+Transformers,” in ICLR, arXiv:2106.08254, 2022.
+[24] J. Devlin et al., “BERT: Pre-training of Deep Bidirectional Transformers
+for Language Understanding,” arXiv:1810.04805, 2018.
+[25] “Grading diabetic retinopathy from stereoscopic color fundus pho-
+tographs—an extension of the modified airlie house classification: Etdrs
+report number 10,” Ophthalmology, vol. 98, no. 5, Supplement, pp. 786–
+806, 1991.
+[26] U. R. Acharya et al., “Automated diabetic macular edema (dme) grading
+system using dwt, dct features and maculopathy index,” Computers in
+biology and medicine, vol. 84, pp. 59–68, 2017.
+[27] K. M. Adal et al., “An automated system for the detection and clas-
+sification of retinal changes due to red lesions in longitudinal fundus
+images,” IEEE transactions on biomedical engineering, vol. 65, no. 6,
+pp. 1382–1390, 2017.
+[28] L. Tang et al., “Splat feature classification with application to retinal
+hemorrhage detection in fundus images,” IEEE Transactions on Medical
+Imaging, vol. 32, no. 2, pp. 364–375, 2013.
+[29] A. Herliana et al., “Feature selection of diabetic retinopathy disease
+using particle swarm optimization and neural network,” in CITSM.
+IEEE, 2018, pp. 1–4.
+[30] S. Sanromà et al., “Assessment of diabetic retinopathy risk with random
+forests.” in ESANN, 2016.
+[31] A. Osareh et al., “A computational-intelligence-based approach for
+detection of exudates in diabetic retinopathy images,” IEEE Transactions
+on Information Technology in Biomedicine, vol. 13, no. 4, pp. 535–545,
+2009.
+[32] P. Chudzik et al., “Microaneurysm detection using deep learning and
+interleaved freezing,” in Medical imaging 2018: image processing, vol.
+10574.
+SPIE, 2018, pp. 379–387.
+[33] R. Gargeya and T. Leng, “Automated identification of diabetic retinopa-
+thy using deep learning,” Ophthalmology, vol. 124, no. 7, pp. 962–969,
+2017.
+[34] S. Wan et al., “Deep convolutional neural networks for diabetic retinopa-
+thy detection by image classification,” Computers & Electrical Engineer-
+ing, vol. 72, pp. 274–282, 2018.
+[35] S. H. Kassani et al., “Diabetic retinopathy classification using a modified
+xception architecture,” in ISSPIT, 2019, pp. 1–6.
+[36] M. D. Alahmadi, “Texture attention network for diabetic retinopathy
+classification,” IEEE Access, vol. 10, pp. 55 522–55 532, 2022.
+[37] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for convo-
+lutional neural networks,” in ICML, 2019, pp. 6105–6114.
+[38] J. Hu et al., “Squeeze-and-excitation networks,” in CVPR, 2018, pp.
+7132–7141.
+[39] S. Zhao et al., “Cot-xnet: Contextual transformer with xception network
+for diabetic retinopathy grading,” Physics in Medicine & Biology, 2022.
+[40] Y. Huang et al., “Ssit: Saliency-guided self-supervised image trans-
+former for diabetic retinopathy grading,” arXiv:2210.10969, 2022.
+[41] I. Goodfellow et al., Deep learning.
+MIT press, 2016.
+[42] V. Stimper et al., “Multidimensional contrast limited adaptive histogram
+equalization,” IEEE Access, vol. 7, pp. 165 437–165 447, 2019.
+[43] D. Sarvamangala et al., “Convolutional neural networks in medical
+image understanding: a survey,” Evolutionary intel., pp. 1–22, 2021.
+[44] T. Wolf et al., “Transformers: State-of-the-Art Natural Language Pro-
+cessing,” in EMNLP: System Demonstrations, 2020, pp. 38–45.
+[45] A. Vaswani et al., “Attention is all you need,” Advances in neural
+information processing systems, vol. 30, 2017.
+[46] J. L. Ba, J. R. Kiros, and G. E. Hinton, “Layer normalization,”
+arXiv:1607.06450, 2016.
+[47] D. Misra, “Mish: A self regularized non-monotonic neural activation
+function,” BMVC, Paper #928, 2020.
+[48] G. Hinton, O. Vinyals, and J. Dean, “Distilling the Knowledge in a
+Neural Network,” in NIPS Deep Learning and Representation Learning
+Workshop, arXiv:1503.02531, 2015.
+[49] A. Ramesh et al., “Zero-shot text-to-image generation,” in ICML, 2021,
+pp. 8821–8831.
+[50] L. Rokach, “Ensemble-based Classifiers,” Artificial intelligence review,
+vol. 33, no. 1, pp. 1–39, 2010.
+[51] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,”
+ICLR, arXiv:1711.05101, 2019.
+[52] R. Selvaraju et al., “Grad-CAM: Visual Explanations from Deep Net-
+works via Gradient-based Localization,” in ICCV, 2017, pp. 618–626.
+[53] M. Grandini, E. Bagli, and G. Visani, “Metrics for multi-class classifi-
+cation: an overview,” arXiv preprint arXiv:2008.05756, 2020.
+[54] K. He et al., “Deep residual learning for image recognition,” in CVPR,
+2016, pp. 770–778.
+[55] C. Szegedy et al., “Rethinking the inception architecture for computer
+vision,” in CVPR, 2016, pp. 2818–2826.
+[56] M. Sandler et al., “Mobilenetv2: Inverted residuals and linear bottle-
+necks,” in CVPR, 2018, pp. 4510–4520.
+[57] F. Chollet, “Xception: Deep learning with depthwise separable convo-
+lutions,” in CVPR, 2017, pp. 1251–1258.
+[58] E. Decencière et al., “Feedback on a publicly distributed image database:
+the Messidor database,” Image Analysis & Stereology, vol. 33, no. 3, pp.
+231–234, 2014.
+[59] J. Krause et al., “Grader variability and the importance of reference stan-
+dards for evaluating machine learning models for diabetic retinopathy,”
+Ophthalmology, vol. 125, no. 8, pp. 1264–1272, 2018.
+[60] P. Porwal et al., “Indian Diabetic Retinopathy Image Dataset (IDRiD),”
+2018. [Online]. Available: https://dx.doi.org/10.21227/H25W98
+

0tAzT4oBgHgl3EQfDPqY/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:cc427607bd91fe7c63af46d8869ec43535372a3a594fd7019d44feeaa3e8c4cc
+size 149953

0tFPT4oBgHgl3EQfTjTq/content/tmp_files/2301.13054v1.pdf.txt

@@ -0,0 +1,1564 @@
+arXiv:2301.13054v1  [cs.FL]  30 Jan 2023
+Monadic Expressions and their Derivatives
+Samira Attou1, Ludovic Mignot2, Clément Miklarz2, and Florent Nicart2
+1 Université Gustave Eiffel,
+5 Boulevard Descartes — Champs s/ Marne
+77454 Marne-la-Vallée Cedex 2
+2 GR2IF,
+Université de Rouen Normandie,
+Avenue de l’Université,
+76801 Saint-Étienne-du-Rouvray, France
+[email protected],
+{ludovic.mignot,clement.miklarz1, florent.nicart}@univ-rouen.fr
+Abstract. We propose another interpretation of well-known derivatives computations from regular expres-
+sions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data
+structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization
+advantage, we first introduce a new derivation technique based on the graded module monad and then show an
+application of this technique to generalize the parsing of expression with capture groups and back references.
+We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations
+(like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights).
+Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression,
+using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us
+to perform this construction in a unified way, whatever the underlying monad.
+Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques
+of functional programming, and we provide a web interface to manipulate concrete examples.
+1
+Introduction
+This paper is an extended version of [2].
+Regular expressions are a classical way to represent associations between words and value sets. As an example,
+classical regular expressions denote sets of words and regular expressions with multiplicities denote formal series.
+From a regular expression, solving the membership test (determining whether a word belongs to the denoted
+language) or the weighting test (determining the weight of a word in the denoted formal series) can be solved,
+following Kleene theorems [11,17] by computing a finite automaton, such as the position automaton [9,3,5,6].
+Another family of methods to solve these tests is the family of derivative computations, that does not require the
+construction of a whole automaton. The common point of these techniques is to transform the test for an arbitrary
+word into the test for the empty word, which can be easily solved in a purely syntactical way (i.e. by induction over
+the structure of expressions). Brzozowski [4] shows how to compute, from a regular expression E and a word w, a
+regular expression dw(E) denoting the set of words w′ such that ww′ belongs to the language denoted by E. Solving
+the membership test hence becomes the membership test for the empty word in the expression dw(E). Antimirov [1]
+modifies this method in order to produce sets of expressions instead of expressions, i.e. defines the partial derivatives
+∂w(E) as a set of expressions the sum of which denotes the same language as dw(E). If the number of derivatives
+is exponential w.r.t. the length |E| of E in the worst case3, the partial derivatives produce at most a linear number
+of expressions w.r.t. |E|. Lombardy and Sakarovitch [13] extends these methods to expressions with multiplicities.
+Finally, Sulzmann and Lu [18] apply these derivation techniques to parse POSIX expressions.
+It is well-known that these methods are based on a common operation, the quotient of languages. Furthermore,
+Antimirov’s method can be interpreted as the derivation of regular expression with multiplicities in the Boolean
+semiring. However, the Brzozowski computation does not produce the same expressions (i.e. equality over the syntax
+trees) as the Antimirov one.
+Main contributions: In this paper, we present a unification of these computations by applying notions of
+category theory to the category of sets, and show how to compute categorical automata as defined in [7], by reinter-
+preting the work started in [15]. We make use of classical monads to model well-known derivatives computations.
+Furthermore, we deal with extended expressions in a general way: in this paper, expressions can support extended
+3 as far as rules of associativity, commutativity and idempotence of the sum are considered, possibly infinite otherwise.
+
+operators like complement, intersection, but also any n-ary function (algebraic mean, extrema multiplications, etc.).
+The main difference with [15] is that we formally state the languages and series that the expressions denote in an
+inherent way w.r.t. the underlying monads.
+More precisely, this paper presents:
+– an extension of expressions to any n-ary function over the value set,
+– a monadic generalization of expressions,
+– a solution for the membership/weight test for these expressions,
+– a computation of categorical derivative automata,
+– a new monad that fits with the extension to n-ary functions,
+– an illustration implemented in Haskell using advanced functional programming,
+– an extension to capture groups and back references expressions.
+Motivation: The unification of derivation techniques is a goal by itself. Moreover, the formal tools used to
+achieve this unification are also useful: Monads offer both theoretical and practical advantages. Indeed, from a
+theoretical point of view, these structures allow the abstraction of properties and focus on the principal mechanisms
+that allow solving the membership and weight problems. Besides, the introduction of exotic monads can also facilitate
+the study of finiteness of derivated terms. From a practical point of view, monads are easy to implement (even in
+some other languages than Haskell) and allow us to produce compact and safe code. Finally, we can easily combine
+different algebraic structures or add some technical functionalities (capture groups, logging, nondeterminism, etc.)
+thanks to notions like monad transformers [10] that we consider in this paper.
+This paper is structured as follows. In Section 2, we gather some preliminary material, like algebraic structures
+or category theory notions. We also introduce some functions well-known to the Haskell community that can allow
+us to reduce the size of our equations. We then structurally define the expressions we deal with, the associated series
+and the weight test for the empty word in Section 3. In order to extend this test to any arbitrary word, we first state
+in Section 4 some properties required by the monads we consider. Once this so-called support is determined, we show
+in Section 5 how to compute the derivatives. The computation of derivative automata is explained in Section 6.
+A new monad and its associated derivatives computation is given in Section 7. An implementation is presented
+in Section 8. Finally, we show how to (alternatively to [18]) compute derivatives of capture group expressions in
+Section 9 and show that as far as the same operators are concerned, the derivative formulae are the same whatever
+the underlying monad is.
+2
+Preliminaries
+We denote by S → S′ the set of functions from a set S to a set S′. The notation λx → f(x) is an equivalent notation
+for a function f.
+A monoid is a set S endowed with an associative operation and a unit element. A semiring is a structure
+(S, ×, +, 1, 0) such that (S, ×, 1) is a monoid, (S, +, 0) is a commutative monoid, × distributes over + and 0 is an
+annihilator for ×. A starred semiring is a semiring with a unary function ⋆ such that
+k⋆ = 1 + k × k⋆ = 1 + k⋆ × k.
+A K-series over the free monoid (Σ∗, ·, ε) associated with an alphabet Σ, for a semiring K = (K, ×, +, 1, 0), is
+a function from Σ∗ to K. The set of K-series can be endowed with the structure of semiring as follows:
+1(w) =
+�
+1
+if w = ε,
+0
+otherwise,
+0(w) = 0,
+(S1 + S2)(w) = S1(w) + S2(w),
+(S1 × S2)(w) =
+�
+u·v=w
+S1(u) × S2(v).
+Furthermore, if S1(ε) = 0 (i.e. S1 is said to be proper), the star of S1 is the series defined by
+(S1)⋆(ε) = 1,
+(S1)⋆(w) =
+�
+n≤|w|,w=u1···un,uj̸=ε
+S1(u1) × · · · × S1(un).
+Finally, for any function f in Kn → K, we set:
+(f(S1, . . . , Sn))(w) = f(S1(w), . . . , Sn(w)).
+(1)
+A functor 4 F associates with each set S a set F(S) and with each function f in S → S′ a function F(f) from
+F(S) to F(S′) such that
+F(id) = id,
+F(f ◦ g) = F(f) ◦ F(g),
+4 More precisely, a functor over a subcategory of the category of sets.
+
+where id is the identity function and ◦ the classical function composition.
+A monad5 M is a functor endowed with two (families of) functions
+– pure, from a set S to M(S),
+– bind, sending any function f in S → M(S′) to M(S) → M(S′),
+such that the three following conditions are satisfied:
+bind(f)(pure(s)) = f(s),
+bind(pure) = id,
+bind(g)(bind(f)(m)) = bind(λx → bind(g)(f(x)))(m).
+Example 1. The Maybe monad associates:
+– any set S with the set Maybe(S) = {Just(s) | s ∈ S} ∪ {Nothing}, where Just and Nothing are two syntactic
+tokens allowing us to extend a set with one value;
+– any function f with the function Maybe(f) defined by
+Maybe(f)(Just(s)) = Just(f(s)),
+Maybe(f)(Nothing) = Nothing
+– is endowed with the functions pure and bind defined by:
+pure(s) = Just(s),
+bind(f)(Just(s)) = f(s),
+bind(f)(Nothing) = Nothing.
+Example 2. The Set monad associates:
+– with any set S the set 2S,
+– with any function f the function Set(f) defined by Set(f)(R) = �
+r∈R{f(r)},
+– is endowed with the functions pure and bind defined by:
+pure(s) = {s},
+bind(f)(R) =
+�
+r∈R
+f(r).
+Example 3. The LinComb(K) monad, for K = (K, ×, +, 1, 0), associates:
+– with any set S the set of K-linear combinations of elements of S, where a linear combination is a finite (formal,
+commutative) sum of couples (denoted by ⊞) in K × S where (k, s) ⊞ (k′, s) = (k + k′, s),
+– with any function f the function LinComb(K)(f) defined by
+LinComb(K)(f)(R) = ⊞
+(k,r)∈R
+(k, f(r)),
+– is endowed with the functions pure and bind defined by:
+pure(s) = (1, s),
+bind(f)(R) = ⊞
+(k,r)∈R
+k ⊗ f(r),
+where k ⊗ R = ⊞
+(k′,r)∈R
+(k × k′, r).
+To compact equations, we use the following operators for any monad M:
+f <$> s = M(f)(s),
+m >>= f = bind(f)(m).
+If <$> can be used to lift unary functions to the monadic level, >>= and pure can be used to lift any n-ary function
+f in S1 × · · · × Sn → S, defining a function liftn sending S1 × · · · × Sn → S to M(S1) × · · · × M(Sn) → M(S) as
+follows:
+liftn(f)(m1, . . . , mn) =m1 >>= (λs1 → . . .
+mn >>= (λsn → pure(f(s1, . . . , sn))) . . .)
+Let us consider the set
+1 = {⊤} with only one element. The images of this set by some previously defined monads
+can be evaluated as value sets classically used to weight words in association with classical regular expressions. As
+an example, Maybe(1) and Set(1) are isomorphic to the Boolean set, and any set LinComb(K)(1) can be converted
+into the underlying set of K. This property allows us to extend in a coherent way classical expressions to monadic
+expressions, where the type of the weights is therefore given by the ambient monad.
+5 More precisely, a monad over a subcategory of the category of sets.
+
+3
+Monadic Expressions
+As seen in the previous section, elements in M(1) can be evaluated as classical value sets for some particular
+monads. Hence, we use these elements not only for the weights associated with words by expressions, but also for
+the elements that act over the denoted series.
+In the following, in addition to classical operators (+, · and ∗), we denote:
+– the action of an element over a series by ⊙,
+– the application of a function by itself.
+Definition 1. Let M be a monad. An M-monadic expression E over an alphabet Σ is inductively defined as follows:
+E = a,
+E = ε,
+E = ∅,
+E = E1 + E2,
+E = E1 · E2,
+E = E∗
+1,
+E = α ⊙ E1,
+E = E1 ⊙ α,
+E = f (E1, . . . , En) ,
+where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
+function from (M(1))n to M(1).
+We denote by Exp(Σ) the set of monadic expressions over an alphabet Σ.
+Example 4. As an example of functions that can be used in our extension of classical operators, one can define the
+function ExtDist(x1, x2, x3) = max(x1, x2, x3) − min(x1, x2, x3) from N3 to N.
+Similarly to classical regular expressions, monadic expressions associate a weight with any word. Such a relation
+can be denoted via a formal series. However, before defining this notion, in order to simplify our study, we choose
+to only consider proper expressions. Let us first show how to characterize them by the computation of a nullability
+value.
+Definition 2. Let M be a monad such that the structure (M(1), +, ×, ⋆, 1, 0) is a starred semiring. The nullability
+value of an M-monadic expression E over an alphabet Σ is the element Null(E) of M(1) inductively defined as
+follows:
+Null(ε) = 1,
+Null(∅) = 0,
+Null(a) = 0,
+Null(E1 + E2) = Null(E1) + Null(E2),
+Null(E1 · E2) = Null(E1) × Null(E2),
+Null(E∗
+1) = Null(E1)⋆,
+Null(α ⊙ E1) = α × Null(E1),
+Null(E1 ⊙ α) = Null(E1) × α,
+Null(f(E1, . . . , En)) = f(Null(E1), . . . , Null(En)),
+where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
+function from (M(1))n to M(1).
+When the considered semiring is not a starred one, we restrict the nullability value computation to expressions
+where a starred subexpression admits a null nullability value. In order to compute it, let us consider the Maybe
+monad, allowing us to elegantly deal with such a partial function.
+Definition 3. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. The partial nullability
+value of an M-monadic expression E over an alphabet Σ is the element PartNull(E) of Maybe(M(1)) defined as
+follows:
+PartNull(ε) = Just(1),
+PartNull(∅) = Just(0),
+PartNull(a) = Just(0),
+PartNull(E1 + E2) = lift2(+)(PartNull(E1), PartNull(E2)),
+PartNull(E1 · E2) = lift2(×)(PartNull(E1), PartNull(E2)),
+PartNull(E∗
+1) =
+�
+Just(1)
+if PartNull(E1) = Just(0),
+Nothing
+otherwise,
+PartNull(α ⊙ E1) = (λE → α × E) <$> PartNull(E1),
+PartNull(E1 ⊙ α) = (λE → E × α) <$> PartNull(E1),
+PartNull(f(E1, . . . , En)) = liftn(f)(PartNull(E1), . . . , PartNull(En)),
+where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
+function from (M(1))n to M(1).
+
+An expression E is proper if its partial nullability value is not Nothing, therefore if it is a value Just(v); in this
+case, v is its nullability value, denoted by Null(E) (by abuse).
+Definition 4. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring, and E be a M-monadic
+proper expression over an alphabet Σ. The series S(E) associated with E is inductively defined as follows:
+S(ε)(w) =
+�
+1
+if w = ε,
+0
+otherwise,
+S(∅)(w) = 0,
+S(a)(w) =
+�
+1
+if w = a,
+0
+otherwise,
+S(E1 + E2) = S(E1) + S(E2),
+S(E1 · E2) = S(E1) × S(E2),
+S(E∗
+1) = (S(E1))⋆,
+S(α ⊙ E1)(w) = α × S(E1)(w),
+S(E1 ⊙ α)(w) = S(E1)(w) × α,
+S(f(E1, . . . , En)) = f(S(E1), . . . , S(En)),
+where a is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
+function from (M(1))n to M(1).
+From now on, we consider the set Exp(Σ) of M-monadic expressions over Σ to be endowed with the structure of
+a semiring, and two expressions denoting the same series to be equal. The weight associated with a word w in Σ∗
+by E is the value weightw(E) = S(E)(w). The nullability of a proper expression is the weight it associates with ε,
+following Definition 3 and Definition 4.
+Proposition 1. Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring. Let E be an M-monadic
+proper expression over Σ. Then:
+Null(E) = weightε(E).
+The previous proposition implies that the weight of the empty word can be syntactically computed (i.e. inductively
+computed from a monadic expression). Now, let us show how to extend this computation by defining the computation
+of derivatives for monadic expressions.
+4
+Monadic Supports for Expressions
+A K-left-semimodule, for a semiring K = (K, ×, +, 1, 0), is a commutative monoid (S, ±, 0) endowed with a function
+⊲ from K × S to S such that:
+(k × k′) ⊲ s = k ⊲ (k′ ⊲ s),
+(k + k′) ⊲ s = k ⊲ s ± k′ ⊲ s,
+k ⊲ (s ± s′) = k ⊲ s ± k ⊲ s′,
+1 ⊲ s = s,
+0 ⊲ s = k ⊲ 0 = 0.
+A K-right-semimodule can be defined symmetrically.
+An operad [12,14] is a structure (O, (◦j)j∈N, id) where O is a graded set (i.e. O = �
+n∈N On), id is an element of
+O1, ◦j is a function defined for any three integers (i, j, k)6 with 0 < j ≤ k in Ok × Oi → Ok+i−1 such that for any
+elements p1 ∈ Om, p2 ∈ On, p3 ∈ Op:
+∀0 < j ≤ m, id ◦1 p1 = p1 ◦j id = p1,
+∀0 < j ≤ m, 0 < j′ ≤ n, p1 ◦j (p2 ◦j′ p3) = (p1 ◦j p2) ◦j+j′−1 p3,
+∀0 < j′ ≤ j ≤ m, (p1 ◦j p2) ◦j′ p3 = (p1 ◦j′ p3) ◦j+p−1 p2.
+Combining these compositions ◦j, one can define a composition ◦ sending Ok × Oi1 × · · · × Oik to Oi1+···+ik: for
+any element (p, q1, . . . , qk) in Ok × Ok,
+p ◦ (q1, . . . , qk) = (· · · ((p ◦k qk) ◦k−1 qk−1 · · · ) · · · ) ◦1 q1.
+Conversely, the composition ◦ can define the compositions ◦j using the identity element: for any two elements (p, q)
+in Ok × Oi, for any integer 0 < j ≤ k:
+p ◦j q = p ◦ (id, . . . , id
+�
+��
+�
+j−1 times
+, q, id, . . . , id
+�
+��
+�
+k−j times
+).
+As an example, the set of n-ary functions over a set, with the identity function as unit, forms an operad.
+A module over an operad (O, ◦, id) is a set S endowed with a function ⋇ from On × Sn to S such that
+f ⋇ (f1 ⋇ (s1,1, . . . , s1,i1), . . . , fn ⋇ (sn,1, . . . , sn,in))
+= (f ◦ (f1, . . . , fn)) ⋇ (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in).
+6 every couple (i, k) unambiguously defines the domain and codomain of a function ◦j
+
+The extension of the computation of derivatives could be performed for any monad. Indeed, any monad could
+be used to define well-typed auxiliary functions that mimic the classical computations. However, some properties
+should be satisfied in order to compute weights equivalently to Definition 4. Therefore, in the following we consider
+a restricted kind of monads.
+A monadic support is a structure (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) satisfying:
+– M is a monad,
+– R = (M(1), +, ×, 1, 0) is a semiring,
+– M = (M(Exp(Σ)), ±, 0) is a monoid,
+– (M, ⋉) is a Exp(Σ)-right-semimodule,
+– (M, ⊲) is a R-left-semimodule,
+– (M, ⊳) is a R-right-semimodule,
+– (M(Exp(Σ)), ⋇) is a module for the operad of the functions over M(1).
+An expressive support is a monadic support (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) endowed with a function toExp from
+M(Exp(Σ)) to Exp(Σ) satisfying the following conditions:
+weightw(toExp(m)) = m >>= weightw
+(2)
+toExp(m ⋉ F) = toExp(m) · F,
+(3)
+toExp(m ± m′) = toExp(m) + toExp(m′),
+(4)
+toExp(m ⊲ x) = toExp(m) ⊙ x,
+(5)
+toExp(x ⊳ m) = x ⊙ toExp(m),
+(6)
+toExp(f ⋇ (m1, . . . , mn)) = f(toExp(m1), . . . , toExp(mn)).
+(7)
+Let us now illustrate this notion with three expressive supports that will allow us to model well-known derivatives
+computations.
+Example 5 (The Maybe support).
+toExp(Nothing) = 0,
+toExp(Just(E)) = E,
+Nothing + m = m,
+m + Nothing = m,
+Just(⊤) + Just(⊤) = Just(⊤),
+Nothing × m = Nothing,
+m × Nothing = Nothing,
+Just(⊤) × Just(⊤) = Just(⊤),
+Nothing ± m = m,
+m ± Nothing = m,
+Just(E) ± Just(E′) = Just(E + E′),
+1 = Just(⊤),
+0 = Nothing,
+0 = Nothing,
+m ⋉ F = (λE → E · F) <$> m,
+m ⊲ m′ = m >>= (λx → m′),
+m ⊳ m′ = m′ >>= (λx → m),
+f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
+Example 6 (The Set support).
+toExp({E1, . . . , En}) = E1 + · · · + En,
++ = ∪,
+× = ∩,
+± = ∪,
+1 = {⊤},
+0 = ∅,
+0 = ∅,
+m ⋉ F = (λE → E · F) <$> m,
+m ⊲ m′ = m >>= (λx → m′),
+m ⊳ m′ = m′ >>= (λx → m),
+f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
+Example 7 (The LinComb(K) support).
+toExp((k1, E1) ⊞ · · · ⊞ (kn, En)) = k1 ⊙ E1 + · · · + kn ⊙ En,
++ = ⊞,
+(k, ⊤) × (k′, ⊤) = (k × k′, ⊤),
+1 = (1, ⊤),
+0 = (0, ⊤),
+± = ⊞,
+0 = (0, ⊤),
+m ⋉ F = (λE → E · F) <$> m,
+m ⊲ m′ = m >>= (λx → m′),
+m ⊳ k = (λE → E ⊙ k) <$> m,
+f ⋇ (m1, . . . , mn) = pure(f(toExp(m1), . . . , toExp(mn))).
+
+5
+Monadic Derivatives
+In the following, (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇, toExp) is an expressive support.
+Definition 5. The derivative of an M-monadic expression E over Σ w.r.t. a symbol a in Σ is the element da(E)
+in M(Exp(Σ)) inductively defined as follows:
+da(ε) = 0,
+da(∅) = 0,
+da(b) =
+�
+pure(ε)
+if a = b,
+0
+otherwise,
+da(E1 + E2) = da(E1) ± da(E2),
+da(E∗
+1) = da(E1) ⋉ E∗
+1,
+da(E1 · E2) = da(E1) ⋉ E2 ± Null(E1) ⊲ da(E2),
+da(α ⊙ E1) = α ⊲ da(E1),
+da(E1 ⊙ α) = da(E1) ⊳ α,
+da(f(E1, . . . , En)) = f ⋇ (da(E1), . . . , da(En))
+where b is a symbol in Σ, (E1, . . . , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a
+function from (M(1))n to M(1).
+The link between derivatives and series can be stated as follows, which is an alternative description of the classical
+quotient.
+Proposition 2. Let E be an M-monadic expression over an alphabet Σ, a be a symbol in Σ and w be a word in
+Σ∗. Then:
+weightaw(E) = da(E) >>= weightw.
+Proof. Let us proceed by induction over the structure of E. All the classical cases (i.e. the function operator left
+aside) can be proved following the classical methods ([1,4,13]). Therefore, let us consider this last case.
+da(f(E1, . . . , En)) >>= weightw
+= weightw(toExp(da(f(E1, . . . , En))))
+(Eq (2))
+= weightw(toExp(f ⋇ (da(E1), . . . , da(En)))
+(Def 5))
+= weightw(f(toExp(da(E1)), . . . , toExp(da(En))))
+(Eq (7))
+= f(weightw(toExp(da(E1))), . . . , weightw(toExp(da(En))))
+(Def 4, Eq (1))
+= f(da(E1) >>= weightw, . . . , da(En) >>= weightw)
+(Eq (2))
+= f(weightaw(E1), . . . , weightaw(En))
+(Ind. hyp.)
+= weightaw(f(E1, . . . , En))
+(Def 4, Eq (1))
+Let us define how to extend the derivative computation from symbols to words, using the monadic functions.
+Definition 6. The derivative of an M-monadic expression E over Σ w.r.t. a word w in Σ∗ is the element dw(E)
+in M(Exp(Σ)) inductively defined as follows:
+dε(E) = pure(E),
+da·v(E) = da(E) >>= dv,
+where a is a symbol in Σ and v a word in Σ∗.
+Finally, it can be easily shown, by induction over the length of the words, following Proposition 2, that the
+derivatives computation can be used to define a syntactical computation of the weight of a word associated with an
+expression.
+Theorem 1. Let E be an M-monadic expression over an alphabet Σ and w be a word in Σ∗. Then:
+weightw(E) = dw(E) >>= Null.
+Notice that, restraining monadic expressions to regular ones,
+– the Maybe support leads to the classical derivatives [4],
+– the Set support leads to the partial derivatives [1],
+– the LinComb support leads to the derivatives with multiplicities [13].
+Example 8. Let us consider the function ExtDist defined in Example 4 and the LinComb(N)-monadic expression
+E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).
+da(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗)
+daa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗)
+
+daaa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 3 ⊙ a∗)
+daab(E) = ExtDist(b∗, b∗, b∗a∗)
+weightaaa(E) = daaa(E) >>= Null
+= ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3
+weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0
+In the next section, we show how to compute the derivative automaton associated with an expression.
+6
+Automata Construction
+A category C is defined by:
+– a class ObjC of objects,
+– for any two objects A and B, a set HomC(A, B) of morphisms,
+– for any three objects A, B and C, an associative composition function ◦C in HomC(B, C) −→ HomC(A, B) −→
+HomC(A, C),
+– for any object A, an identity morphism idA in HomC(A, A), such that for any morphisms f in HomC(A, B) and
+g in HomC(B, A), f ◦C idA = f and idA ◦C g = g.
+Given a category C, a C-automaton is a tuple (Σ, I, Q, F, i, δ, f) where
+– Σ is a set of symbols (the alphabet),
+– I is the initial object, in Obj(C),
+– Q is the state object, in Obj(C),
+– F is the final object, in Obj(C),
+– i is the initial morphism, in HomC(I, Q),
+– δ is the transition function, in Σ −→ HomC(Q, Q),
+– f is the value morphism, in HomC(Q, F).
+The function δ can be extended as a monoid morphism from the free monoid (Σ∗, ·, ε) to the morphism monoid
+(HomC(Q, Q), ◦C, idQ), leading to the following weight definition.
+The weight associated by a C-automaton A = (Σ, I, Q, F, i, δ, f) with a word w in Σ∗ is the morphism weight(w)
+in HomC(I, F) defined by
+weight(w) = f ◦C δ(w) ◦C i.
+If the ambient category is the category of sets, and if I =
+1, the weight of a word is equivalently an element of
+F. Consequently, a deterministic (complete) automaton is equivalently a Set-automaton with
+1 as the initial object
+and B as the final object.
+Given a monad M, the Kleisli composition of two morphisms f ∈ HomC(A, B) and g ∈ HomC(B, C) is the
+morphism (f >=> g)(x) = f(x) >>= g in HomC(A, C). This composition defines a category, called the Kleisli
+category K(M) of M, where:
+– the objects are the sets,
+– the morphisms between two sets A and B are the functions between A and M(B),
+– the identity is the function pure.
+Considering these categories:
+– a deterministic automaton is equivalently a K(Maybe)-automaton,
+– a nondeterministic automaton is equivalently a K(Set)-automaton,
+– a weighted automaton over a semiring K is equivalently a K(LinComb(K))-automaton,
+all with
+1 as both the initial object and the final object.
+Furthermore, for a given expression E, if i = pure(E), δ(a)(E′) = da(E′) and f = Null, we can compute
+the well-known derivative automata using the three previously defined supports, and the accessible part of these
+automata are finite ones as far as classical expressions are concerned [4,1,13].
+More precisely, extended expressions can lead to infinite automata, as shown in the next example.
+
+Example 9. Considering the computations of Example 8, it can be shown that
+dan(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗).
+Hence, there is not a finite number of derivated terms, that are the states in the classical derivative automaton.
+This infinite automaton is represented in Figure 1, where the final weights of the states are represented by double
+edges. The sink states are omitted.
+ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗)
+ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗)
+ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗)
+ExtDist(b∗, b∗, b∗a∗)
+ExtDist(0, 0, a∗)
+ExtDist(b∗ + b∗a∗, b∗a∗b∗ + b∗, b∗a∗)
+ExtDist(b∗ + b∗a∗, b∗a∗b∗ + 2 ⊙ b∗, b∗a∗)
+ExtDist(a∗, a∗b∗, a∗)
+ExtDist(0, b∗, 0)
+ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗)
+ExtDist(b∗ + b∗a∗, b∗a∗b∗ + n ⊙ b∗, b∗a∗)
+1
+1
+2
+n
+1
+1
+2
+1
+n
+a
+b
+b
+a
+b
+a
+b
+a
+b
+a
+b
+a
+b
+a
+b
+a
+b
+a
+Fig. 1. The (infinite) derivative weighted automaton associated with E.
+In the following section, let us show how to model a new monad in order to solve this problem.
+7
+The Graded Module Monad
+Let us consider an operad O = (O, ◦, id) and the association sending:
+– any set S to �
+n∈N On × Sn,
+– any f in S → S′ to the function g in �
+n∈N On × Sn → �
+n∈N On × S′n:
+g(o, (s1, . . . , sn)) = (o, (f(s1), . . . , f(sn)))
+It can be checked that this is a functor, denoted by GradMod(O). Moreover, it forms a monad considering the two
+following functions:
+pure(s) = (id, s),
+(o, (s1, . . . , sn)) >>= f = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in))
+where f(sj) = (oj, sj,1, . . . , sj,ij). However, notice that GradMod(O)(1) cannot be easily evaluated as a value space.
+Thus, let us compose it with another monad. As an example, let us consider a semiring K = (K, ×, +, 1, 0) and
+the operad O of the n-ary functions over K. Hence, let us define the functor7 GradComb(O, K) that sends S to
+GradMod(O)(LinComb(K)(S)).
+7 it is folk knowledge that the composition of two functors is a functor.
+
+To show that this combination is a monad, let us first define a function α sending GradComb(O, K)(S) to
+GradMod(O)(S). It can be easily done by converting a linear combination into an operadic combination, i.e. an
+element in GradMod(O)(S), with the following function toOp:
+toOp((k1, s1) ⊞ · · · ⊞ (kn, sn))
+= (λ(x1, . . . , xn) → k1 × x1 + · · · + kn × xn, (s1, . . . , sn)),
+α(o, (L1, . . . , Ln)) = (o ◦ (o1, . . . , on), (s1,1, . . . , s1,i1, . . . , sn,1, . . . , sn,in))
+where toOp(Lj) = (oj, (sj,1, . . . , sj,ij)).
+Consequently, we can define the monadic functions as follows:
+pure(s) = (id, (1, s)),
+(o, (L1, . . . , Ln)) >>= f = α(o, (L1, . . . , Ln)) >>= f
+where the second occurrence of >>= is the monadic function associated with the monad GradMod(O).
+Let us finally define an expressive support for this monad:
+toExp(o, (L1, . . . , Ln)) = o(toExp(L1), . . . , toExp(Ln)),
+(o, (L1, . . . , Ln)) + (o′, (L′
+1, . . . , L′
+n′)) = (o + o′, (L1, . . . , Ln, L′
+1, . . . , L′
+n′))
+(o, (L1, . . . , Ln)) × (o′, (L′
+1, . . . , L′
+n′)) = (o × o′, (L1, . . . , Ln, L′
+1, . . . , L′
+n′))
+± = +,
+1 = (id, (1, ⊤)),
+0 = (id, (0, ⊤)),
+0 = (id, (0, ⊤)),
+m ⋉ F = pure(toExp(m) · F),
+(o, (M1, . . . , Mk)) ⊲ (o′, (L1, . . . , Ln)) = (o(M1, . . . , Mk) × o′, (L1, . . . , Ln)),
+(o, (L1, . . . , Ln)) ⊳ (o′, (M1, . . . , Mk)) = (o × o′(M1, . . . , Mk), (L1, . . . , Ln))
+f ⋇ ((o1, (L1,1, . . . , L1,i1)), . . . , (on, (Ln,1, . . . , Ln,in)))
+= (f ◦ (o1, . . . , on), (L1,1, . . . , L1,i1, . . . , Ln,1, . . . , Ln,in))
+where (o + o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) + o′(xn+1, . . . , xn+n′)
+(o × o′)(x1, . . . , xn+n′) = o(x1, . . . , xn) × o′(xn+1, . . . , xn+n′)
+Example 10. Let us consider that two elements in GradComb(O, K)(Exp(Σ)) are equal if they have the same image
+by toExp. Let us consider the expression E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) of Example 8.
+da(E) = ExtDist ⋇ ((+, (a∗b∗, a∗)), (id, a∗b∗), (+, (a∗b∗a∗, a∗)))
+= (ExtDist ◦ (+, id, +), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
+daa(E) = (ExtDist ◦ (+, id, + ◦ (+, id)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗, a∗))
+= (ExtDist ◦ (+, id, + ◦ (id, 2×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
+daaa(E) = (ExtDist ◦ (+, id, + ◦ (id, 3×)), (a∗b∗, a∗, a∗b∗, a∗b∗a∗, a∗))
+daab(E) = (ExtDist ◦ (+, id, +), (b∗, ∅, b∗, b∗a∗, ∅))
+= (ExtDist, (b∗, b∗, b∗a∗))
+weightaaa(E) = daaa(E) >>= Null
+= ExtDist ◦ (+, id, +)(1, 1, 1, 1, 3)
+= ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3
+weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0
+Using this monad, the number of derivated terms, that is the number of states in the associated derivative automaton,
+is finite. Indeed, the computations are absorbed in the transition structure. This automaton is represented in
+Figure 2. Notice that the dashed rectangle represent the functions that are composed during the traversal associated
+with a word. The final weights are represented by double edges. The sink states are omitted. The state b∗ is duplicated
+to simplify the representation.
+
+ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗)
+ExtDist
++
++
+ExtDist
++
+b∗a∗b∗
++
+a∗b∗a∗
++
+a∗b∗
+a∗
+b∗
+b∗a∗
+b∗
+1
+1
+1
+1
+1
+1
+1
+1
+a
+b
+a
+b
+b
+a
+b
+a
+a
+b
+b
+b
+Fig. 2. The Associated Derivative Automaton of ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).
+However, notice that not every monadic expression produces a finite set of derivated terms, as shown in the next
+example.
+Example 11. Let us consider the expression E of Example 8 and the expression F = E · c∗. It can be shown that
+dan(F) = toExp(dan(E)) · c∗
+= ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) · c∗.
+The study of the necessary and sufficient conditions of monads that lead to a finite set of derivated terms is one
+of the next steps of our work.
+8
+Haskell Implementation
+The notions described in the previous sections have been implemented in Haskell, as follows:
+– The notion of monad over a sub-category of sets is a typeclass using the Constraint kind to specify a sub-
+category;
+– n-ary functions and their operadic structures are implemented using fixed length vectors, the size of which is
+determined at compilation using type level programming;
+– The notion of graded module is implemented through an existential type to deal with unknown arities: Its
+monadic structure is based on an extension of heterogeneous lists, the graded vectors, typed w.r.t. the list of
+the arities of the elements it contains;
+– The parser and some type level functions are based on dependently typed programming with singletons [8],
+allowing, for example, determining the type of the monads or the arity of the functions involved at run-time;
+– An application is available here [16] illustrating the computations:
+• the backend uses servant to define an API;
+
+• the frontend is defined using Reflex, a functional reactive programming engine and cross compiled in
+JavaScript with GHCJS.
+As an example, the monadic expression of the previous examples can be entered in the web application as the
+input ExtDist(a*.b*+b*.a*,b*.a*.b*,a*.b*.a*).
+9
+Capture Groups
+Capture groups are a standard feature of POSIX regular expressions where parenthesis are used to memorize
+some part of the input string being matched in order to reuse either for substitution or matching. We give here
+an equivalent definition along with derivation formulae and a monadic definition. The semantic of this definition
+conforms to those of POSIX expressions. Precisely, when a capture group has been involved more than one time
+due to a stared subexpression, the value of the corresponding variable corresponds to the last capture.
+9.1
+Syntax of Expressions with Capture Groups
+A capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression for short)
+is inductively defined as
+E = a,
+E = ε,
+E = ∅,
+E = F + G,
+E = F · G,
+E = F ∗,
+E = (F)x,
+E = x,
+where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ. In the POSIX
+syntax, capture groups are implicitly mapped with variables respectively with the order of the opening parenthesis
+of a pair. Here, each capture group is associated explicitly to a variable by indexing the closing parenthesis with
+the name of this variable.
+9.2
+Contextual Expressions and their Contextual Languages
+In order to define the contextual language and the derivation of capture-group expressions, we need to extend the
+syntax of the expressions in order to attach to any capture group the current part of the input string captured
+during an execution.
+A contextual capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression
+for short) is inductively defined as
+E = a,
+E = ε,
+E = ∅,
+E = F + G,
+E = F · G,
+E = F ∗,
+E = (F)u
+x,
+E = x,
+where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ.
+Notice that a Σ, Γ-expression is equivalent to a contextual capture-group expression where u = ε for every
+occurrence of capture group.
+In the following, we consider that a context is a function from Γ to Maybe(Σ∗), modelling the possibility that
+a variable was initialized (or not) during the parsing. The set of contexts is denoted by Ctxt(Γ, Σ).
+Using these notions of contexts, let us now explain the semantics of contextual capture-group expressions. While
+parsing, a context is built to memorize the different affectations of words to variables. Therefore, a (contextual)
+language associated with an expression is a set of couples built from a language and the context that was used to
+compute it.
+The classic atomic cases (a symbol, the empty word or the empty set) are easy to define, preserving the context.
+Another one is the case of a variable x: the context is applied here to compute the associated word (if it exists) and
+is preserved.
+The recursive cases are interpreted as such:
+– The contextual language of a sum of two expressions is the union of their contextual languages, computed
+independently.
+– The contextual language of a catenation of two expressions F and G is computed in three steps. First, the
+contextual language of F is computed. Secondly, for each couple (L, ctxt) of this contextual language, the
+function ctxt is considered as the new context to compute the contextual language of G, leading to new couples
+(L′, ctxt′). Finally, for each of these combinations, a couple (L·L′, ctxt′) is added to form the resulting contextual
+language.
+
+– The contextual language of a starred expression is, classically, the infinite union of the powered contextual
+languages, computed by iterated catenations.
+– The contextual language of a captured expression (F)u
+x is computed in two steps. First, the contextual language
+of F is computed. Then, for each couple (L, ctxt) of it, a word w is chosen in L and the context ctxt must be
+updated coherently.
+More formally, the contextual language of a Σ, Γ-expression E associated with a context ctxt in Ctxt(Γ, Σ) is
+the subset Lctxt(E) of 2Σ∗ × Ctxt(Γ, Σ) inductively defined as follows:
+Lctxt(a) = {({a}, ctxt)},
+Lctxt(ε) = {({ε}, ctxt)},
+Lctxt(∅) = ∅,
+Lctxt(x) =
+�
+∅
+if ctxt(x) = Nothing,
+{({w}, ctxt)}
+otherwise if ctxt(x) = Just(w),
+Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G),
+Lctxt(F · G) =
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1(G)
+{(L1 · L2, ctxt2)},
+Lctxt(F ∗) =
+�
+n∈N
+(Lctxt(F))
+n,
+Lctxt((F)u
+x) =
+�
+(L1,ctxt1)∈Lctxt(F ),
+w∈L1
+{({w}, [ctxt1]x←uw)},
+where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ, u is in Σ∗, Ln is defined, for any
+set L of couples (language, context) by
+Ln =
+
+
+
+
+
+
+
+
+
+
+
+�
+(L,ctxt)∈L
+{({ε}, ctxt)}
+if n = 0,
+�
+(L1,ctxt1)∈L,
+(L2,ctxt2)∈Ln−1
+{(L1 · L2, ctxt2)}
+otherwise,
+and [ctxt]x←w is the context defined by
+[ctxt]x←w(y) =
+�
+Just(w)
+if x = y,
+ctxt(y)
+otherwise.
+The contextual language of an expression E is the set of couples obtained from an uninitialised context, where
+nothing is associated with any variable, that is the set
+Lλ_→Nothing(E).
+Finally, the language denoted by an expression E is the set of words obtained by forgetting the contexts, that is the
+set
+�
+(L,_)∈Lλ_→Nothing(E)
+L.
+Example 12. Let us consider the three following expressions over the symbol alphabet {a, b, c} and the variable
+alphabet {x}:
+E = E1 · E2,
+E1 = ((a∗)xbx)∗,
+E2 = cx.
+The language denoted by E2 is empty, since it is computed from the empty context, where nothing is associated
+with x. However, parsing E1 allows us to compute contexts that define word values to affect to x. Let us thus show
+how is defined the contextual language of E1:
+– the contextual language of (a∗)x is the set�
+n∈N
+{({an}, λx → Just(an))}
+where each word an is recorded in a context;
+– the contextual language of (a∗)xbx is the set
+�
+n∈N
+{({anban}, λx → Just(an))}
+where each word an is recorded in a context applied to evaluate the variable x;
+– the contextual language of E1 is the union of the two following sets S1 and S2:
+S1 = {({��}, λx → Nothing)}
+S2 = {({anban | n ∈ N}∗ · {ambam}, λx → Just(am)) | m ∈ N}
+where each iteration of the outermost star produces a new record for the variable x in the context; however,
+notice that only the last one is recorded at the end of the process.
+
+Finally, the language of E is obtained by considering the contexts obtained from the parsing of E1 to evaluate the
+occurrence of x in E2, leading to the set�
+m∈N
+({anban | n ∈ N}∗ · {ambamcam}).
+Obviously, some classical equations still hold with these computations:
+Lemma 1. Let E, F and G be three Σ, Γ-expressions and ctxt be a context in Ctxt(Γ, Σ). The two following
+equations hold:
+Lctxt(E · (F + G)) = Lctxt(E · F + E · G)
+Lctxt(F ∗) = Lctxt(ε + F · F ∗)
+Proof. Let us proceed by equality sequences:
+Lctxt(E · (F + G)) =
+�
+(L1,ctxt1)∈Lctxt(E),
+(L2,ctxt2)∈Lctxt1 (F +G)
+{(L1 · L2, ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(E),
+(L2,ctxt2)∈Lctxt1 (F )∪Lctxt1(G)
+{(L1 · L2, ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(E),
+(L2,ctxt2)∈Lctxt1 (F )
+{(L1 · L2, ctxt2)}
+∪
+�
+(L1,ctxt1)∈Lctxt(E),
+(L2,ctxt2)∈Lctxt1(G)
+{(L1 · L2, ctxt2)}
+= Lctxt(E · F) ∪ Lctxt(E · G)
+= Lctxt(E · F + E · G)
+Lctxt(F ∗) =
+�
+n∈N
+(Lctxt(F))
+n
+= (Lctxt(F))
+0 ∪
+�
+n∈N,n≥1
+(Lctxt(F))
+n
+= (Lctxt(F))
+0 ∪
+�
+n∈N
+Lctxt(F) · (Lctxt(F))
+n
+= (Lctxt(F))
+0 ∪ Lctxt(F) ·
+�
+n∈N
+(Lctxt(F))
+n
+= Lctxt(ε + F · F ∗)
+In order to solve the membership test for the contextual capture-group expressions, let us extend the classical
+derivation method. But first, let us show how to extend the nullability predicate, needed at the end of the process.
+9.3
+Nullability Computation
+The nullability predicate allows us to determine whether the empty word belongs to the language denoted by
+an expression. As far as capture groups are concerned, a context has to be computed. Therefore, the nullability
+predicate can be represented as a set of contexts the application of which produces a language that contains the
+empty word.
+As we have seen, the nullability depends on the current context. Given an expression and a context ctxt, the
+nullability predicate is a set in 2Ctxt(Γ,Σ), computed as follows:
+Nullctxt(ε) = {ctxt}
+Nullctxt(∅) = ∅
+Nullctxt(a) = ∅
+Nullctxt(x) =
+�
+{ctxt}
+if ctxt(x) = Just(ε)
+∅
+otherwise.
+Nullctxt(E + F) = Nullctxt(E) ∪ Nullctxt(F)
+Nullctxt(E · F) =
+�
+ctxt′∈Nullctxt(F ),
+ctxt′′∈Nullctxt′ (G)
+{ctxt′′}
+Nullctxt(E∗) = {ctxt}
+Nullctxt((E)u
+x) =
+�
+ctxt′∈Nullctxt(F )
+{[ctxt′]x←u}
+where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.
+Example 13. Let us consider the three expressions of Example 12:
+E = E1 · E2,
+E1 = ((a∗)xbx)∗,
+E2 = cx.
+For any context ctxt,
+Nullctxt(E1) = {ctxt},
+Nullctxt(E2) = ∅,
+Nullctxt(E) = ∅.
+The nullability predicate allows us to determine whether there exists a couple in the contextual language of an
+expression such that its first component contains the empty word.
+
+Proposition 3. Let E be a Σ, Γ-expression and ctxt be a context in Ctxt(Γ, Σ). Then the two following conditions
+are equivalent:
+– Nullctxt(E) ̸= ∅,
+– ∃(L, _) ∈ Lctxt(E) | ε ∈ L.
+Proof. By induction over the structure of E:
+– If E = a ∈ Σ or E = ∅, the property holds since Nullctxt(E) is empty and since there is no couple (L, ctxt′) in
+Lctxt(E) with ε in L.
+– If E = ε, the following two conditions hold,
+Nullctxt(E) = {ctxt},
+Lctxt(E) = {({ε}, ctxt)},
+satisfying the stated condition.
+– If E = F + G, the following two conditions hold:
+Nullctxt(F + G) = Nullctxt(F) ∪ Nullctxt(G),
+Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G).
+Since, by induction hypothesis, the following two conditions hold
+Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
+Nullctxt(G) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(G) | ε ∈ L,
+the proposition holds.
+– If E = F · G, the two following conditions hold:
+Nullctxt(F · G) =
+�
+ctxt′∈Nullctxt(F ),
+ctxt′′∈Nullctxt′(G),
+{ctxt′′},
+Lctxt(F · G) =
+�
+(L,ctxt′)∈Lctxt(F ),
+(L′,ctxt′′)∈Lctxt′(G),
+{(L · L′, ctxt′′)}.
+Since, by induction hypothesis, the two following conditions hold,
+Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
+Nullctxt′(G) ̸= ∅ ⇔ ∃(L, ctxt′′) ∈ Lctxt′(G) | ε ∈ L,
+the proposition holds.
+– If E = F ∗, since the two following conditions hold
+Nullctxt(F ∗) = {ctxt},
+Lctxt(F)
+0 = {({ε}, ctxt)} ∈ Lctxt(F ∗),
+the stated condition holds.
+– If E = (F)u
+x, both following conditions hold:
+Nullctxt((F)u
+x) =
+�
+ctxt′∈Nullctxt(F )
+{[ctxt′]x←u},
+Lctxt((F)u
+x) =
+�
+(L,ctxt′)∈Lctxt(F ),
+w∈L
+{({w}, [ctxt′]x←uw)}.
+Then, following induction hypothesis,
+Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L,
+the stated condition holds.
+– If E = x, both following conditions hold:
+Nullctxt(x) =
+�
+{ctxt}
+if ctxt(x) = Just(ε)
+∅
+otherwise,
+Lctxt(x) =
+�
+∅
+if ctxt(x) = Nothing,
+{({w}, ctxt)}
+otherwise if ctxt(x) = Just(w).
+Therefore, the proposition holds.
+9.4
+Derivation formulae
+Similarly to the nullability predicate, the derivation computation builds the context while parsing the expression.
+Therefore, the derivative of an expression with respect to a context is a set of couples (expression, context), induc-
+tively computed as follows, for any Σ, Γ-expression and for any context ctxt in Ctxt(Γ, Σ):
+dctxt
+a
+(ε) = ∅
+dctxt
+a
+(∅) = ∅
+
+dctxt
+a
+(b) =
+�
+∅
+if a ̸= b,
+{(ε, ctxt)}
+otherwise,
+dctxt
+a
+(x) =
+�
+dctxt
+a
+(w)
+if ctxt(x) = Just(w)
+∅
+otherwise
+dctxt
+a
+(F + G) = dctxt
+a
+(F) ∪ dctxt
+a
+(G)
+dctxt
+a
+(F · G) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+{(F ′ · G, ctxt′)}
+∪
+�
+ctxt′∈Nullctxt(F )
+dctxt′
+a
+(G)
+dctxt
+a
+(F ∗) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+{(F ′ · F ∗, ctxt′)}
+dctxt
+a
+((F)u
+x) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+{((F ′)u·a
+x , ctxt′)}
+where F and G are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.
+Example 14. Let us consider the three expressions of Example 12:
+E = E1 · E2,
+E1 = ((a∗)xbx)∗,
+E2 = cx.
+Then, for any context ctxt,
+dctxt
+a
+(E) = {((a∗)a
+xbx((a∗)xbx)∗cx, ctxt)},
+dctxt
+b
+(E) = {(x((a∗)xbx)∗cx, λx → ε)},
+dctxt
+c
+(E) = {(x, ctxt)}.
+The derivation of an expression allows us to syntactically express the computation of the quotient of the language
+components in contextual languages, where the quotient w−1(L) is the set {w′ | ww′ ∈ L}.
+Proposition 4. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and a be a symbol in Σ. Then:
+�
+(E′,ctxt′)∈dctxt
+a
+(E)
+Lctxt′(E′) =
+�
+(L′,ctxt′)∈Lctxt(E)
+{(a−1(L′), ctxt′)}
+Proof. By induction over the structure of E, assimilating ∅ and {(∅, ctxt)} for any context ctxt.
+– If E = ε or E = ∅, the property vacuously holds.
+– If E = b ∈ Σ,
+�
+(E′,ctxt′)∈dctxt
+a
+(b)
+Lctxt′(E′) =
+�
+∅
+if b ̸= a,
+{({ε}, ctxt)}
+otherwise,
+= {(a−1({b}), ctxt)} =
+�
+(L′,ctxt′)∈Lctxt(b)
+{(a−1(L′), ctxt′)}.
+– If E = F + G,�
+(E′,ctxt′)∈dctxt
+a
+(F +G)
+Lctxt′(E′) =
+�
+(E′,ctxt′)∈dctxt
+a
+(F )∪dctxt
+a
+(G)
+Lctxt′(E′)
+=
+�
+(E′,ctxt′)∈dctxt
+a
+(F )
+Lctxt′(E′) ∪
+�
+(E′,ctxt′)∈dctxt
+a
+(G)
+Lctxt′(E′)
+=
+�
+(L′,ctxt′)∈Lctxt(F )
+{(a−1(L′), ctxt′)} ∪
+�
+(L′,ctxt′)∈Lctxt(G)
+{(a−1(L′), ctxt′)}
+=
+�
+(L′,ctxt′)∈Lctxt(F )∪Lctxt(G)
+{(a−1(L′), ctxt′)}
+=
+�
+(L′,ctxt′)∈Lctxt(F +G)
+{(a−1(L′), ctxt′)}.
+– If E = F · G,
+�
+(E′,ctxt′)∈dctxt
+a
+(F ·G)
+Lctxt′(E′) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+Lctxt′(F ′ · G) ∪
+�
+ctxt′∈Nullctxt(F ),
+(G′,ctxt′′)∈dctxt′
+a
+(G)
+Lctxt′′(G′)
+=
+�
+(ctxt′,F ′)∈dctxt
+a
+(F ),
+(L1,ctxt1)∈Lctxt(F ′),
+(L2,ctxt2)∈Lctxt1(G)
+{(L1 · L2, ctxt2)} ∪
+�
+ctxt′∈Nullctxt(F ),
+(G′,ctxt′′)∈dctxt′
+a
+(G)
+Lctxt′′(G′)
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L1) · L2, ctxt2)} ∪
+�
+ctxt1∈Nullctxt(F ),
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L2), ctxt2)}
+
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L1) · L2, ctxt2)} ∪
+�
+∃(L,ctxt1)∈Lctxt(F )|ε∈L,
+(L2,ctxt2)∈Lctxt1(G)
+{(a−1(L2), ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L1) · L2, ctxt2)} ∪
+�
+(L1,ctxt1)∈Lctxt(F ),
+ε∈L1,
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L2), ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1 (G)
+{(a−1(L1 · L2), ctxt2)}
+=
+�
+(L′,ctxt′)∈Lctxt(F ·G)
+{(a−1(L′), ctxt′)}.
+– If E = F ∗,
+�
+(E′,ctxt′)∈dctxt
+a
+(F ∗)
+Lctxt′(E′) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+Lctxt′(F ′ · F ∗)
+=
+�
+(ctxt′,F ′)∈dctxt
+a
+(F ),
+(L1,ctxt1)∈Lctxt(F ′),
+(L2,ctxt2)∈Lctxt1(F ∗)
+{(L1 · L2, ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1(F ∗)
+{(a−1(L1) · L2, ctxt2)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+(L2,ctxt2)∈Lctxt1(F ∗)
+{(a−1(L1 · L2), ctxt2)}
+=
+�
+(L′,ctxt′)∈Lctxt(F ·F ∗)
+{(a−1(L′), ctxt′)}
+=
+�
+(L′,ctxt′)∈Lctxt(ε+F ·F ∗)
+{(a−1(L′), ctxt′)}
+=
+�
+(L′,ctxt′)∈Lctxt(F ∗)
+{(a−1(L′), ctxt′)}
+– If E = (F)u
+x,
+�
+(E′,ctxt′)∈dctxt
+a
+((F )u
+x)
+Lctxt′(E′) =
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+Lctxt′((F ′)u·a
+x )
+=
+�
+(ctxt′,F ′)∈dctxt
+a
+(F )
+(L1,ctxt1)∈Lctxt′(F ′),
+w∈L1
+{({w}, [ctxt1]x←uaw)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+w∈a−1(L1)
+{({w}, [ctxt1]x←uaw)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+aw∈L1
+{({w}, [ctxt1]x←uaw)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+aw∈L1
+{(a−1({aw}), [ctxt1]x←uaw)}
+=
+�
+(L1,ctxt1)∈Lctxt(F ),
+w∈L1
+{(a−1({w}), [ctxt1]x←uw)}
+=
+�
+(L′,ctxt′)∈Lctxt((F )u
+x)
+{(a−1(L′), ctxt′)}
+
+– If E = x,
+�
+(E′,ctxt′)∈dctxt
+a
+(x)
+Lctxt′(E′) =
+
+
+
+
+
+�
+(E′,ctxt′)∈dctxt
+a
+(w)
+Lctxt′(E′)
+if ctxt(x) = Just(w),
+∅
+otherwise,
+=
+
+
+
+
+
+�
+(w,ctxt)∈dctxt
+a
+(aw)
+Lctxt(w)
+if ctxt(x) = Just(aw),
+∅
+otherwise,
+=
+�
+{({w}, ctxt)}
+if ctxt(x) = Just(aw),
+∅
+otherwise,
+=
+�
+{(a−1({aw}), ctxt)}
+if ctxt(x) = Just(aw),
+∅
+otherwise,
+=
+�
+{(a−1({w}), ctxt)}
+if ctxt(x) = Just(w),
+∅
+otherwise,
+=
+�
+(L′,ctxt′)∈Lctxt(x)
+{(a−1(L′), ctxt′)}
+The derivation w.r.t. a word is, as usual, an iterated application of the derivation w.r.t. a symbol, recursively
+defined as follows, for any Σ, Γ-expression E, for any context ctxt in Ctxt(Γ, Σ), for any symbol a in Σ and for
+any word v in Σ∗:
+dctxt
+ε
+(E) = {(E, ctxt)},
+dctxt
+a·v (E) =
+�
+(E′,ctxt′)∈dctxt
+a
+(E)
+dctxt′
+v
+(E′).
+Example 15. Let us consider the three expressions of Example 14:
+E = E1 · E2,
+E1 = ((a∗)xbx)∗,
+E2 = cx.
+Then, for any context ctxt,
+dctxt
+ab (E) = dctxt
+b
+((a∗)a
+xbx((a∗)xbx)∗cx)
+= {(x((a∗)xbx)∗cx, λx → a)}
+dctxt
+aba (E) = dλx→a
+a
+(x((a∗)xbx)∗cx)
+= {(((a∗)xbx)∗cx, λx → a)}
+dctxt
+abac(E) = dλx→a
+c
+(((a∗)xbx)∗cx)
+= {(x, λx → a)}
+dctxt
+abaca(E) = dλx→a
+a
+(x)
+= {(ε, λx → a)}
+Such an operation allows us to syntactically compute the quotient.
+Proposition 5. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then:
+�
+(E′,ctxt′)∈dctxt
+w
+(E)
+Lctxt′(E′) =
+�
+(L′,ctxt′)∈Lctxt(E)
+{(w−1(L′), ctxt′)}
+Proof. By a direct induction over the structure of words.
+Finally, the membership test of a word w can be performed as usual by first computing the derivation w.r.t. w, and
+then by determining the existence of a nullable derivative, as a direct corollary of Proposition 3 and Proposition 5.
+Theorem 2. Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗. Then the two
+following conditions are equivalent:
+– ∃(L, _) ∈ Lctxt(E) | w ∈ L,
+– ∃(E′, ctxt′) ∈ dctxt
+w
+(E) | Nullctxt′(E′) ̸= ∅.
+We have shown how to compute the derivatives and solve the membership test in a classical way. Let us show how
+to embed the context computation in a convenient monad, in order to generalize the definitions to other structure
+than sets.
+
+9.5
+The StateT Monad Transformer
+Monads do not compose well in general. However, ones can consider particular combinations of these objects. Among
+those, well-known patterns are the monad transformers like the StateT Monad Transformer [10]. This combination
+allows us to mimick the use of global variables in a functional way. In our setting, it allows us to embed the context
+computation in an elegant way.
+Let S be a set and M be a monad. We denote by StateT(S, M) following the mapping:
+StateT(S, M)(A) = S → M(A × S).
+In other terms, StateT(S, M)(A) is the set of functions from S to the monadic structure M(A×S) based on couples
+in the cartesian product (A × S).
+The mapping StateT(S, M) can be equipped by a structure of functor, defined for any function f from a set A
+to a set B by
+StateT(S, M)(f)(state)(s) = M(λ(a, s) → (f(a), s))(state(s)).
+It can also be equipped with the structure of monad, defined for any function f from a set A to the set StateT(S, M)(B):
+pure(a) = λs → pure(a, s)
+bind(f)(state)(s) = state(s) >>= λ(a, s′) → f(a)(s′)
+9.6
+Monadic Definitions
+The previous definitions associated with capture-group expressions can be equivalently restated using the StateT
+monad transformer specialised with the Set monad.
+Let us first consider the following claims where M = StateT(Ctxt(Γ, Σ), Set), allowing us to bring closer M
+and the previous notion of monadic support:
+– R = (M(1), +, ×, 1, 0) is a semiring by setting:
+f1 + f2 = λs → f1(s) ∪ f2(s),
+f1 × f2 = f1 >>= λ_ → f2,
+1 = λs → {(⊤, s)} = pure(⊤),
+0 = λs → ∅,
+– M = (M(Exp(Σ)), ±, 0) is a monoid by setting:
+± = +,
+0 = 0,
+– (M, ⋉) is a Exp(Σ)-right-semimodule by setting:
+f ⋉ F = λs →
+�
+(E,ctxt)∈f(s)
+{(E · F, ctxt)},
+– (M, ⊲) is a R-left-semimodule by setting:
+f1 ⊲ f2 = f1 >>= λ_ → f2.
+Then, the nullable predicate formulae can be equivalently restated as an element in StateT(Ctxt(Γ, Σ), Set)(1),
+which is equal by definition to Ctxt(Γ, Σ) → Set(1 × Ctxt(Γ, Σ)), isomorphic to Ctxt(Γ, Σ) → Set(Ctxt(Γ, Σ)). It
+can inductively be computed as follows:
+Null(ε) = 1
+Null(∅) = 0
+Null(a) = 0
+Null(E + F) = Null(E) + Null(F)
+Null(E · F) = Null(E) × Null(F)
+Null(E∗) = 1
+Null(x)(ctxt) =
+�
+pure((⊤, ctxt))
+if ctxt(x) = Just(ε),
+∅
+otherwise,
+Null((E)u
+x)(ctxt) = Set(λ(⊤, ctxt′) → (⊤, [ctxt′]x←u))(Null(F)(ctxt)),
+where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Notice that
+these formulae are the same that the ones in Definition 2 as far as classical operators are concerned, and that these
+formulae can be easily generalized to other convenient monads than Set.
+Moreover, the derivative of an expression is an element in StateT(Ctxt(Γ, Σ), Set)(Exp(Σ, Γ)):
+da(ε) = 0
+da(∅) = 0
+da(b) =
+�
+0
+if a ̸= b,
+pure(ε)
+otherwise,
+da(E + F) = da(E) ± da(F)
+da(E · F) = da(E) ⋉ F + Null(E) ⊲ da(F)
+da(E∗) = da(E) ⋉ E∗
+da((E)u
+x) = StateT(Ctxt(Γ, Σ), Set)(λF → (F)ua
+x )(da(E))
+da(x)(ctxt) =
+�
+pure((w, ctxt))
+if ctxt(x) = Just(aw),
+∅
+otherwise,
+
+where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗. Once again, notice
+that these formulae are the same that the ones in Definition 5 as far as classical operators are concerned, and that
+these formulae can be easily generalized to other convenient monads than Set.
+Finally, the derivation w.r.t. a word is monadically defined as in previous sections:
+dε(E) = pure(E),
+dav(E) = da(E) >>= dv,
+and the membership test of a word w can be equivalently rewritten as follows:
+(dw(E) >>= Null)(λ_ → Nothing) ̸= ∅.
+10
+Conclusion and Perspectives
+In this paper, we achieved the first step of our plan to unify the derivative computation over word expressions.
+Monads are indeed useful tools to abstract the underlying computation structures and thus may allow us to consider
+some other functionalities, such as capture groups via the well-known StateT monad transformer [10]. We aim to
+study the conditions satisfying by monads that lead to finite set of derivated terms, and to extend this method
+to tree expressions using enriched categories. Finally, we plan to extend monadic derivation to other underlying
+monads for capture groups, linear combinations for example.
+References
+1. Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput. Sci.
+155(2) (1996) 291–319
+2. Attou, S., Mignot, L., Miklarz, C., Nicart, F.: Monadic expressions and their derivatives. In: NCMA. Volume 367 of
+EPTCS (2022) 49–64
+3. Berry, G., Sethi, R.:
+From regular expressions to deterministic automata.
+Theoretical computer science 48 (1986)
+117–126
+4. Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4) (1964) 481–494
+5. Caron, P., Flouret, M.: From glushkov wfas to k-expressions. Fundam. Informaticae 109(1) (2011) 1–25
+6. Champarnaud, J., Laugerotte, É., Ouardi, F., Ziadi, D.: From regular weighted expressions to finite automata. Int. J.
+Found. Comput. Sci. 15(5) (2004) 687–700
+7. Colcombet, T., Petrisan, D.: Automata and minimization. SIGLOG News 4(2) (2017) 4–27
+8. Eisenberg, R.A., Weirich, S.: Dependently typed programming with singletons. In: Haskell, ACM (2012) 117–130
+9. Glushkov, V.M.: The abstract theory of automata. Russian Mathematical Surveys 16(5) (1961) 1
+10. Jones, M.P.: Functional programming with overloading and higher-order polymorphism. In: Adv. Func. Prog. Volume
+925 of LNCS, Springer (1995) 97–136
+11. Kleene, S.: Representation of events in nerve nets and finite automata. Automata Studies Ann. Math. Studies 34
+(1956) 3–41 Princeton U. Press.
+12. Loday, J.L., Vallette, B.: Algebraic operads. Volume 346. Springer Science & Business Media (2012)
+13. Lombardy, S., Sakarovitch, J.: Derivatives of rational expressions with multiplicity. Theor. Comput. Sci. 332(1-3) (2005)
+141–177
+14. May, J.P.: The geometry of iterated loop spaces. Volume 271. Springer (2006)
+15. Mignot, L.: Une proposition d’implantation des structures d’automates, d’expressions et de leurs algorithmes associés
+utilisant les catégories enrichies (in french).
+Habilitation à diriger des recherches, Université de Rouen normandie
+(Décembre 2020) 212 pages.
+16. Mignot, L.: Monadic derivatives. https://github.com/LudovicMignot/MonadicDerivatives (2022)
+17. Schützenberger, M.P.: On the definition of a family of automata. Inf. Control. 4(2-3) (1961) 245–270
+18. Sulzmann, M., Lu, K.Z.M.: POSIX regular expression parsing with derivatives. In: FLOPS. Volume 8475 of Lecture
+Notes in Computer Science, Springer (2014) 203–220
+

0tFPT4oBgHgl3EQfTjTq/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:46a711f10f601c1825a35c4c8b3d94027a479a8a4864de43e97b9d3bcd7b1b79
+size 159172

29E2T4oBgHgl3EQfjQdC/content/tmp_files/2301.03966v1.pdf.txt

@@ -0,0 +1,1687 @@
+AdvBiom: Adversarial Attacks on Biometric
+Matchers
+Debayan Deb, Vishesh Mistry, Rahul Parthe
+TECH5,
+Troy, MI, USA
+{debayan.deb, vishesh.mistry, rahul.parthe}@tech5-sa.com
+Abstract
+With the advent of deep learning models, face recognition systems have achieved
+impressive recognition rates. The workhorses behind this success are Convolutional
+Neural Networks (CNNs) and the availability of large training datasets. However,
+we show that small human-imperceptible changes to face samples can evade most
+prevailing face recognition systems. Even more alarming is the fact that the same
+generator can be extended to other traits in the future. In this work, we present how
+such a generator can be trained and also extended to other biometric modalities,
+such as fingerprint recognition systems.
+1
+Introduction
+The last decade has seen a massive influx of deep learning-based technologies that have tackled
+problems which were once thought to be unsolvable. Much of this progress can be attributed to
+Convolutional Neural Networks (CNNs) [1, 2] which are now deployed in a plethora of applications
+ranging from cancer detection to driving autonomous vehicles. Akin to the computer vision domain,
+the use of CNNs have completely changed the face of biometrics due to the availability of powerful
+computing devices (GPUs, TPUs) and deep architectures capable of learning rich features [3–5].
+Automated face recognition systems (AFR) have been proven to achieve accuracies as high as 99%
+True Accept Rate (TAR) @ 0.1% False Accept Rate (FAR) [6], majorly owing to publicly available
+large-scale face datasets.
+Unfortunately, studies have shown that CNN-based networks are vulnerable to adversarial pertur-
+bations1 [7–12]. It is not surprising that AFR systems too are not impervious to these attacks.
+Adversarial attacks to an AFR system can be classified into two categories - (i) impersonation attack
+where the hacker tries to perturb his face image to match it to a target victim, and (ii) obfuscation
+attack where the hacker’s face image is perturbed to match with a random identity. Both the above at-
+tacks involve the hacker adding targeted human-imperceptible perturbations to the face image. These
+adversarial attacks are different from face digital manipulation that include attribute manipulation
+and synthetic faces, and also from presentation attacks which involves the perpetrator wearing a
+physical artifact such as a mask or replaying a photograph/video of a genuine individual which may
+be conspicuous in scenarios where human operators are involved.
+Let us consider, as example, the largest deployment of fingerprint recognition systems - India’s
+Aadhaar Project [13], which currently has an enrolled gallery size of about 1.35 billion faces from
+nearly all of its citizens. In September 2022 alone, Aadhaar received 1.3 billion authentication
+requests2. In order to deny a citizen his/her rightful access to government benefits, healthcare, and
+financial services, an attacker can maliciously perturb enrolled face images such that they do not
+1Adversarial perturbations refer to altering an input image instance with small, human imperceptible changes
+in a manner that can evade CNN models.
+2https://bit.ly/3BzlpZJ
+arXiv:2301.03966v1  [cs.CV]  10 Jan 2023
+
+match to the genuine person during verification. In a typical AFR system, adversarial faces can
+be replaced with a captured face image in order to prevent the probe face from matching to any of
+its corresponding enrolled faces. Additionally, the attacker can compromise the entire gallery by
+inserting adversarial faces in the enrolled gallery, where no probe face will match to the correct
+identity’s gallery.
+Adversarial attacks can further be categorized into two types of attacks based on how the attack vector
+is trained and generated:
+1. White-box attack: Attacks in which the hacker has full knowledge of the recognition system,
+and iteratively perturbs every pixel by various optimization schemes are termed as white-box
+attacks [14–22].
+2. Black-box attack: With no information about the parameters of the recognition system,
+black-box attacks are deployed by either transferring attacks learned from an available AFR
+system [23–28], or querying the the target system for score [29–31] or decision [32, 33].
+3. Semi-whitebox attack: Here, a white-box model is utilized only during training and then ad-
+versarial examples are synthesized during inference without any knowledge of the deployed
+AFR model.
+We propose an automated adversarial synthesis method, named AdvBiom, which generates an ad-
+versarial image for a probe image and satisfies all the above requirements. The contributions of the
+paper are as follows:
+1. GAN-based AdvBiom that learns to generate visually realistic adversarial face images that
+are misclassified by state-of-the-art automated biometric systems.
+2. Adversarial images generated via AdvBiom are model-agnostic and transferable, and achieve
+high success rate on 5 state-of-the-art automated face recognition systems.
+3. Visualizing regions where pixels are perturbed and analyzing the transferability of AdvBiom .
+4. We show that AdvBiom achieves significantly higher attack success rate under current
+defense mechanisms compared to baselines.
+5. With the addition of the proposed Minutiae Displacement and Distortion modules, we show
+thatAdvBiom can also be extended to successfully evade automated fingerprint recognition
+systems.
+2
+Related Work
+2.1
+Adversarial Attacks
+As discussed earlier, adversarial attacks are broadly classified into white-box attacks and black-box
+attacks. A large number of white-box attacks are gradient-based where they analyze the gradients
+during the back-propagation of an available face recognition system and perform pixel-wise per-
+turbations to the target face image. While approaches such as FGSM [14] and PGD [17] exploit
+the high-dimensional space of deep networks to generate adversarial attacks, C&W [18] focuses on
+minimizing objective functions for optimal adversarial perturbations. However, the basic assump-
+tion in white-box attacks that the target recognition system will be available is not plausible. In
+real-life scenarios, the hacker will not have any information regarding the architecture, training and
+deployment of the recognition system.
+Black-box attacks can be classified into three major categories: transfer-based, score-based, and
+decision-based attacks. Transfer-based attacks train their adversarial attack generator using readily
+available recognition systems and then deploy the attacks onto a black-box target system. Dong et
+al. [23] proposed the use of momentum for efficient transferability of the adversarial samples. DI2-
+FGSM [24] suggested to increase input data diversity for improving transferability. Other approaches
+in this category include AI-FGSM [27] and TI-FGSM [28]. Score-based attacks [29–31] query the
+target system for scores and try to estimate its gradients. Decision-based attacks have the most
+challenging setting wherein only the decisions from the target system are queried. Some effective
+methods in this category include Evolutionary attack [32] and Boundary attack [33].
+2
+
+2.2
+Adversarial Attacks on Face Recognition
+Although adversarial attacks on face recognition systems have only been recently explored, there
+has been a significant number of effective approaches for evading AFR systems. Attacks on face
+recognition systems can be broadly categorized into physical attacks and digital attacks. Physical
+attacks involve generating adversarial physical artifacts which are ’worn’ on a face. Sharif et
+al. [34, 35] proposed generating adversarial eye-glass frames for attacking face recognition systems.
+In [36], adversarial printed stickers placed on a hat were generated. However, methods [34–36] are
+implemented in a white-box setting which is unrealistic. Additionally, Nguyen et al. [37] proposed
+an adversarial light projection attack using an on-premise projector. Yin et al. [38] generated and
+printed eye makeup patches to be stuck around the eyes. More recently, authors in [39] proposed an
+adversarial mask for impersonation attacks in a black-box setting. However, all the above methods
+suffer a major drawback of being unrealistic in an operational setting where a human operator is
+present.
+Digital attacks refer to manipulating and perturbing the pixels of a digital face image before being
+passed through a face recognition system. Early works [9, 18, 10, 8, 40] focused on gradient-based
+attacks for face recognition. However, these methods implement lp-norm perturbations to each pixel
+resulting in decreased attack transferability, and vulnerability to denoising models. Cauli et al. [41]
+implemented a backdoor attack where the target face recognition system’s training samples were
+manipulated. Apart from the fact that gaining access to the target AFR’s training samples is highly
+improbable, a thorough visual inspection of the samples can easily identify the digital artifacts. Other
+works employ more stealthy attack approaches against face recognition models. Dong et al. [32]
+proposed an evolutionary optimization method for generating adversarial faces in decision-based
+black-box settings. However, they require a minimum of 1,000 queries to the target face recognition
+system before a realistic adversarial face can be synthesized. [42] added a conditional variational
+autoencoder and attention modules to generate adversarial faces in a transfer-based black-box setting.
+However, they solely focused on impersonation attacks and require at least 5 image samples of the
+target subject for training and inference. Zhong et al. [43] implemented dropout [44] to improve
+the transferability of the adversarial examples. [38] perturbed the eye region of a face to produce
+adversarial eyeshadow artifacts. However, the artifacts are visibly conspicuous under close inspection.
+Deb et al. [25] used a GAN to generate minimal perturbations in salient facial regions. More recently,
+[45] and [46] have focused on manipulating facial attributes for targeted adversarial attacks.
+3
+Adversarial Faces
+3.1
+Preliminaries
+The goal of any attacker is to evade Automated Face Recognition (AFR) systems under either of the
+two settings:
+• Obfuscation Manipulate input face images in a manner such that they cannot be identified
+as the hacker, or
+• Impersonation Edit input face images such that they are identified as a target/desired
+individual (victim).
+While the manipulated face image evades the AFR system, a key requirement in a successful attack is
+such that the input face image should appear as a legitimate face photo of the attacker. In other words,
+the attacker desires an automated method of adding small and human-imperceptible changes to an
+input face image such that it can evade AFR systems while appear benign to human observers. These
+changes are denoted as adversarial perturbations and the manipulated image is hereby referred to as
+adversarial images3. In addition, the automated method of synthesizing adversarial perturbations is
+named as adversarial generator.
+Formally, given an input face image, x, an adversarial generator has two requirements under the
+obfuscation scenario:
+• synthesize an adversarial face image, xadv = x + δ, such that AFR systems fail to match
+xadv and x, and
+3We interchangeably use the terms adversarial images and adversarial faces in this paper.
+3
+
+• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.
+When the attack aims to impersonate a target individual, we need an image of the victim xtarget
+where the identity of x and xtarget are different. Therefore, constraints under the impersonation
+setting are as follows:
+• synthesize an adversarial face image, xadv = x + δ, such that AFR systems erroneously
+match xadv and xtarget, and
+• limit the magnitude of perturbation ||δ||p such that xadv appears very similar to x to humans.
+Obfuscation attempts (faces are perturbed such that they cannot be identified as the attacker) are gen-
+erally more effective [25], computationally efficient to synthesize [14, 17], and widely adopted [47]
+compared to impersonation attacks (perturbed faces can automatically match to a target subject).
+Therefore, this paper focuses on crafting obfuscation attacks, however, we will still show examples
+on synthesizing impersonation attacks.
+3.2
+Gradient-based Attacks
+In white-box attacks, the attacker is assumed to have the knowledge and access to the AFR system’s
+model and parameters. Naturally, we then expect a much better attack success rate under white-box
+settings since the attacker can carefully craft adversarial perturbations that necessarily evade the target
+AFR system. However, these white-box manipulations of face recognition models are impractical in
+real-world scenarios. For instance, assuming access to an airport’s already deployed AFR system
+may be extremely difficult.
+Nevertheless, it is advantageous to understand prevailing white-box methods. That is, if given access
+to a CNN-based AFR system, how could one utilize all of its model parameters to launch a successful
+adversarial attack?
+A common approach is to utilize gradients of the whitebox AFR models. Namely the attackers modify
+the image in the direction of the gradient of the loss function with respect to the input image. There
+are two prevailing approaches to perform such gradient-based attacks:
+• one-shot attacks, in which the attacker takes a single step in the direction of the gradient,
+and
+• iterative attacks where instead of a single step, several steps are taken until we obtain a
+successful adversarial pattern.
+3.2.1
+Fast Gradient Sign Method (FGSM)
+This method computes an adversarial image by adding a pixel-wide perturbation of magnitude in the
+direction of the gradient [14]. Under FGSM attack, we take a single step towards the direction of the
+gradient, and therefore, FGSM is very efficient in terms of computation time. Formally, given an
+input image x, we obtain an adversarial image xadv:
+xadv = x + ϵ · sign (▽xJ (x, y))
+where, J is the loss function used to train the AFR system (typically, softmax cross entropy loss), and
+y is the ground truth class label of x (typically, the subject ID of the identity in x).
+FGSM was first proposed for the object classification domain and therefore, utilizes softmax proba-
+bilities for crafting adversarial perturbations. Therefore, the number of object classes are assumed to
+be known during training and testing. However, face recognition systems do not utilize the softmax
+layer for classification (as the number of identities are not fixed during deployment) instead features
+from the last fully connected layer are used for comparing face images.
+We first modify FGSM appropriately in order to evade AFR systems rather than object classifiers.
+Instead of considering the softmax cross-entropy loss as J, we craft a new loss function that models
+real-world scenario4:
+LfeatureMatch = 1 − Ex
+�
+F(x) · F(xadv)
+||F(x)|| ||F(xadv)||
+�
+.
+4For brevity, we denote Ex ≡ Ex∈Pdata.
+4
+
+where, F is the matcher and F(x) is the feature representation of an input image x. The above feature
+matching loss function computes the cosine distance between a pair of images and ensures that the
+features between adversarial image xadv and input image x are as close as possible. Therefore, the
+gradient of the above loss ensures the features do not match and hence, can be considered as an
+obfuscation adversarial attack.
+In Fig. 1, we show the results of launching our modified FGSM attack on a state-of-the-art AFR
+system, namely ArcFace [3]. We see that with a single step and with minimal perturbations, the real
+and adversarial images of Tiger Woods does not match via ArcFace while humans can easily identity
+both images as pertaining to the same subject.
+(a) Real Input Image
+(b) Perturbation
+(c) FGSM [14]
+Figure 1: Adversarial face synthesized via FGSM [14]. A state-of-the-art face matcher, ArcFace [3], fails to
+match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.27,
+while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
+3.2.2
+Projected Gradient Descent (PGD)
+An extreme case of white-box attacks is the PGD attack [17] where we assume that the attacker also
+has unlimited number of attempts to try and evade the deployed AFR system. Unlike FGSM, PGD is
+an iterative attack. PGD attempts to find the perturbation δ that maximises the loss of a model on a
+particular input while keeping the size of the perturbation smaller than a specified amount referred
+to as ϵ. We keep iterating until such a δ is obtained. Similar to FGSM, we modify the loss function
+of PGD to fit the requirements of AFR system by again considering LfeatureMatch as the loss. Fig.
+2 shows the results of PGD attack on ArcFace matcher. Note that due to multiple iterations, PGD
+attack on AFR systems is more powerful (lower cosine similarity) but also more visible to humans as
+compared to the single-step FGSM attack.
+(a) Real Input Image
+(b) Perturbation
+(c) PGD [17]
+Figure 2: Adversarial face synthesized via PGD [17]. A state-of-the-art face matcher, ArcFace [3], fails to match
+the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.12, while a
+score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
+3.3
+Geometric Perturbations (GFLM)
+Prior efforts in crafting adversarial faces have also tried non-linear deformations as a natural method
+for evading AFR systems [48]. Non-linear deformations are applied by performing geometric warping
+to the input face images.
+Unlike traditional adversarial perturbations that basically add an adversarial perturbation δ, authors
+in [48] propose a fast method of generating adversarial faces by altering the landmark locations of
+the input images. The resulting adversarial faces completely lie on the manifold of natural images,
+which makes it extremely difficult to detect any adversarial perturbations. Results of geometrically
+warped adversarial faces are presented in 3.
+5
+
+(a) Real Input Image
+(b) Perturbation
+(c) GFLM
+Figure 3: Adversarial face synthesized via GFLM [48]. A state-of-the-art face matcher, ArcFace [3], fails to
+match the adversarial and input image. Cosine similarity score (∈ [−1, 1]) between the two images is 0.33,
+while a score above 0.36 (threshold @ 0.1% False Accept Rate) indicates that two faces are of the same subject.
+3.4
+Attribute-based Perturbations
+Unlike geometric-warping and gradient-based attacks that may perturb every pixel in the image, a
+few studies propose manipulating only salient regions in faces, e.g., eyes, nose, and mouth.
+By restricting perturbations to only semantic regions of the face, SemanticAdv [46] generates
+adversarial examples in a more controllable fashion by editing a single semantic aspect through
+attribute-conditioned image editing. Fig. 4 shows results from adversarial manipulating semantic
+attributes. We can see while the attacks are indeed successful, it comes at the cost of altering the
+perceived identity as well as leads to degraded image quality.
+(a) Real Input Image
+(b) Blond
+(c) Bangs
+(d) Mouth Open
+(e) Eyeglasses
+(f) Makeup
+Figure 4: Adversarial face synthesized via manipulating semantic attributes [46]. All adversarial images (b-f)
+fail to match with the real image (a) via ArcFace [3].
+4
+AdvBiom: Learning to Synthesize Adversarial Attacks
+We find that majority of prior efforts on crafting adversarial attacks either degrade the visual quality
+where an observant human can still visually pick out the adversarial patterns. We also identify the
+following challenges with prior efforts:
+• Gradient-based attacks rely on white-box settings where the entire deployed CNN-based
+AFR system is available to the attacker to compute its gradients.
+• Geometrically-warping faces generally do not guarantee adversarial success and greatly
+distort the face image.
+• Semantic attribute manipulation can also degrade visual quality and may lead to greater
+conspicuous changes.
+Instead, we propose to train a network to “learn" the salient regions of the face that can be perturbed
+to evade AFR systems in a semi-whitebox setting. These leads to the following advantages over prior
+efforts:
+6
+
+• Perceptual Realism Given a large enough training dataset, a network can gradually learn to
+synthesize adversarial face images that are perceptually realistic such that a human observer
+can identify the image as a legitimate face image.
+• Higher Attack Success The faces can be learned to be perturbed in a manner such that they
+cannot be identified as the hacker (obfuscation at- tack) or automatically matched to a target
+subject (impersonation attack) by an AFR system.
+• Controllable The amount of perturbation can also be controllable by the attacker so that
+they can examine the success of the learning model as a function of amount of perturbation.
+• Transferability Due to the semi-whitebox setting: once the network learns to generate the
+perturbed instances based on a single face recognition system, attacks can be transferred to
+any black-box AFR systems.
+We propose an automated adversarial biometric synthesis method, named AdvBiom, which generates
+an adversarial image for a probe face image and satisfies all the above requirements.
+4.1
+Methodology
+Our goal is to synthesize a face image that visually appears to pertain to the target face, yet automatic
+face recognition systems either incorrectly matches the synthesized image to another person or does
+not match to target’s gallery images. AdvBiom comprises of a generator G, a discriminator D, and
+face matcher (see Figure 5).
+Probe
+ℒ"#$
+Synthesized
++
+ℒ%&'()%)*
+ℒ+',)-,./)%0(
+Adversarial Mask
+1
+ℱ
+3
+Figure 5: Given a probe face image, AdvBiom automatically generates an adversarial mask that is then added to
+the probe to obtain an adversarial face image.
+Generator
+The proposed generator takes an input face image, x ∈ X, and outputs an image, G(x).
+The generator is conditioned on the input image x; for different input faces, we will get different
+synthesized images.
+Since our goal is to obtain an adversarial image that is metrically similar to the probe in the image
+space, x, it is not desirable to perturb all the pixels in the probe image. For this reason, we treat the
+output from the generator as an additive mask and the adversarial face is defined as x + G(x). If
+the magnitude of the pixels in G(x) is minimal, then the adversarial image comprises mostly of the
+probe x. Here, we denote G(x) as an “adversarial mask". In order to bound the magnitude of the
+adversarial mask, we introduce a perturbation loss during training by minimizing the L2 norm5:
+Lperturbation = Ex [max (ϵ, ∥G(x)∥2)]
+(1)
+where ϵ ∈ [0, ∞) is a hyperparameter that controls the minimum amount of perturbation allowed.
+5For brevity, we denote Ex ≡ Ex∈X .
+7
+
+In order to achieve our goal of impersonating a target subject’s face or obfuscating one’s own identity,
+we need a face matcher, F, to supervise the training of AdvBiom. For obfuscation attack, at each
+training iteration, AdvBiom tries to minimize the cosine similarity between face embeddings of the
+input probe x and the generated image x + G(x) via an identity loss function:
+Lidentity = Ex[F(x, x + G(x))]
+(2)
+For an impersonation attack, AdvBiom maximizes the cosine similarity between the face embeddings
+of a randomly chosen target’s probe, y, and the generated adversarial face x + G(x) via:
+Lidentity = Ex[1 − F(y, x + G(x))]
+(3)
+The perturbation and identity loss functions enforce the network to learn the salient facial regions
+that can be perturbed minimally in order to evade automatic face recognition systems.
+Discriminator
+Akin to previous works on GANs [49, 50], we introduce a discriminator in order
+to encourage perceptual realism of the generated images. We use a fully-convolution network as a
+patch-based discriminator [50]. Here, the discriminator, D, aims to distinguish between a probe, x,
+and a generated adversarial face image x + G(x) via a GAN loss:
+LGAN =
+Ex [log D(x)] +
+Ex[log(1 − D(x + G(x)))]
+(4)
+Finally, AdvBiom is trained in an end-to-end fashion with the following objectives:
+min
+D LD = −LGAN
+(5)
+min
+G LG = LGAN + λiLidentity + λpLperturbation
+(6)
+where λi and λp are hyper-parameters controlling the relative importance of identity and perturbation
+losses, respectively. Note that LGAN and Lperturbation encourage the generated images to be visually
+similar to the original face images, while Lidentity optimizes for a high attack success rate. After
+training, the generator G can generate an adversarial face image for any input image and can be tested
+on any black-box face recognition system.
+The overall algorithm describing the training procedure of AdvBiom can be found in Algorithm 1.
+4.2
+Experimental Results
+Evaluation Metrics
+We quantify the effectiveness of the adversarial attacks generated by Ad-
+vBiom and other state-of-the-art baselines via (i) attack success rate and (ii) structural similarity
+(SSIM).
+The attack success rate for obfuscation attack is computed as,
+Attack Success Rate = (No. of Comparisons < τ)
+Total No. of Comparisons
+(7)
+where each comparison consists of a subject’s adversarial probe and an enrollment image. Here, τ
+is a pre-determined threshold computed at, say, 0.1% FAR6. Attack success rate for impersonation
+attack is defined as,
+Attack Success Rate = (No. of Comparisons ≥ τ)
+Total No. of Comparisons
+(8)
+Here, a comparison comprises of an adversarial image synthesized with a target’s probe and matched
+to the target’s enrolled image. We evaluate the success rate for the impersonation setting via 10-fold
+cross-validation where each fold consists of a randomly chosen target.
+Similar to prior studies [42], in order to measure the similarity between the adversarial example and
+the input face, we compute the structural similarity index (SSIM) between the images. SSIM is a
+normalized metric between −1 (completely different image pairs) to 1 (identical image pairs).
+6For each face matcher, we pre-compute the threshold at 0.1% FAR on all possible image pairs in LFW.
+For e.g., threshold @ 0.1% FAR for ArcFace is 0.28.
+8
+
+Algorithm 1 Training AdvBiom. All experiments in this work use α = 0.0001, β1 = 0.5, β2 = 0.9,
+λi = 10.0, λp = 1.0, m = 32.
+We set ϵ = 3.0 (obfuscation), ϵ = 8.0 (impersonation).
+1: Input
+2:
+X
+Training Dataset
+3:
+F
+Cosine similarity between an image pair obtained by biometric matcher
+4:
+G
+Generator with weights Gθ
+5:
+D
+Discriminator with weights Dθ
+6:
+m
+Batch size
+7:
+α
+Learning rate
+8: for number of training iterations do
+9:
+Sample a batch of probes {x(i)}m
+i=1 ∼ X
+10:
+if impersonation attack then
+11:
+Sample a batch of target images y(i) ∼ X
+12:
+δ(i) = G((x(i), y(i))
+13:
+else if obfuscation attack then
+14:
+δ(i) = G(x(i))
+15:
+end if
+16:
+x(i)
+adv = x(i) + δ(i)
+17:
+Lperturbation = 1
+m
+��m
+i=1 max
+�
+ϵ, ||δ(i)||2
+��
+18:
+if impersonation attack then
+19:
+Lidentity = 1
+m
+��m
+i=1 F
+�
+x(i), x(i)
+adv
+��
+20:
+else if obfuscation attack then
+21:
+Lidentity = 1
+m
+��m
+i=1
+�
+1 − F
+�
+y(i), x(i)
+adv
+���
+22:
+end if
+23:
+LG
+GAN = 1
+m
+��m
+i=1 log
+�
+1 − D(x(i)
+adv)
+��
+24:
+LD = 1
+m
+�m
+i=1
+�
+log
+�
+D(x(i))
+�
++ log
+�
+1 − D(x(i)
+adv)
+��
+25:
+LG = LG
+GAN + λiLidentity + λpLperturbation
+26:
+Gθ = Adam(▽GLG, Gθ, α, β1, β2)
+27:
+Dθ = Adam(▽DLD, Dθ, α, β1, β2)
+28: end for
+Datasets
+We train AdvBiom on CASIA-WebFace [51] and then test on LFW [52]7.
+• CASIA-WebFace [51] is comprised of 494,414 face images belonging to 10,575 different
+subjects. We removed 84 subjects that are also present in LFW and the testing images in
+this paper.
+• LFW [52] contains 13,233 web-collected images of 5,749 different subjects. In order to
+compute the attack success rate, we only consider subjects with at least two face images.
+After this filtering, 9,614 face images of 1,680 subjects are available for evaluation.
+All the testing images in this paper have no identity overlap with the training set, CASIA-
+WebFace [51].
+Data Preprocessing
+All face images are passed through MTCNN face detector [53] to detect five
+landmarks (two eyes, nose, and two mouth corners). Via similarity transformation, the face images
+are aligned. After transformation, the images are resized to 160 × 160. Prior to training and testing,
+each pixel in the RGB image is normalized by subtracting 127.5 and dividing by 128.
+Experimental Settings
+We use ADAM optimizers in Tensorflow with β1 = 0.5 and β2 = 0.9 for
+the entire network. Each mini-batch consists of 32 face images. We train AdvBiom for 200,000 steps
+with a fixed learning rate of 0.0001. Since our goal is to generate adversarial faces with high success
+7Training on CASIA-WebFace and evaluating on LFW is a common approach in face recognition literature [3,
+4]
+9
+
+rate, the identity loss is of utmost importance. We empirically set λi = 10.0 and λp = 1.0. We
+train two separate models and set ϵ = 3.0 and ϵ = 8.0 for obfuscation and impersonation attacks,
+respectively.
+Gallery
+Probe
+Proposed AdvBiom
+GFLM [48]
+PGD [17]
+FGSM [14]
+0.68
+0.14
+0.26
+0.27
+0.04
+0.38
+0.08
+0.12
+0.21
+0.02
+(a) Obfuscation Attack
+Target’s Gallery Target’s Probe
+Probe
+Proposed AdvBiom
+A3GN [42]
+FGSM [14]
+0.78
+0.10
+0.30
+0.29
+0.36
+0.80
+0.15
+0.34
+0.33
+0.42
+(b) Impersonation Attack
+Figure 6: Adversarial face synthesis results on LFW dataset in (a) obfuscation and (b) impersonation attack
+settings (cosine similarity scores obtained from ArcFace [3] with threshold @ 0.1% FAR= 0.28). The proposed
+method synthesizes adversarial faces that are seemingly inconspicuous and maintain high perceptual quality.
+Architecture
+Let c7s1-k be a 7 × 7 convolutional layer with k filters and stride 1. dk denotes a
+4 × 4 convolutional layer with k filters and stride 2. Rk denotes a residual block that contains two
+3 × 3 convolutional layers. uk denotes a 2× upsampling layer followed by a 5 × 5 convolutional
+layer with k filters and stride 1. We apply Instance Normalization and Batch Normalization to the
+generator and discriminator, respectively. We use Leaky ReLU with slope 0.2 in the discriminator
+and ReLU activation in the generator. The architectures of the two modules are as follows:
+• Generator:
+c7s1-64,d128,d256,R256,R256,R256, u128, u64, c7s1-3
+• Discriminator:
+d32,d64,d128,d256,d512
+A 1 × 1 convolutional layer with 3 filters and stride 1 is attached to the last convolutional layer of the
+discriminator for the patch-based GAN loss LGAN.
+We apply the tanh activation function on the last convolution layer of the generator to ensure
+that the generated image ∈ [−1, 1]. In the paper, we denoted the output of the tanh layer as an
+“adversarial mask”, G(x) ∈ [−1, 1] and x ∈ [−1, 1]. The final adversarial image is computed as
+10
+
+Obfuscation Attack
+Proposed AdvBiom
+GFLM [48]
+PGD [17]
+FGSM [14]
+Attack Success Rate (%) @ 0.1% FAR
+FaceNet [5]
+99.67
+23.34
+99.70
+99.96
+SphereFace [4]
+97.22
+29.49
+99.34
+98.71
+ArcFace [3]
+64.53
+03.43
+33.25
+35.30
+COTS-A
+82.98
+08.89
+18.74
+32.48
+COTS-B
+60.71
+05.05
+01.49
+18.75
+Structural Similarity
+0.95 ± 0.01
+0.82 ± 0.12
+0.29 ± 0.06
+0.25 ± 0.06
+Computation Time (s)
+0.01
+3.22
+11.74
+0.03
+Impersonation Attack
+Proposed AdvBiom
+A3GN [42]
+PGD [17]
+FGSM [14]
+Attack Success Rate (%) @ 0.1% FAR
+FaceNet [5]
+20.85 ± 0.40
+05.99 ± 0.19
+76.79 ± 0.26
+13.04 ± 0.12
+SphereFace [4]
+20.19 ± 0.27
+07.94 ± 0.19
+09.03 ± 0.39
+02.34 ± 0.03
+ArcFace [3]
+24.30 ± 0.44
+17.14 ± 0.29
+19.50 ± 1.95
+08.34 ± 0.21
+COTS-A
+20.75 ± 0.35
+15.01 ± 0.30
+01.76 ± 0.10
+01.40 ± 0.08
+COTS-B
+19.85 ± 0.28
+10.23 ± 0.50
+12.49 ± 0.24
+04.67 ± 0.16
+Structural Similarity
+0.92 ± 0.02
+0.69 ± 0.04
+0.77 ± 0.04
+0.48 ± 0.75
+Computation Time (s)
+0.01
+0.04
+11.74
+0.03
+White-box matcher (used for training)
+Black-box matcher (never used in training)
+Table 1: Attack success rates and structural similarities between probe and gallery images for obfus-
+cation and impersonation attacks. Attack rates for obfuscation comprises of 484,514 comparisons and
+the mean and standard deviation across 10-folds for impersonation reported. The mean and standard
+deviation of the structural similarities between adversarial and probe images along with the time
+taken to generate a single adversarial image (on a Quadro M6000 GPU) also reported.
+xadv = 2 × clamp
+�
+G(x) +
+� x+1
+2
+��1
+0 − 1. This ensures G(x) can either add or subtract pixels from
+x when G(x) ̸= 0. When G(x) → 0, then xadv → x.
+Face Recognition Systems
+For all our experiments, we employ 5 state-of-the-art face matchers8.
+Three of them are publicly available, namely, FaceNet [5], SphereFace [4], and ArcFace [3]. We also
+report our results on two commercial-off-the-shelf (COTS) face matchers, COTS-A and COTS-B9.
+We use FaceNet [5] as the white-box face recognition model, F, during training. All the testing
+images in this paper are generated from the same model (trained only with FaceNet) and tested on
+different matchers.
+4.2.1
+Comparison with Prevailing Adversarial Face Generators
+We compare our adversarial face synthesis method with state-of-the-art methods that have specifi-
+cally been implemented or proposed for faces, including GFLM [48], PGD [17], FGSM [14], and
+A3GN [42]10. In Table 1, we find that compared to the state-of-the-art, AdvBiom generates adversarial
+faces that are similar to the probe 6.
+Moreover, the adversarial images attain a high obfuscation attack success rate on 4 state-of-the-art
+black-box AFR systems in both obfuscation and impersonation settings. AdvBiom learns to perturb
+the salient regions of the face, unlike PGD [17] and FGSM [14], which alter every pixel in the
+image. GFLM [48], on the other hand, geometrically warps the face images and thereby, results
+in low structural similarity. In addition, the state-of-the-art matchers are robust to such geometric
+deformation which explains the low success rate of GFLM on face matchers. A3GN, another
+GAN-based method, however, fails to achieve a reasonable success rate in an impersonation setting.
+8All the open-source and COTS matchers achieve 99% accuracy on LFW under LFW protocol.
+9Both COTS-A and COTS-B utilize CNNs for face recognition. COTS-B is one of the top performers in the
+NIST Ongoing Face Recognition Vendor Test (FRVT) [54].
+10We train the baselines using their official implementations (detailed in the supplementary material).
+11
+
+4.2.2
+Ablation Study
+In order to analyze the importance of each module in our system, in Figure 7, we train three variants
+of AdvBiom for comparison by removing the discriminator (D), perturbation loss Lperturbation, and
+identity loss Lidentity, respectively.
+Input
+w/o D
+w/o Lprt
+w/o Lidt
+with all
+Figure 7: Variants of AdvBiom trained without the discriminator, perturbation loss, and identity loss, respectively.
+Every component of AdvBiom is necessary.
+The discriminator helps to ensure the visual quality of the synthesized faces are maintained. With
+the generator alone, undesirable artifacts are introduced. Without the proposed perturbation loss,
+perturbations in the adversarial mask are unbounded and therefore, leads to a lack in perceptual
+quality. The identity loss is imperative in ensuring an adversarial image is obtained. Without the
+identity loss, the synthesized image cannot evade state-of-the-art face matchers. We find that every
+component of AdvBiom is necessary in order to obtain an adversarial face that is not only perceptually
+realistic but can also evade state-of-the-art face matchers.
+4.2.3
+What is AdvBiom Learning?
+Via Lperturbation, during training, AdvBiom learns to perturb only the salient facial regions that can
+evade the face matcher, F (FaceNet [5] in our case). In Figure 8, AdvBiom synthesizes the adversarial
+masks corresponding to the probes. We then threshold the mask to extract pixels with perturbation
+magnitudes exceeding 0.40. It can be inferred that the eyebrows, eyeballs, and nose contain highly
+discriminative information that an AFR system utilizes to identify an individual. Therefore, perturbing
+these salient regions are enough to evade state-of-the-art face recognition systems.
+4.2.4
+Transferability of AdvBiom
+In Table 1, we find that attacks synthesized by AdvBiom when trained on a white-box matcher
+(FaceNet), can successfully evade 5 other face matchers that are not utilized during training in both
+obfuscation and impersonation settings. In order to investigate the transferability property of AdvBiom,
+we extract face embeddings of real images and their corresponding adversarial images, under the
+obfuscation setting, via the white-box matcher (FaceNet) and a black-box matcher (ArcFace). In total,
+we extract feature vectors from 1,456 face images of 10 subjects in the LFW dataset [52]. In Figure 9,
+we plot the correlation heatmap between face features of real images, their corresponding adversarial
+masks and adversarial images. First, we observe that face embeddings of real images extracted by
+FaceNet and ArcFace are correlated in a similar fashion. This indicates that both matchers extract
+features with related pairwise correlations. Consequently, perturbing salient features for FaceNet
+can lead to high attack success rates for ArcFace as well. The similarity among the correlation
+distributions of both matchers can also be observed when adversarial masks and adversarial images
+are input to the matchers. That is, receptive fields for automatic face recognition systems attend to
+similar regions in the face.
+12
+
+Probe
+Adv. Mask
+Visualization
+Adv. Image
+0.12
+0.26
+Figure 8: State-of-the-art face matchers can be evaded by slightly perturbing salient facial regions, such as
+eyebrows, eyeballs, and nose (cosine similarity obtained via ArcFace [3]).
+Figure 9: Correlation between face features extracted via FaceNet and ArcFace from 1,456 images belonging to
+10 subjects.
+To further illustrate the distributions of the embeddings of real and synthesized images, we plot
+the 2D t-SNE visualization of the face embeddings for the 10 subjects in Figure 10. The identity
+clusterings can be clearly observed from both real and adversarial images. In particular, the adversarial
+counterpart of each subject forms a new cluster that draws closer to the adversarial clusterings of
+other subjects. This shows that AdvBiom perturbs only salient pixels related to face identity while
+maintaining a semantic meaning in the feature space, resulting in a similar manifold of synthesized
+faces to that of real faces.
+4.2.5
+Controllable Perturbation
+The perturbation loss, Lperturbation is bounded by a hyper-parameter, ϵ, i.e., the L2 norm of the
+adversarial mask must be at least ϵ. Without this constraint, the adversarial mask becomes a blank
+image with no changes to the probe. With ϵ, we can observe a trade-off between the attack success
+rate and the structural similarity between the probe and synthesized adversarial face (Fig. 11). A
+higher ϵ leads to less perturbation restriction, resulting in a higher attack success rate at the cost of a
+lower structural similarity. For an impersonation attack, this implies that the adversarial image may
+13
+
+Real Image
+Adversarial Mask
+Adversarial Image
+0.8
+FaceNet
+0.4
+0.0
+ArcFace
+-0.4FaceNet
+Real Image
+Adversarial Image (Obfuscation)
+ArcFace
+Figure 10: 2D t-SNE visualization of face representations extracted via FaceNet and ArcFace from 1,456 images
+belonging to 10 subjects.
+contain facial features from both the hacker and the target. In our experiments, we chose ϵ = 8.0 and
+ϵ = 3.0 for impersonation and obfuscation attacks, respectively.
+4
+6
+8
+10
+12
+14
+16
+0.81
+0.92
+0.95
+0.76
+0.69
+5
+13
+21
+39
+52
+60
+66
+Hyper-parameter (ε)
+Success Rate (%)
+Structural Similarity
+ε = 4.0
+ε = 8.0
+ε = 10.0
+ε = 16.0
+Figure 11: Trade-off between attack success rate and structural similarity for impersonation attacks.
+4.2.6
+Attacks via AdvBiom Beyond Faces
+We now show that the AdvBiommethod, coupled with the proposed Minutiae Displacement and
+Distortion Modules, can be extended to effectively generate adversarial fingerprints which are visually
+similar to corresponding probe fingerprints while evading two state-of-the-art COTS fingerprint
+matchers as well as a deep network-based fingerprint matcher.
+14
+
+FS: 0.97
+FS: 0.92
+(a) Enrolled Mate
+VS: 235 | FS: 0.96
+VS: 172 | FS: 0.99
+(b) Input Probe
+VS: 31 | FS: 0.92
+VS: 10 | FS: 0.92
+(c) AdvBiom
+VS: 134 | FS: 0.96
+VS: 104 | FS: 0.96
+(d) DeepFool [16]
+VS: 139 | FS: 0.95
+VS: 104 | FS: 0.96
+(d) PGD [17]
+Figure 12: Example probe and corresponding mate fingerprints along with synthesized adversarial probes. (a)
+Two example mate fingerprints from NIST SD4 [55], and (b) the corresponding mates. Adversarial probe
+fingerprints using different approaches are shown in: (c) proposed synthesis method, AdvBiom; (d-e) state-of-
+the-art methods, DeepFool and PGD respectively. VeriFinger v11.0 match score (probe v. mate) - VS, and the
+fingerprintness score (degree of similarity of a given image to a fingerprint pattern) - FS ∈ [0,1] [56], which
+ranges from 1 (the highest) to 0 (the lowest), are given below each image. A VS of above 48 (at 0.01% FAR)
+indicates a successful match between the probe and the mate. The proposed attack AdvBiom successfully
+evades COTS and deep network-based matchers, while maintaining visual fingerprint perceptibility and high
+fingerprintness scores.
+Grosz et. al [57] showed that random minutiae position displacements and non-linear distortions
+drastically affected the performance of COTS fingerprint matchers. AdvBiom builds upon these two
+perturbations and when given a probe fingerprint, can synthesize an adversarial fingerprint image that
+retains all of the original fingerprint attributes except the identity, i.e. a fingerprint recognition system
+should not match the adversarial fingerprint to the probe fingerprint (obfuscation attack).
+Figure 13 shows the schematic of AdvBiom conditioned for fingerprints. The following subsections
+explain the major components of the approach in detail.
+Minutiae Displacement Module
+While the authors in [57] showed the effectiveness of random
+minutiae position displacements on COTS matchers, they studied the effect of this perturbation by
+directly modifying the minutiae template instead of the fingerprint image (pixel space). However, it
+may be difficult to obtain the minutiae template of a given fingerprint image using COTS minutiae
+extractors rather than the source fingerprint image itself. Thus, we propose a minutiae displacement
+module Gdisp which, given a fingerprint image, displaces its minutiae points in random directions by
+a predefined distance. To extract minutiae points from a fingerprint image, we employ a minutiae map
+extractor (M) from [58]. For a fingerprint image of width w and height h, M outputs a 12 channel
+heat map H ∈ Rh×w×12, where if H(i,j,c), value of the heat map at position (i,j) and channel c, is
+greater than a threshold mt and is the local maximum in its 5 × 5 × 3 neighboring cube, a minutiae is
+marked at (i,j). The minutiae direction θ is calculated by maximising the quadratic interpolation with
+respect to:
+f
+�
+(c − 1) × π
+6
+�
+= H (i, j, (c − 1)%12)
+(9)
+f
+�
+c × π
+6
+�
+= H(i, j, c)
+(10)
+f
+�
+(c + 1) × π
+6
+�
+= H(i, j, (c + 1)%12)
+(11)
+Figure 14 shows a fingerprint image and its corresponding 12 channel minutiae map. Once M
+extracts a minutiae map Hprobe from the input probe fingerprint x, we detect minutiae points by
+applying a threshold of 0.2 on Hprobe and finding closed contours. Each detected contour, at say
+15
+
+𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝
+Extractor (ℳ)
+𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑡𝑜𝑟
+(𝒟)
+GAN Loss
+ℒgan
+𝑀𝑖𝑛. 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
+Module (𝒢𝑑𝑖𝑠𝑝)
+Probe Fingerprint
+Displaced Fingerprint
+Original M.Map
+𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑃𝑖𝑥.
+Displacement
+Target M.Map
+M.Map Sim Loss
+ℒmmap_sim
+Pixel Loss
+ℒpixel
+Predicted M.Map
+M.Map Dis Loss
+ℒmmap_dis
+𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛
+Module (𝒢𝑑𝑖𝑠𝑡)
+𝑀𝑖𝑛𝑢𝑡𝑖𝑎𝑒 𝑀𝑎𝑝
+Extractor (ℳ)
+Adversarial Fingerprint
+Figure 13: Schematic of AdvBiom for generating adversarial fingerprints. Given a probe fingerprint image, it is
+passed to Gdisp which randomly displaces its minutiae points. The distortion module (Gdist) identifies control
+points on the displaced fingerprint and non-linearly distorts the image to output the adversarial fingerprint. The
+solid black arrows show the forward pass of the network while the dotted black arrows show the propagation of
+the losses.
+location (i, j), is displaced by a predefined L1 distance d = |∆i|+|∆j|, giving us the target minutiae
+map Htarget.
+0
+𝜋/6
+𝜋/3
+𝜋/2
+2𝜋/3
+5𝜋/6
+𝜋
+7𝜋/6
+4𝜋/3
+3𝜋/2
+5𝜋/3
+11𝜋/6
+Figure 14: The 12 channel minutiae map of an example fingerprint image shown on the left. The minutiae points
+(shown in red) are marked by a COTS minutiae extractor. The bright spots in each channel image indicate the
+spatial location of minutiae points while the kth channel (k ∈ [0, 11]) indicate the contributions of minutiae
+points to the kπ/6 orientation.
+The minutiae displacement module Gdisp is essentially an autoencoder conditioned on the probe
+fingerprint x and the target minutiae map Htarget. It learns to generate a displaced fingerprint xdisp
+whose predicted minutiae map Hpred is as close as possible to the target minutiae map Htarget in the
+pixel space. To achieve this, we have three losses that govern Gdisp:
+Lmmap_sim = ||Htarget − Hpred||1
+(12)
+Lmmap_dis =
+1
+||Htarget − Hprobe||1
+(13)
+, where Lmmap_sim is the minutiae map similarity loss which minimises the distance between the
+predicted and target minutiae map, while the minutiae map dissimilarity loss Lmmap_dis maximises
+16
+
+G
+QU
+Q
+Q
+QQ
+Q
+Gthe distance between the predicted and probe minutiae map. In figure 15, we show two example
+probe fingerprints and their corresponding displaced fingerprints after passing through Gdisp.
+Distortion Module
+One of the most noteworthy conclusions from [57] was that non-linear distor-
+tions to minutiae points was one of the most successful perturbations to lower the similarity scores
+between perturbed and corresponding unperturbed fingerprints. Again, the non-linear distortion
+was applied to all the minutiae points in the template and not to the image. Thus, our next step in
+generating adversarial fingerprints consists of a distortion module Gdist which learns to distort salient
+points in a fingerprint image.
+The architecture of Gdist consists of an encoder conditioned on the input probe fingerprint x and the
+target minutiae map Hprobe. The output from the encoder is a predefined number of control points11 c.
+The non-linear distortion model proposed in [59], learned using a thin plate spline (TPS) model [60]
+from 320 already distorted fingerprint videos, was employed to calculate the displacements of the
+predicted control points. The hyper-parameter σ is used to indicate the extent of the distortion. The
+control points and their displacements are then fed to a differentiable warping module [61] to get the
+resultant adversarial fingerprint xadv.
+To limit the magnitude of non-linear distortion and to ensure that xdisp and xadv are close to the
+probe fingerprint x, we introduce pixel loss between the image pairs (x, xdisp) and (x, xadv):
+Lpixel = 1
+n
+�
+i,j
+|xi,j − xdispi,j|+ 1
+n
+�
+i,j
+|xi,j − xadvi,j|
+(14)
+Figure 16 shows two displaced fingerprints and their corresponding output from Gdist.
+Discriminator
+In order to guide the generative modules Gdisp and Gdist to synthesize realistic
+fingerprint images, we introduce a fully convolutional network as a patch-based discriminator D.
+The job of the discriminator is to distinguish between real fingerprint images x and the generated
+adversarial fingerprint images xadv. This is accomplished through the GAN loss:
+Lgan = logD(x) + log(1 − D(xadv))
+(15)
+The proposed approach AdvBiom is trained in an end-to-end manner with respect to the following
+objective function:
+(16)
+L = Lgan + λmmap_simLmmap_sim + λmmap_disLmmap_dis + λpixelLpixel
+where the hyper-parameters λmmap_sim, λmmap_dis, and λpixel denote the relative importance of
+their respective losses. Once trained, AdvBiom can generate an adversarial fingerprint image for
+any input probe fingerprint and can be tested on any fingerprint matcher regardless of the feature
+extraction method (minutiae or deep-features).
+11Control points are points in an image to which non-linear distortion is applied.
+Probe Fingerprint
+Displaced Fingerprint
+Probe Fingerprint
+Displaced Fingerprint
+Figure 15: Example probe fingerprints from NIST SD4 [55] and their corresponding output from the minutiae
+displacement module Gdisp. The minutiae points (shown in red) are marked using a COTS minutiae extractor.
+17
+
+Displaced Fingerprint
+Distorted Fingerprint
+Displaced Fingerprint
+Distorted Fingerprint
+Figure 16: Fingerprints in the left column are example displaced fingerprints from Gdisp. The distortion module
+Gdist predicts control points (marked in blue) and distorts the images based on their displacements (red arrows)
+using the non-linear distortion model from [59]. The resultant distorted fingerprint images are shown in the right
+column.
+Successful
+Attacks
+Failed
+Attacks
+Original 
+Probe
+Adv. Probe
+(AdvFinge)
+Mate
+Original 
+Probe
+Adv. Probe
+(AdvFinge)
+Mate
+VS: 36 | FS: 0.82 
+VS: 47 | FS: 0.88 
+VS: 109 | FS: 0.87 
+VS: 76 | FS: 0.89 
+(AdvBiom)
+(AdvBiom)
+Figure 17: Example successful and failed adversarial fingerprints attack using AdvBiom on NIST SD4 [55]. The
+VeriFinger matching scores (probe v. mate): VS, and fingerprintness [56] scores: FS, of adversarial probes are
+shown below their respective triplet. Note that the VeriFinger matching threshold is 48 at 0.01% FAR.
+Evaluation Metrics:
+The requirement of a good adversarial fingerprints generator is to evade
+fingerprint matchers while preserving fingerprint attributes and being model-agnostic. Thus, in order
+to quantify the performance of adversarial attacks generated by AdvBiom and other state-of-the-art
+baselines, we employ the following evaluation metrics:
+• True Accept Rate (TAR): The extent to which an adversarial attack can evade a fingerprint
+matcher is measured by the drop in TAR at an operational setting, say 0.01% False Accept
+Rate (FAR).
+• Fingerprintness: Soweon and Jain [56] proposed a domain-specific metric called finger-
+printness to measure the degree of similarity of a given image to a fingerprint pattern.
+Fingerprintness ranges from [0,1] where higher the score, higher the probability of the
+pattern in the image corresponding to a fingerprint pattern.
+• NFIQ 2.0: Lastly, we use NFIQ 2.0 [62] quality scores to evaluate the fingerprint quality
+of adversarial fingerprint images. NFIQ scores range from [0,100] where a score of 100
+depicts the highest fingerprint quality.
+Note that since non-linear distortions change the structure of the image, using the structural similarity
+index (SSIM) metric is inappropriate as it essentially measures the local change in structures of the
+image pairs.
+Datasets: We train AdvBiom on an internal dataset of 120,000 rolled fingerprint images. Furthermore,
+we evaluate the performance of the proposed fingerprint adversarial attack and other baselines on:
+• 2,000 fingerprint pairs from NIST SD4 [55]
+• 27,000 fingerprint pairs from NIST SD14 [63]
+18
+
+111@
+8
+Q
+Q
+QQ
+Q
+de
+G
+G母
+Q
+%
+d
+TG
+%
+Q
+Qd
+Q
+d
+00
+&
+QQ
+%
+CQ
+d
+3
+d
+&
+QQQ
+@山
+G
+%Q
+QQ
+Q
+q Q
+bu
+G
+P dppdQ
+@
+d
+QQ-
+Q?
+&
+d
+&
+?
+de
+qQ
+&
+GQ QC
+QQ
+Q
+Q
+?
+多
+&
+G
+lG
+Q
+QQ
+%
+Q
+@
+G
+QQQQ
+G
+G20-
+Q
+Q
+Q
+q
+QQ
+G
+Q
+G8
+a
+QQ
+QQ
+cs
+G
+Q
+G
+QQ
+0甲Accuracy
+Adversarial Attacks
+Original
+Probes
+FGSM
+I-FGSM
+Deep
+Fool
+PGD
+Adv
+Biom
+TAR
+(%)
+at
+0.01%
+FAR
+NIST
+SD4
+VeriFinger
+99.05
+95.20
+98.30
+95.00
+97.60
+56.25
+Innovatrics
+97.00
+93.00
+95.50
+92.65
+94.75
+41.35
+DeepPrint
+94.55
+36.20
+64.15
+30.40
+68.75
+46.35
+NIST
+SD14
+VeriFinger
+99.42
+95.20
+98.30
+95.00
+97.60
+37.67
+Innovatrics
+98.24
+90.84
+95.68
+91.32
+94.01
+25.69
+DeepPrint
+96.52
+48.70
+84.48
+31.44
+64.28
+69.42
+FVC
+2004
+DB1 A
+VeriFinger
+94.89
+91.60
+91.53
+86.92
+92.69
+22.31
+Innovatrics
+94.15
+87.36
+85.68
+82.32
+88.75
+5.52
+DeepPrint
+75.36
+13.22
+33.31
+6.87
+27.39
+20.62
+Table 2: True Accept Rate (TAR) @ 0.01% FAR of AdvBiom along with state-of-the-art baselines attacks on
+three datasets - NIST SD4 [55], NIST SD14 [63], and FVC 2004 DB1 A [64]. 2 COTS fingerprint matchers -
+VeriFinger v11.0 [65] and Innovatrics v7.6.0.627 [66], and a deep network-based matcher DeepPrint [67] were
+employed for the evaluation. It is observed that DeepPrint, a deep network-based matcher, is susceptible to all
+types of adversarial attacks while VeriFinger and Innovatrics are more robust.
+• 558 fingerprints from DB1 A of FVC 2004 [64], consisting of 1,369 genuine pairs.
+Experimental Settings: AdvBiom was trained using the Adam optimizer with β1 as 0.5 and β2 as
+0.9. The hyper-parameters were empirically set to λmmap_sim = 0.05, λmmap_dis = 500000, and
+λpixel = 1000 for convergence. Based on the conclusions drawn in [57], d, c, and λ were set to
+20, 16, and 2.0 respectively for optimal effectiveness against fingerprint matchers while ensuring
+fingerprint realism. AdvBiom was trained for 16,000 steps using Tensorflow r1.14.0 on an Intel Core
+i7-11700F @ 2.50GHz CPU with a RTX 3070 GPU. On the same machine, AdvBiom can synthesize
+an adversarial fingerprint within 0.35 seconds.
+Fingerprint Authentication Systems: Since AdvBiom is a black-box attack, we do not require any
+fingerprint authentication system while training the network. However, we evaluate AdvBiom and
+other baseline attacks on two COTS fingerprint matchers and one deep network-based matcher:
+• VeriFinger v11.0 [65]
+• Innovatrics v7.6.0.627 [66]
+• DeepPrint [67]
+Comparison with Prevailing Fingerprint Adversarial Generators
+We show the performance
+of our method AdvBiom as compared to other state-of-the-art attacks in Table 2. We observe that
+the TAR of two COTS and a deep network-based fingerprint matcher for the aforementioned three
+datasets. It is to note that all the baseline attacks [14, 15, 17, 16] are white-box attacks and were
+trained using DeepPrint [67]. It is evident from Table 2 that AdvBiom is the most successful attack on
+COTS matchers VeriFinger and Innovatrics, and is also able to effectively evade a deep network-based
+fingerprint matcher, namely DeepPrint. It can also be observed that while COTS fingerprint matchers
+are robust to most adversarial attacks, DeepPrint is very susceptible to the same attacks since it
+heavily relies on the texture of the fingerprint which is majorly affected by adversarial attacks.
+A successful adversarial attack should not only evade fingerprint matchers but should also preserve
+fingerprint attributes. In order to observe the effect of adversarial attacks on fingerprint pattern
+in images, we plot the fingerprintness [56] distribution of 2,000 probes from NIST SD4 [55] for
+AdvBiom as well as for other baseline attacks. Since all the state-of-the-art baselines essentially add
+noise to each pixel in the image, they do not change the structure of the fingerprint and thus do not
+19
+
+0.6
+0.7
+0.8
+0.9
+1.0
+Fingerprintedness Scores
+0
+2
+4
+6
+8
+10
+12
+Probability of occurence
+Original Probes (  = 0.91)
+AdvFinge (  = 0.86)
+FGSM (  = 0.89)
+I-FGSM (  = 0.91)
+PGD (  = 0.90)
+DeepFool (  = 0.90)
+AdvBiom
+Figure 18: Fingerprintness [56] distribution of 2,000
+probes from NIST SD4 with respect to AdvBiom and
+other state-of-the-art baselines attacks.
+0
+20
+40
+60
+80
+100
+NFIQ Score
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+Probability of occurence
+Original Probes (  = 41.70)
+AdvFinge (  = 31.46)
+FGSM (  = 43.30)
+I-FGSM (  = 44.23)
+PGD (  = 41.28)
+DeepFool (  = 43.42)
+AdvBiom
+Figure 19: NFIQ 2.0 [62] quality scores distribution
+of 2,000 probes from NIST SD4 [55] with respect to
+AdvBiom and other baselines attacks.
+affect fingerprintness scores. AdvBiom , on the other hand, displaces minutiae points and non-linearly
+distorts the image, and still maintains a high mean fingerprintness score of µ = 0.86.
+Furthermore, we also compute the NFIQ 2.0 [62] quality scores distribution (figure 19) of the original
+and adversarial probes from NIST SD4 [55]. As shown in figure 12, baseline attacks tend to minutely
+perturb image pixels to generate adversarial fingerprints and as a result do not have much of an effect
+on the quality scores. AdvBiom , on the other hand, provides an optimal solution by successfully
+attacking fingerprint matchers while maintaining high fingerprintness and NFIQ scores.
+Genuine and Imposter Scores Distribution
+To determine the effect of adversarial fingerprint
+on both genuine and imposter pairs, we plot the genuine and imposter scores distribution of NIST
+SD4 [55] in figure 20 before and after applying AdvBiom . We computed a total of 2,000 genuine
+and 20,000 imposter scores for the evaluation. It can be observed that the genuine scores drastically
+decrease and shift to the left of the axis as their mean drops from 183.87 to 55.55 after the attack.
+However, the imposter scores remain unaffected with the mean imposter score changing by only 0.53.
+0
+100
+200
+300
+400
+Matching Scores
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+Probability of Occurence
+Genuine Scores | Before Attack (  = 183.88)
+Genuine Scores | After Attack (  = 55.55)
+Imposter Scores | Before Attack (  = 6.00)
+Imposter Scores | After Attack (  = 6.52)
+48: Matching Threshold at 0.01% FAR
+(a) Using VeriFinger SDK [65]
+(b) Using Innovatrics SDK [66]
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Matching Scores
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+Probability of Occurence
+Genuine Scores | Before Attack (  = 0.94)
+Genuine Scores | After Attack (  = 0.78)
+Imposter Scores | Before Attack (  = 0.10)
+Imposter Scores | After Attack (  = 0.09)
+0.837: Matching Threshold at 0.01% FAR
+(c) using DeepPrint [67]
+Figure 20: Genuine and imposter scores distribution of NIST SD4 [55] before and after the adversarial attack
+AdvBiom using three state-of-the-art fingerprint matchers - VeriFinger v11.0 [65], Innovatrics v7.6.0.627 [66],
+and DeepPrint [67]. Here, µ refers to the mean of the scores distribution. In all the three cases, the genuine
+scores shift towards the left while the imposter scores do not get affected by the attack.
+Is AdvBiom Biased Towards Certain Fingerprint Types?
+The generated adversarial fingerprint
+from AdvBiom is conditioned on the input probe fingerprint. Thus, it is essential to check if there is a
+relation between the amount of perturbation applied and the fingerprint type. The confusion matrix
+for the five fingerprint types (left loop, right loop, whorl, arch, tented arch) before and after applying
+AdvBiom on the 2,000 probes of NIST SD4 [55] is shown in table 3. Note that we use NIST SD4 for
+this evaluation since it has a uniform number of fingerprint images per each type (400 fingerprints
+per type). It is evident from the table that all five fingerprint types are almost equally susceptible to
+the attack, and thus the attack crafted AdvBiom is not biased towards a particular fingerprint type.
+20
+
+Genuine Scores LBefore Attack (u = 590.00)
+Genuine Scores 1After Attack (μu = 48.72)
+0.6
+Imposter Scores /Before Attack (μu = 1.34)
+Imposter Scores l After Attack (μ = 1.4o)
+0.5
+Probability of Occurence
+40:Matching Threshold at 0.01% FAR
+0.4
+0.3
+0.2
+0.1
+0.0
+200
+600
+0
+400
+800
+1000
+Matching Scores0.012
+0.010
+0.008
+0.006
+0.004
+0.002
+0.000
+0
+200
+400
+600
+800
+1000Before Attack
+After Attack
+TAR
+L: 99.75%
+R: 99.25%
+W: 99.50%
+T: 99.25%
+A: 97.50%
+FAR
+L: 0%
+R: 0%
+W: 0%
+T: 0%
+A: 0%
+FRR
+L: 0.25%
+R: 0.75%
+W: 0.50%
+T: 0.75%
+A: 2.50%
+TRR
+L: 100%
+R: 100%
+W: 100%
+T: 100%
+A: 100%
+TAR
+L: 59.00%
+R: 56.50%
+W: 58.75%
+T: 56.00%
+A: 57.00%
+FAR
+L: 0%
+R: 0%
+W: 0%
+T: 0%
+A: 0%
+FRR
+L: 41.00%
+R: 43.50%
+W: 41.25%
+T: 44.00%
+A: 43.00%
+TRR
+L: 100%
+R: 100%
+W: 100%
+T: 100%
+A: 100%
+Table 3: Confusion matrix for five fingerprint types (left loop: L, right loop: R, whorl: W, tented arch: T, arch:
+A) from NIST SD4 [55] before and after the adversarial attack using AdvBiom. Here, TAR = True Accept Rate,
+FAR = False Accept Rate, FRR = False Reject Rate, and TRR = True Reject Rate. Note that the matching
+threshold was 48 at 0.01% FAR using the COTS fingerprint matcher VeriFinger. AdvBiom is not biased towards
+any fingerprint type.
+5
+Conclusions
+We show that a new method of adversarial synthesis, namely AdvBiom, that automatically generates
+adversarial face images with imperceptible perturbations evading state-of-the-art biometric matchers.
+With the help of a GAN, and the proposed perturbation and identity losses, AdvBiom learns the set
+of pixel locations required by face matchers for identification and only perturbs those salient facial
+regions (such as eyebrows and nose). Once trained, AdvBiom generates high quality and perceptually
+realistic adversarial examples that are benign to the human eye but can evade state-of-the-art black-
+box face matchers, while outperforming other state-of-the-art adversarial face methods. Beyond
+faces, we show for the first time that such a method with the proposed Minutiae Displacement and
+Distortion Modules can also evade state-of-the-art automated fingerprint recognition systems.
+References
+[1] D. E. Rumelhart and J. L. McClelland, Learning Internal Representations by Error Propagation,
+pp. 318–362. 1987.
+[2] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional
+neural networks,” Commun. ACM, vol. 60, p. 84–90, may 2017.
+[3] J. Deng, J. Guo, N. Xue, and S. Zafeiriou, “Arcface: Additive angular margin loss for deep
+face recognition,” in Proceedings of the IEEE/CVF conference on computer vision and pattern
+recognition, pp. 4690–4699, 2019.
+[4] W. Liu, Y. Wen, Z. Yu, M. Li, B. Raj, and L. Song, “Sphereface: Deep hypersphere embedding
+for face recognition,” in Proceedings of the IEEE conference on computer vision and pattern
+recognition, pp. 212–220, 2017.
+[5] F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: A unified embedding for face recogni-
+tion and clustering,” in Proceedings of the IEEE conference on computer vision and pattern
+recognition, pp. 815–823, 2015.
+[6] P. Grother, G. Quinn, and P. Phillips, “Report on the evaluation of 2d still-image face recognition
+algorithms,” 2010-06-17 2010.
+[7] N. Carlini and D. Wagner, “Towards evaluating the robustness of neural networks,” in 2017 ieee
+symposium on security and privacy (sp), pp. 39–57, Ieee, 2017.
+[8] Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu, and J. Li, “Boosting adversarial attacks with
+momentum,” in Proceedings of the IEEE conference on computer vision and pattern recognition,
+pp. 9185–9193, 2018.
+[9] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,”
+arXiv preprint arXiv:1412.6572, 2014.
+[10] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models
+resistant to adversarial attacks,” arXiv preprint arXiv:1706.06083, 2017.
+21
+
+[11] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus,
+“Intriguing properties of neural networks,” arXiv preprint arXiv:1312.6199, 2013.
+[12] S.-M. Moosavi-Dezfooli, A. Fawzi, O. Fawzi, and P. Frossard, “Universal adversarial pertur-
+bations,” in Proceedings of the IEEE conference on computer vision and pattern recognition,
+pp. 1765–1773, 2017.
+[13] UIDAI, “Unique Identification Authority of India.” https://uidai.gov.in, 2022.
+[14] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,”
+in 3rd International Conference on Learning Representations, ICLR, 2015.
+[15] A. Kurakin, I. J. Goodfellow, and S. Bengio, “Adversarial examples in the physical world,” in
+5th International Conference on Learning Representations, ICLR, 2017.
+[16] S. Moosavi-Dezfooli, A. Fawzi, and P. Frossard, “Deepfool: A simple and accurate method to
+fool deep neural networks,” in IEEE Conference on Computer Vision and Pattern Recognition,
+CVPR, pp. 2574–2582, IEEE Computer Society, 2016.
+[17] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models
+resistant to adversarial attacks,” in 6th International Conference on Learning Representations,
+ICLR, 2018.
+[18] N. Carlini and D. A. Wagner, “Towards evaluating the robustness of neural networks,” in IEEE
+Symposium on Security and Privacy, SP, pp. 39–57, IEEE Computer Society, 2017.
+[19] C. Xiao, J. Zhu, B. Li, W. He, M. Liu, and D. Song, “Spatially transformed adversarial examples,”
+in 6th International Conference on Learning Representations, ICLR, 2018.
+[20] K. Eykholt, I. Evtimov, E. Fernandes, B. Li, A. Rahmati, C. Xiao, A. Prakash, T. Kohno, and
+D. Song, “Robust physical-world attacks on deep learning visual classification,” in IEEE/CVF
+Conference on Computer Vision and Pattern Recognition, pp. 1625–1634, 2018.
+[21] N. Papernot, P. D. McDaniel, S. Jha, M. Fredrikson, Z. B. Celik, and A. Swami, “The limitations
+of deep learning in adversarial settings,” in IEEE European Symposium on Security and Privacy,
+EuroS&P, pp. 372–387, IEEE, 2016.
+[22] A. Kurakin, I. J. Goodfellow, and S. Bengio, “Adversarial machine learning at scale,” in 5th
+International Conference on Learning Representations, ICLR, 2017.
+[23] Y. Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu, and J. Li, “Boosting adversarial attacks
+with momentum,” in IEEE Conference on Computer Vision and Pattern Recognition, CVPR,
+pp. 9185–9193, IEEE Computer Society, 2018.
+[24] C. Xie, Z. Zhang, Y. Zhou, S. Bai, J. Wang, Z. Ren, and A. L. Yuille, “Improving transferability
+of adversarial examples with input diversity,” in Proceedings of the IEEE/CVF Conference on
+Computer Vision and Pattern Recognition, pp. 2730–2739, 2019.
+[25] D. Deb, J. Zhang, and A. K. Jain, “Advfaces: Adversarial face synthesis,” in 2020 IEEE
+International Joint Conference on Biometrics (IJCB), pp. 1–10, IEEE, 2020.
+[26] Y. Liu, X. Chen, C. Liu, and D. Song, “Delving into transferable adversarial examples and
+black-box attacks,” in 5th International Conference on Learning Representations, ICLR, 2017.
+[27] W. Xiang, H. Su, C. Liu, Y. Guo, and S. Zheng, “Improving the robustness of adversarial attacks
+using an affine-invariant gradient estimator,” Available at SSRN 4095198, 2021.
+[28] Y. Dong, T. Pang, H. Su, and J. Zhu, “Evading defenses to transferable adversarial examples by
+translation-invariant attacks,” in Proceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pp. 4312–4321, 2019.
+[29] S. Cheng, Y. Dong, T. Pang, H. Su, and J. Zhu, “Improving black-box adversarial attacks with a
+transfer-based prior,” Advances in neural information processing systems, vol. 32, 2019.
+22
+
+[30] A. Ilyas, L. Engstrom, A. Athalye, and J. Lin, “Black-box adversarial attacks with limited
+queries and information,” in International Conference on Machine Learning, pp. 2137–2146,
+PMLR, 2018.
+[31] Y. Li, L. Li, L. Wang, T. Zhang, and B. Gong, “Nattack: Learning the distributions of adver-
+sarial examples for an improved black-box attack on deep neural networks,” in International
+Conference on Machine Learning, pp. 3866–3876, PMLR, 2019.
+[32] Y. Dong, H. Su, B. Wu, Z. Li, W. Liu, T. Zhang, and J. Zhu, “Efficient decision-based black-
+box adversarial attacks on face recognition,” in Proceedings of the IEEE/CVF Conference on
+Computer Vision and Pattern Recognition, pp. 7714–7722, 2019.
+[33] W. Brendel, J. Rauber, and M. Bethge, “Decision-based adversarial attacks: Reliable attacks
+against black-box machine learning models,” arXiv preprint arXiv:1712.04248, 2017.
+[34] M. Sharif, S. Bhagavatula, L. Bauer, and M. K. Reiter, “Accessorize to a crime: Real and stealthy
+attacks on state-of-the-art face recognition,” CCS ’16, (New York, NY, USA), p. 1528–1540,
+Association for Computing Machinery, 2016.
+[35] M. Sharif, S. Bhagavatula, L. Bauer, and M. K. Reiter, “A general framework for adversarial
+examples with objectives,” ACM Transactions on Privacy and Security (TOPS), vol. 22, no. 3,
+pp. 1–30, 2019.
+[36] S. Komkov and A. Petiushko, “Advhat: Real-world adversarial attack on arcface face id system,”
+in 2020 25th International Conference on Pattern Recognition (ICPR), pp. 819–826, IEEE,
+2021.
+[37] D.-L. Nguyen, S. S. Arora, Y. Wu, and H. Yang, “Adversarial light projection attacks on face
+recognition systems: A feasibility study,” in Proceedings of the IEEE/CVF conference on
+computer vision and pattern recognition workshops, pp. 814–815, 2020.
+[38] B. Yin, W. Wang, T. Yao, J. Guo, Z. Kong, S. Ding, J. Li, and C. Liu, “Adv-makeup: A new
+imperceptible and transferable attack on face recognition,” arXiv preprint arXiv:2105.03162,
+2021.
+[39] X. Liu, F. Shen, J. Zhao, and C. Nie, “Rstam: An effective black-box impersonation attack on
+face recognition using a mobile and compact printer,” arXiv preprint arXiv:2206.12590, 2022.
+[40] A. J. Bose and P. Aarabi, “Adversarial attacks on face detectors using neural net based con-
+strained optimization,” in 2018 IEEE 20th International Workshop on Multimedia Signal
+Processing (MMSP), pp. 1–6, IEEE, 2018.
+[41] N. Cauli, A. Ortis, and S. Battiato, “Fooling a face recognition system with a marker-free
+label-consistent backdoor attack,” in Image Analysis and Processing – ICIAP 2022: 21st Inter-
+national Conference, Lecce, Italy, May 23–27, 2022, Proceedings, Part II, (Berlin, Heidelberg),
+p. 176–185, Springer-Verlag, 2022.
+[42] L. Yang, Q. Song, and Y. Wu, “Attacks on state-of-the-art face recognition using attentional
+adversarial attack generative network,” Multimedia tools and applications, vol. 80, no. 1,
+pp. 855–875, 2021.
+[43] Y. Zhong and W. Deng, “Towards transferable adversarial attack against deep face recognition,”
+IEEE Transactions on Information Forensics and Security, vol. 16, pp. 1452–1466, 2020.
+[44] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A
+simple way to prevent neural networks from overfitting,” Journal of Machine Learning Research,
+vol. 15, no. 56, pp. 1929–1958, 2014.
+[45] S. Jia, B. Yin, T. Yao, S. Ding, C. Shen, X. Yang, and C. Ma, “Adv-attribute: Inconspicuous and
+transferable adversarial attack on face recognition,” arXiv preprint arXiv:2210.06871, 2022.
+[46] H. Qiu, C. Xiao, L. Yang, X. Yan, H. Lee, and B. Li, “Semanticadv: Generating adversarial
+examples via attribute-conditioned image editing,” in European Conference on Computer Vision,
+pp. 19–37, Springer, 2020.
+23
+
+[47] S. Shan, E. Wenger, J. Zhang, H. Li, H. Zheng, and B. Y. Zhao, “Fawkes: protecting privacy
+against unauthorized deep learning models,” in USENIX, pp. 1589–1604, 2020.
+[48] A. Dabouei, S. Soleymani, J. Dawson, and N. Nasrabadi, “Fast geometrically-perturbed adver-
+sarial faces,” in WACV, 2019.
+[49] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and
+Y. Bengio, “Generative adversarial networks,” Communications of the ACM, vol. 63, no. 11,
+pp. 139–144, 2020.
+[50] E. L. Denton, S. Chintala, R. Fergus, et al., “Deep generative image models using a laplacian
+pyramid of adversarial networks,” Advances in neural information processing systems, vol. 28,
+2015.
+[51] D. Yi, Z. Lei, S. Liao, and S. Z. Li, “Learning face representation from scratch,” arXiv preprint
+arXiv:1411.7923, 2014.
+[52] G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: A database
+for studying face recognition in unconstrained environments,” Tech. Rep. 07-49, University of
+Massachusetts, Amherst, October 2007.
+[53] K. Zhang, Z. Zhang, Z. Li, and Y. Qiao, “Joint face detection and alignment using multitask
+cascaded convolutional networks,” IEEE SPL, vol. 23, no. 10, pp. 1499–1503, 2016.
+[54] P. J. Grother, M. Ngan, and K. Hanaoka, “Ongoing Face Recognition Vendor Test (FRVT), Part
+2: Identification,” NIST Interagency Report, 2018.
+[55] NIST,
+“NIST
+Special
+Database
+4.”
+https://www.nist.gov/srd/
+nist-special-database-4, 2022.
+[56] S. Yoon and A. K. Jain, “Is there a fingerprint pattern in the image?,” in 2013 International
+Conference on Biometrics (ICB), pp. 1–8, 2013.
+[57] S. A. Grosz, J. J. Engelsma, N. G. P. Jr., and A. K. Jain, “White-box evaluation of fingerprint
+matchers,” CoRR, vol. abs/1909.00799, 2019.
+[58] K. Cao, D. Nguyen, C. Tymoszek, and A. K. Jain, “End-to-end latent fingerprint search,” IEEE
+Trans. Inf. Forensics Secur., vol. 15, pp. 880–894, 2020.
+[59] X. Si, J. Feng, J. Zhou, and Y. Luo, “Detection and rectification of distorted fingerprints,” IEEE
+Trans. Pattern Anal. Mach. Intell., vol. 37, no. 3, pp. 555–568, 2015.
+[60] F. L. Bookstein, “Principal warps: Thin-plate splines and the decomposition of deformations,”
+IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 6, pp. 567–585, 1989.
+[61] F. Cole, D. Belanger, D. Krishnan, A. Sarna, I. Mosseri, and W. T. Freeman, “Synthesizing
+normalized faces from facial identity features,” in Proceedings of the IEEE Conference on
+Computer Vision and Pattern Recognition (CVPR), July 2017.
+[62] E. Tabassi, “NFIQ 2.0: NIST Fingerprint image quality,” NISTIR 8034, 2016.
+[63] NIST,
+“NIST
+Special
+Database
+14.”
+https://www.nist.gov/srd/
+nist-special-database-14, 2022.
+[64] D. Maio, D. Maltoni, R. Cappelli, J. L. Wayman, and A. K. Jain, “Fvc2004: Third fingerprint
+verification competition,” in International conference on biometric authentication, pp. 1–7,
+Springer, 2004.
+[65] Neurotechnology, “VeriFinger SDK.” https://www.neurotechnology.com, 2022.
+[66] Innovatrics, “Innovatrics SDK.” https://www.innovatrics.com/innovatrics-abis/,
+2022.
+[67] J. J. Engelsma, K. Cao, and A. K. Jain, “Learning a fixed-length fingerprint representation,”
+IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1–1, 2019.
+24
+

2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3159fafd204453ed29f98d9337d708a94b879cc82191127cd18eec840b448685
+size 381697

2tFKT4oBgHgl3EQf7y43/content/tmp_files/2301.11946v1.pdf.txt

@@ -0,0 +1,2027 @@
+On the motion of an electron through vacuum fluctuations
+Anirudh Gundhi1, 2, ∗ and Angelo Bassi1, 2, †
+1Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy
+2Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy
+(Dated: January 31, 2023)
+We study the effects of the electromagnetic vacuum on the motion of a non-relativistic electron.
+To this end, the vacuum is treated as the environment and the electron as the system within the
+framework of open quantum systems.
+After tracing over the environmental degrees of freedom,
+we obtain the time evolution of the reduced density matrix of the electron in the position basis.
+Using the master equation, in the first part of the article we derive the equation of motion for the
+expectation value of the position operator. In the presence of an external potential, the equation
+turns out to be the same as its classical counterpart: the Abraham-Lorentz equation. However, in
+its absence, the dynamics is free of the runaway solution. In the second part of the article we study
+decoherence induced by vacuum fluctuations. We show that decoherence that appears at the level
+of the reduced density matrix does not correspond to actual irreversible loss of coherence.
+Numerous physical phenomena such as the Casimir ef-
+fect [1–3], the Unruh effect [4–6] and the Lamb shift [7–
+10] are attributed to the presence of vacuum fluctuations.
+The possibility of decoherence due to vacuum fluctua-
+tions, as being fundamental and unavoidable, has also
+been discussed in various works [11–18] without arriving
+at a general consensus.
+The interaction of an electron with the vacuum fluctu-
+ations can be studied within the framework of open quan-
+tum systems. We use this formalism to study two spe-
+cific phenomena. First, we derive the equation of motion
+(EOM) for the electron in the presence of an external po-
+tential that provides a quantum mechanical description
+of radiation emission by an accelerated electron.
+Sec-
+ond, we investigate if the interactions with the vacuum
+fluctuations alone can lead to spatial decoherence of the
+electron.
+The quantum mechanical version of the classical
+Abraham-Lorentz (AL) equation, which describes the re-
+coil force experienced by an accelerated electron due to
+the emission of radiation [19–22], has been previously
+derived, for example, in [10]. Instead of the electron’s
+position, the equation was obtained for the position op-
+erator and it was then argued why this operator equa-
+tion is fundamentally different from the classical one.
+The difficulties in making a direct connection with the
+classical dynamics were attributed to the presence of the
+additional transverse electric field operator of the electro-
+magnetic vacuum, which is zero classically. Similar prob-
+lem persists concerning the interpretation of the quantum
+Langevin equation obtained in [17] for an electron inter-
+acting with vacuum fluctuations.
+In our work, we use the path-integral formalism to ob-
+tain the explicit expression of the reduced density matrix
+in the position basis. The formalism used is adopted from
+[23].
+Within this framework, instead of the Langevin
+∗ [email protected]
+† [email protected]
+equation, we derive the master equation which yields the
+EOM for the expectation value of the position operator
+which provides a direct correspondence with the classical
+dynamics. In the presence of an arbitrary potential, we
+show that the classical EOM is the same as the one ob-
+tained from the reduced quantum dynamics. Moreover,
+the equation that emerges after a quantum mechanical
+treatment appears to be free of the problems associated
+with the AL equation: the existence of the runaway so-
+lution which leads to an exponential increase of the elec-
+tron’s acceleration, even in the absence of an external
+potential [19–21].
+Concerning decoherence, we show that the loss of co-
+herence due to vacuum fluctuations at the level of the re-
+duced density matrix is only apparent and reversible. To
+this end we show that by ‘switching off’ the interactions
+with the EM field, the original coherence is restored at
+the level of the system. Moreover, the expression for the
+decoherence factor that we obtain differs from the ones
+obtained in [17, 18] where the authors argue for a finite
+loss of coherence for momentum superpositions, due to
+vacuum fluctuations, but with different estimates for the
+magnitude of decoherence.
+The action. We work in the Coulomb gauge in which the
+Lagrangian relevant for the dynamics of a non-relativistic
+electron in the presence of an external potential and an
+external radiation field is given by [24]
+L(t) = 1
+2m˙r2
+e − V0(re) +
+�
+d3rLEM − ereE⊥(re) .
+(1)
+Here, re denotes the position of the electron, m the
+bare mass, e the electric charge, V0(re) an arbitrary
+bare external potential (acting only on the electron) and
+LEM := (ϵ0/2)
+�
+E2
+⊥(r) − c2B2(r)
+��
+in which E⊥ denotes
+the transverse electric field, B the magnetic field, ϵ0 the
+permittivity of free space and c the speed of light. As
+detailed in Appendix A, Eq. (1) is obtained from the
+general Lagrangian for electrodynamics under the non-
+relativistic approximation.
+Following the standard prescription, the EM field is
+quantized by quantizing the transverse vector potential
+arXiv:2301.11946v1  [quant-ph]  27 Jan 2023
+
+2
+ˆA⊥. In terms of its conjugate momentum ˆΠ (which is
+not proportional to E⊥ due to the form of the interaction
+term in Eq. (1), c.f. Appendix A), we define and work
+with ˆΠE = − ˆΠ/ϵ0, since it appears repeatedly in the
+calculations. Further, the quantized EM field is initially
+assumed to be in its vacuum state.
+The master equation via path integral formalism.
+The
+position basis representation of the full density matrix
+within the path integral formalism is given by [23, 25]
+⟨x′
+f| ˆρ(t) |xf⟩ =
+�
+D[x, x′]e
+i
+ℏ (S′
+T−ST)ρ(x′
+i, xi, ti) .
+(2)
+Eq. (2) describes the density matrix at some final time t,
+starting from an initial time ti, such that xi := x(ti),
+x′
+i := x′(ti), with S′
+T := ST[x′] (and similarly ST :=
+ST[x]) denoting the full action describing some general
+dynamics along the x-axis. The path integral in Eq. (2)
+is computed with the boundary conditions xf = x(t),
+x′
+f = x′(t), and includes the integral over xi and x′
+i.
+In our case, the quantized radiation field is treated as
+the environment, initially assumed to be in its vacuum
+state and the electron as the system. We are interested
+in the reduced effective dynamics of the electron having
+taken into account its interaction with the environment.
+This is described by the reduced density matrix ˆρr which
+is obtained after tracing over the environmental degrees
+of freedom. After performing the trace, by assuming the
+initial density matrix to be in the product state ˆρ(ti) =
+ˆρS(ti) ⊗ ˆρEM(ti), ˆρr takes the form [23] (c.f. Appendix B)
+ρr(x′
+f, xf, t) =
+�
+D[x, x′]e
+i
+ℏ (S′
+S−SS+SIF[x,x′])ρr(x′
+i, xi, ti) ,
+with, SIF = 1
+2
+� t
+ti
+dt1dt2xa(t1)Mab(t1; t2)xb(t2) .
+(3)
+Here, SS denotes the action corresponding to the sys-
+tem Hamiltonian (c.f. Appendix A) and, under the Ein-
+stein summation convention, we have introduced the vec-
+tor notation xa(t1) = x(t1) for a = 1 and xa(t1) = x′(t1)
+for a = 2 such that the matrix elements Mab are re-
+lated to the two-point correlations of the canonical trans-
+verse electric field operator ˆΠE (c.f. Appendix B). Since
+the electron’s motion is considered to be along the x-
+axis only, the two-point correlations involve only the x-
+component of ˆΠE. In terms of the creation and annihi-
+lation operators, and the x-component of the unit polar-
+ization vector εx
+k, it is given by [26]
+ˆΠE(r, t) = iC
+�
+d3k
+√
+k
+�
+ε
+ˆaε(k)ei(k·r−ωt)εx
+k + c.c , (4)
+with the constant prefactor C :=
+�
+ℏc/(2ϵ0(2π)3)
+� 1
+2 . By
+making a change of basis to (X(t), u(t)) with X(t) =
+(x(t) + x′(t))/2 and u(t) = x′(t) − x(t), the so-called
+influence functional SIF [27] takes the simplified form
+SIF[X, u](t) =
+� t
+ti
+dt1dt2
+�
+iu(t1)N(t1; t2)u(t2)
+2
++
+u(t1)D(t1; t2)X(t2)] ,
+(5)
+where the noise kernel N(t1; t2) and the dissipation ker-
+nel D(t1; t2) are defined to be
+N(t1; t2) := e2
+2ℏ ⟨0| {ˆΠE(t1), ˆΠE(t2)} |0⟩ ,
+D(t1; t2) :=ie2
+ℏ ⟨0|
+�
+ˆΠE(t1), ˆΠE(t2)
+�
+|0⟩ θ(t1 − t2) .
+(6)
+Here, |0⟩ is the vacuum state of the free radiation field
+and θ(τ) is the Heaviside step function. As in [17, 18],
+we have also used the standard non-relativistic dipole ap-
+proximation in which one ignores the spatial dependence
+of the EM fields. From the definitions in Eq. (6) and the
+expression for ˆΠE in Eq. (4), the explicit expressions for
+the noise and the dissipation kernels can be obtained.
+It is important to note that the evaluation of the ker-
+nels necessiates the introduction of a high frequency cut-
+off in the calculations. This is due to the fact that the
+expressions for the kernels, which only depend upon the
+difference τ := t1 − t2, diverge at τ = 0. A cure is pro-
+vided by the standard Hadamard finite part prescription
+[23] which introduces the convergence factor e−k/kmax in-
+side the integrals appearing in the vacuum expectation
+values of the commutator and the anti-commutator. In
+terms of ϵ = 1/ωmax, with ωmax = kmaxc being the high
+frequency cut-off, the kernels read (c.f. Appendix C)
+N(t1; t2) = N(τ) =
+e2
+π2ϵ0c3
+�
+ϵ4 − 6ϵ2τ 2 + τ 4�
+(ϵ2 + τ 2)4
+,
+(7)
+D(t1; t2) = D(τ) =
+e2
+3πϵ0c3 θ(τ) d3
+dτ 3 δϵ(τ) .
+(8)
+The function πδϵ(τ) := ϵ/(τ 2 + ϵ2) appearing in Eq. (8)
+behaves like a Dirac delta for τ ≫ ϵ but is non-singular
+at τ = 0 due to the finite cut-off. We refer to Appendix C
+for more details.
+Following [23], starting from Eq. (3) and using the ex-
+plicit functional form of SIF in Eq. (5), the master equa-
+tion for the reduced density matrix can be derived. Upto
+second order in the interactions, we obtain its expression
+to be (c.f. Appendix B for a detailed derivation)
+∂tˆρr(t) = − i
+ℏ
+�
+ˆHs, ˆρr(t)
+�
+− 1
+ℏ
+� t−ti
+0
+dτN(t; t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]]
++ i
+2ℏ
+� t−ti
+0
+dτD(t; t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] .
+(9)
+The first line of the master equation is the usual Liouville-
+von Neuman evolution and involves only the system
+Hamiltonian ˆHs. In the second and the third lines, which
+encode the system’s interaction with the environment,
+the operator ˆxHs(−τ) is used as a place holder for the
+expression
+ˆxHs(−τ) := ˆU −1
+s
+(t − τ; t)ˆx ˆUs(t − τ; t) ,
+(10)
+
+3
+where ˆUs(t − τ; t) is the unitary operator that evolves
+the statevector of the system from time t to t − τ via the
+system Hamiltonian ˆHs only.
+The operator ˆx without
+the subscript is the usual Schr¨odinger operator such that
+ˆxHs(0) = ˆx.
+Note that due to the coupling between the position of
+the electron and the transverse electric field in Eq. (1),
+the system Hamiltonian receives an additional contribu-
+tion such that ˆHs = ˆp2/(2m) + ˆV0(x) + ˆVEM(x), where,
+having introduced a cut-off scale in the calculations and
+considering the motion of the electron along the x-axis
+only, ˆVEM(x) =
+e2ω3
+max
+3π2ϵ0c3 ˆx2 (c.f. Appendix A). We point
+out that since the master equation is valid upto second
+order in the interactions and since the operator ˆxHs(−τ)
+appears alongside the dissipation and the noise kernels
+(which are already second order in e), the time evolu-
+tion governed by ˆUs(t − τ; t) in Eq. (10) is understood
+to involve only ˆV0 and not ˆVEM. Therefore, upto second
+order in the interactions, ˆVEM only contributes via the
+Liouville-von Neuman term.
+The equation of motion. Using the master equation (9),
+we obtain the coupled equations for the time evolution
+of ⟨ˆx⟩ and ⟨ˆp⟩. It is interesting to compare the quantum
+mechanical EOM with the one derived classically.
+Within classical electrodynamics, a charged spherical
+shell of radius R which is accelerated by an external force
+Fext, experiences an extra recoil force (radiation reaction)
+due to the emission of radiation. By taking the limit R →
+0 in the equation describing its dynamics, one obtains the
+Abraham-Lorentz formula
+mR¨x = Fext + 2ℏα
+3c2
+...x ,
+(11)
+where mR denotes the observed renormalized mass. See
+for example [20, 28] and the references therein for the
+derivation of the AL formula. The triple derivative term
+appearing in Eq. (11) can be interpreted as the friction
+term that leads to energy loss due to radiation emission.
+For instance, when the external potential is taken to be
+V0(x) = (1/2)mω2
+0x2, one has ...x ≈ −ω2
+0 ˙x [22]. However,
+the issue with Eq. (11) is that the same triple derivative
+term persists even when the external potential is switched
+off, leading to an exponentially increase of the particle’s
+acceleration.
+A discussion of the AL formula and the
+problems associated with it can be found in [19–21, 28]
+and the references therein.
+In the case that we are considering, the rate of change
+of the expectation values is calculated from Eq. (9). The
+coupled differential equations for ⟨ˆx⟩ and ⟨ˆp⟩ are given
+by (c.f. Appendix E)
+d
+dt⟨ˆx⟩ =Tr(ˆx ˙ˆρr) = ⟨ˆp⟩
+m ,
+(12)
+d
+dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ + Tr
+�
+ˆρr(t)
+� t−ti
+0
+dτD(τ)ˆxHs(−τ)
+�
+− 2e2ω3
+max⟨ˆx⟩/(3π2ϵ0c3) .
+(13)
+While it might not be apparent at the first glance,
+Eq. (13) is actually local in time due the form of the
+dissipation kernel in Eq. (8). To see this explicitly, the
+integral involving the dissipation kernel needs to be eval-
+uated. In order to do so, we integrate by parts such that
+the derivatives acting on δϵ (which appear in the expres-
+sion obtained for the dissipation kernel in Eq. (8)) are
+shifted onto the adjacent function. The integral is calcu-
+lated explicitly in Appendix D and the following identity
+is derived
+� t
+0
+dτD(τ)f(τ) = − 2αℏ
+3c2 f ′′′(0) − 4αℏωmax
+3πc2
+f ′′(0)
++ 2e2ω3
+maxf(0)/(3π2ϵ0c3) .
+(14)
+Here, the prime denotes the derivative taken with respect
+to τ and α = e2/(4πϵ0ℏc) the fine structure constant.
+Using the identity (14), Eq. (13) becomes
+d
+dt⟨ˆp⟩ = − ⟨ ˆV0,x ⟩ − 4αℏωmax
+3πc2
+Tr
+�
+ˆρr(t) d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+�
+− 2αℏ
+3c2 Tr
+�
+ˆρr(t) d3
+dτ 3 ˆxHs(−τ)
+����
+τ=0
+�
+.
+(15)
+We see that in the EOM (15) only the original bare po-
+tential ˆV0 remains, because the contribution coming from
+ˆVEM in the last line of Eq. (13) is canceled by the term
+in the last line of the integral (14), after one introduces
+the cut-off consistently throughout the calculations. For
+more details we refer to Appendices A and E, or Ref. [17]
+where the same cancellation was argued for.
+The time derivatives of ˆxHs in Eq. (15) can be easily
+computed, since from Eq. (10) we have the relation (upto
+leading order in the interactions)
+d
+dτ ˆxHs(−τ) = − i
+ℏ
+�
+ˆV0(x) + ˆp2
+2m, ˆxHs(−τ)
+�
+.
+(16)
+First we consider the situation when the external poten-
+tial is switched off. From Eq. (16), with ˆV0(x) = 0, taking
+another time derivative of ˆxHs we get
+d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+=
+�−i
+ℏ
+�2 � ˆp2
+2m,
+� ˆp2
+2m, ˆx
+��
+= 0 ,
+(17)
+where, in Eq. (17), we have also used the relation
+ˆxHs(0) = ˆx. Similarly, the third derivative term appear-
+ing in Eq. (15) also vanishes. Therefore, when ˆV0(x) = 0,
+Eq. (15) simply reduces to
+d
+dt⟨ˆp⟩ = 0 .
+(18)
+Unlike the AL formula in Eq. (11), we see that upto sec-
+ond order in the interactions there are no solutions which
+allow for an exponential increase of the particle’s accel-
+eration in the absence of an external potential.
+Next we consider the case when the external potential
+is switched on. When the potential does not depend ex-
+plicitly on time, the double and triple derivative terms
+
+4
+in Eq. (15) yield double and triple commutators with re-
+spect to the system Hamiltonian respectively (discarding
+ˆVEM upto second order). Eq. (15) can then be written as
+d
+dt⟨ˆp⟩ =Fext + 4αℏωmax
+3πc2
+Tr
+� 1
+ℏ2 ˆρr(t)
+�
+ˆHs,
+�
+ˆHs, ˆx
+���
+− 2αℏ
+3c2 Tr
+� i
+ℏ3 ˆρr(t)
+�
+ˆHs,
+�
+ˆHs,
+�
+ˆHs, ˆx
+����
+. (19)
+Here, we have defined Fext := −⟨ ˆV0(x),x ⟩. Due to the
+presence of ˆV0(x), the commutators of ˆHs with ˆx no
+longer vanish. To simplify the equation further, we shift
+the commutators onto the density matrix using the cyclic
+property Tr(ˆa · [ˆb, ˆc]) = Tr([ˆa,ˆb] · ˆc) such that
+Tr
+�
+ˆρr
+�
+ˆHs,
+�
+ˆHs, ˆx
+���
+= Tr
+�
+ˆx
+�
+ˆHs,
+�
+ˆHs, ˆρr
+���
+.
+(20)
+The same relationship is also obtained for the triple com-
+mutator term, with an additional minus sign. Remem-
+bering that the master equation is only valid upto second
+order in the interaction, it is sufficient to evaluate the
+trace in Eq. (19) at 0th order. This implies that within
+the trace, the time dependence of the density matrix can
+be evaluated only by retaining the Liouville-von Neuman
+term in Eq. (9). The right hand side of Eq. (20) thus
+becomes proportional to Tr(ˆx¨ˆρr). With these simplifica-
+tions, Eq. (19) can be written as
+mR
+d2
+dt2 ⟨ˆx⟩ = Fext + 2αℏ
+3c2
+d3
+dt3 ⟨ˆx⟩ .
+(21)
+After identifying the observed electron mass with the re-
+normalized mass mR := m + (4αℏωmax)/(3πc2), Eq. (21)
+reduces to the Abraham-Lorentz formula (11). The same
+result is also obtained for the general case in which the
+bare potential ˆV0(x, t) depends explicitly on time, as
+shown in Appendix E. We remark that the equation of
+motion derived quantum mechanically only reduces to
+Eq. (11) in the presence of an external potential. When
+the external potential is switched off, the EOM reduces
+to Eq. (18) and is therefore free of the runaway solution.
+Decoherence. In this final part of the article, we are inter-
+ested in assessing if the spatial superposition of a charged
+particle at rest can be suppressed via its interaction with
+the vacuum fluctuations alone. We begin by writing the
+position space representation of the master equation (9)
+relevant for decoherence
+∂tρr =
+�
+−(x′ − x)2N1(t)
+ℏ
+�
+ρr ,
+(22)
+where N1(τ) is defined to be N1(τ) :=
+� τ
+0 dτ ′N(τ ′) =
+−4αℏ(τ 3 − 3τϵ2)(τ 2 +ϵ2)−3(3πc2)−1 . We have set ti = 0
+and only retained the second term involving the noise ker-
+nel in Eq. (9). This is because the other terms typically
+give subdominant contributions when the question of in-
+terest is to evaluate the rate of decay of the off-diagonal
+elements of the density matrix at late times [23, 29]. We
+have also used the expression of the noise kernel in Eq. (7)
+inside the integral to obtain the expression for N1. Inte-
+grating Eq. (22) we get
+ρr(x′, x, t) = exp
+�
+−(x′ − x)2
+ℏ
+N2(t)
+�
+ρr(x′, x, 0) , (23)
+where N2(t) :=
+� t
+0 dτN1(τ). The function N2(t) is in-
+versely proportional to the coherence length lx(t) defined
+by lx(t) := (ℏ/N2(t))
+1
+2 .
+After performing the integral
+over N1 the expression for the coherence length is ob-
+tained to be
+lx(t) =
+�
+3πc2
+2αω2
+max
+· (t2 + ϵ2)2
+t4 + 3t2ϵ2
+t≫ϵ
+=
+�
+3π
+2α
+1
+kmax
+.
+(24)
+We see that the coherence length approaches a constant
+value on time scales much larger than ϵ = 1/ωmax and
+that its value scales inversely with the UV cut-off. Taken
+literally, if one sets kmax = 1/λdb, where λdb is the de
+Broglie wavelength of the electron, one would arrive at
+the conclusion that vacuum fluctuations lead to decoher-
+ence with the coherence length of the charged particle
+asymptotically reducing to lx ≈ 25λdb within the time
+scales t ≈ λdb/c.
+False Decoherence. It is clearly unsatisfactory to have
+an observable effect scale explicitly with the UV cut-off,
+since the precise numerical value of the cut-off is, strictly
+speaking, arbitrary. A similar situation was encountered
+in [30] in a different context of a harmonic oscillator cou-
+pled to a massive scalar field. However, it was argued in
+[30] that the reduced density matrix of the harmonic os-
+cillator described false decoherence. In such a situation,
+the off-diagonal elements of the density matrix are sup-
+pressed simply because the state of the environment goes
+into different configurations depending upon the spatial
+location of the system. However, these changes in the
+environmental states remain locally around the system
+and are reversible. For the electron interacting with vac-
+uum fluctuations, we therefore take the point of view
+that if the reduced density matrix describes false deco-
+herence, then after adiabatically switching off the inter-
+actions with the environment (after having adiabatically
+switched it on initially), the original coherence must be
+fully restored at the level of the system.
+To formulate the argument we consider a time depen-
+dent coupling q(t) = −ef(t) such that f(t) = 1 for
+most of the dynamics between the initial time t = 0
+and the final time t = T, while f(0) = f(T) = 0.
+The quantity relevant for decoherence is the noise kernel
+which, under the time-dependent coupling, transforms as
+N → ˜
+N = f(t1)f(t2)N(t1; t2) = f(t1)f(t2)N(t1 − t2) .
+The decoherence factor in the double commutator in
+Eq. (9) involves replacing t2 with t1 − τ and then in-
+tegrating over τ. Therefore, the function N1 transforms
+as N1 → ˜
+N1, with ˜
+N1 given by
+˜
+N1(t1) = f(t1)
+� t1
+0
+dτf(t1 − τ)N(τ) .
+(25)
+
+5
+From the definitions of N1 and N2 we have N1 =
+(d/dτ)N, N2 = (d/dτ)N1 and N1(0) = N2(0) = 0. Using
+these relations and integrating by parts, Eq. (25) becomes
+˜
+N1(t1) =f(t1)N1(t1)f(0) + f(t1)N2(t1) ˙f(0)
++ f(t1)
+� t1
+0
+dτN2(τ) d2
+dτ 2 f(t1 − τ) .
+(26)
+In the limit ϵ → 0 (taking the UV cut-off to infinity), we
+see from Eq. (24) that N2 looses any time dependence.
+We can therefore bring N2 outside the integral such that
+˜
+N1(t1) = f(t1)N1(t1)f(0)+f(t1)N2 ˙f(0)−f(t1)N2( ˙f(0)−
+˙f(t1)). The terms involving ˙f(0) cancel out and we get
+˜
+N1(t1) = f(t1)N1(t1)f(0) + f(t1)N2 ˙f(t1) .
+(27)
+After integrating by parts Eq. (27), in order to obtain
+˜
+N2(T) =
+� T
+0 dt1 ˜
+N1(t1), we get
+˜
+N2(T) =f(0) (f(T)N2(T) − f(0)N2(0))
+− f(0)N2
+� T
+0
+dt1 ˙f + N2
+2
+� T
+0
+dt1
+d
+dt1
+f 2 .
+(28)
+In the limit ϵ → 0, as we noted earlier, N2(t) takes a
+constant value for any time t > 0 but is zero at t = 0
+from the way it is defined. Therefore, after completing
+the remaining integrals, we get
+˜
+N2(T) = N2
+2
+�
+f 2(0) + f 2(T)
+�
+.
+(29)
+Since we assume that the interactions are switched off
+in the very beginning and at the very end, we see that
+˜
+N2(T) = 0 such that Eq. (23) becomes ˜ρr(x′, x, T) =
+ρr(x′, x, 0). Therefore, by adiabatically switching off the
+interactions we recover the original coherence within the
+system.
+This is different from standard collisional decoherence
+where, for example, one originally has ∂tρr(x′, x, t) =
+−Λ(x′ − x)2ρr(x′, x, t) [29].
+When in this case we
+send
+Λ
+→
+˜Λ
+=
+f(t)Λ,
+we
+get
+˜ρr(x′, x, t)
+=
+exp
+�
+−Λ(x′ − x)2 � t
+0 dt′f(t′)
+�
+ρr(x′, x, 0).
+The density
+matrix depends on the integral of f(t) rather than its
+end points and we see that coherence is indeed lost ir-
+reversibly.
+We interpret this result to imply that the
+vacuum fluctuations alone do not lead to irreversible loss
+of coherence. Moreover, our results imply that the ap-
+parent decoherence cannot be due to emission of photons
+as otherwise one would not be able to retrieve the coher-
+ence back into the system simply by switching off the
+interactions with the environment at late times.
+Discussion.
+We formulated the interaction of a non-
+relativistic electron with the radiation field within the
+framework of open quantum systems and obtained the
+master equation for the reduced electron dynamics in the
+position basis. We showed that the classical limit of the
+quantum dynamics is free of the problems associated with
+the purely classical derivation of the Abraham-Lorentz
+formula. With respect to possible decoherence induced
+by vacuum fluctuations alone, we showed that the ap-
+parent decoherence at the level of the reduced density
+matrix is reversible and is an artifact of the formalism
+used. In mathematically tracing over the environment,
+one traces over the degrees of freedom that physically
+surround the system being observed. These degrees of
+freedom must be considered part of the system being ob-
+served, rather than the environment [16, 30]. We formu-
+lated this interpretation by showing that one restores full
+initial coherence back into the system after switching off
+the interactions with the environment adiabatically. The
+formulation is fairly general and might also be used in
+other situations to distinguish true decoherence from a
+false one. The analysis therefore brings together various
+works in the literature [15–18, 30] and addresses some of
+the conflicting results.
+Acknowledgements.
+A.G. thanks Davide Bason and
+Lorenzo Di Pietro for numerous discussions. We thank
+Oliviero Angeli for cross checking some of the results ob-
+tained in the manuscript and Lajos Di´osi for discussions
+concerning false decoherence. A.B. acknowledges finan-
+cial support from the EIC Pathfinder project QuCoM
+(GA no. 101046973) and the PNRR PE National Quan-
+tum Science and Technology Institute (PE0000023). We
+thank the University of Trieste and INFN for financial
+support.
+Appendix A: The Lagrangian and the Hamiltonian formulation
+In the Coulomb gauge, the standard Lagrangian for electrodynamics is given by [24]
+L = 1
+2m˙r2
+e − V0(re) −
+�
+1/2
+d3k |ρ|2
+ϵ0k2 + ϵ0
+2
+�
+d3r
+�
+E2
+⊥(r) − c2B2(r)
+�
++
+�
+d3rj(r) · A⊥(r) .
+(A1)
+In addition to the terms that have been described in the main article, Eq. (A1) also includes the Coulomb potential
+between different particles. It is given by the third term in which ρ(r) denotes the charge density and the symbol
+�
+1/2 means that the integral is taken over half the volume in the reciprocal space. For a single particle, it reduces to
+the particle’s Coulomb self energy ECoul. After the introduction of a suitable cut-off it takes a finite value given by
+ECoul = αℏωmax/π [22]. The transverse vector potential is denoted by A⊥(r, t) whose negative partial time derivative
+yields the transverse electric field E⊥(r, t) while its curl gives the magnetic field B(r, t). For an electron, the current
+
+6
+density is given by j(r) = −e˙rδ(r−re) and the interaction term becomes −e˙reA⊥(re, t). For a non-relativistic charged
+particle, the time derivative can be shifted from the position of the particle onto the transverse vector potential. This
+is because in addition to a total derivative term, a term of the form erevi∂iA⊥(r, t) appears (where vi := ˙ri). After
+the wave expansion of A⊥, this term is seen to be negligible with respect to ere ˙A⊥(re, t) = −ereE⊥(re, t) as long as
+ωk ≫ vk or v ≪ c. Therefore, for the non-relativistic electron, the Lagrangian relevant for the dynamics reduces to
+L(t) ≈ 1
+2m˙r2
+e − V0(re) + ϵ0
+2
+�
+d3r
+�
+E2
+⊥(r) − c2B2(r)
+�
+− ereE⊥(re) .
+(A2)
+In Eq. (A2) the total derivative d/dt(reA⊥(re)) and the constant Coulomb self energy term have been omitted as
+these do not affect the electron’s dynamics.
+The Hamiltonian corresponding to the Lagrangian (A2) can now be obtained. In terms of the canonical variables
+re, p, A⊥ and ΠE := − 1
+ϵ0 Π, it takes the form
+H = HS + HEM + Hint ,
+(A3)
+where HEM = ϵ0
+2
+�
+d3r(Π2
+E(r) + c2B2(r)) is the free field Hamiltonian of the radiation field, Hint = ereΠE(re) the
+interaction term and HS the system Hamiltonian given by
+HS = p2
+2m + V0(re) + e2
+2ϵ0
+�
+d3rriδ⊥
+im(r − re)δ⊥
+mj(r − re)rj .
+(A4)
+Here, the transverse Dirac delta δ⊥
+ij(r − re), which appears due to the coupling of the position of the electron with
+the transverse electric field, is defined to be [22]
+δ⊥
+ij(r − re) :=
+1
+(2π)3
+�
+d3k
+�
+δij − kikj
+k2
+�
+eik·(r−re) .
+(A5)
+The form of HS calls for an identification of the full effective potential V (re) governing the dynamics of the electron
+such that
+V (re) := V0(re) + VEM(re) ,
+VEM(re) = e2
+2ϵ0
+�
+d3rriδ⊥
+im(r − re)δ⊥
+mj(r − re)rj .
+(A6)
+Note that the extra term VEM(re) is not added to the bare potential by hand, but arises naturally due to the reE⊥
+coupling [17]. Although it gives a divergent contribution
+e2
+2ϵ0 δ⊥
+ij(0)ri
+erj
+e, after regularizing the transverse delta function
+on a minimum length scale rmin = 1/kmax, the contribution coming from this term scales as O( e2
+2ϵ0 r2
+ek3
+max). To be
+more precise, we impose the cut-off consistently throughout the calculations by introducing the convergence factor
+e−k/kmax inside the integral in the reciprocal space (c.f. Appendix C). Using this procedure, the expression for δ⊥
+ij(0)
+is obtained to be
+δ⊥
+ij(0) =
+1
+(2π)3
+�
+dkk2e−k/kmax
+�
+dΩ
+�
+δij − kikj
+k2
+�
+.
+(A7)
+First evaluating the angular integral, which gives a factor 8π
+3 δij, and then the radial integral, we get
+VEM(re) = e2ω3
+max
+3π2ϵ0c3 r2
+e .
+(A8)
+Since the contribution of VEM(re) is canceled exactly by another term, as shown in the discussion around Eq. (15) of
+the main text, for all practical purposes, VEM(re) has no consequences on the dynamics of the electron.
+Appendix B: The master equation
+The probability amplitude for a particle to be at the position xf at some final time t, starting from the position xi
+at some initial time ti, is given by [25]
+⟨xf| ˆU(t; ti) |xi⟩ =
+�
+x(t)=xf,
+x(ti)=xi
+D[x, p]e��� i
+ℏ
+� t
+ti dt′(HT[x,p]−p ˙x) =
+�
+x(t)=xf,
+x(ti)=xi
+D[x]e
+i
+ℏ ST[x] ,
+(B1)
+
+7
+where HT is the full Hamiltonian and ST is the corresponding action describing some general dynamics. From Eq. (B1)
+the expression for the density matrix at time t can be written as [23]
+⟨x′
+f| ˆρ(t) |xf⟩ =
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]e
+i
+ℏ (ST[x′]−ST[x])ρ(x′
+i, xi, ti) ,
+(B2)
+where the integrals over xi and x′
+i are included within the path integral. The expression analogous to Eq. (B1) also
+exists for ⟨pf| ˆU(t; ti) |pi⟩ in which the boundary conditions are fixed on p(t) and the phase-space weighing function is
+instead given by exp{ −i
+ℏ
+� t
+ti dt′ (HT [x, p] + x ˙p)} such that
+⟨pf| ˆU(t; ti) |pi⟩ =
+�
+p(t)=pf,
+p(ti)=pi
+D[x, p]e− i
+ℏ
+� t
+ti dt′(HT[x,p]+x ˙p) .
+(B3)
+For computing the path integral over the EM field, with a slight abuse of notation, we understand exp{ i
+ℏSEM}
+to be simply the appropriate phase-space weighing function appearing inside the path integral with SEM :=
+−
+� t
+ti dt′d3r(HEM − Π ˙A⊥) or SEM := −
+� t
+ti dt′d3r(HEM + A⊥ ˙Π) depending upon the basis states between which the
+transition amplitudes are calculated. We are interested in the dynamics of the electron, having taken into account its
+interaction with the radiation field environment. With this distinction, the total phase-space function can be written
+as ST = SS[x] + SEM[µ] + Sint[x, ΠE], where SS denotes the system action, SEM[µ] := SEM[A⊥, ΠE] the phase-space
+function governing the time evolution of the free radiation field in which µ denotes its phase-space degrees of freedom
+and Sint[x, ΠE] := −e
+� t
+ti dt′xΠE encodes the interaction between the two. The expression for the system-environment
+density matrix can then be written as
+�
+x′
+f; Πf′
+E
+�� ˆρ(t)
+��xf; Πf
+E
+�
+=
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]e
+i
+ℏ (SS[x′]−SS[x])ρS(x′
+i, xi, ti)×
+×
+�
+ΠE(t)=Πf
+E,
+Π′
+E(t)=Πf′
+E
+D[µ, µ′]e
+i
+ℏ (SEM[µ′]+Sint[x′,Π′
+E]−SEM[µ]−Sint[x,ΠE])ρEM(Π′
+E(ti), ΠE(ti), ti) ,
+(B4)
+where
+���Πf
+E
+�
+denotes the basis state of the environment. Note that the precise choice of the environmental basis states
+is unimportant since the reduced density matrix is obtained by tracing over the environment. In writing Eq. (B4) we
+have also assumed the full density matrix ˆρ(ti) to be in the product state ˆρ(ti) = ˆρS(ti) ⊗ ˆρEM(ti) at the initial time
+ti. We notice that SEM[µ] is quadratic in the environmental degrees of freedom while Sint[x, ΠE] is linear in both x
+and ΠE. After tracing over the environment, that is integrating over ΠE(t) = Π′
+E(t), the term in the second line of
+Eq. (B4) yields a Gaussian in x such that [23]
+�
+ΠE(t)=Π′
+E(t)
+dΠE(t)D[µ, µ′]e
+i
+ℏ (SEM[µ′]+Sint[x′,Π′
+E]−SEM[µ]−Sint[x,ΠE])ρi
+EM = e
+i
+2ℏ
+��
+dt1dt2Mab(t1;t2)xa(t1)xb(t2) ,
+(B5)
+where ρi
+EM := ρEM(Π′
+E(ti), ΠE(ti), ti). We have also introduced the vector notation with the convention xa = x for
+a = 1, xa = x′ for a = 2 and xa = ηabxb with ηab = diag(−1, 1). It is the matrix elements Mab which determine the
+effective action of the system and contain the information about its interaction with the environment. They can be
+obtained by acting with ℏ
+i
+δ
+δxa
+δ
+δxb |xa=xb=0 (where xa and xb are set to zero after taking the derivatives) on Eq. (B5)
+such that
+M ab(t1; t2) = ie2
+ℏ
+�
+ΠE(t)=Π′
+E(t)
+dΠE(t)D[µ, µ′]Πa
+E (t1) Πb
+E (t2) e
+i
+ℏ (SEM[µ′]−SEM[µ])ρi
+EM .
+(B6)
+Here, in the light of the standard non-relativistic dipole approximation, we have ignored the spatial dependence of the
+canonical fields (c.f. Appendix C). Depending upon the value of the indices a and b, the matrix elements correspond
+to the expectation values of the time-ordered or path-ordered correlations in the Heisenberg picture [23]. For the
+dynamics of the non-relativistic electron that we are considering, the expression for Mab reads
+Mab(t1; t2) = ie2
+ℏ
+�
+�
+�
+˜T {ˆΠE(t1)ˆΠE(t2)}
+�
+0
+−
+�
+ˆΠE(t1)ˆΠE(t2)
+�
+0
+−
+�
+ˆΠE(t2)ˆΠE(t1)
+�
+0
+�
+T {ˆΠE(t1)ˆΠE(t2)}
+�
+0
+�
+� .
+(B7)
+
+8
+The zero in the subscript denotes that the expectation values are calculated by disregarding the interaction with the
+system, while T and ˜T denote the time-ordered and the anti-time ordered products respectively. It is also understood
+that since the electron’s motion is considered to be along the x-axis, the canonical field operator that enters Mab is
+only the x-component given by [26]
+ˆΠE(r, t) = i
+�
+ℏc
+2ϵ0(2π)3
+� 1
+2 �
+d3k
+√
+k
+�
+ε
+ˆaε(k)ei(k·r−ωt)εx
+k + c.c .
+(B8)
+In our case, the initial state of the environment is taken to be the vacuum state |0⟩ of the radiation field such that
+⟨·⟩0 = ⟨0| · |0⟩. After tracing over the environment, the reduced density matrix of the electron is obtained from
+Eq. (B4) to be
+⟨x′
+f| ˆρr(t) |xf⟩ =
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]e
+i
+ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′
+i, xi, ti) ,
+(B9)
+where
+SIF[x, x′] = ie2
+2ℏ
+� t
+ti
+dt1dt2
+��
+˜T {ˆΠE(t1)ˆΠE(t2)}
+�
+0 x(t1)x(t2) −
+�
+ˆΠE(t1)ˆΠE(t2)
+�
+0 x(t1)x′(t2)
+−
+�
+ˆΠE(t2)ˆΠE(t1)
+�
+0 x′(t1)x(t2) +
+�
+T {ˆΠE(t1)ˆΠE(t2)}
+�
+0 x′(t1)x′(t2)
+�
+.
+(B10)
+The integral
+� t
+ti stands for both the t1 and the t2 integrals which run from ti to t.
+Alternatively, the influence
+functional SIF can be written in the matrix notation as
+SIF[x, x′] = 1
+2
+� t
+ti
+dt1dt2
+�x(t1) x′(t1)�
+·
+�
+M11 M12
+M21 M22
+�
+·
+�
+x(t2)
+x′(t2)
+�
+.
+(B11)
+As it is more convenient, we make a change of basis to (X , u) defined by
+X(t) :=(x′(t) + x(t))/2 ,
+u(t) = x′(t) − x(t) ,
+(B12)
+in which the influence functional transforms as
+SIF[X, u] = 1
+2
+� t
+ti
+dt1dt2
+�X(t1) u(t1)�
+·
+� ˜
+M11
+˜
+M12
+˜
+M21
+˜
+M22
+�
+·
+�
+X(t2)
+u(t2)
+�
+,
+(B13)
+where
+� ˜
+M11
+˜
+M12
+˜
+M21
+˜
+M22
+�
+=
+�
+M11 + M12 + M21 + M22
+1
+2 ((M12 − M21) + (M22 − M11))
+1
+2 (−(M12 − M21) + (M22 − M11))
+1
+4((M11 + M22) − (M12 + M21))
+�
+.
+(B14)
+From Eq. (B7) we obtain the following relations
+M11 + M22 = −(M12 + M21) = ie2
+ℏ
+�
+{ˆΠE(t1), ˆΠE(t2)}
+�
+0 ,
+(B15)
+M12 − M21 = ie2
+ℏ
+��
+ˆΠE(t2), ˆΠE(t1)
+��
+0 ,
+(B16)
+M22 − M11 = ie2
+ℏ
+��
+ˆΠE(t1), ˆΠE(t2)
+��
+0 sgn(t1 − t2) .
+(B17)
+Using these relations, ˜
+M takes the simplified form
+� ˜
+M11
+˜
+M12
+˜
+M21
+˜
+M22
+�
+= ie2
+ℏ
+�
+�
+0
+��
+ˆΠE(t2), ˆΠE(t1)
+��
+0 θ(t2 − t1)
+��
+ˆΠE(t1), ˆΠE(t2)
+��
+0 θ(t1 − t2)
+1
+2
+�
+{ˆΠE(t1), ˆΠE(t2)}
+�
+0
+�
+� ,
+(B18)
+where θ(t) is the Heaviside step function. Thus, in the (X, u) basis, the influence functional in Eq. (B10) takes the
+compact form
+SIF[X, u](t) =
+� t
+ti
+dt1dt2
+�
+iu(t1)N(t1; t2)u(t2)
+2
++ u(t1)D(t1; t2)X(t2)
+�
+,
+(B19)
+
+9
+where the noise kernel N and the dissipation kernel D are defined as
+N(t1; t2) := e2
+2ℏ
+�
+{ˆΠE(t1), ˆΠE(t2)}
+�
+0 ,
+D(t1; t2) :=ie2
+ℏ
+��
+ˆΠE(t1), ˆΠE(t2)
+��
+0 θ(t1 − t2) .
+(B20)
+Having determined the full effective action for the electron, in terms of the influence functional, we can now derive
+the master equation. From Eq. (B9), it can be seen that the time derivative of the reduced density matrix will have,
+in addition to the standard Liouville-von Neuman term, the contribution coming from the influence functional. In
+order to compute that we need to evaluate the rate of change of SIF. It is given by
+δtSIF[X, u] = u(t)
+� t
+ti
+dt1 (iN(t; t1)u(t1) + D(t; t1)X(t1)) .
+(B21)
+In terms of the original (x, x′) basis, the full expression for the master equation can now be written as
+∂tρr(x′
+f, xf, t) = − i
+ℏ ⟨x′
+f|
+�
+ˆHs, ˆρr
+�
+|xf⟩ + i
+ℏ
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]δtSIF[x′, x]e
+i
+ℏ (SS[x′]−SS[x]+SIF[x,x′])ρr(x′
+i, xi, ti)
+≈ − i
+ℏ ⟨x′
+f|
+�
+ˆHs, ˆρr
+�
+|xf⟩ + i
+ℏ
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]δtSIF[x′, x]e
+i
+ℏ (SS[x′]−SS[x])ρr(x′
+i, xi, ti)
+≈ − i
+ℏ ⟨x′
+f|
+�
+ˆHs, ˆρr
+�
+|xf⟩
+− 1
+ℏ(x′
+f − xf)
+� t
+ti
+dt1N(t; t1)
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′](x′(t1) − x(t1))e
+i
+ℏ (SS[x′]−SS[x])ρr(x′
+i, xi, ti)
++ i
+2ℏ(x′
+f − xf)
+� t
+ti
+dt1D(t; t1)
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′](x′(t1) + x(t1))e
+i
+ℏ (SS[x′]−SS[x])ρr(x′
+i, xi, ti) .
+(B22)
+The Liouville-von Neuman evolution is governed by the system Hamiltonian ˆHs alone. For the second term on the
+right hand side in the second line of Eq. (B22), we have omitted SIF in the exponential. This is because SIF is second
+order in the coupling constant and is already present adjacent to the exponential. Since we limit our calculations to
+second order in the interactions, SIF can be neglected inside the exponential.
+To simplify the master equation further, we note that the last two lines of Eq. (B22) can be written much more
+compactly. This is due to the following identity [23]
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]x′(t1)e
+i
+ℏ (SS[x′]−SS[x])ρr(x′
+i, xi, ti) =
+=
+�
+dx′(t1) ⟨x′
+f| ˆUs(t; t1) |x′(t1)⟩ x′(t1) ⟨x′(t1)| ˆUs(t1; ti)ˆρr(ti) ˆU −1
+s
+(t; ti) |xf⟩
+= ⟨x′
+f| ˆUs(t; t1)ˆx ˆUs(t1; ti)ˆρr(ti) ˆU −1
+s
+(t; ti) |xf⟩ = ⟨x′
+f| ˆUs(t; t1)ˆx ˆUs(t1; ti) ˆU −1
+s
+(t; ti) ˆUs(t; ti)ˆρr(ti) ˆU −1
+s
+(t; ti) |xf⟩
+= ⟨x′
+f| ˆUs(t; t1)ˆx ˆU −1
+s
+(t; t1)ˆρr(t) |xf⟩ = ⟨x′
+f| ˆxHs(−τ)ˆρr(t) |xf⟩ ,
+(B23)
+where
+ˆxHs(−τ) := ˆU −1
+s
+(t − τ; t)ˆx ˆUs(t − τ; t) ,
+τ := t − t1 .
+(B24)
+Similarly, we also have
+�
+x(t)=xf,
+x′(t)=x′
+f
+D[x, x′]x(t1)e
+i
+ℏ (SS[x′]−SS[x])ρr(x′
+i, xi, ti) = ⟨x′
+f| ˆρr(t)ˆxHs(−τ) |xf⟩ .
+(B25)
+The operator ˆxHs(−τ) is understood to be simply the placeholder for the expression that appears on the right hand
+side of the first equality in Eq. (B24) such that
+ˆxHs(0) = ˆx .
+(B26)
+
+10
+Here, the operator ˆx without the subscript Hs is the usual position operator in the Schr¨odinger picture. Using these
+relations, and replacing the t1 integral with the τ integral (t1 = t − τ), the master equation takes the compact form
+∂tρr(x′
+f, xf, t) = − i
+ℏ ⟨x′
+f|
+�
+ˆHs, ˆρr(t)
+�
+|xf⟩
+− 1
+ℏ(x′
+f − xf)
+� t−ti
+0
+dτN(t; t − τ) ⟨x′
+f| [ˆxHs(−τ), ˆρr(t)] |xf⟩
++ i
+2ℏ(x′
+f − xf)
+� t−ti
+0
+dτD(t; t − τ) ⟨x′
+f| {ˆxHs(−τ), ˆρr(t)} |xf⟩ .
+(B27)
+The eigenvalues outside of the integrals in Eq. (B27) can be obtained by acting with the position operator ˆx such that
+⟨x′
+f| ∂tˆρr |xf⟩ = − i
+ℏ ⟨x′
+f|
+�
+ˆHs, ˆρr(t)
+�
+|xf⟩
+− 1
+ℏ
+� t−ti
+0
+dτN(t; t − τ) ⟨x′
+f| [ˆx, [ˆxHs(−τ), ˆρr(t)]] |xf⟩
++ i
+2ℏ
+� t−ti
+0
+dτD(t; t − τ) ⟨x′
+f| [ˆx, {ˆxHs(−τ), ˆρr(t)}] |xf⟩ .
+(B28)
+The master equation in the operator form can therefore be written as
+∂tˆρr = − i
+ℏ
+�
+ˆHs, ˆρr
+�
+− 1
+ℏ
+� t−ti
+0
+dτN(t; t − τ) [ˆx, [ˆxHs(−τ), ˆρr(t)]] + i
+2ℏ
+� t−ti
+0
+dτD(t; t − τ) [ˆx, {ˆxHs(−τ), ˆρr(t)}] .
+(B29)
+Appendix C: The dissipation and the noise kernels
+In order to solve the master equation (B29), the kernels need to be evaluated explicitly. To achieve that, we begin
+with the expression for the vacuum expectation value of the correlator
+⟨0| ˆΠE(x(t1), t1)ˆΠE(x(t2), t2) |0⟩ =
+−iℏc
+2ϵ04π2 ˆ□
+�1
+r
+� ∞
+0
+dke−ikcτ �
+eikr − e−ikr��
+,
+(C1)
+where
+r := |x(t1) − x(t2)| ,
+τ := t1 − t2 ,
+ˆ□ := − 1
+c2 ∂2
+τ + ∂2
+r .
+(C2)
+Here, the right hand side of Eq. (C1) is obtained with the help of the expression of the quantized canonical transverse
+electric field operator in Eq. (B8). The expression in Eq. (C1) becomes convergent after resorting to the standard
+Hadamard finite part prescription [23], in which the convergence factor e−ωk/ωmax is introduced inside the integral
+(with ωk = kc).
+Physically, this prescription cuts off the contribution coming from the modes ωk ≫ ωmax and
+mathematically it is the same as using the iϵ prescription where one sends τ → τ − iϵ, with ϵ = 1/ωmax. After
+completing the integral by using this prescription we get
+⟨0| ˆΠE(1)ˆΠE(2) |0⟩ =
+ℏc
+4π2ϵ0
+ˆ□
+�
+1
+r2 − c2(τ − iϵ)2
+�
+=
+ℏc
+π2ϵ0
+1
+(r2 − c2(τ − iϵ)2)2 .
+(C3)
+For the correlator in Eq. (C3), we ignore the spatial dependence of the fields in the spirit of the non-relativistic
+approximation r ≪ cτ. In this limit, the correlator becomes
+⟨0| ˆΠE(1)ˆΠE(2) |0⟩ ≈
+ℏ
+π2ϵ0c3 (τ − iϵ)4 .
+(C4)
+Using Eq. (C4), we obtain the explicit functional form of the noise and the dissipation kernels to be
+N(τ) =
+e2
+π2ϵ0c3
+�
+ϵ4 − 6ϵ2τ 2 + τ 4�
+(ϵ2 + τ 2)4
+,
+(C5)
+D(τ) =
+8e2
+π2ϵ0c3
+ϵτ(ϵ2 − τ 2)
+(ϵ2 + τ 2)4 θ(τ) .
+(C6)
+
+11
+With some algebraic manipulation, the dissipation kernel can be expressed more compactly as
+D(τ) =
+e2
+3π2ϵ0c3 θ(τ) d3
+dτ 3
+�
+ϵ
+τ 2 + ϵ2
+�
+.
+(C7)
+Noticing that
+ϵ
+τ 2 + ϵ2 = d
+dτ tan−1(τ/ϵ) = πδϵ(τ) ,
+(C8)
+we arrive at the expression
+D(τ) =
+e2
+3πϵ0c3 θ(τ) d3
+dτ 3 δϵ(τ) .
+(C9)
+The last equality in Eq. (C8) can be understood in the limit ϵ → 0 when the function tan−1(τ/ϵ) takes the shape of
+a step function. Such an expression for D would yield infinite results. For that, we keep in mind that these functions
+are always well behaved for a finite ϵ and that δϵ only behaves like a Dirac delta for τ ≫ ϵ.
+Appendix D: Integrals involving the dissipation kernel
+In this section we derive an identity involving the integrals of the form
+�
+dτD(τ)f(τ). To proceed, we keep in mind
+the situation where ϵ is small but finite so that all the derivatives of the smoothed Dirac delta are large but finite.
+However, for times τ ≫ ϵ, we have δϵ(τ) = δ′
+ϵ(τ) = δ′′
+ϵ (τ) = 0. In addition, since the derivative of the Dirac delta is an
+odd function of τ, we also have δ′
+ϵ(0) = 0. In computing the integral of D(τ) multiplying an arbitrary function f(τ),
+we shift the derivatives acting on δϵ one by one onto f(τ) by integrating by parts. Since the calculations of interest
+involve integrating
+� t
+0 dτD(τ)f(τ), where τ takes only non-negative values from 0 to t, the step function θ(τ) can be
+omitted inside the integral.
+The first integration by parts gives (the constant pre-factors appearing in Eq. (C9) will be plugged in at the end)
+� t
+0
+dτδ′′′
+ϵ (τ)f(τ) = −
+� t
+0
+dτδ′′
+ϵ (τ)f ′(τ) + δ′′
+ϵ (τ)f(τ)|t
+0 .
+(D1)
+Since δ′′
+ϵ (t) = 0, only the boundary term −δ′′
+ϵ (0)f(0) survives. Further,
+−
+� t
+0
+dτδ′′
+ϵ (τ)f ′(τ) =
+� t
+0
+dτδ′
+ϵ(τ)f ′′(τ) − δ′
+ϵ(τ) f ′(τ)|t
+0 .
+(D2)
+Since δ′
+ϵ(t) = δ′
+ϵ(0) = 0 (δ′
+ϵ(τ) being an odd function of τ), both the boundary terms vanish. Proceeding further we
+get
+� t
+0
+dτδ′
+ϵ(τ)f ′′(τ) = −
+� t
+0
+dτδϵ(τ)f ′′′(τ) + δϵ(τ) f ′′(τ)|t
+0 .
+(D3)
+As before, the boundary term at τ = t is zero and only the term −δϵ(0)f ′′(0) survives. Finally, since δϵ(τ) goes to
+zero much faster than a generic function f(τ) for a small ϵ, it can be treated like a Dirac delta such that
+−
+� t
+0
+dτδϵ(τ)f ′′′(τ) = −f ′′′(0)
+2
+.
+(D4)
+The factor of half comes because the integral is performed from 0 to t. Collecting the two boundary terms we get the
+result
+� t
+0
+dτδ′′′
+ϵ (τ)f(τ) = −f ′′′(0)
+2
+− δϵ(0)f ′′(0) − δ′′
+ϵ (0)f(0) .
+(D5)
+From Eq. (C8) we have δϵ(0) = 1/(πϵ) = ωmax/π and δ′′
+ϵ (0) = −2ω3
+max/π such that
+� t
+0
+dτD(τ)f(τ) = −2αℏ
+3c2 f ′′′(0) − 4αℏωmax
+3πc2
+f ′′(0) + 2e2ω3
+max
+3π2ϵ0c3 f(0) .
+(D6)
+Here, we have now plugged in the constant prefactor appearing in Eq. (C9).
+
+12
+Appendix E: The Abraham-Lorentz equation as a classical limit
+The rate of change of the expectation values can be obtained with the help of the master equation (B29). For the
+position operator it is given by
+d
+dt⟨ˆx⟩ = Tr (ˆx∂tˆρr) = − i
+ℏTr
+�
+ˆx ·
+�
+ˆHs, ˆρr
+��
++ i
+2ℏ
+� t−ti
+0
+dτD(t; t − τ)Tr (ˆx · [ˆx, {ˆxHs(−τ), ˆρr(t)}])
+− 1
+ℏ
+� t−ti
+0
+dτN(t; t − τ)Tr (ˆx · [ˆx, [ˆxHs(−τ), ˆρr(t)]]) .
+(E1)
+Due to the identity
+Tr
+�
+ˆA ·
+�
+ˆB, ˆC
+��
+= Tr
+��
+ˆA, ˆB
+�
+· ˆC
+�
+,
+(E2)
+the terms involving the dissipation and the noise kernels vanish and we get
+d
+dt⟨ˆx⟩ = − i
+ℏTr
+�
+ˆρr ·
+�
+ˆx, ˆHs
+��
+= ⟨ˆp⟩
+m .
+(E3)
+Here, we remember that the system Hamiltonian ˆHs receives a contribution from ˆVEM in addition to the bare potential
+ˆV0 such that (c.f. the discussion between Eqs. (A4) and (A8))
+ˆHs(t) = ˆp2
+2m + ˆV0(x, t) + e2ω3
+max
+3π2ϵ0c3 ˆx2 .
+(E4)
+Similarly, for the momentum operator we obtain the relation
+d
+dt⟨ˆp⟩ = Tr (ˆp∂tˆρr) = − i
+ℏTr
+��
+ˆp, ˆHs
+�
+· ˆρr
+�
++ i
+2ℏ
+� t−ti
+0
+dτD(t; t − τ)Tr ([ˆp, ˆx] · {ˆxHs(−τ), ˆρr(t)})
+− 1
+ℏ
+� t−ti
+0
+dτN(t; t − τ)Tr ([ˆp, ˆx] · [ˆxHs(−τ), ˆρr(t)]) .
+(E5)
+Since [ˆx, ˆp] = iℏ1, the term involving the noise kernel vanishes and Eq. (E5) simplifies to
+d
+dt⟨ˆp⟩ = −⟨ ˆV0,x ⟩ − 2e2ω3
+max
+3π2ϵ0c3 ⟨ˆx⟩ + Tr
+�
+ˆρr(t)
+� t−ti
+0
+dτD(τ)ˆxHs(−τ)
+�
+.
+(E6)
+Evaluating the integral using Eq. (D6), we see that the last term in the integral gives the contribution 2e2ω3
+max
+3π2ϵ0c3 ⟨ˆx⟩ to
+d
+dt⟨ˆp⟩ in Eq. (E6) and cancels the contribution coming from ˆVEM. The EOM therefore reduces to
+d
+dt⟨ˆp⟩ = −⟨ ˆV0(x),x ⟩ − 2αℏ
+3c2 Tr
+�
+ˆρr(t) d3
+dτ 3 ˆxHs(−τ)
+����
+τ=0
+�
+− 4αℏωmax
+3πc2
+Tr
+�
+ˆρr(t) d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+�
+.
+(E7)
+As shown in the main article, when ˆV0(x, t) = 0, the double and the triple derivatives acting on ˆxHs(−τ) vanish upto
+second order in the interactions. Here, we only focus on the general case in which the external (time-dependent)
+potential is switched on. To simplify the equation further, we begin by evaluating the second order derivative in
+Eq. (E7). From Eq. (B24) we have
+d2
+dτ 2 ˆxHs(−τ) = ˆU −1
+s
+(t − τ; t)ˆx ˆU ′′
+s (t − τ; t) + 2 ˆU −1′
+s
+(t − τ; t)ˆx ˆU ′
+s(t − τ; t) + ˆU −1′′
+s
+(t − τ; t)ˆx ˆUs(t − τ; t) ,
+(E8)
+where the prime denotes the derivative with respect to τ. From the Schr¨odinger equation
+ˆU ′
+s(t − τ; t) = i
+ℏ
+ˆHs(t − τ) ˆUs(t − τ; t) ,
+(E9)
+the derivatives acting on the unitary operator can be expressed in terms of the Hamiltonian. It is clear that taking
+higher derivatives of ˆUs(t − τ; t) would result in higher powers of the Hamiltonian or the partial derivative of the
+Hamiltonian with respect to τ, multiplied with only a single unitary operator on the very right. However, if in the
+
+13
+end τ is set to zero, the Hamiltonian and its explicit time derivatives will be evaluated at time t, and the unitary
+operator on the very right disappears since ˆUs(t; t) = 1. We therefore have the following identities
+ˆU (′n)
+s
+(t − τ; t)
+���
+τ=0 = (−1)n
+� dn
+dtn ˆUs(t; ti)
+�
+ˆU −1
+s
+(t; ti) ,
+(E10)
+ˆU −1(′n)
+s
+(t − τ; t)
+���
+τ=0 = (−1)n ˆUs(t; ti)
+� dn
+dtn ˆU −1
+s
+(t; ti)
+�
+.
+(E11)
+The additional time parameter ti that appears in Eqs. (E10) and (E11) is only apparent.
+As discussed before,
+evaluating the time derivatives on the right hand side of Eq. (E10) would result in powers of ˆHs(t) and its derivatives
+evaluated at t. The remaining unitary matrix ˆUs(t; ti) would be canceled by the additional ˆU −1
+s
+(t; ti) on the very
+right such that ti disappears from the equation. Using Eqs. (E10) and (E11) in Eq. (E8) we get
+Tr
+�
+ˆρr(t) d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+�
+= Tr
+��� d2
+dt2 ˆUs(t; ti)
+�
+ˆU −1
+s
+(t; ti)ˆρr(t)
++2
+�
+− d
+dt
+ˆUs(t; ti)
+�
+ˆU −1
+s
+(t; ti)ˆρr(t) ˆUs(t; ti)
+�
+− d
+dt
+ˆU −1
+s
+(t; ti)
+�
++ˆρr(t) ˆUs(t; ti)
+� d2
+dt2 ˆU −1
+s
+(t; ti)
+��
+ˆx
+�
+.
+(E12)
+Here, we have used the cyclic property within the trace to shift the unitary operators ˆUs and its derivatives on the
+right of ˆx in Eq. (E8) onto the very left within the trace. To proceed further we note that the terms involving the
+trace in Eq. (E7) are multiplied by α. It is therefore sufficient to evaluate the trace at 0th order in the interactions as
+the master equation is valid only upto second order in the interactions. This implies that within the trace the time
+dependence of the density matrix can be evaluated by keeping only the Liouville-von Neuman term such that
+ˆρr(t) = ˆUs(t; ti)ˆρr(ti) ˆU −1
+s
+(t; ti) .
+(E13)
+Eq. (E12) then simplifies to
+Tr
+�
+ˆρr(t) d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+�
+= Tr
+��� d2
+dt2 ˆUs(t; ti)
+�
+ˆρr(ti) ˆU −1
+s
+(t; ti) + 2
+� d
+dt
+ˆUs(t; ti)
+�
+ˆρr(ti)
+� d
+dt
+ˆU −1
+s
+(t; ti)
+�
++ ˆUs(t; ti)ˆρr(ti)
+� d2
+dt2 ˆU −1
+s
+(t; ti)
+��
+ˆx
+�
+= Tr
+� d2
+dt2 ˆρr(t)ˆx
+�
+.
+(E14)
+Thus, we have the relation
+Tr
+�
+ˆρr(t) d2
+dτ 2 ˆxHs(−τ)
+����
+τ=0
+�
+= Tr
+� d2
+dt2 ˆρr(t)ˆx
+�
+= d2
+dt2 ⟨ˆx⟩ .
+(E15)
+Similar line of reasoning also leads to the identity
+Tr
+�
+ˆρr(t) d3
+dτ 3 ˆxHs(−τ)
+����
+τ=0
+�
+= −Tr
+� d3
+dt3 ˆρr(t)ˆx
+�
+= − d3
+dt3 ⟨ˆx⟩ .
+(E16)
+Using Eqs. (E15) and (E16) in Eq. (E7), the EOM for the expectation value of the position operator in the presence
+of an external potential is obtained to be
+mR
+d2
+dt2 ⟨ˆx⟩ = −⟨ ˆV0(x),x ⟩ + 2αℏ
+3c2
+d3
+dt3 ⟨ˆx⟩ ,
+where
+mR := m + 4αℏωmax
+3πc2
+.
+(E17)
+[1] H. B. G. Casimir, Indag. Math. 10, 261 (1948).
+
+14
+[2] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics
+(Cambridge Univ. Press, Cambridge, UK, 1984).
+[3] L. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge Mono-
+graphs on Mathematical Physics (Cambridge University Press, 2009).
+[4] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
+[5] S. A. Fulling, Phys. Rev. D 7, 2850 (1973).
+[6] S. Takagi, Prog. Theor. Phys. Suppl. 88, 1 (1986).
+[7] H. A. Bethe, Phys. Rev. 72, 339 (1947).
+[8] W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241 (1947).
+[9] T. A. Welton, Phys. Rev. 74, 1157 (1948).
+[10] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji, Journal de Physique 43, 1617 (1982).
+[11] E.
+Joos,
+H.
+Zeh,
+D.
+Giulini,
+C.
+Kiefer,
+J.
+Kupsch,
+and
+I.
+Stamatescu,
+Decoherence and the Appearance of a Classical World in Quantum Theory,
+Physics
+and
+astronomy
+online
+library
+(Springer, 2003).
+[12] C. Kiefer, Phys. Rev. D 46, 1658 (1992).
+[13] L. H. Ford, Phys. Rev. D 47, 5571 (1993).
+[14] G. Baym and T. Ozawa, Proceedings of the National Academy of Sciences 106, 3035 (2009).
+[15] E. Santos, Physics Letters A 188, 198 (1994).
+[16] L. Di´osi, Physics Letters A 197, 183 (1995).
+[17] P. M. V. B. Barone and A. O. Caldeira, Phys. Rev. A 43, 57 (1991).
+[18] H.-P. Breuer and F. Petruccione, in Relativistic Quantum Measurement and Decoherence, edited by H.-P. Breuer and
+F. Petruccione (Springer Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 31–65.
+[19] S. Coleman, “Classical electron theory from a modern standpoint,” in Electromagnetism: Paths to Research, edited by
+D. Teplitz (Springer US, Boston, MA, 1982) pp. 183–210.
+[20] P. Pearle, “Classical electron models,” in Electromagnetism: Paths to Research, edited by D. Teplitz (Springer US, Boston,
+MA, 1982) pp. 211–295.
+[21] D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Cambridge University Press, 2017).
+[22] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, “Classical electrodynamics: The fundamental equations and the
+dynamical variables,” in Photons and Atoms (John Wiley and Sons, Ltd, 1997) Chap. 1, pp. 5–77.
+[23] E. A. Calzetta and B.-L. B. Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs on Mathematical Physics
+(Cambridge University Press, 2008).
+[24] “Lagrangian and hamiltonian approach to electrodynamics, the standard lagrangian and the coulomb gauge,” in
+Photons and Atoms (John Wiley & Sons, Ltd, 1997) Chap. 2, pp. 79–168.
+[25] A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010).
+[26] C. Cohen-Tannoudji, J. Dupont-Roc,
+and G. Grynberg, “Quantum electrodynamics in the coulomb gauge,” in
+Photons and Atoms (John Wiley & Sons, Ltd, 1997) Chap. 3, pp. 169–252.
+[27] R. Feynman and F. Vernon, Annals of Physics 24, 118 (1963).
+[28] D. J. Griffiths, T. C. Proctor, and D. F. Schroeter, American Journal of Physics 78, 391 (2010).
+[29] M. A. Schlosshauer, Decoherence and the Quantum-To-Classical Transition (Springer-Verlag Berlin Heidelberg, 2007).
+[30] W. G. Unruh, in Relativistic Quantum Measurement and Decoherence, edited by H.-P. Breuer and F. Petruccione (Springer
+Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 125–140.
+

4tE1T4oBgHgl3EQf6QWC/content/tmp_files/2301.03521v1.pdf.txt

@@ -0,0 +1,1047 @@
+arXiv:2301.03521v1  [math.SP]  9 Jan 2023
+GREEN’S FUNCTIONS FOR FIRST-ORDER SYSTEMS OF
+ORDINARY DIFFERENTIAL EQUATIONS WITHOUT THE
+UNIQUE CONTINUATION PROPERTY
+STEVEN REDOLFI AND RUDI WEIKARD
+Abstract. This paper is a contribution to the spectral theory associated with
+the differential equation Ju′ + qu = wf on the real interval (a, b) when J is
+a constant, invertible skew-Hermitian matrix and q and w are matrices whose
+entries are distributions of order zero with q Hermitian and w non-negative.
+Under these hypotheses it may not be possible to uniquely continue a solution
+from one point to another, thus blunting the standard tools of spectral theory.
+Despite this fact we are able to describe symmetric restrictions of the maximal
+relation associated with Ju′ + qu = wf and show the existence of Green’s
+functions for self-adjoint relations even if unique continuation of solutions fails.
+1. Introduction
+This paper is a contribution to the spectral theory for the differential equation
+Ju′ + qu = wf
+posed on the real interval (a, b) when J is a constant, invertible, and skew-Hermitian
+n × n-matrix while the entries of the matrices q and w are distributions of order
+zero1 with q Hermitian and w non-negative. Ghatasheh and Weikard [7] studied this
+equation under the additional hypothesis that initial value problems have unique
+balanced2 solutions in the space of functions of locally bounded variation.
+The equation Ju′ + qu = wf has, of course, been investigated by many people
+when the coefficients q and w are locally integrable. In that situation initial value
+problems always have unique solutions. This is not necessarily the case when the
+measures induced by q or w have discrete components. It appears that an equation
+with measure coefficients was first considered in 1952, when Krein [8] modelled a
+vibrating string. In 1964 Atkinson [2] suggested to unify the treatment of differen-
+tial and difference equations by writing them as systems of integral equation where
+integrals were to be viewed as matrix-valued Riemann-Stieltjes integrals. Atkinson
+explained that the presence of point masses may prevent the continuation of so-
+lutions across such points and posed a condition avoiding that problem but more
+Date: 11. May 2022.
+This is a preprint of an article published in Integral Equations and Operator Theory which is
+available online at https://doi.org/10.1007/s00020-022-02703-6.
+©2022.
+This manuscript version is made available under the CC-BY-NC-ND 4.0 license
+http://creativecommons.org/licenses/by-nc-nd/4.0/.
+1Recall that distributions of order 0 are distributional derivatives of functions of locally
+bounded variation and hence may be thought of, on compact subintervals of (a, b), as measures.
+For simplicity we might use the word measure instead of distribution of order 0 below.
+2A function of locally bounded variation is called balanced, if its values at any given point are
+averages of its left- and right-hand limits at that point.
+1
+
+2
+STEVEN REDOLFI AND RUDI WEIKARD
+restrictive than the one posed in [7]. In 1999 Savchuk and Shkalikov [10] treated
+Schr¨odinger equations with potentials in the Sobolev space W −1,2
+loc
+. Their paper was
+very influential and spurred many further developments. Nevertheless, Eckhardt
+et al. [5] showed in 2013, with the help of quasi-derivatives or, equivalently, by
+writing the equation as a system, that a treatment without leaving the realm of
+locally integrable coefficients is possible. In the same year Eckhardt and Teschl [6]
+investigated 2×2-systems with diagonal measure-valued matrices q and w requiring
+essentially Atkinson’s condition.
+A more thorough account of the subject’s history is given in [7]. The papers
+[5] and [6], mentioned above, may also serve as excellent sources, with perhaps
+different emphases, of this history.
+One feature of systems of first-order equations is that, generally, they are repre-
+sented by linear relations rather than linear operators. There is a well-developed
+spectral theory for linear relations initiated by Arens [1], see also Orcutt [9], and
+Bennewitz [3]. The most important results (for our purposes) are also surveyed in
+Appendix B of [7].
+Existence or uniqueness of solutions of an initial value problem for Ju′+qu = wf
+fails when, for some x ∈ (a, b), the matrices
+B±(x, 0) = J ± 1
+2∆q(x)
+are not invertible. Here ∆q(x) = Q+(x)−Q−(x) when Q denotes an anti-derivative
+of q. Equivalently, ∆q(x) = dQ({x}) where dQ is the measure (locally) generated
+by q. Assuming the unique continuation property for solutions of Ju′ + qu = wf
+Ghatasheh and Weikard defined maximal and minimal relations Tmax and Tmin
+associated with the differential equation Ju′ + qu = wf and showed that Tmax
+is the adjoint of Tmin. They characterized the self-adjoint restrictions of Tmax, if
+any, with the aid of boundary conditions and proved that resolvents are given as
+integral operators, i.e., the existence of a Green’s function for any such self-adjoint
+relation T . Under even more restrictive conditions they also showed the existence
+of a Fourier transform diagonalizing T .
+Campbell, Nguyen, and Weikard [4] defined maximal and minimal relations and
+showed that Tmax = T ∗
+min without the hypothesis of unique continuation of so-
+lutions. Our goal here is to advance their ideas. In particular, even though the
+equation Ju′ + qu = w(λu + f) may have infinitely many linearly independent
+solutions the deficiency indices, i.e., the number of linearly independent solutions
+of Ju′ + qu = ±iwu of finite positive norm, is still bounded by n, the size of the
+system. We show that symmetric restrictions of Tmax, in particular the self-adjoint
+ones, are still given by posing boundary conditions and we show that the resolvents
+of self-adjoint restrictions are integral operators by proving the existence of Green’s
+functions.
+We will not approach the problem of Fourier transforms and eigenfunction ex-
+pansions but hope to return to it in future work.
+The material in this paper is arranged as follows. In Section 2 we recall the
+circumstances under which existence and uniqueness of solutions to initial value
+problems does hold and investigate the sets of those x ∈ (a, b) and λ ∈ C giving
+rise to trouble.
+Then, in Section 3 we discuss the manifold of solutions of our
+differential equation in the special case when a and b are regular endpoints. These
+results are instrumental in Section 4 where we investigate the deficiency indices
+
+GREEN’S FUNCTIONS
+3
+of the minimal relation and its symmetric extensions but without the assumption
+that a and b are regular. Before we prove the existence of Green’s functions for
+self-adjoint restrictions of the maximal relation in Section 6 we discuss the role
+played by non-trivial solutions of zero norm in Section 5.
+Let us add a few words about notation. D′0((a, b)) is the space of distributions of
+order 0, i.e., the space of distributional derivatives of functions of locally bounded
+variation. Any function u of locally bounded variation has left- and right-hand
+limits denoted by u− and u+, respectively. Also, u is called balanced if u = u# =
+(u+ + u−)/2. The space of balanced functions of bounded variation defined on
+(a, b) is denoted by BV#((a, b)) while BV#
+loc((a, b)) stands for the space of balanced
+functions of locally bounded variation.
+We use
+1 to denote an identity matrix
+of appropriate size and superscripts ⊤ and ∗ indicate transposition and adjoint,
+respectively. The sum of two closed only trivially intersecting subspaces S and T of
+some Hilbert space (i.e., their direct sum) is denoted by S ⊎ T ; if S and T are even
+orthogonal we may use ⊕ instead of ⊎. The orthogonal complement of a subspace
+S of a Hilbert space H is denoted by H ⊖ S or by S⊥. For c1, ..., cN ∈ Cn we
+abbreviate the column vector (c⊤
+1 , ..., c⊤
+N)⊤ ∈ CnN by (c1, ..., cN)⋄.
+2. Preliminaries
+Throughout this paper we assume the following hypothesis to be in force.
+Hypothesis 2.1. J is a constant, invertible and skew-Hermitian n × n-matrix.
+Both q and w are in D′0((a, b))n×n, w is non-negative and q Hermitian.
+Given that w is non-negative it gives rise to a positive measure on (a, b) and we
+denote the space of functions f which satisfy
+�
+f ∗wf < ∞ by L2(w). This space
+permits the semi-inner product ⟨f, g⟩ =
+�
+f ∗wg (note that ⟨f, f⟩ may be 0 without
+f being 0).
+Consider the differential equation
+Ju′ + (q − λw)u = wf
+(2.1)
+where λ is a complex parameter and f an element of L2(w). The latter condition
+guarantees that wf is in D′0((a, b))n. We will search for solutions in BV#
+loc((a, b))n.
+In this case each term in (2.1) is a distribution of order 0 so that it makes sense to
+pose the equation.
+The point a is called a regular endpoint for Ju′ + qu = wf, if there is a point
+c ∈ (a, b) such that the left-continuous anti-derivatives Q and W of q and w are
+of bounded variation on (a, c). In this case q and w may be thought of as finite
+measures on (a, c). Similarly, b is called regular, if Q and W are of bounded variation
+on (c, b). If an endpoint is not regular, it is called singular. Not surprisingly, the
+study of our problem is less complicated when the endpoints are regular and we
+will use this fact to our advantage.
+Despite our earlier denigration of the existence and uniqueness theorem of so-
+lutions of initial value problems it continues to play a crucial role. The following
+theorem was proved in [7].
+Theorem 2.2. Suppose r ∈ D′0((a, b))n×n, g ∈ D′0((a, b))n and that the matrices
+1 ± ∆r(x)/2 are invertible for all x ∈ (a, b). Let x0 be a point in (a, b). Then the
+initial value problem u′ = ru + g, u(x0) = u0 ∈ Cn has a unique balanced solution
+u ∈ BV#
+loc((a, b))n.
+
+4
+STEVEN REDOLFI AND RUDI WEIKARD
+If a is a regular endpoint we may pose an initial condition (for u+) at a. Simi-
+larly, if b is regular we may prescribe u−(b) as the initial condition.
+Suppose now that u is a solution of (2.1). Treating either side of this equation
+as a measure (restricted to a compact subset of (a, b)) evaluation at a singleton {x}
+shows that
+J(u+(x) − u−(x)) + ∆q−λw(x)u#(x) = ∆w(x)f(x)
+or, equivalently,
+B+(x, λ)u+(x) − B−(x, λ)u−(x) = ∆w(x)f(x)
+(2.2)
+when we define
+B±(x, λ) = J ± 1
+2
+�
+∆q(x) − λ∆w(x)
+�
+.
+Note that, if B+(x, λ) is not invertible, we could be in one of the following two
+situations: (i) a solution given on (a, x) may fail to exist on (x, b) or (ii) there
+are infinitely many ways to continue a solution on (a, x) to (x, b). An analogous
+statement holds, of course, if B−(x, λ) is not invertible.
+Let us now investigate the circumstances when a pair (x, λ) gives such trouble.
+Define the sets Λx = {λ ∈ C : det(B+(x, λ)) det(B−(x, λ)) = 0} and Ξλ = {x ∈
+(a, b) : det(B+(x, λ)) det(B−(x, λ)) = 0}. First note, since B−(x, λ) = −B+(x, λ)∗,
+we have that Ξλ = Ξλ and that each Λx is symmetric with respect to the real axis.
+Also, Λx is empty unless at least one of ∆q(x) and ∆w(x) is different from 0 and
+hence for all but countably many x. Next, we claim that Λx is finite as soon as it
+misses one point. To see this suppose that B+(x, λ0) is invertible and that λ ̸= λ0.
+Since
+B+(x, λ) = (λ0 − λ)B+(x, λ0)
+�1
+2B+(x, λ0)−1∆w(x) − 1/(λ − λ0)
+�
+we see that B+(x, λ) fails to be invertible only if 1/(λ−λ0) is an eigenvalue of some
+n × n-matrix. A similar statement holds, of course, for B− proving our claim.
+The really bad points x, namely those where Λx = C, are thus contained in Ξ0.
+Here we wish to remove the hypothesis Ξ0 = ∅ posed in [7]. On any subinterval of
+(a, b) on which q gives rise to a finite measure we find that �∞
+k=1 ∥∆q(xk)∥ must be
+finite, when k �→ xk is a sequence of distinct points in that interval. It follows now
+that Ξ0 is a discrete set. One shows similarly that, for any fixed complex number
+λ the set Ξλ is discrete.
+Lemma 2.3. Suppose [s, t] ⊂ (a, b) and (s, t)∩Ξ0 = ∅. Then we have that Λ(s,t) =
+�
+x∈(s,t) Λx is a discrete subset of C.
+Proof. There are only finitely many points x in (s, t) where ∥J−1∆q(x)∥ > 1. Using
+a Neumann series one sees that only at such points the norm of B+(x, 0)−1 can be
+larger than 2∥J−1∥. Thus there is a positive number C such that ∥B+(x, 0)−1∥ ≤ C
+for all x ∈ (s, t). Now suppose that B+(x, λ) is not invertible and that |λ| ≤ R.
+Then 1/λ is an eigenvalue of 1
+2B+(x, 0)−1∆w(x). This requires that ∥∆w(x)∥ ≥
+2/(RC) and thus can happen only for finitely many x ∈ (s, t). Since similar argu-
+ments work for B− the number of points in �
+x∈(s,t) Λx which lie in a disk of radius
+R centered at 0 must be finite.
+□
+We remark that, when one of the anti-derivatives of q and w is only locally of
+bounded variation, the set �
+x∈(a,b) Λx need not be discrete even if every Λx is finite.
+
+GREEN’S FUNCTIONS
+5
+Theorem 2.4. Suppose [s, t] ⊂ (a, b) and (s, t) ∩ Ξ0 = ∅. If u0 ∈ Cn and λ ∈
+C \ Λ(s,t), then the initial value problem Ju′ + qu = λwu, u+(s) = u0 has a unique
+balanced solution in (s, t). Moreover, u(x, ·) for x ∈ (s, t) as well as u−(t, ·) are
+analytic in C \ Λ(s,t) and meromorphic on C. An analogous statement holds when
+the initial condition is posed at t.
+Proof. The first claim is simply a consequence of Theorem 2.2. When x ∈ (s, t)
+the analyticity of u(x, ·) in C \ Λ(s,t), which is an open set, was proved in Section
+2.3 of [7]. If we modify q and w by setting them 0 on [t, b) we do not change the
+solution on (s, t). The solution for the modified problem evaluated at t is analytic
+and coincides with u−(t, ·) proving its analyticity. It remains to show that a point
+λ0 ∈ Λ(s,t) can merely give rise to poles.
+We know already that there are only finitely many points x in (s, t) where one of
+B±(x, λ0) fails to be invertible. Suppose x′ and x′′ are two consecutive such points.
+If we know the solution on (s, x′) and that u−(x′, ·) has, at worst, a pole at λ0,
+then the solution in (x′, x′′) is determined by the initial value
+u+(x′, λ) = B+(x′, λ)−1B−(x′, λ)u−(x′, λ)
+which also has, at worst, a pole at λ0 since this is true for B+(x′, λ)−1. For x ∈ (s, t)
+the claim follows now by induction. To prove that u−(t, ·) is also meromorphic we
+proceed as before and modify q and w on [t, b).
+□
+3. Solving the differential equation
+Our goal in this section is to investigate the set of solutions of the differential
+equation Ju′ + (q − λw)u = wf on (a, b) under a strengthened hypothesis.
+Hypothesis 3.1. In addition to Hypothesis 2.1 we ask that a and b are regular
+endpoints for Ju′ + qu = wf.
+Moreover, given the partition
+a = x0 < x1 < x2 < ... < xN < xN+1 = b
+(3.1)
+of (a, b) we require that Ξ0 ⊂ {x1, ..., xN}. We then consider only λ for which both
+B+(x, λ) and B−(x, λ) are invertible unless x is in {x1, ..., xN}.
+This hypothesis is in force throughout this section but later only if explicitly
+mentioned. We emphasize that Ξ0 is finite when a and b are regular. Also, the set
+of permissible λ, which we call Ω0, is symmetric with respect to the real axis and
+avoids only a discrete set.
+On each interval (xj, xj+1) we let Uj(·, λ) be a fundamental matrix of balanced
+solutions of the homogeneous differential equation Ju′ + (q − λw)u = 0 such that
+limx↓xj Uj(x, λ) =
+1. The existence of these fundamental matrices is guaranteed by
+Theorem 2.2. The general balanced solution u of the non-homogeneous equation
+Ju′ + (q − λw)u = wf on (xj, xj+1) satisfies, according to Lemma 3.3 in [7],
+u−(x) = U −
+j (x, λ)
+�
+cj + J−1
+�
+(xj,x)
+Uj(·, λ)∗wf
+�
+for any cj ∈ Cn. Define
+Uj(xj+1, λ) =
+lim
+x↑xj+1 Uj(x, λ)
+and
+Ij(f, λ) =
+�
+(xj,xj+1)
+Uj(·, λ)∗wf.
+
+6
+STEVEN REDOLFI AND RUDI WEIKARD
+Using u+(xj) = cj and u−(xj) = Uj−1(xj, λ)(cj−1 + J−1Ij−1(f, λ)) in equation
+(2.2) gives
+(−B−(xj, λ)Uj−1(xj, λ), B+(xj, λ))
+�cj−1
+cj
+�
+= ∆w(xj)f(xj) + B−(xj, λ)Uj−1(xj, λ)J−1Ij−1(f, λ).
+We need to consider these equations for j = 1, ..., N simultaneously. This gives rise
+to the system
+B(λ)˜u = F0(f, λ)
+(3.2)
+where ˜u = (c0, ..., cN)⋄, B(λ), to be specified presently, is in CnN×n(N+1), and
+F0(f, λ) is in CnN.
+The two-diagonal block-matrix structure of B suggests the
+introduction of matrices E⊤ and E⊥, which, respectively, strip the first and last
+n components off a vector in their domain Cn(N+1). If we also define the block-
+matrices
+B(λ) = diag(B+(x1, λ), ..., B+(xN, λ)),
+U(λ) = diag(U0(x1, λ), ..., UN−1(xN, λ)),
+and J = diag(J, ..., J) and when we note that
+B(λ)∗ = diag(−B−(x1, λ), ..., −B−(xN, λ)),
+we obtain
+B(λ) = B(λ)∗U(λ)E⊥ + B(λ)E⊤.
+(3.3)
+The vector F0(f, λ) is given by
+F0(f, λ) = R(f) − B(λ)∗U(λ)J −1I(f, λ)
+with R(f) = ((∆wf)(x1), ..., (∆wf)(xN))⋄ and I(f, λ) = (I0(f, λ), ..., IN−1(f, λ))⋄.
+We now have the following theorem.
+Theorem 3.2. The differential equation Ju′ + (q − λw)u = wf has a solution u
+on (a, b) if and only if ˜u = (u+(x0), ..., u+(xN))⋄ is a solution of equation (3.2).
+In particular, in the homogeneous case, where f = 0, the space of solutions has
+dimension n(N + 1) − rk B(λ) ≥ n.
+We note that rk B(λ) = n when N = 1 so that the space of solutions of Ju′ +
+(q − λw)u = 0 is then exactly n-dimensional. For N = 2, however, consider the
+example (a, b) = R, J =
+� 0 −1
+1
+0
+�
+, q =
+� 0 2
+2 0
+�
+(δ1 − δ2), w =
+� 2 0
+0 0
+�
+(δ1 + δ2), where the
+δk are Dirac point measures concentrated on {k}. It shows that the dimension of
+the space of solutions of Ju′ + (q − λw)u = 0 may be strictly larger than n.
+Next we investigate the connection between the right-hand limits of a solution u
+of the homogeneous equation Ju′ + (q − λw)u = 0 at the points x0, ..., xN (given
+by the vector ˜u) and the vector ˆu = (u(x1), ..., u(xN))⋄. We have ˆu = D(λ)˜u where
+D(λ) = 1
+2(U(λ)E⊥ + E⊤)
+(3.4)
+is again a two-diagonal block-matrix. If N ≥ 2 we will also introduce the matrices
+Bm(λ) and Dm(λ) which are obtained by deleting the first and last n columns from
+B(λ) and D(λ), respectively. If N = 1 we should think of Bm(λ) and Dm(λ) as
+maps from the trivial vector space to Cn. Their adjoints are the map from Cn to
+{0}. With this understanding the following results hold also for N = 1 even though
+they then involve “matrices” with no rows or columns.
+
+GREEN’S FUNCTIONS
+7
+Lemma 3.3. D(λ)∗B(λ) − B(λ)∗D(λ) = diag(−J, 0, ..., 0, J) and Dm(λ)∗B(λ) −
+Bm(λ)∗D(λ) = 0.
+Proof. This follows since U(λ)∗J U(λ) = J which, in turn, follows from Lemma 3.2
+in [7].
+□
+Lemma 3.4. The map v �→ B(λ)v, restricted to ker D(λ), is a bijection onto
+ker Dm(λ)∗. Similarly, the map v �→ D(λ)v, restricted to ker B(λ), is a bijection
+onto ker Bm(λ)∗. In particular, dim ker D(λ) = dim ker Dm(λ)∗ and dim ker B(λ) =
+dim ker Bm(λ)∗.
+Proof. The identity Dm(λ)∗B(λ)−Bm(λ)∗D(λ) = 0 shows that B(λ) maps ker D(λ)
+to ker Dm(λ)∗ as well as that D(λ) maps ker B(λ) to ker Bm(λ)∗.
+If v ∈ ker B(λ) ∩ ker D(λ) one shows that E⊥v = E⊤v = 0 using the definitions
+(3.3) and (3.4) of B and D and the fact that B(λ) − B(λ)∗ = 2J . This, of course,
+implies that v = 0 and hence the injectivity of both B(λ)|ker D(λ) and D(λ)|ker B(λ).
+Clearly, both D(λ) and Dm(λ)∗, having invertible matrices along their main
+diagonal, are of full rank.
+The rank-nullity theorem shows therefore that their
+kernels both have dimension n. This proves surjectivity of B(λ)|ker D(λ).
+Finally, assume that v ∈ ker Bm(λ)∗. Then v = D(λ)x for some x ∈ Cn(N+1)
+which implies that 0 = Bm(λ)∗D(λ)x = Dm(λ)∗B(λ)x. The first part of the proof
+shows that there is a y ∈ ker D(λ) such that B(λ)y = B(λ)x. Hence v = D(λ)(x���y)
+where x − y ∈ ker B(λ).
+□
+The following theorem establishes a connection between solutions of the differ-
+ential equation Ju′ + (q − λw)u = 0 and elements of ker Bm(λ)∗.
+Theorem 3.5. If u is a solution of Ju′ + (q − λw)u = 0 on (a, b), then ˆu =
+(u(x1), ..., u(xN))⋄ is in ker Bm(λ)∗.
+If, in addition, u+(a) = u−(b) = 0, then
+ˆu ∈ ker B(λ)∗ (a subspace of ker Bm(λ)∗).
+Conversely, if ˆu ∈ ker Bm(λ)∗, then Ju′ + (q − λw)u = 0 has a unique solution
+u on (a, b) such that (u(x1), ..., u(xN))⋄ = ˆu. If, indeed, ˆu ∈ ker B(λ)∗, we further
+have u+(a) = u−(b) = 0.
+Let us emphasize that supp u ⊂ [x1, xN] when u+(a) = u−(b) = 0.
+Proof. If u solves Ju′ + (q − λw)u = 0, then, by Theorem 3.2, ˜u ∈ ker B(λ).
+Lemma 3.4 shows then that ˆu = D(λ)˜u is in ker Bm(λ)∗. If u+(a) = u−(b) = 0,
+then Lemma 3.3 gives 0 = B(λ)∗D(λ)˜u = B(λ)∗ˆu.
+Conversely, assume that ˆu ∈ ker Bm(λ)∗ = D(λ)(ker B(λ)).
+Then there is a
+unique vector ˜u ∈ ker B(λ) such that ˆu = D(λ)˜u, which, in turn, defines a unique
+solution u of Ju′+(q−λw)u = 0 such that (u(x1), ..., u(xN))⋄ = ˆu. If ˆu ∈ ker B(λ)∗,
+then, according to Lemma 3.3, diag(−J, 0, ..., 0, J)˜u = 0 which shows that u+(a) =
+u−(b) = 0.
+□
+Given an algebraic system Ax = b we know that there exist solutions only if
+b ∈ ran A = (ker A∗)⊥. For the differential equation Ju′ + (q − λw)u = wf with
+integrable coefficients q and w the unique continuation property for the solutions
+gives rise to the variation of constants formula, which then guarantees the existence
+of solutions for any non-homogeneity f (within reason). In the present situation,
+
+8
+STEVEN REDOLFI AND RUDI WEIKARD
+however, the problem of existence raises its head and we now set out to give neces-
+sary and sufficient conditions for f guaranteeing the existence of a solution in the
+spirit of Linear Algebra.
+Lemma 3.6. If ˜v ∈ ker B(λ) and ˆv = D(λ)˜v, then
+˜v∗E∗
+⊥ = −ˆv∗B(λ)∗U(λ)J −1
+and
+˜v∗E∗
+⊤ = ˆv∗B(λ)J −1.
+Moreover, if f ∈ L2(w) and Jv′ + (q − λw)v = 0, then
+�
+v∗wf = ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ) = ˆv∗F0(f, λ) + ˜v∗E∗
+⊤˜I(f, λ)
+where (v(x1), ..., v(xN))⋄ = ˆv = D(λ)˜v and ˜I(f, λ) = (0, ..., 0, IN(f, λ))⋄ ∈ CnN.
+Proof. Using the definitions (3.3) and (3.4) of B and D and the identities B(λ) −
+B(λ)∗ = 2J and U(λ)∗J U(λ) = J we obtain that B(λ)˜v = 0 implies
+B(λ)D(λ)˜v = U(λ)∗−1J E⊥˜v
+and
+B(λ)∗D(λ)˜v = −J E⊤˜v.
+Taking adjoints gives the first claim since ˆv = D(λ)˜v.
+The second claim is an immediate consequence of this, since
+�
+v∗wf = ˆv∗R(f) + ˜v∗(I0(f, λ), ..., IN (f, λ))⋄
+= ˆv∗R(f) + ˜v∗E∗
+⊥I(f, λ) + ˜v∗E∗
+⊤˜I(f, λ)
+= ˆv∗R(f) − ˆv∗B(λ)∗U(λ)J −1I(f, λ) + ˆv∗B(λ)J −1˜I(f, λ)
+= ˆv∗F0(f, λ) + ˆv∗B(λ)J −1˜I(f, λ).
+□
+Theorem 3.7. The differential equation Ju′ + (q − λw)u = wf has a solution on
+(a, b) if and only if
+�
+v∗wf = 0 for every solution v of Jv′ + (q − λw)v = 0 which
+vanishes at a and b.
+Proof. By Theorem 3.2 the solution u exists if and only if the system (3.2) has a
+solution ˜u = (u+(x0), ..., u+(xN))⋄. This, in turn, happens if and only if F0(f, λ) ∈
+ran B(λ) = (ker B(λ)∗)⊥.
+By Theorem 3.5 the solutions of Jv′ +(q −λw)v = 0 which vanish at a and b are
+in one-to-one correspondence with elements of ker B(λ)∗. Since v+(xN) = 0 we have
+˜v∗E∗
+⊤˜I(f, λ) = 0 and then, from Lemma 3.6, we obtain ˆv∗F0(f, λ) =
+�
+v∗wf.
+□
+In the case of unique continuation of solutions the condition that v vanishes at a
+or b implies, of course, that v = 0. Consequently, Ju′ + (q − λw)u = wf has then a
+solution for any f ∈ L2(w). The set of all solutions is thus obtained by adding the
+general solution of Ju′ +(q −λw)u = 0 whose dimension is n(N + 1)−rkB(λ) ≥ n.
+Theorem 3.8. The differential equation Ju′ + (q − λw)u = wf has a solution on
+(a, b) which vanishes at a and b if and only if
+�
+v∗wf = 0 for every solution v of
+Jv′ + (q − λw)v = 0.
+Proof. For u to vanish at a and b it is required that u+(x0) = 0 and u+(xN) =
+−J−1IN(f, λ). The system (3.2) is therefore equivalent to
+Bm(λ)(c1, ..., cN−1)⋄ = F0(f, λ) + B(λ)J −1˜I(f, λ).
+The proof is now analogous to the one for Theorem 3.7.
+□
+
+GREEN’S FUNCTIONS
+9
+We conclude this section by “counting” the solutions of Ju′ + qu = λwu which
+are not compactly supported.
+More precisely, we will determine the dimension
+of the quotient space of all solutions of Ju′ + qu = λwu modulo the space of
+compactly supported solutions. Theorem 3.5 shows that the space of all solutions of
+Ju′+qu = λwu is in one-to-one correspondence with ker Bm(λ)∗ and that the space
+of compactly supported solutions of Ju′+qu = λwu is in one-to-one correspondence
+with ker B(λ)∗. We therefore define
+˜n(λ) = dim(ker Bm(λ)∗/ ker B(λ)∗) = dim ker Bm(λ)∗ − dim ker B(λ)∗.
+Lemma 3.9. ˜n(λ) + ˜n(λ) = 2n.
+Proof. Since rk B(λ) = rk B(λ)∗, the rank-nullity theorem implies
+dim ker B(λ) = n(N + 1) − rk B(λ)∗ = n + dim ker B(λ)∗.
+Hence, using also the analogous equation for λ,
+dim ker B(λ) − dim ker B(λ)∗ + dim ker B(λ) − dim ker B(λ)∗ = 2n.
+Lemma 3.4 gives that dim ker B(λ) = dim ker Bm(λ)∗ yielding the claim.
+□
+From Theorem 2.4 we know that the matrices Uj(xj+1, ·) are meromorphic on C
+with poles at most at points in the complement of Ω0. It follows that the entries of
+B are also meromorphic. Since the meromorphic functions on C form a field there
+is a row-echelon matrix ˜B with meromorphic entries such that B˜u = 0 has the same
+solutions as ˜B˜u = 0. Now define a set Ω as Ω0 without the set of all poles of ˜B as
+well as their complex conjugates, and the set of zeros and their conjugates of any
+of the pivots of ˜B.
+Theorem 3.10. If λ ∈ Ω, then dim ker B(λ) = dim ker B(λ) and ˜n(λ) = n.
+Proof. The construction of Ω entails that rk B(λ) = rk ˜B(λ) = rk B(λ) if λ ∈ Ω.
+Since ˜n(λ) = dim ker B(λ) − dim ker B(λ)∗ = dim ker B(λ) + n − dim ker B(λ) we
+obtain ˜n(λ) = n.
+□
+4. Symmetric restrictions of Tmax
+Given a differential equation Ju′ + qu = wf we now define associated minimal
+and maximal relations. Recall that L2(w) is the space of functions f such that
+�
+f ∗wf < ∞. First we define
+Tmax = {(u, f) ∈ L2(w) × L2(w) : u ∈ BV#
+loc((a, b))n, Ju′ + qu = wf}.
+Subsequently we will always tacitly assume that u ∈ BV#
+loc((a, b))n, when we use
+u′. Next, let
+Tmin = {(u, f) ∈ Tmax : supp u is compact in (a, b)}.
+Note that these are spaces of pairs of functions. To employ the power of functional
+analysis we need to realize these relations in Hilbert spaces. Therefore we introduce,
+as usual, the space L2(w) as the quotient of L2(w) modulo the subspace of all u ∈
+L2(w) for which ∥u∥2 =
+�
+u∗wu = 0. Denoting the equivalence class corresponding
+to u by [u] we now set
+Tmax = {([u], [f]) ∈ L2(w) × L2(w) : (u, f) ∈ Tmax}
+
+10
+STEVEN REDOLFI AND RUDI WEIKARD
+and
+Tmin = {([u], [f]) ∈ Tmax : (u, f) ∈ Tmin}.
+Here (and elsewhere) we choose brevity over precision: whenever we have a pair
+([u], [f]) in Tmax we choose u and f such that (u, f) ∈ Tmax.
+Define the vector space
+L0 = {u ∈ BV#
+loc((a, b))n : Ju′ + qu = 0 and ∥u∥ = 0}.
+In many cases this space is trivial and some authors restrict their attention to the
+case where it is; this is then called the definiteness condition. However, we will
+not do so here. Note that ∥u∥ = 0 if and only if wu is the zero distribution. The
+significance of L0 stems from the following fact. Suppose ([u], [f]) ∈ Tmax and that
+there are u, v ∈ [u] and f, g ∈ [f] such that Ju′ + qu = wf and Jv′ + qv = wg.
+Then J(u − v)′ + q(u − v) = w(f − g) = 0 as well as w(u − v) = 0, i.e., u − v ∈ L0.
+In other words, in the presence of a non-trivial space L0, the class [u] has many
+representatives of locally bounded variation satisfying the differential equation for a
+given class [f] (the choice of a representative of [f], on the other hand, is irrelevant).
+In Section 5 we will describe a procedure to choose a representative of [u] in a
+distinctive way.
+In [4] it was proved that Tmin is symmetric, indeed that T ∗
+min = Tmax. In this case
+it is well-known that von Neumann’s theorem holds. Setting Dλ = {([u], λ[u]) ∈
+Tmax} it states that
+Tmax = Tmin ⊎ Dλ ⊎ Dλ
+when Im λ ̸= 0. Moreover, when λ = ±i, these direct sums are even orthogonal. It
+is also known that the dimension of Dλ does not change as λ varies in either the
+upper or the lower half plane. The numbers n± = dim D±i are called deficiency
+indices of Tmin and we are now setting out to investigate these.
+If u is a solution of Ju′ +qu = λwu which is compactly supported then (u, λu) ∈
+Tmin and ([u], λ[u]) ∈ Tmin ∩ Dλ. If λ is not real, then Tmin ∩ Dλ is trivial and it
+follows that compactly supported solutions of Ju′ + qu = λwu do not contribute to
+the corresponding deficiency index. We now have, as a corollary of Theorem 3.10,
+that the deficiency indices of Tmin cannot be more than n if a and b are regular
+endpoints. We do not state this result separately since it is included in the next
+theorem about the general case.
+Thus, to emphasize, we allow in the following a and b to be either regular or
+singular endpoints. Let τk, k ∈ Z, be a strictly increasing sequence in (a, b) having
+a and b as its only limit points and such that all points in Ξ0 are among the
+τk. Considering now only the interval Ik = (τ−k, τk) we set xj = τ−k+j for j =
+0, ..., N + 1 = 2k. We can then introduce the objects from Section 3. To emphasize
+their dependence on k we will add a superscript (k) to those objects. We have then,
+in particular, the matrices B(k), B(k)
+m and the sets Ω(k) of permissible values of λ.
+We now define Ω = �∞
+k=1 Ω(k) and note that Ω is symmetric with respect to the
+real axis and misses only countably many values from C.
+Now fix a non-real λ ∈ Ω. If u is a solution of Ju′+qu = λwu on (a, b) we denote
+its restriction to the interval Ik by u(k). We are interested in the quotient space Xk
+of all solutions of Ju′ +qu = λwu on Ik modulo the compactly supported solutions.
+If u is a solution of Ju′+qu = λwu on Ik we denote the associated equivalence class
+in Xk by ⌊u⌋k. A compactly supported solution u of Ju′ + qu = λwu on Ik can be
+extended by 0 to all of (a, b) yielding an element in Tmin ∩ Dλ. This implies, since
+
+GREEN’S FUNCTIONS
+11
+Im λ ̸= 0, that ∥u∥2 =
+�
+Ik u∗wu = 0 and shows that Xk is a normed space with the
+norm given by ∥u∥2
+k =
+�
+Ik u∗wu. According to Theorem 3.5 the quotient space Xk
+is isomorphic to ker B(k)
+m (λ)∗/ ker B(k)(λ)∗ and, by Theorem 3.10, its dimension is
+equal to n since λ ∈ Ω ⊂ Ω(k).
+Theorem 4.1. The deficiency indices of Tmin are less than or equal to n.
+Proof. Fix a non-real λ ∈ Ω. Suppose u1, ..., um are solutions of Ju′ + qu = λwu
+such that [u1], ..., [um] are linearly independent elements of Dλ. We will show below
+that there is an interval Ip = (τ−p, τp) such that ⌊u(p)
+1 ⌋p, ..., ⌊u(p)
+m ⌋p are linearly
+independent elements of Xp. Hence m ≤ n, the dimension of Xp. Since deficiency
+indices are constant in either half-plane they cannot be larger than n.
+We will now prove the existence of Ip by induction. That is we prove that, for
+every k ∈ {1, ..., m}, there is an interval Iℓk such that the restrictions of u1, ..., uk
+to Iℓk generate linearly independent elements ⌊u(ℓk)
+1
+⌋ℓk, ..., ⌊u(ℓk)
+k
+⌋ℓk of Xℓk. Once
+this is achieved we set p = ℓm.
+Suppose k = 1 and let Iℓ1 be an interval such that ∥u(ℓ1)
+1
+∥ > 0. By what we
+argued above we know that u(ℓ1)
+1
+is not compactly supported in Iℓ1 and thus gives
+rise to a non-zero (and hence linearly independent) element of Xℓ1.
+Now suppose we had already shown our claim for some k < m. If ⌊u(ℓk)
+1
+⌋ℓk, ...,
+⌊u(ℓk)
+k+1⌋ℓk are already linearly independent as elements of Xℓk we choose ℓk+1 = ℓk
+and our induction step is complete. Otherwise, there are unique complex numbers
+α1, ..., αk such that
+∥(α1u1 + ... + αkuk + uk+1)(ℓk)∥ℓk = 0.
+However, there must be an interval Iℓk+1 ⊃ Iℓk where
+∥(α1u1 + ... + αkuk + uk+1)(ℓk+1)∥ℓk+1 > 0
+on account that [u1], ..., [uk+1] are linearly independent. It follows now that, as ele-
+ments of Xℓk+1 the vectors ⌊u(ℓk+1)
+1
+⌋ℓk+1, ..., ⌊u(ℓk+1)
+k+1
+⌋ℓk+1 are linearly independent.
+This completes our induction step also in this case.
+□
+Corollary 4.2. If a and b are regular, then n+ = n−.
+Proof. Fix a non-real λ in Ω. Since a and b are regular, the set Ξλ = Ξλ is finite.
+Thus we may assume that it is contained in Ik = (τ−k, τk) for some appropriate k.
+Then dim ker B(k)(λ) is the number of linearly independent solutions of Ju′ + qu =
+λwu. Theorem 3.10 shows that Ju′ + qu = λwu has the same number of linearly
+independent solutions. Any of these solutions has finite norm but some may have
+norm 0. Now note, that if u is a solution of Ju′+qu = λwu of norm 0, then we have
+wu = 0, so that u is also a solution of Ju′ + qu = λwu. Therefore n+ = n−.
+□
+As mentioned above, it is well-known, even in the case of relations, that von
+Neumann’s theorem E∗ = E⊕Di⊕D−i holds when E is a closed symmetric relation
+in H × H when H is a Hilbert space. In our case, when d = dim Di ⊕ D−i is finite,
+as we just showed, we can use Theorem B.5 in [7] to characterize the symmetric
+restriction of Tmax in terms of boundary conditions. We state that theorem here
+for easy reference. The operator J appearing there is defined by J (u, f) = (f, −u)
+for u, f ∈ H.
+
+12
+STEVEN REDOLFI AND RUDI WEIKARD
+Theorem 4.3. Suppose E is a closed symmetric relation in H × H with d =
+dim Di ⊕ D−i < ∞ and that m ≤ d/2 is a natural number or 0. If A : E∗ → Cd−m
+is a surjective linear operator such that E ⊂ ker A and AJ A∗ has rank d−2m then
+ker A is a closed symmetric restriction of E∗ for which the dimension of (ker A)⊖E
+is m. Conversely, every closed symmetric restriction of E∗ is the kernel of such a
+linear operator A. Finally, ker A is self-adjoint if and only if AJ A∗ = 0 (entailing
+m = d/2).
+A second ingredient for our next considerations is Lagrange’s identity (or Green’s
+formula). If (u, f) and (v, g) are in Tmax, then v∗wf and g∗wu are finite measures.
+Therefore v∗Ju′ + v′∗Ju = v∗wf − g∗wu is also a finite measure. Its antiderivative
+v∗Ju is of bounded variation and thus has limits at a and b. Integration now gives
+Lagrange’s identity
+(v∗Ju)−(b) − (v∗Ju)+(a) = ⟨v, f⟩ − ⟨g, u⟩.
+(4.1)
+Note the right-hand side, and hence the left-hand side, does not change upon choos-
+ing different representatives in place of u, f, v, or g.
+Now, if (v, g) is an element of Di⊕D−i, then (u, f) �→ ⟨(v, g), (u, f)⟩ is a bounded
+linear functional on Tmax. Conversely, since Tmax is a Hilbert space, a bounded
+linear functional on Tmax is given by (u, f) �→ ⟨(v, g), (u, f)⟩ for some (v, g) ∈ Tmax.
+When it is also known that Tmin is in the kernel of this functional, (v, g) may be
+chosen in Di ⊕ D−i. Hence, in our situation, the operator A from Theorem 4.3
+is given by d − m linearly independent elements in Di ⊕ D−i. Lagrange’s identity
+implies that the entries of the matrix AJ A∗ are then given by
+(AJ A∗)k,ℓ = ⟨(vk, gk), (gℓ, −vℓ)⟩ = (g∗
+kJgℓ)−(b) − (g∗
+kJgℓ)+(a).
+(4.2)
+Therefore we arrive at the following theorem.
+Theorem 4.4. Let d = n+ + n− and suppose that m ≤ min{n+, n−}. If (v1, g1),
+..., (vd−m, gd−m) are linearly independent elements of Di⊕D−i such that the matrix
+defined in (4.2) has rank d − 2m, then
+T = {(u, f) ∈ Tmax : (g∗
+j Ju)−(b) − (g∗
+j Ju)+(a) = 0 for j = 1, ..., d − m}
+(4.3)
+is a closed symmetric restriction of Tmax.
+Conversely, if T is a closed symmetric restriction of Tmax and m is the dimen-
+sion of T ⊖ Tmin, then T is given by (4.3) for appropriate elements (v1, g1), ...,
+(vd−m, gd−m) of Di ⊕ D−i for which the matrix defined in (4.2) has rank d − 2m.
+For self-adjoint restrictions of Tmax it is hence necessary and sufficient that n+ =
+n− = m = d−m and that (g∗
+kJgℓ)−(b)−(g∗
+kJgℓ)+(a) = 0 for all 1 ≤ k, ℓ ≤ m = d/2.
+5. The space L0
+We mentioned earlier that the class [u] does not have a unique balanced repre-
+sentative when ([u], [f]) ∈ Tmax, if the space L0 has non-trivial elements. In this
+section we describe a procedure to choose a representative in a distinctive way.
+To this end we assume, without loss of generality, that B+(τ0, 0) = B−(τ0, 0) = J
+so that solutions of our differential equations are continuous at τ0. Define N0 =
+{h(τ0) : h ∈ L0} and for each k ∈ N both Nk = {h+(τk) : h ∈ L0, supp h ⊂ [τk, b)}
+and N−k = {h−(τ−k) : h ∈ L0, supp h ⊂ (a, τ−k]}. Then, for k ∈ N0, we say that a
+function u ∈ BV#
+loc((a, b))n satisfies condition (±k), if u±(τ±k) is perpendicular to
+N±k (using always the upper sign or always the lower sign).
+
+GREEN’S FUNCTIONS
+13
+Lemma 5.1. Suppose ([u], [f]) ∈ Tmax. Then there is a unique balanced v ∈ [u]
+such that (v, f) ∈ Tmax and v satisfies condition (k) for every k ∈ Z.
+Proof. First consider uniqueness. Suppose u and v are two functions satisfying the
+given conditions. Then u − v ∈ L0 and hence (u − v)(τ0)∗t(τ0) = 0 for t = u and
+t = v. Subtract these equations to find (u−v)(τ0) = 0, and thus u = v on (τ−1, τ1).
+Moreover, h1 = (u − v)χ[τ1,b) and h−1 = (u − v)χ(a,τ−1] are in L0. Conditions (1)
+and (−1) show therefore that (u−v)+(τ1) and (u−v)−(τ−1) are also 0 which proves
+that u = v on (τ−2, τ2). Induction informs us now that u = v everywhere.
+We now turn to existence. Pick a balanced representative u ∈ [u] such that
+(u, f) ∈ Tmax. There is an element h0 ∈ L0 such that the orthogonal projection of
+u(τ0) onto N0 equals h0(τ0). Thus v0 = u − h0 satisfies (v0, f) ∈ Tmax, v0 ∈ [u],
+and condition (0).
+Next, there is an element h1 ∈ L0 with support in [τ1, b) such that the orthogonal
+projection of v+
+0 (τ1) onto N1 equals h+
+1 (τ1). We now define v1 = v0 − h1. Then
+(v1, f) ∈ Tmax, v1 ∈ [u], and v1 satisfies condition (1). Notice that v1 = v0 on
+(a, τ1) implying that v1 also satisfies condition (0).
+Proceeding recursively, we may define, for each k ∈ N, functions hk ∈ L0 sup-
+ported in [τk, b) such that vk = u−�k
+j=0 hj satisfies conditions (0), ..., (k), vk ∈ [u],
+and (vk, f) ∈ Tmax.
+Since, for a fixed x ∈ [τ0, b), only finitely many of the numbers hk(x) are different
+from 0, we find that the sequence k �→ vk converges pointwise to a function ˜v ∈ [u]
+satisfying conditions (k) for all k ∈ N0 and (˜v, f) ∈ Tmax. We can now repeat
+this process for negative integers starting from the function ˜v instead of u arriving
+eventually at a function v ∈ [u] satisfying conditions (k) for all k ∈ Z and (v, f) ∈
+Tmax.
+□
+We denote the operator which assigns the function v just constructed to a given
+element ([u], [f]) ∈ Tmax by E. If Im = (τ−m, τm) we also define Em : Tmax →
+BV#(Im)n by composing E with the restriction to the interval Im.
+Note that
+BV#(Im)n is a Banach space with the norm |||u|||m defined as the sum of the
+variation of u over Im and the norm of u(τ0).
+Theorem 5.2. The operator Em : Tmax → BV#(Im)n is bounded.
+Proof. Due to the closed graph theorem we merely have to show that Em is a
+closed operator. Thus assume that the sequence ([uj], [fj]) converges to ([u], [f]) in
+Tmax and that Em([uj], [fj]) converges to v in BV#(Im)n and hence pointwise. To
+simplify notation we assume that Em([uj], [fj])) and Em([u], [f]) are the restrictions
+of uj and u, respectively, to the interval Im. We need to show that u = v on Im.
+First note that u±
+j (τ±k) ∈ N ⊥
+±k and
+��u±
+j (τ±k) − v±(τ±k)
+�� → 0 imply that v
+satisfies conditions (±k) for each k ∈ {0, ..., m − 1}. For ℓ ∈ {−m, m − 1} and
+x ∈ (τℓ, τℓ+1) we have
+u−
+j (x) = U −
+ℓ (x)
+�
+u+
+j (τℓ) + J−1
+�
+(τℓ,x)
+U ∗
+ℓ wfj
+�
+when Uℓ denotes the fundamental matrix of Ju′ + qu = 0 on the interval (τℓ, τℓ+1)
+satisfying U +
+ℓ (τℓ) =
+1. Taking the limit as j → ∞ gives
+v−(x) = U −
+ℓ (x)
+�
+v+(τℓ) + J−1
+�
+(τℓ,x)
+U ∗
+ℓ wf
+�
+
+14
+STEVEN REDOLFI AND RUDI WEIKARD
+since the integral may be considered as a vector of scalar products which are, of
+course, continuous. The variation of constants formula shows that v is a balanced
+solution for Jv′ + qv = wf on (τℓ, τℓ+1). We also have
+J(u+
+j (τℓ) − u−
+j (τℓ)) + ∆q(τℓ)uj(τℓ) = ∆w(τℓ)fj(τℓ).
+(5.1)
+The fact that [fj] converges to [f] in L2(w) implies, on account of the positivity
+of w, that ∆w(τℓ)fj(τℓ) converges to ∆w(τℓ)f(τℓ).
+Therefore taking a limit in
+(5.1) shows, in conjunction with the previous observations, that Jv′ + qv = wf on
+the interval Im. Since u satisfies the same equation we have that u − v satisfies
+J(u − v)′ + q(u − v) = 0 on Im.
+Next we show w(u − v) = 0 on Im. Fatou’s lemma implies
+0 ≤
+�
+Im
+(u − v)∗w(u − v) ≤ lim inf
+j→∞
+�
+Im
+(u − uj)∗w(u − uj) = 0.
+It follows that w(u − v) = 0 on Im.
+Finally, a variant of Lemma 5.1 shows now that u = v.
+□
+6. Green’s function
+Now suppose that we have a self-adjoint restriction T of Tmax. The resolvent set
+of T is the set of those λ for which T − λ : dom(T ) → L2(w) is bijective, i.e.,
+̺(T ) = {λ ∈ C : ker(T − λ) = {0}, ran(T − λ) = L2(w)}
+which is an open set. We denote its complement, the spectrum of T , by σ(T ).
+Since T is self-adjoint, σ(T ) is a subset of R.
+If λ ∈ ̺(T ), then the resolvent
+Rλ = (T − λ)−1 is a bounded linear operator from L2(w) to dom(T ). We now
+define Rλ : L2(w) → BV#
+loc((a, b))n by
+Rλ[f] = E((Rλ[f], λRλ[f] + [f])).
+Thus Rλ[f] is the unique solution of Ju′ + qu = w(λu + f) in L2(w) satisfying
+condition (k) for every k ∈ Z.
+We will now show that Rλ is an integral operator. Its kernel G is called a Green’s
+function for T .
+Theorem 6.1. If T is a self-adjoint restriction of Tmax, then there exists, for given
+x ∈ (a, b) and λ ∈ ̺(T ), a matrix G(x, ·, λ) such that the columns of G(x, ·, λ)∗ are
+in L2(w) and
+(Rλ[f])(x) =
+�
+G(x, ·, λ)wf.
+(6.1)
+Proof. Fix x ∈ Im and λ ∈ ̺(T ). Consider the restriction of Rλ[f] to the interval
+Im. Since Em and Rλ are bounded operators the map [f] �→ (Rλ[f])(x) is a bounded
+linear map from L2(w) to Cn. Hence there are elements [g1], ..., [gn] ∈ L2(w) such
+that the k-th component of (Rλ[f])(x) equals ⟨[gk], [f]⟩. Let these be the columns
+of the matrix-valued function G(x, ·, λ)∗. Then we obtain (6.1).
+□
+One wishes to complement this fairly abstract existence result by a more concrete
+one where Green’s function is given in terms of solutions of the differential equation
+as is done in the classical case, see, for instance, Zettl [11]. This was also achieved
+in [7] under the assumption that Ξ0 is empty and minor generalizations of this
+are certainly possible. Such an explicit construction of Green’s function, where
+possible, is the cornerstone of many other results in spectral theory, in particular
+
+GREEN’S FUNCTIONS
+15
+the development of a spectral transformation and more detailed information about
+the resolvent, e.g., the compactness of the resolvent in the regular case. Due to
+the difficulties posed by the absence of an existence and uniqueness theorem for
+initial value problems we have, so far, not been able to obtain such a construction
+in general. However, we hope to return to this issue in the future.
+7. Example
+In this section we treat an example where the matrices B±(x, λ) fail to be invert-
+ible for infinitely many x and all λ, in other words where Ξ0 is infinite and Λx = C
+for all x ∈ Ξ0 (recall that in [7] the hypothesis Ξ0 = ∅ was made causing each Λx
+to be finite). The example is Ju′ + qu = wf on (a, b) = R where
+J =
+�
+0
+−1
+1
+0
+�
+, q =
+�
+0
+2
+2
+0
+� �
+k∈Z
+(δ2k − δ2k+1), and, w =
+�
+2
+0
+0
+0
+� �
+k∈Z
+δk
+with δk denoting the Dirac point measure concentrated on {k}. Since we are seeking
+balanced solutions we need the matrices
+B−(2k − 1, λ) =
+�λ
+0
+2
+0
+�
+and
+B+(2k − 1, λ) =
+�−λ
+−2
+0
+0
+�
+as well as
+B−(2k, λ) =
+�
+λ
+−2
+0
+0
+�
+and
+B+(2k, λ) =
+�
+−λ
+0
+2
+0
+�
+.
+If x is not an integer we have B±(x, λ) = J. Note that f ∈ L2(w) if and only if
+k �→ f1(k) is in ℓ2(Z) and any element in L2(w) is uniquely determined by these
+values (here f1 denotes the first component of f).
+In any interval (k, k + 1) solutions of Ju′ + qu = w(λu + f) are constant, say
+(αk, βk)⊤. At x = 2k − 1 the equation
+B+(2k − 1, λ)u+(2k − 1) − B−(2k − 1, λ)u−(2k − 1) = (2f1(2k − 1), 0)⊤
+implies α2k−2 = 0 and
+− λα2k−1 − 2β2k−1 = 2f1(2k − 1).
+(7.1)
+Similarly, at x = 2k we get α2k = 0 and
+− λα2k−1 + 2β2k−1 = 2f1(2k).
+(7.2)
+We can now describe the space Tmax. A pair (u, f) is in Tmax if and only if the
+sequences k �→ f1(k) and k �→ u1(k) are in ℓ2(Z), f1(2k) = −f1(2k − 1), u1(2k) =
+u1(2k − 1), and
+u =
+�
+k∈Z
+� �2u1(2k)
+f1(2k)
+�
+χ#
+(2k−1,2k) +
+� 0
+β2k
+�
+χ#
+(2k,2k+1)
+�
+with arbitrary numbers β2k. Note that ∥u∥2 = 4 �
+k∈Z |u1(2k)|2.
+Choosing here f = 0 shows that 0 is an eigenvalue of Tmax with infinite multi-
+plicity. Choosing f = 0 and requiring ∥u∥ = 0 determines the space L0. Indeed,
+L0 =
+� �
+k∈Z
+�
+0
+β2k
+�
+χ#
+(2k,2k+1) : β2k ∈ C
+�
+
+16
+STEVEN REDOLFI AND RUDI WEIKARD
+which is infinite-dimensional. We now define the sequence τ setting τ0 = 1/2 and,
+for k ∈ N, τk = k and τ−k = 1 − k. A solution u of Ju′ + qu = w(λu + f) always
+satisfies condition (2k + 1) and it satisfies condition (2k) exactly when β2k = 0.
+For f = 0 equations (7.1) and (7.2) show that no non-zero λ can be an eigenvalue
+of Tmax. In particular, the deficiency indices n± are 0, i.e., Tmax is self-adjoint. Now
+choose λ ̸= 0 and f arbitrary in L2(w). Then
+(Rλf)(x) = − 1
+2λ
+�
+k∈Z
+�2f1(2k − 1) + 2f1(2k)
+λf1(2k − 1) − λf1(2k)
+�
+χ#
+(2k−1,2k)(x)
+(7.3)
+is the unique solution of Ju′ + qu = w(λu + f) satisfying condition (k) for any
+k ∈ Z. Since
+∥Rλf∥2 =
+�
+k∈Z
+2|(Rλf)1(k)|2 =
+1
+|λ|2
+�
+k∈Z
+|f1(2k − 1) + f1(2k)|2
+(7.4)
+is finite we have that C \ {0} is the resolvent set of Tmax.
+We now define H = {u ∈ L2(w) : u1(2k − 1) = u1(2k)} and H∞ = {f ∈ L2(w) :
+f1(2k − 1) = −f1(2k)}. These spaces are orthogonal to each other and their direct
+sum is L2(w). Equation (7.4) shows that ker Rλ = H∞. Moreover, we have
+Tmax = (H × {0}) ⊕ ({0} × H∞).
+This is an instance of a general feature for a self-adjoint linear relation T : if H is
+the closure of the domain of T , H∞ the orthogonal complement of H, and T0 =
+T ∩ (H × H), then T = T0 ⊕ ({0} × H∞). The former summand is then a linear
+operator densely defined in H called the operator part of T . The latter summand
+is called the multi-valued part of T .
+We end this example by identifying Green’s function for our example. It may
+be guessed by looking at equation (7.3). In any case one can check directly that
+(Rλf)(x) =
+�
+G(x, ·, λ)wf. Note that the second column of G is irrelevant since
+the second row of w is 0. When x is not integer G(x, y, λ) is given by
+�
+k∈Z
+�
+− 1
+λ
+�1
+0
+0
+0
+�
++ 1
+2
+� 0
+1
+−1
+0
+�
+sgn(x − y)
+�
+χ#
+(2k−1,2k)(x)χ#
+(2k−1,2k)(y).
+If x is an integer we have instead
+G(2k − 1, y, λ) = 1
+2
+lim
+x↓2k−1 G(x, y, λ)
+and
+G(2k, y, λ) = 1
+2 lim
+x↑2k G(x, y, λ).
+References
+[1] Richard Arens. Operational calculus of linear relations. Pacific J. Math., 11:9–23, 1961.
+[2] F. V. Atkinson. Discrete and continuous boundary problems. Mathematics in Science and
+Engineering, Vol. 8. Academic Press, New York-London, 1964.
+[3] Christer Bennewitz. Symmetric relations on a Hilbert space. Pages 212–218. Lecture Notes
+in Math., Vol. 280, 1972.
+[4] Kevin Campbell, Minh Nguyen, and Rudi Weikard. On the spectral theory for first-order
+systems without the unique continuation property. Linear Multilinear Algebra, 69(12):2315–
+2323, 2021. Published online: 04 Oct 2019.
+[5] Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl. Weyl-Titchmarsh the-
+ory for Sturm-Liouville operators with distributional potentials. Opuscula Math., 33(3):467–
+563, 2013.
+[6] Jonathan Eckhardt and Gerald Teschl. Sturm-Liouville operators with measure-valued coef-
+ficients. J. Anal. Math., 120:151–224, 2013.
+
+GREEN’S FUNCTIONS
+17
+[7] Ahmed Ghatasheh and Rudi Weikard. Spectral theory for systems of ordinary differential
+equations with distributional coefficients. J. Differential Equations, 268(6):2752–2801, 2020.
+[8] M. G. Kre˘ın. On a generalization of investigations of Stieltjes. Doklady Akad. Nauk SSSR
+(N.S.), 87:881–884, 1952.
+[9] Bruce Call Orcutt. Canonical differential equations. PhD thesis, University of Virginia, 1969.
+[10] A. M. Savchuk and A. A. Shkalikov. Sturm-Liouville operators with singular potentials. Math-
+ematical Notes, 66(6):741–753, 1999. Translated from Mat. Zametki, Vol. 66, pp. 897–912
+(1999).
+[11] Anton Zettl. Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs.
+American Mathematical Society, Providence, RI, 2005.
+Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL
+35226-1170, USA
+Email address: [email protected], [email protected]
+

7dE4T4oBgHgl3EQf2Q2c/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:be8a5013faccdd81061f3c8f73f926d2610b93bff75439220031b3032a7c9b77
+size 111673

89FST4oBgHgl3EQfajh_/content/tmp_files/2301.13796v1.pdf.txt

@@ -0,0 +1,1672 @@
+1
+Scalable Grid-Aware Dynamic Matching using
+Deep Reinforcement Learning
+Majid Majidi Student Member, IEEE, Deepan Muthirayan Member, IEEE, Masood Parvania Senior Member,
+IEEE, Pramod P. Khargonekar Fellow, IEEE
+Abstract—This paper proposes a two-level hierarchical match-
+ing framework for Integrated Hybrid Resources (IHRs) with grid
+constraints. An IHR is a collection of Renewable Energy Sources
+(RES) and flexible customers within a certain power system zone,
+endowed with an agent to match. The key idea is to pick the
+IHR zones so that the power loss effects within the IHRs can
+be neglected. This simplifies the overall matching problem into
+independent IHR-level matching problems, and an upper-level
+optimal power flow problem to meet the IHR-level upstream
+flow requirements while respecting the grid constraints. Within
+each IHR, the agent employs a scalable Deep Reinforcement
+Learning algorithm to identify matching solutions such that
+the customer’s service constraints are met. The central agent
+then solves an optimal power flow problem with the IHRs as
+the nodes, with their active power flow and reactive power
+limits, and grid constraints to determine the final flows such
+that matched power can be delivered to the extent the grid
+constraints are satisfied. The proposed framework is implemented
+on a test power distribution system, and multiple case studies are
+presented to substantiate the welfare efficiency of the proposed
+solution and the satisfaction of the grid and customers’ servicing
+constraints.
+Index Terms—Hierarchical dynamic matching, integrated hy-
+brid resources, deep reinforcement learning, uncertainty.
+I. INTRODUCTION
+D
+RIVEN by the advances in communication technologies
+and supporting policies, Distributed Energy Resources
+(DERs) and flexible loads are going to be highly penetrated in
+power grids. Federal Energy Regulatory Commission (FERC)
+order 2222 requires power system operators to facilitate the
+participation of demand-side resources in the electricity mar-
+kets, reflecting their significant potential to provide energy
+flexibility [1]. The DERs and flexible loads, if coordinated and
+controlled carefully, can make the grid flexible and energy-
+efficient [2], [3]. However, a large number of DERs and
+flexible loads might challenge the structure and capacity of
+power distribution grids. Hence, it is essential to develop
+intelligent energy management solutions that can manage
+different sorts of DERs and flexible loads in a scalable manner
+without compromising the power grid’s stability.
+In recent years, several efforts have been made to develop
+energy management solutions for the coordination of DERs
+in power distribution systems. One promising solution is
+This work is supported in part by the National Science Foundation under
+Grant ECCS-1839429.
+M. Majidi, and M. Parvania are with the Department of Electrical and
+Computer Engineering, the University of Utah, Salt Lake City, UT 84112 USA
+(e-mails: {majid.majidi, masood.parvania,}@utah.edu). Deepan Muthirayan
+and Pramod P. Khargonekar are with the Department of Electrical Engineering
+and Computer Science, University of California, Irvine, CA 92697 USA (e-
+mails: {deepan.m, pramod.khargonekar,}@uci.edu).
+matching, which is a Peer-to-Peer (P2P) solution. Unlike
+traditional energy management solutions based on pooling re-
+sources, matching offers optimal use of energy flexibility while
+accounting for the energy preferences of flexible customers.
+This is the key feature that makes matching very promising to
+future power grids. However, developing a matching solution
+for power grids still has several challenges: (i) the solution
+has to be online and capable of adapting the matching strategy
+with the state of the whole grid, (ii) it has to cope with a large
+number of DERs and flexible loads, and (iii) it has to satisfy
+the security constraints of the grid. Although [4], [5] propose
+online solutions for dynamic matching, the proposed solutions
+are heuristics and therefore can be sub-optimal. Alternatively,
+data-driven approaches like Reinforcement Learning (RL) can
+be used to discover high-performing dynamic matching poli-
+cies. However, RL approaches have severe limitations when
+it comes to learning policies for power grids with constraints
+such as power flow limits. For instance, [6] proposes a Deep
+Reinforcement Learning (DRL) solution for dynamic matching
+but it fails to account for the grid constraints. Moreover, a
+central matching structure might not be efficient or feasible
+for managing a large number of DERs and flexible loads in
+the power grid. Hence, an efficient learning-based hierarchical
+matching model with Integrated Hybrid Resources (IHRs) is of
+interest in addressing dynamic matching in power grids with
+grid constraints.
+IHRs present a viable solution to facilitate efficient energy
+management and control of uncertain DERs in various appli-
+cations [7]–[10]. In a power grid, different types of renewable
+and non-renewable DERs and flexible can be combined and
+operated as an IHR to supply distributed energy flexibility.
+The key feature of an IHR is that the resources within the IHR
+can be treated as an integrated set of resources with a single
+interconnection point. Therefore, from a matching solution
+point of view, each IHR can be treated as a separate matching
+market with a single interconnection to the distribution grid.
+This then enables the use of an RL-type algorithm to determine
+the optimal way to manage the DERs within each IHR,
+where the RL solution does not need to take into account
+the grid constraints. Once the IHR-level matching results are
+determined, a central controller can re-dispatch the IHRs using
+a reduced-dimension Optimal Power Flow (OPF) model with
+each IHR as a node to balance the excesses while ensuring that
+the grid constraints across the distribution system are satisfied.
+This paper proposes a hierarchical framework for dynamic
+matching markets in power distribution systems composed of
+IHRs. The schematic of the proposed framework is shown
+in Fig. 1. An IHR consists of an agent that employs DRL
+arXiv:2301.13796v1  [eess.SY]  31 Jan 2023
+
+2
+to locally match the distributed Renewable Energy Sources
+(RES) and flexible customers in each IHR while satisfying
+the quality of service constraints of customers, i.e., critical-
+ity, servicing deadline, etc. Such a learning-based approach
+allows for developing a very effective online matching pol-
+icy, which is otherwise very difficult to design. Once the
+IHR-level matching results are determined, each IHR agent
+communicates the net active power flow (i.e., net active
+power consumption/generation), as well as the reactive power
+limits of the IHR to a central agent. The central agent then
+formulates a reduced-dimension OPF model with the IHRs as
+the nodes to determine the final flows (set-points) such that
+the grid constraints are met. In this stage, the central agent
+can curtail the IHR-level matching decisions and control the
+reactive power flow from/into each IHR in order to make sure
+that power flows in the grid and voltage levels across the
+distribution nodes are not violated.
+A. Related Works and Contributions
+Several works in the literature have explored the control
+and management of DERs in distribution systems. P2P energy
+trading markets of different types are developed in [11]–
+[20]. The authors in [12] studied the operation and benefits
+of centralized and decentralized battery energy storage under
+different P2P market designs. A P2P market design based
+on bilateral contracts is proposed in [13] for energy trading
+between multiple DERs and flexible loads, with the objective
+of minimizing peak load in the low-voltage power distribution
+system. A decentralized P2P optimization framework is pre-
+sented in [14] to enable local energy sharing between DER
+agents in the low-voltage distribution systems. A local energy
+sharing framework is proposed for prosumers in low-voltage
+distribution systems in [15], where the voltage regulation capa-
+bility of the proposed energy sharing framework is highlighted.
+An iterative sequential approach is implemented in [16] to
+enable P2P energy and reserve sharing between prosumers in
+power distribution systems with grid constraints. A negotiation
+algorithm is proposed to facilitate energy sharing between
+interconnected DER owners in [17]. The use of game theory
+to determine the interaction strategy of DERs in P2P energy
+sharing is investigated in [18]–[20].
+The application of data-driven approaches to energy man-
+agement of DERs is studied in [21]–[25]. The authors in [21]
+implemented a multi-agent learning framework to determine
+real-time local energy trading strategies for DER owners in
+regional microgrids. A multi-energy sharing model based on
+RL is proposed in [22], [23] for local heat and power sharing
+in energy microgrids equipped with DERs. In [24], [25], the
+authors proposed a specific price-based market framework for
+coordinating the prosumers in the market to minimize the
+peak load. In [26], a hierarchical energy management model
+based on DRL is proposed for local energy management of
+energy storage systems to improve the resilience of the power
+distribution system.
+Although the works reviewed here study the management
+and coordination of DERs in distribution systems, their P2P
+solutions address specific scenarios, and many do not account
+for the grid constraints. Moreover, the existing hierarchical
+energy management models for DERs in power distribution
+systems neglect the preference and dynamic characteristics
+of the DERs and flexible loads, which impacts the energy
+flexibility available to the grid. In contrast, this paper develops
+a broadly applicable online matching solution that (i) is de-
+signed to maximize the integration of local RES and the overall
+welfare in a very generic real-time operating scenario that can
+be riddled with uncertainties, (ii) takes into consideration the
+flexible loads’ servicing deadline, as well as their dynamic
+criticality, and willingness to pay for a unit of energy, and (iii)
+at the same time accounts for the power grid constraints. The
+key contributions of the paper can be summarized as follows:
+• A hierarchical dynamic matching framework for power
+distribution systems constituted by IHRs.
+• An efficient and scalable learning-based solution to match
+the flexible loads and uncertain RES within each IHR
+with no need for prior experience or expert supervision
+or elaborate design. An independent DRL algorithm for
+each IHR agent that matches the flexible loads and supply
+sources to improve the utilization of RES and therefore
+maximize the social welfare in each IHR such that the
+quality of service constraints of the loads are satisfied
+prior to their servicing deadline, and their dynamic will-
+ingness to pay for a unit of energy is taken into account.
+• An upper-level optimization model to fix the excesses or
+imbalances within the IHRs. Once the matching results in
+each IHR are determined, a central agent runs a reduced-
+dimension OPF problem to fix the possible imbalances
+within the IHRs and, at the same time, ensures the
+power flow limits and voltage security constraints of the
+distribution system are met.
+The remaining of the paper is categorized as follows: Hi-
+erarchical matching market framework is presented in Section
+II. The proposed learning-based solution approach is explained
+in Section III. The simulation results are presented in Section
+IV, and the paper is concluded in Section V.
+II. HIERARCHICAL MATCHING MARKET
+The proposed hierarchical matching framework is composed
+of (i) a market operator or central agent, and (ii) multiple
+IHRs, with each IHR operating as a separate matching market
+and the central agent acting as the coordinating agent between
+the IHRs. In the proposed framework, the grid is divided into
+multiple IHRs, where each IHR is an integrated unit of several
+types of DERs that are treated as a single resource with a single
+interconnection point to the grid. This is feasible to do when
+the region of the grid representing the IHR is such that the
+voltage variation within each IHR is within a small δ as [26]:
+|Vit − Vjt| < δ,
+∀t, ∀b, b′, ∈ Bh, ∀h ∈ H,
+(1)
+where H is the set of all IHRs and b, b′, ∈ Bh represent any
+pair of buses in IHR h ∈ H. Hence, each IHR is treated as a
+regular matching market with no power flow constraints, and
+the rest of the grid as the upstream supply source, to balance
+any imbalances or excesses in the IHRs.
+
+3
+IHR Agent
+Matching 
+Policy
+Matching 
+Policy
+Matching Market
+Constraints
+Supply-Demand Balance
+RES Generation Availability
+Objective
+Maximize Social Welfare
+Customers Deadline
+IHR Matching Problem
+Active Customers
+RES Availability
+Customers Criticality
+Wait
+State
+Match
+Action
+Match
+Reward
+Social Welfare
+Constraints
+Optimal Power Flow
+Integrated Hybrid Resource
+Objective
+Minimize System Energy Cost
+System-Level Power Flow Constraints
+- Net Active Power 
+- Min & Max Reactive Power Capacity
+Fixed Mismatches 
+Fig. 1. Structure of hierarchical dynamic matching model in power distribu-
+tion systems using deep reinforcement learning.
+The matching market within each IHR is an online market,
+with customers and RES that are characterized by uncertain
+arrivals and generation over time. In the proposed model,
+each IHR is endowed with an agent which can adapt its
+decision according to the state of the local IHR market, which
+includes the history of customers and renewable generation.
+This ensures that the decisions can be optimized with respect
+to the underlying state and the expected future conditioned
+on the current state. But, such state-dependent solutions are
+difficult to characterize for an online market. Alternatively,
+data-driven approaches like RL can be used to derive state-
+dependent solutions for systems like electricity markets, which
+are dynamic and uncertain. Given this, the IHR agents are
+endowed with DRL algorithms to discover state-dependent
+matching policies from their respective operational data.
+The design of a DRL model for a matching market has many
+practical limitations like scalability because of the size of the
+action space, which can grow exponentially with the number
+of customers, in addition to the servicing constraints that the
+matching outputs from the DRL model are required to satisfy.
+In this study, the scalable DRL-based solution proposed in
+our prior work [6] is adopted as the DRL model for each
+of the IHRs. This model is designed specifically to address
+the scalability and convergence of DRL applied to matching
+markets. The DRL model is discussed in the next section.
+The central agent plays the role of managing the whole
+grid, coordinating the upstream demand of the IHRs such
+that the grid constraints are satisfied. At any moment, after
+the IHR-level matching decisions are determined, the IHR
+agents send their net active power and reactive power limits
+to the central agent. The central agent then solves a reduced-
+dimension OPF problem with IHR as the nodes to compute the
+active and reactive power flows for the nodes such that the grid
+constraints (i.e., overall power flow constraints and voltage
+boundary limits derived from (1) are met. The voltage bound-
+ary constraints ensure that the condition in (1) is satisfied,
+and therefore the matched power is delivered without much
+loss. The inclusion of the power flow constraints ensures that
+the overall flow delivers the matched power to the extent the
+constraints can be satisfied. Now, a single centralized market
+can perform matching, and be adaptable to the changing
+market condition, but it is typically hard to compute a solution
+with DRL that satisfies stability constraints, i.e., power flow
+constraints. This is the key benefit of the proposed hierarchical
+approach, which uses DRL to identify state-dependent policies
+and optimization to ensure the feasibility of the policies for
+grid operation. The rest of the section describes the supply
+and customer models, the market state, the matching market
+formulation, and finally the IHR-level matching and upper-
+level OPF problems.
+A. Supply Model
+Lets denote the time within a day by t. The matching market
+comprises two sources of supply: 1) grid supply (type g),
+which it can be drawn from its interconnection to the grid
+and 2) RES (type s). The upstream grid supply at a time t,
+denoted by gt ∈ R, is priced at the retail price of electricity,
+c/kWh, while the unit cost of RES generation, rt ∈ R, is
+assumed to be zero.
+B. Flexible Customer Model
+Each customer (or load) is characterized by three param-
+eters, {ai, pi, di}, where ai ∈ N is the arrival time of the
+customer, pi ∈ R is the load requested by the customer, and
+di ∈ N is the servicing deadline by which the customer is
+to be served. Moreover, each customer has a criticality rate
+bi, at which its willingness to pay decreases from ai until di.
+The heterogeneity of customers lies in the differing deadlines
+and their criticality. Hence, the utility function of customer i
+representing its willingness to pay for a unit of energy can be
+defined as follows:
+πi
+t = c − bi(t − ai),
+πi
+t ≥ 0, ai ≤ t ≤ di,
+bi = ϕc/(di − ai),
+(2)
+where ϕ ∈ [0, 1] determines the reduction rate in customers’
+willingness to pay for a unit of energy. The utility function
+for different values of criticality rate is shown in Fig. 2. In
+addition to the flexible customers, the market can also have
+non-flexible loads.
+In Fig. 2, customer’s willingness to pay is less than or equal
+to the grid supply price c/kWh. This is reasonable considering
+that the grid supply is available at this price at all times. A
+customer with ϕ = 1 will only be willing to pay zero if it
+is served at its deadline. On the other hand, a customer with
+ϕ = 0 can be served at any time without any change to the
+
+4
+πt
+t
+b > 0
+a
+d
+c
+πt
+t
+b = 0
+a
+d
+c
+flexible load c at bus i in time t, the state equation of queuing
+system can be expressed as:
+˙Qc
+t,i = Ac
+t,i − P c
+t,i
+(4)
+Qc
+t,i = Qc
+t−∆t,i +
+�
+Ac
+t,i − P c
+t,i
+�
+∆t
+(5)
+According to the state equations expressed in In (4)-(5), the
+flexible load queue backlog in each time interval t is equal to
+customer i at time k by qj (k). Then, the function χk is given
+by
+χk(j, i, Z) = qi
+j(k), χk(j, st, Z) = qst
+j (k)
+(8)
+Given these definitions, the matching problem for the distri-
+bution system is given by the following optimization problem
+Fig. 2. Illustration of utility function for different values of b.
+willingness to pay. This model captures a variety of customers
+in the market, where customers’ willingness to pay can remain
+fixed or decay with time and at distinct rates. As the number
+of customers is finite in real markets, the number of customers
+arriving on the platform at any time t, nt ∈ N, is assumed to
+be upper bound by a constant n.
+C. IHR Market State
+Let zt
+:=
+[a⊤
+t , p⊤
+t , b⊤
+t , d⊤
+t , rt]1 be the vector of state
+parameters, where at ∈ Nn is the vector of the arrival times of
+the customers which arrive at time t, pt ∈ Nn is the vector of
+their respective requested loads, bt ∈ [0, 1] is the criticality rate
+of customer at time t, dt ∈ Nn is the vector of their respective
+deadlines, and rt ∈ R is the amount of RES generation at time
+t. The scenario at time t is given by
+Z⊤
+t = [z⊤
+1 , z⊤
+2 , ..., z⊤
+t−1, z⊤
+t ].
+The probability that zt = z is given by the stochastic process
+modeled by P (zt = z|Zt−1). This process is not known to
+the market operator. Let xt := [a⊤
+t , p⊤
+t , b⊤
+t , d⊤
+t , p⊤
+u,t, b⊤
+u,t, rt],
+where pu,t denotes the vector of the portion of the requested
+load that has not been served to the customers who arrived at
+t and expressed the criticality b⊤
+u,t. Let denote the set of all
+possible states at time t by Ωt and the state of the market by
+Xt. Then Xt is given by
+X⊤
+t = [x⊤
+1 , x���
+2 , ..., x⊤
+t−1, x⊤
+t ].
+Note that the state Xt depends on the scenario Zt and the
+matching decisions till time t − 1. Given that the state of the
+market evolves, the matching solution will have to be able to
+adapt to the changing market state.
+D. Matching Market for Integrated Hybrid Resources
+This part formulates the IHR-level dynamic matching mar-
+ket problem for a duration of T, divided into time periods
+spaced equally at an interval ∆t. The IHR-level dynamic
+matching market problem aims to match the load request
+of flexible and inflexible customers to maximize the social
+welfare in IHR h subject to satisfying the supply-demand
+balance constraint for non-flexible loads and the quality of
+service constraints for flexible loads arriving sequentially. Let
+define Ah,t as the set of all active customers at time t and IHR
+h and define Sh,t = {g, s} as the set of supply types. The IHR
+agent, at each time, can decide to match and supply or skip the
+load requests. Let define pi
+h,t as the skipped and unsupplied
+1[.]⊤ denotes the transpose operation.
+load request of the customer i and Mh,t(j, i, Xt) ∈ R define
+the amount of supply of type j matched to customer i at time
+t and IHR h, at the unit cost of ch,t. The matching market
+problem can be then stated as:
+sup
+T
+�
+t=1
+�
+i∈Ah,t
+�
+j∈Sh,t
+(πi
+h,t − ch,j)Mh,t(j, i, Xt), s.t.
+(3)
+�
+j∈Sh,t
+Mh,t(j, i, Xt) ≤ pi
+h,t, ∀h ∈ H, ∀i ∈ Ah
+t , t ̸= di
+h,
+(4)
+�
+j∈Sh,t
+Mh,t(j, i, Xt) = pi
+h,t, ∀h ∈ H, ∀i ∈ Ah,t, t = di
+h, (5)
+�
+i∈Ah,t
+Mh,t(r, i, Xt) ≤ rp
+h,t, ∀h ∈ H, ∀t.
+(6)
+pNet
+h,t =
+�
+i∈Ah,t
+Mh,t(g, i, Xt), ∀h ∈ H, ∀t,
+(7)
+where the dependency of Mt on Xt accounts for the depen-
+dency of the matching decision on the full state information in
+each IHR. Here, the power balance constraint for the flexible
+loads is given in (4). The power balance for the non-flexible
+loads and the critical flexible loads at their departure (t=di
+h)
+is given in (5). The constraint (6) limits the matching power
+from RES to the active power output of RES rp
+h,t. Finally,
+the net active power flow exchanged between the IHR and
+upstream grid, pNet
+h,t , is given by (7).
+The output of the above problem is a matching policy M1:T
+for the entire duration of a day. Because of the interdependence
+across time, the optimal policy Mt is dependent on the load
+arrivals and RES generation for the full day. This makes
+the computation of the optimal policy through the above
+approach infeasible in real-time operation. There are also no
+known explicit characterizations for Mt. This is what makes
+approaches like DRL very appealing, since they are general-
+purpose methods that can be used to discover state-dependent
+policies, such as Mt, from just operational data. Therefore, we
+use a DRL algorithm to compute the matching decisions. The
+proposed DRL model for a specific IHR is designed to output
+a matching decision at any point of time depending on the
+market state of the IHR. The DRL framework for the IHRs
+matching is described in the next section.
+Once the matching decisions are computed by the IHRs,
+the agent determines the net active power flow, i.e., the active
+power to be taken or injected from and to the upstream grid,
+as well as the reactive power limits of its zone to the central
+agent. The reactive power limits are utilized by the central
+agent to adjust the nodal reactive power demands such that
+the constraint (1) is satisfied. The reactive power limits for
+each IHR, denoted by qNet
+h,t
+and qNet
+h,t , are obtained through
+(8)-(10):
+rq
+h,t=−
+�
+rs
+h,t
+2−rp
+h,t
+2 ,
+rq
+h,t=
+�
+rs
+h,t
+2−rp
+h,t
+2, ∀h∈H,∀t, (8)
+qNet
+h,t =
+�
+i∈Ah,t
+qi
+h,t − rq
+h,t,
+∀h ∈ H, ∀t,
+(9)
+qNet
+h,t =
+�
+i∈Ah,t
+qi,
+h,t − rq
+h,t,
+∀h ∈ H, ∀t,
+(10)
+
+5
+where rq
+h,t, rq
+h,t are the minimum and maximum reactive
+power output of RES, rs
+h,t is the available RES generation and
+the term qi
+h,t represents the reactive power load of the IHR,
+determined based on the non-flexible reactive power load and
+matched power to flexible loads.
+E. Reduced-Dimension Optimal Power Flow
+This section describes the reduced-dimension OPF problem
+solved by the central agent to determine the final flows for the
+nodes in the network to deliver the matched power to the extent
+it does not violate the grid constraints. The agent specifically
+solves a quadratic optimization model with the IHRs as the
+nodes, where the constraints are the power flow constraints
+with the active power flow demand and reactive power limits
+of the IHRs, and the voltage limit constraints, defined based on
+(1). The voltage limit constraints ensure that the final flows are
+consistent with the matched power in each of the IHRs. The
+central agent also curtails the active power flow demand by ph
+C
+to the extent that the flow constraints are satisfied. The central
+agent’s objective function, which is the distribution system
+cost, is given in (11):
+min
+�
+λRT P G−
+H
+�
+h=1
+λCpC
+h
+�
+,
+(11)
+where P G is the active power taken from the transmission
+system at the real-time market price λRT and pC
+h is the active
+load request curtailment with the unit cost of λC.
+1) Power Balance Constraints: The active and reactive
+energy balance equations for the slack buses in the distribution
+system are given in (12)-(13), where P1h, Q1h are the active
+and reactive power flows from the substation bus to the IHR
+h, V sq
+1
+is the squared voltage on the substation bus, g1, b1 are
+the shunt conductance and susceptance at the substation bus
+and L is the set of lines in the distribution grid.
+P G =
+�
+1h∈L
+P1h + g1V sq
+1 ,
+(12)
+QG =
+�
+1h∈L
+Q1h + b1V sq
+1 .
+(13)
+The energy balance constraints for the IHR nodes are
+presented in (14)-(15), where pNet
+h
+is the net active power
+flow submitted by the IHR, Phh′′, Ph′h and Qhh′′, Qh′h are
+the active and reactive power flows in lines hh′′ and h′h, V sq
+h
+is the squared voltage on IHR node h, Isq
+h′h is the squared
+current flow in line h′h, and rh′h, xh′h are the resistance and
+reactance of the line h′h and gh, bh are the shunt conductance
+and susceptance at the IHR node h. In the active power balance
+constraint, the curtailment pC
+h ensures that the final net active
+power flow is consistent with the power flow constraints. Here,
+the curtailment is only made to the extent that the constraints
+are satisfied. The reactive power flow of the IHRs, denoted by
+qNet
+h
+, is limited to the reactive power limits of IHRs in (16).
+Phh′′ +pNet
+h
+−pC
+h =
+�
+h′h∈L
+(Ph′h−rh′hIsq
+h′h)+ghV sq
+h , ∀h∈B, (14)
+Qhh′′ + qNet
+h
+=
+�
+h′h∈L
+(Qh′h− xh′hIsq
+h′h)+ bhV sq
+h , ∀h ∈ B, (15)
+qNet
+h
+≤ qNet
+h
+≤ qNet
+h
+, ∀h.
+(16)
+2) Voltage and Power Flow Limits: The voltage drop across
+the grid is given by (17). The limits on the squared voltage
+level and the limits on the current flow are given in Eq. (18)
+and Eq. (19), where V sq
+h , V
+sq
+h are the minimum and maximum
+squared voltage boundaries, defined based on the nominal node
+voltage and δ in (1) and I
+sq
+h′h is the squared current flow limit.
+Finally, the complex power flow constraint is given in (20).
+V sq
+h − V sq
+h′ = −2 (rh′hPh′h + xh′hQh′h)
++
+�
+r2
+h′h + x2
+h′h
+�
+Isq
+h′h,
+∀(h′h) ∈ L,
+(17)
+V sq
+h ≤ V sq
+h
+≤ V
+sq
+h ,
+∀h ∈ B,
+(18)
+Isq
+h′h ≤ I
+sq
+h′h,
+∀(h′h) ∈ L,
+(19)
+V sq
+h,tIsq
+h′h ≥ P 2
+h′h + Q2
+h′h,
+∀(h′h) ∈ L.
+(20)
+Any feasible solution to the online OPF problem in (11)-
+(20) ensures the matched power in each of the IHRs is
+delivered to the extent the flow and voltage constraints are
+met. In case a solution is feasible without any curtailment,
+then the matched power will be delivered to the customers.
+III. DEEP REINFORCEMENT LEARNING FOR IHRS
+In the proposed hierarchical framework, each IHR agent
+is endowed with a trainable policy that outputs a probability
+distribution over the set of matching decisions for the flexible
+loads and RES with the IHR. A policy gradient RL algorithm
+is applied to train the matching policy given the load and
+generation data for multiple instances of the market. This
+algorithm does not require supervision or expert knowledge as
+it measures its own performance for the training process. The
+following subsections briefly discuss the matching policy’s
+structure for an IHR and then the learning algorithm. Note
+that the expectation with respect to all sources of randomness
+is denoted by E[.]. It is implicit that all the descriptions in this
+section are confined to a single IHR.
+A. General Discrete Matching Policy
+Each IHR agent in the proposed study learns an online
+matching policy given by χ = {χ1, χ2, χ3, ..., χT }. The
+discrete matching policy for time t, denoted by χt, indicates
+whether a customer is to be matched to a supply or not,
+regardless of the amount of matching. Let define Mt as the
+space of discrete matching at time t. Each component in this
+set m ∈ Mt, is a feasible discrete matching that specifies
+whether a customer is matched or not (i.e., mi,k ∈ {0, 1} with
+one indicating “matched” and zero indicating “not matched”).
+Hence, the general matching policy χt can be given by:
+χt : Ωt → Mt.
+Aside from the fact that the matching problem is an
+online decision-making procedure with a future ridden with
+uncertainties, there are still several general challenges from
+an RL point of view. Firstly, the action space of the matching
+problem is large and specifically exponential in the number
+of customers. For example, if there are m supplies and n
+customers, then there are mn ways of matching; thus it is
+
+6
+exponential in the number of active customers. Secondly,
+not all the actions from this space are feasible as supply
+unavailability might limit the matching decisions. There might
+also be some restrictions enforced by the customers servicing
+constraints. Hence, some actions are infeasible, and their
+infeasibility is state-dependent. Thirdly, RL algorithms can
+converge to a local optimum, a general challenge that applies
+to the matching problem. Therefore, the goal is to develop
+a framework based on RL that is simple and efficient to
+learn, simultaneously satisfies the action constraints, and can
+converge to a good solution. The proposed framework in this
+paper simplifies the output of the policy to be trained by RL
+to just “match” or “not to match” for each active customer,
+regardless of the supply type and action feasibility. Thus, the
+action space of the output of the component that is trained
+is linear in the number of active consumers. Further details
+regarding the proposed matching policy are given below.
+B. Proposed Matching Policy
+The proposed discrete matching policy is characterized by
+a learnable and fixed component [6]. The first component, de-
+noted by µt, determines the probability of matching customers,
+and the latter makes sure that the customers are matched before
+their deadline. Let PMt be the space of probability measures
+over the set Mt. Then, the policy µt can be defined:
+µt : Ωt → PMt.
+Let mt ∈ Mt be given by mt ∼ µt. The component of mt
+corresponding to the customer i is defined by mi,t, where
+mi,t ∈ {0, 1}. The output mi is input to a second function, ϕ.
+The function ϕ matches the customers with mi,t = 1 to the
+available RES in each IHR. When total matching implied by
+the discrete matching is in excess of the RES, the remaining
+customers with mi,t = 1 are matched to the grid supply. When
+total matching implied by the discrete matching is less than the
+available RES, the excess RES generation is assigned to the
+remaining customers. Denote the component of ϕ that specifies
+whether customer i is matched to supply type j by ϕj,i.
+The output ϕj,i is input to a third function, ν, that overturns
+the matching decision for the customers with an immediate
+deadline and ensures that the flexible customers in IHRs are
+served by their deadline:
+νj,i =
+�
+�
+�
+1
+if di = t, i is active, ϕs,i = 0 ∀s
+and j = g,
+ϕj,i
+otherwise.
+Thus, the overall discrete matching policy for time t, χt, is
+given by:
+χt = ν ◦ ϕ ◦ mt, mt ∼ µt.
+(21)
+The proposed policy is parameterized by θt, where the pa-
+rameterization is denoted by µt(.; θt). The learning algorithm,
+presented next, uses the observations from load and generation
+data of the IHR and trains θt for every time step t by evaluating
+its own performance.
+C. Policy Gradient Learning Algorithm
+This part describes the proposed policy gradient learning
+algorithm. EXt∼Pt(.) is used as a shorthand for expectation
+over Xt ∼ P(.|Xt−1, χt−1), where P(.|Xt−1, χt−1) denotes
+the transition probability from state Xt−1 under the pol-
+icy χt−1. Let mt:T
+= {mt, ml+1, ..., mT } and µl:T
+=
+{µt, µl+1, ..., µT }. Denoting Em∼µ as a shortened form of
+Emt+1:T ∼µt+1:T , the market welfare can be defined as:
+V χ
+t+(Xt+1):=Em∼µ
+T
+�
+l=t+1
+�
+j
+�
+k∈At
+(πi
+l − cj)χt,j,i(Xt).
+(22)
+Let:
+vχ
+t :=
+�
+j
+�
+i∈At
+(πi
+l − cj)χt,j,i(Xt),
+(23)
+V χ
+t (Xt|mt) := vχ
+t + EXt+1∼Pt+1(.)
+�
+V χ
+t+(Xt+1)
+�
+.
+(24)
+Then, from the definitions of Vχ and V χ
+t (Xt), the gradient
+of the value function with respect to the policy parameter θt
+can be calculated as follows:
+∂Vχ
+∂θt
+= EXt
+�∂V χ
+t (Xt)
+∂θt
+�
+,
+(25)
+∂Vχ
+∂θt
+= EXt
+�
+mt∈Ht
+∂µt(mt; θt)
+∂θt
+[vχ
+t
++EXt+1∼Pt+1(.)V χ
+t+(Xt+1)
+�
+.
+(26)
+The gradient of the value function with respect to the policy
+parameter derived in (25)-(26) can be written as follows:
+∂Vχ
+∂θt
+= EXt,mt∼µt
+�∂ log µt(mt; θt)
+∂θt
+V χ
+t (Xt|mt)
+�
+,
+(27)
+where an unbiased estimate of this relationship can defined as
+follows:
+δθ
+t =
+�∂ log µt(mt; θt)
+∂θt
+V χ
+t (Xt|mt)
+�
+.
+(28)
+Since the term V χ
+t (Xt|mt) is unknown, the gradient in (28)
+is not computable. Therefore, this term is replaced with the
+social welfare from t to T for a sample epoch under policy χ:
+δθ
+t,r = ∂ log µt(mt; θt)
+∂θt
+� T
+�
+l=t
+vχ
+l
+�
+,
+(29)
+where the gradient is computable using the data from a sample
+epoch (Z = {Z1, Z2, ..., ZT }) and matching decisions under
+the policy χ for the same epoch. Furthermore, the gradient
+estimate is unbiased as
+∂Vχ
+∂θt
+= E[δθ
+t,r]. The vanilla policy
+gradient learning algorithm learns the policy parameter θt for
+each time step using the following stochastic gradient ascent
+algorithm:
+θt+1 ← θt + γθδθ
+t,r,
+(30)
+where θt is updated using the computed gradient δθ
+t,r for
+multiple sample epochs in every update step.
+In addition to the vanilla policy gradient learning algorithm
+described above, the actor-critic algorithm AC−k is also
+proposed for the dynamic matching of IHRs. This algorithm
+learns both the matching policy µ and an approximation of
+
+7
+value function V χ
+t (Xt). This function, which is also called
+the critic function, is parameterized by φk and expressed by
+V χ
+k (Xk; φk). Hence, the approximate policy gradient for the
+actor-critic algorithm AC−k can be defined as follows:
+δθ
+t,k = ∂ log µt(mt; θt)
+∂θt
+�t+k−1
+�
+l=t
+vχ
+l + V χ
+t+k(Xt+k; φt+k)
+�
+,
+(31)
+where the policy parameters are learned using the following
+stochastic gradient ascent algorithm in (32) and similarly the
+parameter φk of the critic function is learned by stochastic
+gradient descent for its least-squares error in (33).
+θt+1 ← θt + γθδθ
+t,k,
+(32)
+φk+1 ← φk − γφ
+�
+V χ
+k (.; φk) −
+T
+�
+l=k
+vχ
+l
+�
+.
+(33)
+Further details regarding the actor-critic algorithm are given
+in Algorithm 1, where the ADAM gradient algorithm of the
+gradient updates in (32) and (33) is implemented.
+Algorithm 1 Actor-Critic (AC−k) Policy Gradient Learn-
+ing Algorithm for an IHR
+1) Initialize D = ∅, j = 0
+2) Initialize θk ∀ k ∈ [1, ..., T]. N: number of epochs
+3) for i = 1 : N
+a) j = j + 1
+b) Set Di = {{Xk, mk, vχ
+k } ∀ k ∈ [1, ..., T]}
+c) Include Di into D
+d) if j == M
+Update θk by ADAM of Eq. (32) using D
+Update φk by ADAM of Eq. (33) using D
+j = 0; D = ∅
+end
+end
+4) end
+The matching policy in the proposed study is trained
+with the Temporal Convolution Network (TCN), denoted
+by TCNµ. Let
+˜Xt denote the input sequence to the TCN
+at each time step, where
+˜X⊤
+t
+= [˜x⊤
+1 , ˜x⊤
+2 , ˜x⊤
+3 , ..., ˜x⊤
+t ], and
+˜xt = [a⊤
+t , p⊤
+t , b⊤
+t , d⊤
+t , p⊤
+u,t, b⊤
+u,t, rt]. The vector of matching
+probabilities P m
+µ
+∈ [0, 1]n×T as the output of TCN can be
+determined as P m
+µ = TCNµ( ˜Xt). The output of TCN in the
+proposed study is fixed and capped at the maximum number
+of active customers at any time, n × T. Let P m
+µ,i denote the
+probability of matching for the active customer i at time step
+t. Hence, the distribution µt is constructed as:
+P(mt,i = 1) = P m
+µ,i, P(mt,i = 0) = 1 − P m
+µ,i.
+IV. SIMULATIONS AND RESULTS
+The proposed hierarchical matching framework is imple-
+mented on the IEEE 33-bus test distribution system using the
+30-minute real-time California Independent System Operator
+(CAISO) load and solar generation data from January 1, 2021
+to September 28, 2021. The distribution system is divided
+into 5 IHRs, each consisting of a learning agent to control
+and match DERs with flexible loads. The structure of the
+distribution system with IHRs is shown in Fig. 3, where
+the electric vehicle (EV) charging stations supply charging
+requests of 6.6 kWh to 24 EVs in IHR 1, 30 EVs in IHR
+2, 8 EVs in IHR 3 and 30 EVs in each one of IHRs 4 and
+5. The CAISO solar power data is scaled to the inverter’s
+nominal capacity of 105 kW in IHR 1, 150 kW in IHR 2, 45
+kW in IHR 3, and 150 kW in each one of IHRs 4 and 5. The
+distribution system active and reactive loads are scaled to 50 %
+of their nominal rates, 3715 kW and 2300 kVAr, respectively.
+The electricity tariff is assumed to be 120 $/MWh, and the
+curtailment penalty for the central agent is assumed to be 500
+$/MWh. To validate the efficiency of the proposed hierarchical
+framework, the following scenarios are considered:
+• Scenario 1: This is a scenario where the EVs are char-
+acterized by earlier arrival times and longer departure
+times. In this scenario, waiting to match will fetch higher
+welfare. This scenario tests the capability of the IHR
+agents to learn to let the customers wait in the market
+and not match them immediately upon their arrival.
+• Scenario 2: This is a scenario where the EVs are charac-
+terized by moderate arrival and longer departure times. In
+this scenario, waiting may not result in improved welfare.
+Here, a strategy that partially waits and partially matches
+upon arrival might be needed. This scenario tests the
+capability of the algorithm to learn such hybrid strategies.
+For illustration, two matching algorithms are considered,
+one is the Learning Algorithm (LA) described in Section III,
+and the other is the standard matching heuristic, Matching on
+Arrival (MA). These matching algorithms are implemented in
+scenarios 1 and 2 under the following market models:
+• Centralized Model: In this model, a single agent manages
+the matching of the whole distribution system.
+• Decentralized Model: In this model, the distribution sys-
+tem is divided into multiple IHRs as described earlier,
+with each IHR employing a separate matching algorithm.
+The central agent solves the reduced-dimension OPF
+model described earlier to meet the respective IHRs flow
+requirements and ensure that the grid constraints are met.
+This model with the learning algorithm is the proposed
+hierarchical framework.
+The best hyper-parameters for the TCN model were iden-
+tified to be 3 for the number of blocks, 4 for the number
+of filters, 3 for the filter size, 0.1 for the dropout factor,
+and 4 for the dilation factor. Sigmoid function is utilized
+as the activation function for each output of TCN and the
+following values are considered as the parameters of the
+ADAM algorithm: α = [0.25, 0.99], β1 = 0.9, β2 = 0.999,
+ϵ = 10−8, where α is the learning rate and β1, β2 are
+exponential decay rates for the moment estimates. The best
+batch size for the LA is 20.
+A. Numerical Results
+The average social welfare achieved in scenarios 1 and 2
+for the centralized and decentralized models is summarized in
+Table I. In scenario 1, it can be seen that the MA algorithm
+achieves a welfare of 17.98$ and 16.17$ in both the centralized
+
+8
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+26
+27 28
+29
+30
+31
+32
+33
+24
+25
+L1
+L2
+L3
+L4
+L5
+L6
+L7
+L8
+L9
+L10 L11
+L12
+L13
+L14
+L15
+L16
+L17
+L23
+L24
+L26
+L27 L28
+L29 L30
+L31
+L32
+L18
+L19
+L20
+L21
+L22
+L25
+IHR 4
+IHR 5
+IHR 1
+IHR 3 
+IHR 2
+Fig. 3.
+Structure of the 33-bus power distribution system, divided into 5
+integrated hybrid resources.
+TABLE I
+AVERAGE SOCIAL WELFARE ($)
+Algorithm
+Scenario
+Centralized Model
+Decentralized Model
+LA
+Scenario1
+218.59
+232.70
+Scenario2
+866.33
+893.52
+Average
+542.46
+563.11
+MA
+Scenario1
+17.98
+16.17
+Scenario2
+742.75
+808.20
+Average
+380.365
+412.185
+and decentralized models, while the LA algorithm achieves
+a higher welfare of 218.59$ and 232.7$ in the centralized
+and decentralized models respectively. This shows that the LA
+algorithm leverages the flexibility better to match the excess
+RES during the middle of the day. In scenario 2, the results
+reveal that the optimal matching policy is not to match all
+the loads on their arrival but to only match the critical ones
+on their arrival, so as to efficiently utilize the RES that is
+available during the middle of the day. In this scenario, the
+LA algorithm is the top-performing, achieving a social welfare
+of 866.33$ and 893.52$ in the centralized and decentralized
+models, followed by the MA algorithm, which achieves a
+social welfare of 742.75$ and 808.2$ in the centralized and
+decentralized models, respectively.
+Comparing the performance of LA and MA algorithms in
+the centralized and decentralized models, it can be seen that
+the decentralized model with the learning algorithm is the best
+performing, substantiating the efficacy of our approach. Table
+II summarizes the social welfare achieved by LA and MA
+in the decentralized matching model. Comparing the social
+welfare, it can be found that the LA algorithm outperforms the
+MA algorithm in each of the IHRs, showing the superiority
+of the learning-based approach.
+TABLE II
+AVERAGE SOCIAL WELFARE IN THE DECENTRALIZED MODEL ($)
+Algorithm
+Model
+IHR1
+IHR2
+IHR3
+IHR4
+IHR5
+LA
+Scenario1
+41.65
+55.06
+12.3
+60.02
+63.67
+Scenario2
+184.15
+216.67
+49.47
+221.36
+221.87
+Average
+112.9
+135.865
+30.885
+140.69
+142.77
+MA
+Scenario1
+3.06
+3.89
+1.02
+4.164
+4.04
+Scenario2
+163.08
+202.4
+43.53
+199.46
+199.73
+Average
+83.07
+103.145
+22.275
+101.81
+101.885
+B. Matching Market Analysis
+This section analyzes the performance of the matching
+algorithms under the centralized and decentralized matching
+markets in scenarios 1 and 2.
+1) Scenario 1: EVs with Earlier Arrival and Longer Depar-
+ture Times: In this scenario, the flexible loads are character-
+ized by earlier arrival and longer departure (deadline) times,
+the RES generation is available during the middle of the day.
+Thus, the market operator (agent) can queue the load requests
+and match them to the RES available during the middle of
+the day. The matching by the LA and MA of IHR 2 in this
+scenario is shown in Fig. 4. The results clearly show that MA
+fails to wait to avail the RES during the middle of the day, and
+instead matches the loads to the grid supply. On the contrary,
+the LA learns to queue the non-critical loads and shift them
+to the periods where RES generation is available.
+-40
+-20
+0
+20
+40
+60
+80
+100
+1
+21
+41
+61
+81
+101
+121
+141
+161
+181
+201
+221
+Social Welfare ($)
+Epoch
+Average: LA
+Average: MA
+Actual: LA
+Actual: MA
+Fig. 4. Average and actual social welfare of LA and MA for IHR 2 under
+scenario 1.
+Figure 5 shows the matching market results for the LA of
+IHR 2 for a representative epoch of scenario 1. In Fig. 5,
+the initial load request of critical loads is supplied using the
+grid power, while a significant portion of non-critical loads
+is shifted to the middle of the day and matched to the RES
+generation, indicating the efficacy of the fixed and trainable
+components of the matching policy to match flexible loads
+with RES, while satisfying the quality of service constraints of
+the loads. This is evident in Fig. 5, where all the requested load
+is supplied without any curtailment and the RES generation
+is efficiently allocated to supply the queued flexible loads and
+the non-flexible loads when the flexible loads are unavailable.
+0
+30
+60
+90
+120
+150
+180
+210
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+Matching Results (kW)
+Time (h)
+RES Generation
+Matched Flexible Load to RES
+Matched Flexible Load to Grid
+Curtailed Flexible Load
+Matched Non-Flexible Load to RES
+Curtailed RES Generation
+Requested Flexible Load
+Total Supplied Flexible Load
+Fig. 5. Matching market results for LA of IHR 2 for a representative epoch
+of scenario 1.
+
+9
+The performance of the decentralized and centralized mod-
+els with LA is compared in Fig. 6. As shown, the centralized
+model obtains higher welfare in the initial epochs, but the
+decentralized model achieves a higher average social welfare
+with experience.
+-30
+0
+30
+60
+90
+120
+150
+180
+210
+240
+1
+21
+41
+61
+81
+101
+121
+141
+161
+181
+201
+221
+Average Social Welfare ($)
+Epoch
+IHR 1
+IHR 2
+IHR 3
+IHR 4
+IHR 5
+Centralized
+Fig. 6. Average social welfare of decentralized and centralized models with
+LA in scenario 1.
+2) Scenario 2: EVs with Moderate Arrival and Longer
+Departure Times: In this scenario, the flexible loads are
+characterized by moderate arrival and longer departure times
+and the RES generation is available during the middle of the
+day. The social welfare achieved by the LA and MA of IHR 5
+in this scenario is shown in Fig. 7. The results show that both
+the LA and MA achieve a similar performance. However, as
+shown, the LA is superior to the MA in that it doesn’t match
+all the flexible loads on their arrival. This is evident in Fig. 8,
+where a portion of the load request is shifted from their arrival
+and matched to the RES generation during the middle of the
+day. As in the previous scenario, the LA is able to meet the
+quality of service constraints of the loads and utilize the RES
+generation, while ensuring that the outcomes are economically
+efficient.
+25
+75
+125
+175
+225
+275
+325
+375
+1
+21
+41
+61
+81
+101
+121
+141
+161
+181
+Social Welfare ($)
+Epoch
+Average: LA
+Average: MA
+Actual: LA
+Actual: MA
+Fig. 7. Average and actual social welfare of LA and MA of IHR 5 in scenario
+2.
+C. Distribution System Constraints
+In the proposed hierarchical matching framework, the cen-
+tral agent solves a reduced-dimension OPF model with the
+IHRs as the nodes to deliver the IHR flow requirements while
+ensuring that the grid constraints are met. The agent can
+also curtail the flow to each IHR to the extent that the grid
+constraints are not violated. Figure 9 shows the voltage profiles
+of IHRs in the decentralized model and scenario 1.
+0
+100
+200
+300
+400
+500
+600
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+Matching Results (kW)
+Time (h)
+RES Generation
+Matched Flexible Load to RES
+Matched Flexible Load to Grid
+Matched Non-Flexible Load to RES
+Curtailed Flexible Load
+Curtailed RES Generation
+Requested Flexible Load
+Total Supplied Flexible Load
+Fig. 8.
+Matching results for LA of IHR 5 for a representative epoch in
+scenario 2.
+12.5
+12.55
+12.6
+12.65
+12.7
+1
+5
+9
+13
+17
+21
+25
+29
+33
+37
+41
+45
+Voltage Level (kV)
+Time (h)
+Substation Bus
+IHR 1
+IHR 2
+IHR 3
+IHR 4
+IHR 5
+Fig. 9. Voltage profiles of IHR nodes in the decentralized model, scenario 1.
+In this epoch, the lower and upper voltage boundaries of
+IHRs are respectively V h = [12.37, 12.2, 12.05, 12.08, 12.22]
+and V h =[12.948, 13.11, 13.25, 13.22, 13.09]. As shown, the
+voltage level of all IHRs is within the safe lower-bound and
+upper-bound limits in all the time periods. The extent to which
+the matching decisions are met in each IHR depends on the
+power flow in the grid operation, which can be curtailed by
+the central agent to ensure that the grid constraints are met.
+Figure 10 shows the matching curtailment of different IHRs
+in the decentralized model and scenario 1. The results show
+that the initial IHR-level matching decisions are not curtailed
+in most of the epochs, though the matching decisions in some
+initial epochs are curtailed to ensure the safe operation of the
+power grid.
+0
+100
+200
+300
+400
+500
+600
+1
+21
+41
+61
+81
+101
+121
+141
+161
+181
+201
+221
+Load Curtailment (kWh)
+Epoch
+IHR 1
+IHR 2
+IHR 3
+IHR 4
+IHR 5
+Fig. 10. Matching curtailment of the central agent in the decentralized model,
+scenario 1.
+
+10
+V. CONCLUSIONS
+This paper proposes a learning-based hierarchical frame-
+work for dynamic matching in power distribution systems.
+In the proposed framework, the power distribution system is
+divided into multiple IHRs, each consisting of flexible loads
+and RES. The IHR agents employ DRL to output an efficient
+and scalable online matching policy to match the available
+RES and active customers as the day progresses such that
+their quality of service constraints are not violated. Once
+the IHR-level matching decisions are determined, a central
+agent uses the net active power flow, as well as the reactive
+power limits of each IHR to formulate a reduced-dimension
+OPF model to determine the final flows such that the flow
+requirements of the IHRs are met and the grid constraints are
+not violated. The hierarchical approach offers a very effective
+way to combine the ability of DRL to learn state-dependent (or
+online) matching policies and that of optimization to ensure
+safe grid operation. The proposed hierarchical framework was
+implemented on the IEEE 33-bus test distribution system and
+tested on multiple scenarios with different matching algo-
+rithms, including the proposed learning algorithm. The results
+show that the hierarchical framework utilizes the flexible loads
+better, resulting in higher social welfare compared to the
+centralized approach that matches across the whole distribution
+system.
+REFERENCES
+[1] “Ferc
+order
+no.
+2222:
+A
+new
+day
+for
+distributed
+energy
+resources,”
+2020.
+[Online].
+Available:
+https://www.ferc.gov/media/
+ferc-order-no-2222-fact-sheet
+[2] K. Oikonomou, M. Parvania, and R. Khatami, “Deliverable energy flex-
+ibility scheduling for active distribution networks,” IEEE Transactions
+on Smart Grid, vol. 11, no. 1, pp. 655–664, 2019.
+[3] ——, “Coordinated deliverable energy flexibility and regulation capacity
+of distribution networks,” International Journal of Electrical Power &
+Energy Systems, vol. 123, p. 106219, 2020.
+[4] D. Muthirayan, M. Parvania, and P. P. Khargonekar, “Online algorithms
+for dynamic matching markets in power distribution systems,” IEEE
+Control Systems Letters, vol. 5, no. 3, pp. 995–1000, 2020.
+[5] M. Majidi, D. Muthirayan, M. Parvania, and P. P. Khargonekar, “Dy-
+namic matching in power systems using model predictive control,” in
+2021 North American Power Symposium (NAPS). IEEE, 2021, pp. 1–6.
+[6] ——, “Dynamic matching markets in power grid: Concepts and solution
+using deep reinforcement learning,” arXiv preprint arXiv:2104.05654,
+2021.
+[7] “Unlocking
+the
+flexibility
+of
+hybrid
+resources,”
+2022.
+[On-
+line]. Available: https://www.esig.energy/wp-content/uploads/2022/03/
+ESIG-Hybrid-Resources-report-2022.pdf
+[8] “Hybrid resources white paper,” 2021. [Online]. Available: https://www.
+ferc.gov/sites/default/files/2021-05/white-paper-hybrid-resources.pdf
+[9] M. Majidi, L. Rodriguez-Garcia, and M. Parvania, “Risk-based operation
+of power networks with hybrid energy systems,” in 2022 17th Interna-
+tional Conference on Probabilistic Methods Applied to Power Systems
+(PMAPS).
+IEEE, 2022, pp. 1–6.
+[10] R. Li, X. Yan, and N. Liu, “Hybrid energy sharing considering network
+cost for prosumers in integrated energy systems,” Applied Energy, vol.
+323, p. 119627, 2022.
+[11] W. Tushar, C. Yuen, T. K. Saha, T. Morstyn, A. C. Chapman, M. J. E.
+Alam, S. Hanif, and H. V. Poor, “Peer-to-peer energy systems for
+connected communities: A review of recent advances and emerging
+challenges,” Applied Energy, vol. 282, p. 116131, 2021.
+[12] A. L¨uth, J. M. Zepter, P. C. del Granado, and R. Egging, “Local
+electricity market designs for peer-to-peer trading: The role of battery
+flexibility,” Applied energy, vol. 229, pp. 1233–1243, 2018.
+[13] T. Morstyn, A. Teytelboym, and M. D. McCulloch, “Bilateral contract
+networks for peer-to-peer energy trading,” IEEE Transactions on Smart
+Grid, vol. 10, no. 2, pp. 2026–2035, 2018.
+[14] J. Guerrero, A. C. Chapman, and G. Verbiˇc, “Decentralized p2p energy
+trading under network constraints in a low-voltage network,” IEEE
+Transactions on Smart Grid, vol. 10, no. 5, pp. 5163–5173, 2018.
+[15] Y. Liu, C. Sun, A. Paudel, Y. Gao, Y. Li, H. B. Gooi, and J. Zhu, “Fully
+decentralized p2p energy trading in active distribution networks with
+voltage regulation,” IEEE Transactions on Smart Grid, 2022.
+[16] D. Botelho, L. de Oliveira, B. Dias, T. Soares, and C. Moraes, “In-
+tegrated prosumers–dso approach applied in peer-to-peer energy and
+reserve tradings considering network constraints,” Applied Energy, vol.
+317, p. 119125, 2022.
+[17] L. A. Soriano, M. Avila, P. Ponce, J. de Jes´us Rubio, and A. Molina,
+“Peer-to-peer energy trades based on multi-objective optimization,”
+International Journal of Electrical Power & Energy Systems, vol. 131,
+p. 107017, 2021.
+[18] W. Amin, Q. Huang, K. Umer, Z. Zhang, M. Afzal, A. A. Khan, and
+S. A. Ahmed, “A motivational game-theoretic approach for peer-to-peer
+energy trading in islanded and grid-connected microgrid,” International
+Journal of Electrical Power & Energy Systems, vol. 123, p. 106307,
+2020.
+[19] X. Luo, W. Shi, Y. Jiang, Y. Liu, and J. Xia, “Distributed peer-to-
+peer energy trading based on game theory in a community microgrid
+considering ownership complexity of distributed energy resources,”
+Journal of Cleaner Production, vol. 351, p. 131573, 2022.
+[20] B. Zheng, W. Wei, Y. Chen, Q. Wu, and S. Mei, “A peer-to-peer energy
+trading market embedded with residential shared energy storage units,”
+Applied Energy, vol. 308, p. 118400, 2022.
+[21] X. Fang, Q. Zhao, J. Wang, Y. Han, and Y. Li, “Multi-agent deep
+reinforcement learning for distributed energy management and strategy
+optimization of microgrid market,” Sustainable Cities and Society,
+vol. 74, p. 103163, 2021.
+[22] L. Li and S. Zhang, “Peer-to-peer multi-energy sharing for home
+microgrids: An integration of data-driven and model-driven approaches,”
+International Journal of Electrical Power & Energy Systems, vol. 133,
+p. 107243, 2021.
+[23] X. Wang, Y. Liu, J. Zhao, C. Liu, J. Liu, and J. Yan, “Surrogate model
+enabled deep reinforcement learning for hybrid energy community
+operation,” Applied Energy, vol. 289, p. 116722, 2021.
+[24] Y. Ye, Y. Tang, H. Wang, X.-P. Zhang, and G. Strbac, “A scalable
+privacy-preserving multi-agent deep reinforcement learning approach for
+large-scale peer-to-peer transactive energy trading,” IEEE Transactions
+on Smart Grid, 2021.
+[25] D. Qiu, Y. Ye, D. Papadaskalopoulos, and G. Strbac, “Scalable coordi-
+nated management of peer-to-peer energy trading: A multi-cluster deep
+reinforcement learning approach,” Applied Energy, vol. 292, p. 116940,
+2021.
+[26] M. M. Hosseini and M. Parvania, “Hierarchical intelligent operation of
+energy storage systems in power distribution grids,” IEEE Transactions
+on Sustainable Energy, 2022.
+

99FJT4oBgHgl3EQfpCw2/content/2301.11598v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3909851f657db119e0c1d90718d9b4ae49d9a8602af2ce63c86108b8784323f5
+size 1654502

99FJT4oBgHgl3EQfpCw2/content/tmp_files/2301.11598v1.pdf.txt

@@ -0,0 +1,1960 @@
+arXiv:2301.11598v1  [math.NA]  27 Jan 2023
+Practical Sketching Algorithms for Low-Rank
+Tucker Approximation of Large Tensors
+Wandi Dong1, Gaohang Yu1*, Liqun Qi2,1,3 and Xiaohao Cai4
+1Department of Mathematics, Hangzhou Dianzi University,
+Hangzhou, 310018, China.
+2Huawei Theory Research Lab, Hong Kong, China.
+3Department of Applied Mathematics, Hongkong Polytechnic
+University, Hong Kong, China.
+4School of Electronics and Computer Science, University of
+Southampton, Southampton, SO17 1BJ, UK.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: [email protected];
+[email protected]; [email protected];
+Abstract
+Low-rank approximation of tensors has been widely used in high-
+dimensional data analysis. It usually involves singular value decom-
+position (SVD) of large-scale matrices with high computational com-
+plexity. Sketching is an effective data compression and dimension-
+ality reduction technique applied to the low-rank approximation of
+large matrices. This paper presents two practical randomized algo-
+rithms for low-rank Tucker approximation of large tensors based
+on sketching and power scheme, with a rigorous error-bound analy-
+sis. Numerical experiments on synthetic and real-world tensor data
+demonstrate the competitive performance of the proposed algorithms.
+Keywords: tensor sketching, randomized algorithm, Tucker decomposition,
+subspace power iteration, high-dimensional data
+MSC Classification: 68W20 , 15A18 , 15A69
+1
+
+2
+Sketching Algorithms for Low-Rank Tucker Approximation
+1 Introduction
+In practical applications, high-dimensional data, such as color images, hyper-
+spectral images and videos, often exhibit a low-rank structure. Low-rank
+approximation of tensors has become a general tool for compressing and
+approximating high-dimensional data and has been widely used in scientific
+computing, machine learning, signal/image processing, data mining, and many
+other fields [1]. The classical low-rank tensor factorization models include,
+e.g., Canonical Polyadic decomposition (CP) [2, 3], Tucker decomposition [4–
+6], Hierarchical Tucker (HT) [7, 8], and Tensor Train decomposition (TT)
+[9]. This paper focuses on low-rank Tucker decomposition, also known as
+the low multilinear rank approximation of tensors. When the target rank
+of Tucker decomposition is much smaller than the original dimensions, it
+will have good compression performance. For a given Nth-order tensor X ∈
+RI1×I2×...×IN , the low-rank Tucker decomposition can be formulated as the
+following optimization problem, i.e.,
+min
+Y ∥X − Y∥2
+F ,
+(1)
+where Y ∈ RI1×I2×...×IN , with rank(Y(n)) ≤ rn for n = 1, 2, . . ., N, Y(n) is the
+mode-n unfolding matrix of Y, and rn is the rank of the mode-n unfolding
+matrix of X.
+For the Tucker approximation of higher-order tensors, the most fre-
+quently used non-iterative algorithms are the improved algorithms for the
+higher-order singular value decomposition (HOSVD) [5], the truncated higher-
+order SVD (THOSVD) [10] and the sequentially truncated higher-order SVD
+(STHOSVD) [11]. Although the results of THOSVD and STHOSVD are usu-
+ally sub-optimal, they can use as reasonable initial solutions for iterative
+methods such as higher-order orthogonal iteration (HOOI) [10]. However, both
+algorithms rely directly on SVD when computing the singular vectors of inter-
+mediate matrices, requiring large memory and high computational complexity
+when the size of tensors is large.
+Strikingly, randomized algorithms can reduce the communication among
+different levels of memories and are parallelizable. In recent years, many schol-
+ars have become increasingly interested in randomized algorithms for finding
+approximation Tucker decomposition of large-scale data tensors [12–17, 19, 20].
+For example, Zhou et al. [12] proposed a randomized version of the HOOI
+algorithm for Tucker decomposition. Che and Wei [13] proposed an adaptive
+randomized algorithm to solve the multilinear rank of tensors. Minster et al.
+[14] designed randomized versions of the THOSVD and STHOSVD algorithms,
+i.e., R-STHOSVD. Sun et al. [17] presented a single-pass randomized algorithm
+to compute the low-rank Tucker approximation of tensors based on a practical
+matrix sketching algorithm for streaming data, see also [18] for more details.
+Regarding more randomized algorithms proposed for Tucker decomposition,
+please refer to [15, 16, 19, 20] for a detailed review of randomized algorithms
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+3
+for solving Tucker decomposition of tensors in recent years involving, e.g., ran-
+dom projection, sampling, count-sketch, random least-squares, single-pass, and
+multi-pass algorithms.
+This paper presents two efficient randomized algorithms for finding the
+low-rank Tucker approximation of tensors, i.e., Sketch-STHOSVD and sub-
+Sketch-STHOSVD summarized in Algorithms 6 and 8, respectively. The main
+contributions of this paper are threefold. Firstly, we propose a new one-pass
+sketching algorithm (i.e., Algorithm 6) for low-rank Tucker approximation,
+which can significantly improve the computational efficiency of STHOSVD.
+Secondly, we present a new matrix sketching algorithm (i.e., Algorithm 7) by
+combining the two-sided sketching algorithm proposed by Tropp et al. [18]
+with subspace power iteration. Algorithm 7 can accurately and efficiently com-
+pute the low-rank approximation of large-scale matrices. Thirdly, the proposed
+Algorithm 8 can deliver a more accurate Tucker approximation than sim-
+pler randomized algorithms by combining the subspace power iteration. More
+importantly, sub-Sketch-STHOSVD can converge quickly for any data tensors
+and independently of singular value gaps.
+The rest of this paper is organized as follows. Section 2 briefly introduces
+some basic notations, definitions, and tensor-matrix operations used in this
+paper and recalls some classical algorithms, including THOSVD, STHOSVD,
+and R-STHOSVD, for low-rank Tucker approximation. Our proposed two-
+sided sketching algorithm for STHOSVD is given in Section 3. In Section 4,
+we present an improved algorithm with subspace power iteration. The effec-
+tiveness of the proposed algorithms is validated thoroughly in Section 5 by
+numerical experiments on synthetic and real-world data tensors. We conclude
+in Section 6.
+2 Preliminary
+2.1 Notations and basic operations
+Some common symbols used in this paper are shown in the following Table 1.
+Table 1 Common symbols used in this paper.
+Symbols
+Notations
+a
+scalar
+A
+matrix
+X
+tensor
+X(n)
+mode-n unfolding matrix of X
+×n
+mode-n product of tensor and matrix
+In
+identity matrix with size n × n
+σi(A)
+the ith largest singular value of A
+A⊤
+transpose of A
+A†
+pseudo-inverse of A
+
+4
+Sketching Algorithms for Low-Rank Tucker Approximation
+We denote an Nth-order tensor X ∈ RI1×I2×...×IN with entries given by
+xi1,i2,...,iN, 1 ≤ in ≤ In, n = 1, 2, ..., N. The Frobenius norm of X is defined as
+∥X∥F =
+�
+�
+�
+�
+I1,I2,...,IN
+�
+i1,i2,...,iN
+x2
+i1,i2,...,iN .
+The mode-n tensor-matrix multiplication is a frequently encountered operation
+in tensor computation. The mode-n product of a tensor X ∈ RI1×I2×...×IN
+by a matrix A ∈ RK×In (with entries ak,in) is denoted as Y = X ×n A ∈
+RI1×...×In−1×K×In+1×...×IN, with entries
+yi1,...,in−1,k,in+1,...,iN =
+In
+�
+in=1
+xi1,...,in−1,in,in+1,...,iNak,in.
+The mode-n matricization of higher-order tensors is the reordering of ten-
+sor elements into a matrix. The columns of mode-n unfolding matrix X(n) ∈
+RIn×(�
+N̸=n IN ) are the mode-n fibers of X. More specifically, a element
+(i1, i2, ..., iN) of X is maps on a element (in, j) of X(n), where
+j = 1 +
+N
+�
+k=1,k̸=n
+[(ik − 1)
+k−1
+�
+m=1,m̸=n
+Im].
+Let the rank of mode-n unfolding matrix X(n) is rn, the integer array
+(r1, r2, ..., rN) is Tucker-rank of Nth-order tensor X, also known as the mul-
+tilinear rank. The Tucker decomposition of X with rank (r1, r2, ..., rN) is
+expressed as
+X = G ×1 U (1) ×2 U (2) . . . ×N U (N),
+(2)
+where G ∈ Rr1×r2×...×rN is the core tensor, and {U (n)}N
+n=1 with U (n) ∈ RIn×rn
+is the mode-n factor matrices. The graphical illustration of Tucker decom-
+position for a third-order tensor shows in Figure 1. We denote an optimal
+rank-(r1, r2, ..., rN) approximation of a tensor X as ˆ
+Xopt, which is the optimal
+Tucker approximation by solving the minimization problem in (1). Below we
+Fig. 1 Tucker decomposition of a third-order tensor.
+present the definitions of some concepts used in this paper.
+
+B
+9
+3
+2
+ASketching Algorithms for Low-Rank Tucker Approximation
+5
+Definition 1 (Kronecker products) The Kronecker product of matrices A ∈ Rm×n
+and B ∈ Rk×l is defined as
+A ⊗ B =
+
+
+a11B
+a12B
+... a1nB
+a21B
+a22B
+... a2nB
+:
+:
+...
+:
+am1B am2B ... amnB
+
+ ∈ Rmk×nl.
+The Kronecker product helps express Tucker decomposition. The Tucker
+decomposition in (2) implies
+X(n) = U (n)G(n)(U (N) ⊗ ... ⊗ U (n+1) ⊗ U (n−1) ⊗ ... ⊗ U (1))⊤.
+Definition 2 (Standard normal matrix) The elements of a standard normal matrix
+follow the real standard normal distribution (i.e., Gaussian with mean zero and
+variance one) form an independent family of standard normal random variables.
+Definition 3 (Standard Gaussian tensor) The elements of a standard Gaussian
+tensor follow the standard Gaussian distribution.
+Definition 4 (Tail energy) The jth tail energy of a matrix X is defined as
+τ 2
+j (X) :=
+min
+rank(Y )<j ∥X − Y ∥2
+F =
+�
+i≥j
+σ2
+i (X).
+2.2 Truncated higher-order SVD
+Since the actual Tucker rank of large-scale higher-order tensor is hard to com-
+pute, the truncated Tucker decomposition with a pre-determined truncation
+(r1, r2, ..., rN) is widely used in practice. THOSVD is a popular approach to
+computing the truncated Tucker approximation, also known as the best low
+multilinear rank-(r1, r2, ..., rN) approximation, which reads
+min
+G; U(1),U(2),··· ,U(N) ∥X − G ×1 U (1) ×2 U (2) · · · ×N U (N)∥2
+F
+s.t.
+U (n)⊤U (n) = Irn, n ∈ {1, 2, ..., N}.
+Algorithm 1 THOSVD
+Require: tensor X ∈ RI1×I2×...×IN and target rank (r1, r2, . . . , rN)
+Ensure: Tucker approximation ˆ
+X = G ×1 U (1) ×1 U (2) · · · ×N U (N)
+1: for n = 1, 2, . . ., N do
+2:
+(U (n), ∼, ∼) ← truncatedSVD(X(n), rn)
+3: end for
+4: G ← X×1U (1)⊤ ×2 U (2)⊤ · · · ×N U (N)⊤
+
+6
+Sketching Algorithms for Low-Rank Tucker Approximation
+Algorithm 1 summarizes the THOSVD approach. Each mode is processed
+individually in Algorithm 1. The low-rank factor matrices of mode-n unfolding
+matrix X(n) are computed through the truncated SVD, i.e.,
+X(n) =
+�
+U (n)
+˜
+U (n)
+� �S(n)
+˜
+S(n)
+� �V (n)⊤
+˜
+V (n)⊤
+�
+∼= U (n)S(n)V (n)⊤,
+where U (n)S(n)V (n)⊤ is a rank-rn approximation of X(n), the orthogonal
+matrix U (n) ∈ RIn×rn is the mode-n factor matrix of X in Tucker decomposi-
+tion, S(n) ∈ Rrn×rn and V (n) ∈ RI1...In−1In+1...IN×rn. Once all factor matrices
+have been computed, the core tensor G can be computed as
+G = X×1U (1)⊤ ×2 U (2)⊤ · · · ×N U (N)⊤ ∈ Rr1×r2×...×rN.
+Then, the Tucker approximation ˆ
+X of X can be computed as
+ˆ
+X = G ×1 U (1) ×2 U (2) · · · ×N U (N)
+= X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×N (U (N)U (N)⊤).
+With the notation ∆2
+n(X) ≜ �In
+i=rn+1 σ2
+i (X(n)) and ∆2
+n(X) ≤ ∥X − ˆ
+Xopt∥2
+F
+[14], the error-bound for Algorithm 1 can be stated in the following Theorem 1.
+Theorem 1 ([11], Theorem 5.1) Let ˆ
+X = G ×1 U(1) ×2 U(2) · · · ×N U(N) be the
+low multilinear rank-(r1, r2, ..., rN) approximation of a tensor X ∈ RI1×I2×...×IN by
+THOSVD. Then
+∥X − ˆ
+X ∥2
+F ≤
+N
+�
+n=1
+∥X ×n (IIn − U(n)U(n)⊤)∥2
+F =
+N
+�
+n=1
+In
+�
+i=rn+1
+σ2
+i (X(n))
+=
+N
+�
+n=1
+∆2
+n(X ) ≤ N∥X − ˆ
+Xopt∥2
+F .
+2.3 Sequentially truncated higher-order SVD
+Vannieuwenhoven et al.[11] proposed one more efficient and less computation-
+ally complex approach for computing approximate Tucker decomposition of
+tensors, called STHOSVD. Unlike THOSVD algorithm, STHOSVD updates
+the core tensor simultaneously whenever a factor matrix has computed.
+Given the target rank (r1, r2, . . . , rN) and a processing order sp
+:
+{1, 2, ..., N}, the minimization problem (1) can be formulated as the following
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+7
+optimization problem
+min
+U(1),··· ,U(N) ∥X − X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×N (U (N)U (N)⊤)∥2
+F
+=
+min
+U(1),··· ,U(N)(∥X ×1 (I1 − U (1)U (1)⊤)∥2
+F + ∥ ˆ
+X (1) ×2 (I2 − U (2)U (2)⊤)∥2
+F +
+· · · + ∥ ˆ
+X (N−1) ×N (IN − U (N)U (N)⊤)∥2
+F )
+= min
+U(1)(∥X ×1 (I1 − U (1)U (1)⊤)∥2
+F + min
+U(2)(∥ ˆ
+X (1) ×2 (I2 − U (2)U (2)⊤)∥2
+F +
+min
+U(3)(· · · + min
+U(N) ∥ ˆ
+X (N−1) ×N (IN − U (N)U (N)⊤)∥2
+F ))),
+(3)
+where
+ˆ
+X (n) = X ×1 (U (1)U (1)⊤) ×2 (U (2)U (2)⊤) · · · ×n (U (n)U (n)⊤), n =
+1, 2, ..., N − 1, denote the intermediate approximation tensors.
+Algorithm 2 STHOSVD
+Require: tensor X ∈ RI1×I2×...×IN , target rank (r1, r2, . . . , rN), and process-
+ing order sp : {i1, i2, . . . , iN}
+Ensure: Tucker approximation ˆ
+X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
+1: G ← X
+2: for n = i1, i2, . . . , iN do
+3:
+(U (n), S(n), V (n)⊤) ← truncatedSVD(G(n), rn)
+4:
+G ← foldn(S(n)V (n)⊤) (% forming the updated tensor from its mode-n
+unfolding)
+5: end for
+In Algorithm 2, the solution U (n) of problem (3) can be obtained via
+truncatedSVD(G(n), rn), where G(n) is mode-n unfolding matrix of the (n−1)-
+th intermediate core tensor G = X ×n−1
+i=1 U (i)⊤ ∈ Rr1×r2×...×rn−1×In×...×IN,
+i.e.,
+G(n) =
+�
+U (n)
+˜
+U (n)
+� �S(n)
+˜
+S(n)
+� �V (n)⊤
+˜
+V (n)⊤
+�
+∼= U (n)S(n)V (n)⊤,
+where the orthogonal matrix U (n)
+is the mode-n factor matrix, and
+S(n)V (n)⊤ ∈ Rrn×r1...rn−1In+1...IN is used to update the n-th intermediate core
+tensor G. Function foldn(S(n)V (n)⊤) tensorizes matrix S(n)V (n)⊤ into ten-
+sor G ∈ Rr1×r2×...×rn×In+1×...×IN. When the target rank rn is much smaller
+than In, the size of the updated intermediate core tensor G is much smaller
+than original tensor. This method can significantly improve computational
+performance. STHOSVD algorithm possesses the following error-bound.
+Theorem 2 ([11], Theorem 6.5) Let ˆ
+X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the
+low multilinear rank-(r1, r2, ..., rN) approximation of a tensor X ∈ RI1×I2×...×IN by
+
+8
+Sketching Algorithms for Low-Rank Tucker Approximation
+STHOSVD with processsing order sp : {1, 2, . . . , N}. Then
+∥X − ˆ
+X ∥2
+F =
+N
+�
+n=1
+∥ ˆ
+X (n−1) − ˆ
+X (n)∥2
+F ≤
+N
+�
+n=1
+∥X ×n (IIn − U(n)U(n)⊤)∥2
+F
+=
+N
+�
+n=1
+∆2
+n(X ) ≤ N∥X − ˆ
+Xopt∥2
+F .
+Although STHOSVD has the same error-bound as THOSVD, it is less com-
+putationally complex and requires less storage. As shown in Section 5 for the
+numerical experiment, the running (CPU) time of the STHOSVD algorithm
+is significantly reduced, and the approximation error has slightly better than
+that of THOSVD in some cases.
+2.4 Randomized STHOSVD
+When the dimensions of data tensors are enormous, the computational cost
+of the classical deterministic algorithm TSVD for finding a low-rank approx-
+imation of mode-n unfolding matrix can be expensive. Randomized low-rank
+matrix algorithms replace original large-scale matrix with a new one through
+a preprocessing step. The new matrix contains as much information as possi-
+ble about the rows or columns of original data matrix. Its size is smaller than
+original matrix, allowing the data matrix to be processed efficiently and thus
+reducing the memory requirements for solving low-rank approximation of large
+matrix.
+Algorithm 3 R-SVD
+Require: matrix A ∈ Rm×n, target rank r, and oversampling parameter p ≥ 0
+Ensure: low-rank approximation matrix ˆA = ˆU ˆS ˆV ⊤ of A
+1: Ω ← randn(n, r + p)
+2: Y ← AΩ
+3: (Q, ∼) ← thinQR(Y )
+4: B ← Q⊤A
+5: (U, S, V ⊤) ← thinSVD(B)
+6: ˆU ← QU(:, 1 : r)
+7: ˆS ← S(1 : r, 1 : r), ˆV ← V (:, 1 : r)
+N. Halko et al. [21] proposed a randomized SVD (R-SVD) for matrices. The
+preprocessing stage of the algorithm is performed by right multiplying original
+data matrix A ∈ Rm×n with a random Gaussian matrix Ω ∈ Rn×r. Each
+column of the resulting new matrix Y = AΩ ∈ Rm×r is a linear combination
+of the columns of original data matrix. When r < n, the size of matrix Y
+is smaller than A. The oversampling technique can improve the accuracy of
+solutions. Subsequent computations are summarised in Algorithm 3, where
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+9
+randn generates a Gaussian random matrix, thinQR produces an economy-size
+of the QR decomposition, and thinSVD is the thin SVD decomposition. When
+A is dense, the arithmetic cost of Algorithm 3 is O(2(r + p)mn + r2(m + n))
+flops, where p > 0 is the oversampling parameter satisfying r+p ≤ min{m, n}.
+Algorithm 3 is an efficient randomized algorithm for computing rank-r
+approximations to matrices. Minster et al. [14] applied Algorithm 3 directly
+to the STHOSVD algorithm and then presented a randomized version of
+STHOSVD (i.e., R-STHOSVD), see Algorithm 4.
+Algorithm 4 R-STHOSVD
+Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
+order sp : {i1, i2, . . . , iN}, and oversampling parameter p ≥ 0
+Ensure: Tucker approximation ˆ
+X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
+1: G ← X
+2: for n = i1, i2, . . . , iN do
+3:
+( ˆU, ˆS, ˆV ⊤) ← R-SVD(G(n), rn, p) (cf. Algorithm 3)
+4:
+U (n) ← ˆU
+5:
+G ← foldn( ˆS ˆV ⊤)
+6: end for
+3 Sketching algorithm for STHOSVD
+A drawback of R-SVD algorithm is that when both dimensions of the inter-
+mediate matrices are enormous, the computational cost can still be high. To
+resolve this problem, we could resort to the two-sided sketching algorithm for
+low-rank matrix approximation proposed by Joel A. Tropp et al. [22]. The
+preprocessing of sketching algorithm needs two sketch matrices to contain
+information regarding the rows and columns of input matrix A ∈ Rm×n. Thus
+we should choose two sketch size parameters k and l, s.t. , r ≤ k ≤ min{l, n},
+0 < l ≤ m. The random matrices Ω ∈ Rn×k and Ψ ∈ Rl×m are fixed indepen-
+dent standard normal matrices. Then we can multiply matrix A left and right
+respectively to obtain random sketch matrices Y ∈
+Rm×k and W ∈ Rl×n,
+which collect sufficient data about the input matrix to compute the low-rank
+approximation. The dimensionality and distribution of the random sketch
+matrices determine the approximation’s potential accuracy, with larger values
+of k and l resulting in better approximations but also requiring more storage
+and computational cost.
+The sketching algorithm for low-rank approximation is given in Algorithm
+5. Function orth(A) in Step 2 produces an orthonormal basis of A. Using
+orthogonalization matrices will achieve smaller errors and better numerical
+stability than directly using the randomly generated Gaussian matrices. In
+particular, when A is dense, the arithmetic cost of Algorithm 5 is O((k +
+l)mn + kl(m + n)) flops. Algorithm 5 is simple, practical, and possesses the
+sub-optimal error-bound as stated in the following Theorem 3. In Theorem 3,
+
+10
+Sketching Algorithms for Low-Rank Tucker Approximation
+Algorithm 5 Sketch for low-rank approximation
+Require: matrix A ∈ Rm×n, and sketch size parameters k, l
+Ensure: rank-k approximation ˆA = QX of A
+1: Ω ← randn(n, k), Ψ ← randn(l, m)
+2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤)
+3: Y ← AΩ
+4: W ← ΨA
+5: (Q, ∼) ← thinQR(Y )
+6: X ← (ΨQ)†W
+function f(s, t) := s/(t − s − 1)(t > s + 1 > 1). The minimum in Theorem
+3 reveals that the low rank approximation of given matrix A automatically
+exploits the decay of tail energy.
+Theorem 3 ([22], Theorem 4.3) Assume that the sketch size parameters satisfy
+l > k + 1, and draw random test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently
+forming the standard normal distribution. Then the rank-k approximation ˆA obtained
+from Algorithm 5 satisfies
+E ∥ A − ˆA ∥2
+F ≤ (1 + f(k, l)) · min
+̺<k−1(1 + f(̺, k)) · τ 2
+̺+1(A)
+=
+k
+l − k − 1 · min
+̺<k−1
+k
+k − ̺ − 1 · τ 2
+̺+1(A).
+Using the two-sided sketching algorithm to leverage STHOSVD algorithm,
+we propose a practical sketching algorithm for STHOSVD named Sketch-
+STHOSVD. We summarize the procedures of Sketch-STHOSVD algorithm in
+Algorithm 6, with its error analysis stated in Theorem 4.
+Algorithm 6 Sketch-STHOSVD
+Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
+order sp : {i1, i2, . . . , iN}, and sketch size parameters {l1, l2, ..., lN}
+Ensure: Tucker approximation ˆ
+X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
+1: G ← X
+2: for n = i1, i2, . . . , iN do
+3:
+(Q, X) ← Sketch(G(n), rn, ln) (cf. Algorithm 5)
+4:
+U (n) ← Q
+5:
+G ← foldn(X)
+6: end for
+Theorem 4 Let ˆ
+X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the Tucker approximation
+of a tensor X ∈ RI1×I2×...×IN by the Sketch-STHOSVD algorithm (i.e., Algorithm
+6) with target rank rn < In, n = 1, 2, ..., N, sketch size parameters {l1, l2, ..., lN} and
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+11
+processing order sp : {1, 2, . . . , N}. Then
+E{Ωj}N
+j=1∥X − �
+X ∥2
+F ≤
+N
+�
+n=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1∆2
+n(X )
+≤
+N
+�
+n=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1∥X − ˆ
+Xopt∥2
+F .
+Proof Combining Theorem 2 and Theorem 3, we have
+E{Ωj}N
+j=1∥X − �
+X ∥2
+F
+=
+N
+�
+n=1
+E{Ωj}N
+j=1∥ ˆ
+X (n−1) − ˆ
+X (n)∥2
+F
+=
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥ ˆ
+X (n−1) − ˆ
+X (n)∥2
+F
+�
+=
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥G(n−1) ×n−1
+i=1 U(i)×n(I − U(n)U(n)⊤)∥2
+F
+�
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥(I − U(n)U(n)⊤)Gn−1
+n
+)∥2
+F
+�
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1
+In
+�
+i=rn+1
+σ2
+i (G(n−1)
+(n)
+)
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1∆2
+n(X )
+=
+N
+�
+n=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1∆2
+n(X )
+≤
+N
+�
+n=1
+rn
+ln − rn − 1
+min
+̺n<rn−1
+rn
+rn − ̺n − 1∥X − ˆ
+Xopt∥2
+F .
+□
+We assume the processing order for STHOSVD, R-STHOSVD, and Sketch-
+STHOSVD algorithms is sp : {1, 2, ..., N}. Table 2 summarises the arithmetic
+cost of different algorithms for the cases related to the general higher-order
+tensor X ∈ RI1×I2×...×IN with target rank (r1, r2, . . . , rN) and the special
+cubic tensor X ∈ RI×I×...×I with target rank (r, r, ..., r). Here the tensors are
+dense and the target ranks rj ≪ Ij, j = 1, 2, . . ., N.
+
+12
+Sketching Algorithms for Low-Rank Tucker Approximation
+Table 2 Arithmetic cost for the algorithms THOSVD, STHOSVD, R-STHOSVD, and
+the proposed Sketch-STHOSVD.
+Algorithm
+X ∈ RI1×I2×...×IN
+X ∈ RI×I×...×I
+THOSVD
+O(
+N
+�
+j=1
+Ij I1:N + �N
+j=1 r1:j Ij:N )
+O(NIN+1 +
+N
+�
+j=1
+rj IN−j+1)
+STHOSVD
+O(
+N
+�
+j=1
+Ij r1:j−1Ij:N +
+N
+�
+j=1
+r1:j Ij+1:N )
+O(
+N
+�
+j=1
+rj−1IN−j+2 + rj IN−j)
+R-STHOSVD
+O(
+N
+�
+j=1
+r1:jIj:N +
+N
+�
+j=1
+r1:j Ij+1:N )
+O(
+N
+�
+j=1
+rj IN−j+1 + rj IN−j )
+Sketch-STHOSVD
+O(
+N
+�
+j=1
+rj lj(Ij + r1:j−1Ij+1:N ) +
+N
+�
+j=1
+r1:j Ij+1:N )
+O(
+N
+�
+j=1
+rl(I + rj−1IN−j ) + rj IN−j )
+4 Sketching algorithm with subspace power
+iteration
+When the size of original matrix is very large or the singular spectrum of
+original matrix decays slowly, Algorithm 5 may produce a poor basis in many
+applications. Inspired by [23], we suggest using the power iteration technique
+to enhance the sketching algorithm by replacing A with (AA⊤)qA, where q
+is a positive integer. According to the SVD decomposition of matrix A, i.e.,
+A = USV ⊤, we know that (AA⊤)qA = US2q+1V ⊤. It can see that A and
+(AA⊤)qA have the same left and right singular vectors, but the latter has a
+faster decay rate of singular values, making its tail energy much smaller.
+Algorithm 7 Sketching algorithm with subspace power iteration (sub-
+Sketch)
+Require: matrix A ∈ Rm×n, sketch size parameters k, l, and integer q > 0
+Ensure: rank-k approximation ˆA = QX of A
+1: Ω ← randn(n, k), Ψ ← randn(l, m)
+2: Ω ← orth(Ω), Ψ⊤ ← orth(Ψ⊤)
+3: Y = AΩ, W = ΨA
+4: Q0 ← thinQR(Y )
+5: for j = 1, . . . , q do
+6:
+ˆYj = A⊤Qj−1
+7:
+( ˆQj, ∼) ← thinQR( ˆYj)
+8:
+Yj = A ˆQj
+9:
+(Qj, ∼) ← thinQR(Yj)
+10: end for
+11: Q = Qq
+12: X ← (ΨQ)†W
+Although power iteration can improve the accuracy of Algorithm 5 to some
+extent, it still suffers from a problem, i.e., during the execution with power
+iteration, the rounding errors will eliminate all information about the singular
+modes associated with the singular values. To address this issue, we propose an
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+13
+improved sketching algorithm by orthonormalizing the columns of the sample
+matrix between each application of A and A⊤, see Algorithm 7. When A is
+dense, the arithmetic cost of Algorithm 7 is O((q + 1)(k + l)mn + kl(m + n))
+flops. Numerical experiments show that a good approximation can achieve
+with a choice of 1 or 2 for subspace power iteration parameter [21].
+Algorithm 8 sub-Sketch-STHOSVD
+Require: tensor X ∈ RI1×I2×...×IN , targer rank (r1, r2, . . . , rN), processing
+order sp : {i1, i2, . . . , iN}, sketch size parameters {l1, l2, ..., lN}, and integer
+q > 0
+Ensure: Tucker approximation ˆ
+X = G ×1 U (1) ×2 U (2) . . . ×N U (N)
+1: G ← X
+2: for n = i1, i2, . . . , iN do
+3:
+(Q, X) ← sub-Sketch(G(n), rn, ln, q) (cf. Algorithm 7)
+4:
+U (n) ← Q
+5:
+G ← foldn(X)
+6: end for
+Using Algorithm 7 to compute the low-rank approximations of intermedi-
+ate matrices, we can obtain an improved sketching algorithm for STHOSVD,
+called sub-Sketch-STHOSVD, see Algorithm 8. The error-bound for Algorithm
+8 states in the following Theorem 5. Its proof is deferred in Appendix.
+Theorem 5 Let ˆ
+X = G ×1 U(1) ×2 U(2) . . . ×N U(N) be the Tucker approximation
+of a tensor X ∈ RI1×I2×...×IN obtained by the sub-Sketch-STHOSVD algorithm
+(i.e., Algorithm 8) with target rank rn < In, n = 1, 2, ..., N, sketch size parameters
+{l1, l2, ..., lN} and processing order p : {1, 2, . . . , N}. Let ̟k ≡
+σk+1
+σk
+denote the
+singular value gap, then
+E{Ωj}N
+j=1∥X − �
+X ∥2
+F ≤
+N
+�
+n=1
+(1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r4q) · τ 2
+̺+1(X(n))
+≤
+N
+�
+n=1
+(1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r4q)∥X − ˆ
+Xopt∥2
+F .
+Proof See Appendix.
+□
+5 Numerical experiments
+This section conducts numerical experiments with synthetic data and
+real-world data, including comparisons between the traditional THOSVD,
+STHOSVD algorithms, the R-STHOSVD algorithm proposed in [14], and our
+
+14
+Sketching Algorithms for Low-Rank Tucker Approximation
+proposed algorithms Sketch-STHOSVD and sub-Sketch-STHOSVD. Regard-
+ing the numerical settings, the oversampling parameter p = 5 is used in
+Algorithm 3, the sketch parameters ln = rn + 2, n = 1, 2, . . ., N, are used
+in Algorithms 6 and 8, and the power iteration parameter q = 1 is used in
+Algorithm 8.
+5.1 Hilbert tensor
+Hilbert tensor is a synthetic and supersymmetric tensor, with each entry
+defined as
+Xi1i2...in =
+1
+i1 + i2 + ... + in
+, 1 ≤ in ≤ In, n = 1, 2, ..., N.
+In the first experiment, we set N = 5 and In = 25, n = 1, 2, . . . , N. The target
+rank is chosen as (r, r, r, r, r), where r ∈ [1, 25]. Due to the supersymmetry of
+the Hilbert tensor, the processing order in the algorithms does not affect the
+final experimental results, and thus the processing order can be directly chosen
+as sp : {1, 2, 3, 4, 5}.
+0
+5
+10
+15
+20
+25
+Target rank
+10-15
+10-10
+10-5
+100
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+0
+5
+10
+15
+20
+25
+Target rank
+10-1
+100
+101
+102
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+Fig. 2 Results comparison on the Hilbert tensor with a size of 25 × 25 × 25 × 25 × 25 in
+terms of numerical error (left) and CPU time (right).
+The results of different algorithms are given in Figure 2. It shows that our
+proposed algorithms (i.e., Sketch-STHOSVD and sub-Sketch-STHOSVD) and
+algorithm R-STHOSVD outperform the algorithms THOSVD and STHOSVD.
+In particular, the error of the proposed algorithms Sketch-STHOSVD and sub-
+Sketch-STHOSVD is comparable to R-STHOSVD (see the left plot in Figure
+2), while they both use less CPU time than R-STHOSVD (see the right plot in
+Figure 2). This result demonstrates the excellent performance of the proposed
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+15
+algorithms and indicates that the two-sided sketching method and the subspace
+power iteration used in our algorithms can indeed improve the performance of
+STHOSVD algorithm.
+For a large-scale test, we use a Hilbert tensor with a size of 500×500×500
+and conduct experiments using ten different approximate multilinear ranks. We
+perform the tests ten times and report the algorithms’ average running time
+and relative error in Table 3 and Table 4, respectively. The results show that
+the randomized algorithms can achieve higher accuracy than the deterministic
+algorithms. The proposed Sketch-STHOSVD algorithm is the fastest, and the
+sub-Sketch-STHOSVD algorithm achieves the highest accuracy efficiently.
+Table 3 Results comparison in terms of the CPU time (in second) on the Hilbert tensor
+with a size of 500 × 500 × 500 as the target rank increases.
+Target rank
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(10,10,10)
+17.18
+7.49
+0.92
+0.86
+0.98
+(20,20,20)
+23.13
+8.87
+1.25
+1.05
+1.48
+(30,30,30)
+24.91
+9.35
+1.66
+1.53
+2.16
+(40,40,40)
+28.05
+10.41
+1.94
+1.44
+2.11
+(50,50,50)
+29.44
+11.39
+2.07
+1.67
+2.43
+(60,60,60)
+30.14
+11.07
+2.37
+1.90
+2.77
+(70,70,70)
+29.44
+11.18
+2.57
+2.10
+3.02
+(80,80,80)
+29.65
+12.30
+3.05
+2.54
+3.75
+(90,90,90)
+31.11
+12.80
+3.80
+2.80
+4.33
+(100,100,100)
+32.22
+13.51
+4.04
+3.07
+4.61
+Table 4 Results comparison in terms of the relative error on the Hilbert tensor with a
+size of 500 × 500 × 500 as the target rank increases.
+Target rank
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(10,10,10)
+2.7354e-06
+2.7347e-06
+2.7347e-06
+1.1178e-05
+2.7568e-06
+(20,20,20)
+1.1794e-12
+1.1793e-12
+1.1794e-12
+7.1408e-12
+1.2677e-12
+(30,30,30)
+4.6574e-15
+3.2739e-15
+3.2201e-15
+4.0641e-15
+2.0182e-15
+(40,40,40)
+4.4282e-15
+3.4249e-15
+2.8212e-15
+2.1562e-15
+1.7860e-15
+(50,50,50)
+4.1628e-15
+3.2342e-15
+2.6823e-15
+2.3205e-15
+1.8625e-15
+(60,60,60)
+4.1214e-15
+3.1271e-15
+2.3652e-15
+2.2920e-15
+1.7472e-15
+(70,70,70)
+4.1085e-15
+3.0000e-15
+2.1761e-15
+2.0499e-15
+1.6370e-15
+(80,80,80)
+4.0956e-15
+3.1350e-15
+1.8382e-15
+1.8209e-15
+1.6424e-15
+(90,90,90)
+4.0792e-15
+3.3742e-15
+1.8102e-15
+1.7193e-15
+1.5264e-15
+(100,100,100)
+4.0390e-15
+3.0571e-15
+1.7323e-15
+1.6304e-15
+1.4957e-15
+5.2 Sparse tensor
+In this experiment, we test the performance of different algorithms on a sparse
+tensor X ∈ R200×200×200, i.e.,
+X =
+10
+�
+i=1
+γ
+i2 xi ◦ yi ◦ zi +
+200
+�
+i=11
+1
+i2 xi ◦ yi ◦ zi.
+Where xi, yi, zi ∈ Rn are sparse vectors all generated using the sprand com-
+mand in MATLAB with 5% nonzeros each, and γ is a user-defined parameter
+
+16
+Sketching Algorithms for Low-Rank Tucker Approximation
+20
+40
+60
+80
+100
+Target rank
+10-3
+10-2
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-4
+10-3
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-6
+10-5
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+Fig. 3 Results comparison on a sparse tensor with a size of 200 × 200 × 200 in terms of
+numerical error (first row) and CPU time (second row).
+which determines the strength of the gap between the first ten terms and the
+rest terms. The target rank is chosen as (r, r, r), where r ∈ [20, 100]. The exper-
+imental results show in Figure 3, in which three different values γ = 2, 10, 200
+are tested. The increase of gap means that the tail energy will be reduced, and
+the accuracy of the algorithms will be improved. Our numerical experiments
+also verified this result.
+Figure 3 demonstrates the superiority of the proposed sketching algo-
+rithms. In particular, we see that the proposed Sketch-STHOSVD is the fastest
+algorithm, with a comparable error against R-STHOSVD; the proposed sub-
+Sketch-STHOSVD can reach the same accuracy as the STHOSVD algorithm
+but in much less CPU time; and the proposed sub-Sketch-STHOSVD achieves
+much better low-rank approximation than R-STHOSVD with similar CPU
+time.
+Now we consider the influence of noise on algorithms’ performance. Specif-
+ically, the sparse tensor X with noise is designed in the same manner as in
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+17
+20
+40
+60
+80
+100
+Target rank
+0.19
+0.195
+0.2
+0.205
+0.21
+0.215
+0.22
+0.225
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+0.045
+0.05
+0.055
+0.06
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+1.45
+1.5
+1.55
+1.6
+1.65
+1.7
+1.75
+1.8
+1.85
+1.9
+1.95
+Relative Error
+10-3
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+20
+40
+60
+80
+100
+Target rank
+10-1
+100
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+Fig. 4 Results comparison on a 200×200×200 sparse tensor with noise in terms of numerical
+error (first row) and CPU time (second row).
+[24], i.e.,
+ˆ
+X = X + δK,
+where K is a standard Gaussian tensor and δ is used to control the noise
+level. Let δ = 10−3 and keep the rest parameters the same as the settings
+in the previous experiment. The relative error and running time of different
+algorithms are shown in Figure 4. In Figure 4, we see that noise indeed affects
+the accuracy of the low-rank approximation, especially when the gap is small.
+However, the influence of noise does not change the conclusion obtained on
+the case without noise. The accuracy of our sub-Sketch-STHOSVD algorithm
+is the highest among the randomized algorithms. As γ increases, sub-Sketch-
+STHOSVD can achieve almost the same accuracy as that of THOSVD and
+STHOSVD in a comparable CPU time against R-STHOSVD.
+
+18
+Sketching Algorithms for Low-Rank Tucker Approximation
+5.3 Real-world data tensor
+In this experiment, we test the performance of different algorithms on a colour
+image, called HDU picture1, with a size of 1200 × 1800 × 3. We also evaluate
+the proposed sketching algorithms on the widely used YUV Video Sequences2.
+Taking the ‘hall monitor’ video as an example and using the first 30 frames, a
+three order tensor with a size of 144 × 176 × 30 is then formed for this test.
+Firstly, we conduct an experiment on the HDU picture with target rank
+(500, 500, 3), and compare the PSNR and CPU time of different algorithms.
+The experimental result is shown in Figure 5, which shows that the PSNR
+of sub-Sketch-STHOSVD, THOSVD and STHOSVD is very similar (i.e.,
+∼ 40) and that sub-Sketch-STHOSVD is more efficient in terms of CPU
+time. R-STHOSVD and Sketch-STHOSVD are also very efficient compared to
+sub-Sketch-STHOSVD; however, the PSNR they achieve is 5 dB less than sub-
+Sketch-STHOSVD. Then we conduct separate numerical experiments on the
+HDU picture and the ‘hall monitor’ video clip as the target rank increases, and
+compare these algorithms in terms of the relative error, CPU time and PSNR,
+see Figure 6 and Figure 7. These experimental results again demonstrate
+the superiority (i.e., low error and good approximation with high efficiency)
+of the proposed sub-Sketch-STHOSVD algorithm in computing the Tucker
+decomposition approximation.
+Original
+THOSVD (2.62; 40.61)
+STHOSVD (1.89; 40.65)
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(0.61; 34.72)
+(0.55; 34.63)
+(0.84; 39.97)
+Fig. 5 Results comparison on a HDU picture with a size of 1200 × 1800 × 3 in terms of
+PSNR (i.e., peak signal-to-noise ratio) and CPU time. The target rank is (500,500,3). The
+two values in e.g. (2.62; 40.61) represent the CPU time and the PSNR, respectively.
+In the last experiment, a larger-scale real-world tensor data is used. We
+choose a color image (called the LONDON picture) with a size of 4775×7155×3
+as the test image and consider the influence of noise. The LONDON picture
+1https://www.hdu.edu.cn/landscape
+2http://trace.eas.asu.edu/yuv/index.html
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+19
+0
+200
+400
+600
+800
+1000
+Target rank
+-11
+-10
+-9
+-8
+-7
+-6
+-5
+-4
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+0
+200
+400
+600
+800
+1000
+Target rank
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+0
+200
+400
+600
+800
+1000
+Target rank
+20
+25
+30
+35
+40
+45
+50
+55
+PSNR
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+Fig. 6 Results comparison on a HDU picture with size of 1200 × 1800 × 3 in terms of
+numerical error (left), CPU time (middle) and PSNR (right). The HDU picture is with target
+rank (r, r, 3), r ∈ [50, 1000].
+0
+20
+40
+60
+80
+100
+Target rank
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+Relative Error
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+0
+20
+40
+60
+80
+100
+Target rank
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+0.055
+Running Time
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+0
+20
+40
+60
+80
+100
+Target rank
+10
+15
+20
+25
+30
+35
+PSNR
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+Fig. 7 Results comparison on the ‘hall monitor’ grey video with size of 144 × 176 × 30 in
+terms of numerical error (left), CPU time (middle) and PSNR (right). The ‘hall monitor’
+grey video is with target rank (r, r, 10), r ∈ [5, 100].
+with white Gaussian noise is generated using the awgn(X,SNR) built-in function
+in MATLAB. We set the target rank as (50,50,3) and SNR to 20. The results
+comparisons without and with white Gaussian noise are respectively shown in
+Figure 8 and Figure 9 in terms of the CPU time and PSNR. Moreover, we also
+test the algorithms on the LONDON picture as the target rank increases. The
+results regarding the relative error, the CPU time and the PSNR are reported
+in Tables 5, 6 and 7, respectively. On the whole, the results again show the
+consistent performance of the proposed methods.
+
+20
+Sketching Algorithms for Low-Rank Tucker Approximation
+Original
+THOSVD (154.95; 24.07)
+STHOSVD (49.34; 24.09)
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(1.29; 21.27)
+(1.17; 21.09)
+(1.29; 23.65)
+Fig. 8 Results comparison on LONDON picture with a size of 4775 × 7155 × 3 in terms of
+CPU time and PSNR. The target rank is (50,50,3).
+Noisy picture(PSNR=16.92)
+THOSVD (160.59; 20.54)
+STHOSVD (50.16; 20.54)
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(1,25; 19.37)
+(1.15; 19.25)
+(1.45; 20.45)
+Fig. 9 Results comparison on LONDON picture with a size of 4775 × 7155 × 3 and white
+Gaussian noise in terms of CPU time and PSNR. The target rank is (50,50,3).
+In summary, the numerical results show the superiority of the sub-sketch
+STHOSVD algorithm for large-scale tensors with or without noise. We can see
+that sub-Sketch-STHOSVD could achieve close approximations to that of the
+deterministic algorithms in a time similar to other randomized algorithms.
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+21
+Table 5 Results comparison in terms of the relative error on the LONDON picture with a
+size of 4775 × 7155 × 3 as the target rank increases.
+Target rank
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(10,10,10)
+0.019037
+0.019025
+0.031000
+0.040006
+0.020756
+(20,20,20)
+0.012669
+0.012644
+0.023467
+0.027398
+0.013703
+(30,30,30)
+0.010168
+0.010124
+0.018354
+0.020451
+0.010965
+(40,40,40)
+0.008630
+0.008599
+0.015792
+0.017029
+0.009443
+(50,50,50)
+0.007576
+0.007532
+0.013917
+0.015333
+0.008286
+(60,60,60)
+0.006778
+0.006710
+0.012967
+0.013589
+0.007359
+(70,70,70)
+0.006119
+0.006049
+0.011813
+0.011886
+0.006687
+(80,80,80)
+0.005532
+0.005491
+0.010658
+0.011148
+0.006123
+(90,90,90)
+0.005076
+0.005023
+0.010018
+0.010378
+0.005602
+(100,100,100)
+0.004669
+0.004619
+0.009249
+0.009578
+0.005172
+Table 6 Results comparison in terms of the CPU time (in second) on the LONDON
+picture with a size of 4775 × 7155 × 3 as the target rank increases.
+Target rank
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(10,10,10)
+156.13
+49.22
+0.94
+0.99
+1.12
+(20,20,20)
+165.22
+77.64
+1.24
+1.48
+1.56
+(30,30,30)
+241.11
+76.57
+1.69
+1.39
+1.69
+(40,40,40)
+242.08
+74.25
+1.57
+1.45
+1.68
+(50,50,50)
+268.71
+72.85
+1.51
+1.45
+1.80
+(60,60,60)
+265.52
+77.80
+1.75
+1.51
+2.26
+(70,70,70)
+241.95
+77.82
+1.93
+1.78
+2.24
+(80,80,80)
+264.86
+73.53
+1.86
+1.74
+2.31
+(90,90,90)
+274.73
+72.67
+1.93
+1.83
+2.16
+(100,100,100)
+283.88
+86.42
+2.24
+2.20
+2.46
+Table 7 Results comparison in terms of the PSNR on the LONDON picture with a size
+of 4775 × 7155 × 3 as the target rank increases.
+Target rank
+THOSVD
+STHOSVD
+R-STHOSVD
+Sketch-STHOSVD
+sub-Sketch-STHOSVD
+(10,10,10)
+20.06
+20.07
+17.96
+16.86
+19.70
+(20,20,20)
+21.84
+21.84
+19.18
+18.51
+21.50
+(30,30,30)
+22.79
+22.81
+20.25
+19.78
+22.46
+(40,40,40)
+23.50
+23.52
+20.90
+20.57
+23.11
+(50,50,50)
+24.07
+24.09
+21.45
+21.03
+23.68
+(60,60,60)
+24.55
+24.60
+21.76
+21.55
+24.20
+(70,70,70)
+25.00
+25.05
+22.16
+22.13
+24.61
+(80,80,80)
+25.43
+25.47
+22.61
+22.41
+25.00
+(90,90,90)
+25.81
+25.85
+22.87
+22.72
+25.38
+(100,100,100)
+26.17
+26.22
+23.22
+23.07
+25.73
+6 Conclusion
+In this paper we proposed efficient sketching algorithms, i.e., Sketch-
+STHOSVD and sub-Sketch-STHOSVD, to calculate the low-rank Tucker
+approximation of tensors by combining the two-sided sketching technique with
+the STHOSVD algorithm and using the subspace power iteration. Detailed
+error analysis is also conducted. Numerical results on both synthetic and real-
+world data tensors demonstrate the competitive performance of the proposed
+algorithms in comparison to the state-of-the-art algorithms.
+Acknowledgements
+We would like to thank the anonymous referees for their comments and sug-
+gestions on our paper, which lead to great improvements of the presentation.
+
+22
+Sketching Algorithms for Low-Rank Tucker Approximation
+G. Yu’s work was supported in part by National Natural Science Foundation
+of China (No. 12071104) and Natural Science Foundation of Zhejiang Province
+(No. LD19A010002).
+Appendix
+Lemma 1 [[25], Theorem 2] Let ̺ < k − 1 be a positive natural number and Ω ∈
+Rk×n be a Gaussian random matrix. Suppose Q is obtained from Algorithm 7. Then
+∀A ∈ Rm×n, we have
+EΩ∥A − QQ⊤A∥2
+F ≤ (1 + f(̺, k)̟4q
+k ) · τ 2
+̺+1(A).
+(4)
+Lemma 2 [[22], Lemma A.3] Let A ∈ Rm×n be an input matrix and ˆA = QX
+be the approximation obtained from Algorithm 7. The approximation error can be
+decomposed as
+∥A − ˆA∥2
+F = ∥A − QQ⊤A∥2
+F + ∥X − Q⊤A∥2
+F .
+(5)
+Lemma 3 [[22], Lemma A.5] Assume Ψ ∈ Rl×n is a standard normal matrix
+independent from Ω. Then
+EΨ∥X − Q⊤A∥2
+F = f(k, l) · ∥A − QQ⊤A∥2
+F .
+(6)
+The error-bound for Algorithm 7 can be shown in Lemma 4 below.
+Lemma 4 Assume the sketch size parameter satisfies l > k + 1. Draw random
+test matrices Ω ∈ Rn×k and Ψ∈ Rl×m independently from the standard normal
+distribution. Then the rank-k approximation ˆA obtained from Algorithm 7 satisfies
+E ∥ A − ˆA ∥2
+F ≤ (1 + f(k, l)) · min
+̺<k−1(1 + f(̺, k)̟k
+4q) · τ 2
+̺+1(A).
+Proof Using equations (4), (5) and (6), we have
+E ∥ A − ˆA ∥2
+F = EΩ∥A − QQ⊤A∥2
+F + EΩEΨ∥X − Q⊤A∥2
+F
+= (1 + f(k, l)) · EΩ∥A − QQ⊤A∥2
+F
+≤ (1 + f(k, l)) · (1 + f(̺, k)̟k
+4q) · τ 2
+̺+1(A).
+After minimizing over eligible index ̺ < k − 1, the proof is completed.
+□
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+23
+We are now in the position to prove Theorem 5. Combining Theorem 2
+and Lemma 4, we have
+E{Ωj}N
+j=1∥X − �
+X ∥2
+F
+=
+N
+�
+n=1
+E{Ωj}N
+j=1∥ ˆ
+X (n−1) − ˆ
+X (n)∥2
+F
+=
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥ ˆ
+X (n−1) − ˆ
+X (n)∥2
+F
+�
+=
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥G(n−1) ×n−1
+i=1 U (i)×n(I − U (n)U (n)⊤)∥2
+F
+�
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1
+�
+EΩn∥(I − U (n)U (n)⊤)G(n−1)
+(n)
+)∥2
+F
+�
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1 (1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r
+4q)
+In
+�
+i=rn+1
+σ2
+i (G(n−1)
+(n)
+)
+≤
+N
+�
+n=1
+E{Ωj}n−1
+j=1 (1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r4q)∆2
+n(X)
+=
+N
+�
+n=1
+(1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r
+4q)∆2
+n(X)
+≤
+N
+�
+n=1
+(1 + f(rn, ln)) ·
+min
+̺n<rn−1(1 + f(̺n, rn)̟r
+4q)∥X − ˆ
+Xopt∥2
+F ,
+which completes the proof of Theorem 5.
+References
+[1] Comon, P.: Tensors: A brief introduction. IEEE Signal Processing Maga-
+zine. 31(3), 44-53(2014)
+[2] Hitchcock, F. L.: Multiple Invariants and Generalized Rank of a P-
+Way Matrix or Tensor. Journal of Mathematics and Physics. 7(1-4),
+39-79(1928)
+[3] Kiers, H. A. L.: Towards a standardized notation and terminology in
+multiway analysis. Journal of Chemometrics Society. 14(3), 105-122(2000)
+[4] Tucker, L. R.: Implications of factor analysis of three-way matrices for
+measurement of change. Problems in measuring change. 15, 122-137(1963)
+
+24
+Sketching Algorithms for Low-Rank Tucker Approximation
+[5] Tucker, L. R.: Some mathematical notes on three-mode factor analysis.
+Psychometrika. 31(3), 279-311(1966)
+[6] De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singu-
+lar value decomposition. SIAM journal on Matrix Analysis Applications.
+21(4), 1253-1278(2000)
+[7] Hackbusch, W., K¨uhn, S.: A new scheme for the tensor representation.
+Journal of Fourier analysis applications. 15(5), 706-722(2009)
+[8] Grasedyck, L.: Hierarchical Singular Value Decomposition of Tensors.
+SIAM journal on Matrix Analysis Applications. 31(4), 2029-2054 (2010)
+[9] Oseledets, I. V.: Tensor-train decomposition. SIAM Journal on Scientific
+Computing. 33(5), 2295-2317(2011)
+[10] De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and
+rank-(r1, r2,...,rn) approximation of higher-order tensors. SIAM journal
+on Matrix Analysis Applications. 21(4), 1324-1342(2000)
+[11] Vannieuwenhoven, N., Vandebril, R., Meerbergen, K.: A new truncation
+strategy for the higher-order singular value decomposition. SIAM Journal
+on Scientific Computing. 34(2), A1027-A1052(2012)
+[12] Zhou, G., Cichocki, A., Xie, S.: Decomposition of big tensors with low
+multilinear rank. arXiv preprint, arXiv:1412.1885(2014)
+[13] Che, M., Wei, Y.: Randomized algorithms for the approximations of
+Tucker and the tensor train decompositions. Advances in Computational
+Mathematics. 45(1), 395-428(2019)
+[14] Minster, R., Saibaba, A. K., Kilmer, M. E.: Randomized algorithms for
+low-rank tensor decompositions in the Tucker format. SIAM Journal on
+Mathematics of Data Science. 2(1), 189-215 (2020)
+[15] Che, M., Wei, Y., Yan, H.: The computation of low multilinear rank
+approximations of tensors via power scheme and random projection.
+SIAM Journal on Matrix Analysis Applications. 41(2), 605-636 (2020)
+[16] Che, M., Wei, Y., Yan, H.: Randomized algorithms for the low multilin-
+ear rank approximations of tensors. Journal of Computational Applied
+Mathematics. 390(2), 113380(2021)
+[17] Sun, Y., Guo, Y., Luo, C., Tropp, J., Udell, M.: Low-rank tucker approx-
+imation of a tensor from streaming data. SIAM Journal on Mathematics
+of Data Science. 2(4), 1123-1150(2020)
+
+Sketching Algorithms for Low-Rank Tucker Approximation
+25
+[18] Tropp, J. A., Yurtsever, A., Udell, M., Cevher, V.: Streaming low-rank
+matrix approximation with an application to scientific simulation. SIAM
+Journal on Scientific Computing. 41(4), A2430-A2463(2019)
+[19] Malik, O. A., Becker, S.: Low-rank tucker decomposition of large tensors
+using tensorsketch. Advances in neural information processing systems.
+31, 10116-10126 (2018)
+[20] Ahmadi-Asl, S., Abukhovich, S., Asante-Mensah, M. G., Cichocki, A.,
+Phan, A. H., Tanaka, T.: Randomized algorithms for computation of
+Tucker decomposition and higher order SVD (HOSVD). IEEE Access. 9,
+28684-28706(2021)
+[21] Halko, N., Martinsson, P.-G., Tropp, J. A.: Finding structure with ran-
+domness: Probabilistic algorithms for constructing approximate matrix
+decompositions. SIAM review. 53(2), 217-288 (2011)
+[22] Tropp, J. A., Yurtsever, A., Udell, M., Cevher, V.: Practical sketching
+algorithms for low-rank matrix approximation. SIAM Journal on Matrix
+Analysis Applications. 38(4), 1454-1485(2017)
+[23] Rokhlin, V., Szlam, A., Tygert, M.: A randomized algorithm for princi-
+pal component analysis. SIAM Journal on Matrix Analysis Applications,
+31(3), 1100-1124(2009)
+[24] Xiao, C., Yang, C., Li, M.: Efficient Alternating Least Squares Algorithms
+for Low Multilinear Rank Approximation of Tensors. Journal of Scientific
+Computing. 87(3), 1-25(2021)
+[25] Zhang, J., Saibaba, A. K., Kilmer, M. E., Aeron, S.: A randomized tensor
+singular value decomposition based on the t-product. Numerical Linear
+Algebra with Applications. 25(5), e2179(2018)
+

99FJT4oBgHgl3EQfpCw2/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:99160d4abd7469d43569b83852673022f7f391d8e721189d99b48392a28ab396
+size 170524

9tAyT4oBgHgl3EQfQ_bX/content/tmp_files/2301.00059v1.pdf.txt

@@ -0,0 +1,1576 @@
+1 
+ 
+Describing NMR chemical exchange by effective phase diffusion approach 
+Guoxing Lin* 
+Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, USA 
+ 
+*Email: [email protected] 
+ 
+Abstract 
+        This paper proposes an effective phase diffusion method to analyze chemical exchange in nuclear 
+magnetic resonance (NMR). The chemical exchange involves spin jumps around different sites where the 
+spin angular frequencies vary, which leads to a random phase walk viewed from the rotating frame 
+reference. Therefore, the random walk in phase space can be treated by the effective phase diffusion 
+method. Both the coupled and uncoupled phase diffusions are considered; additionally, it includes normal 
+diffusion as well as fractional diffusion. Based on these phase diffusion equations, the line shape of NMR 
+exchange spectrum can be analyzed. By comparing these theoretical results with the conventional theory, 
+this phase diffusion approach works for fast exchange, ranging from slightly faster than intermediate 
+exchange to very fast exchange. For normal diffusion models, the theoretically predicted curves agree 
+with those predicted from traditional models in the literature, and the characteristic exchange time 
+obtained from phase diffusion with a fixed jump time is the same as that obtained from the conventional 
+model. However, the phase diffusion with a monoexponential time distribution gives a characteristic 
+exchange time constant which is half of that obtained from the traditional model. Additionally, the 
+fractional diffusion obtains a significantly different line shape than that predicted based on normal 
+diffusion.  
+Keywords: NMR, chemical exchange, Mittag-Leffler function, phase diffusion 
+ 
+1. 
+Introduction 
+       Chemical exchange is a powerful nuclear magnetic resonance (NMR) technique to detect dynamics 
+behavior in biological and polymer systems at the atomic level [1,2,3,4]. Chemical exchange NMR 
+monitors spin jumping around different environmental sites due to changes in conformational or chemical 
+states. In chemical exchange, the angular frequency of the spin precession changes, which results in 
+observable line shape changes in the NMR spectrum. Although chemical exchange has been a established 
+tool, theoretical developments are still needed to better understand the chemical exchange NMR, 
+particularly for complex systems.  
+      The characteristic exchange time could follow a complicated distribution. Many theoretical models 
+have been developed to analyze chemical exchanges in NMR [3, 5, 6]. The two-site exchange model based 
+on the modified-Bloch equation successfully interprets many NMR lines hape [5], where the jump time in 
+the exchange equation is a fixed constant. However, in a real system, the exchange time could follow a 
+distribution such as the exponential function in the Gaussian exchange model reported in [7]. 
+Additionally, a complex exchange distribution could exist in complicated conformational change or 
+diffusion-induced exchange, such as Xenon diffusing in the heterogeneous system [8].   In a complex 
+system, a monoexponential time distribution may not be sufficient to explain the dynamics behavior.  For 
+complicated system  the time distribution function could be the Mittag-Leffler function (MLF) 𝐸𝛼 (− (
+𝑡
+𝜏)
+𝛼
+)  
+[9,10], or a stretched exponential function (SEF) exp (− (
+𝑡
+𝜏)
+𝛼
+) , where α is the time-fractional derivative 
+order, and 𝜏 is the characteristic time. The MLF 𝐸𝛼(−𝑡𝛼) = ∑
+(−𝑡𝛼)𝑛
+Γ(𝑛𝛼+1)
+∞
+𝑛=0
+, can be reduced to a SEF 
+
+2 
+ 
+exp (−
+𝑡𝛼
+Γ(1+𝛼)) when 𝑡 is small. The SEF exp (− (
+𝑡
+𝜏)
+𝛼
+) is the same as the Kohlrausch-Williams-Watts (KWW) 
+function [11,12,13,14]], a well-known time correlation function in macromolecular systems. The Mittag-
+Leffler function-based distribution is heavy-tailed. Mittag Leffler function has been employed to analyze 
+anomalous NMR dynamics processes such as PFG anomalous diffusion [15,16], and anomalous NMR 
+relaxation [17,18,19,20]. Currently chemical theories of NMR are still difficult to handle these complex 
+distributions.    
+       A phase diffusion method is proposed in this paper to explain the NMR chemical exchange. Most 
+current methods are real space approaches, such as the modified Bloch exchange equations [3,4,6], while 
+the Gaussian exchange model is a phase space method based on evaluating the accumulated phase 
+variance for the random phase process during the exchange process. As the spin phase in the chemical 
+exchange undergoes a random walk in the rotating frame reference, an effective phase diffusion method 
+will be employed in this paper to analyze the exchange. Effective phase diffusion has been applied to 
+analyze PFG diffusion and NMR relaxation. It has advantages over the traditional methods: It can provide 
+the exact phase distribution that cannot be obtained by conventional real space theoretical method; 
+additionally, the NMR signal can be directly obtained from vector sum by Fourier transform in phase 
+space, which makes the analysis intuitive and often simplifies the solving process; furthermore, the phase 
+diffusion method could be straightforwardly applied to anomalous dynamics process based on fractional 
+diffusion [18, 21, 22, 23,24,25].   Both the normal and fractional phase diffusion are considered, where the 
+exchange time can be a simple constant or a certain type of distribution. 
+      Additionally, each the individual phase jump length is proportional to the jump time and the angular 
+frequency. Because both the jump time and angular frequency fluctuate and obey certain types of 
+distributions, the distribution of phase jump length could be either strongly or weakly correlated to the 
+jump time distribution. The phase walk with weak phase time correlation can be treated by the uncoupled 
+diffusion, while the strong correlation may require coupled diffusion model [26]. The traditional 
+uncoupled diffusion has been successful in explaining many transport phenomena. However, it is 
+insufficient to account for the divergence of the second moment of Levy flight processes [26], where a 
+coupled diffusion is needed. 
+    The rest of the paper is organized as follows. Section 2.1 treats the simplest normal diffusion with a 
+fixed jump time: The obtained exchange time agrees well with the traditional two-site exchange. While 
+the phase diffusion with jump time distribution is presented in Section 2.2: Firstly, the general expressions 
+for phase evolution in chemical exchange are derived in Section 2.2.1; secondly, the normal diffusion is 
+presented in Section 2.2.2, with both uncoupled and coupled diffusion, and it is found that the exchange 
+time constant is two-time faster than that of the traditional model and the fixed jump time diffusion result; 
+thirdly, the fractional diffusion with MLF based jump time distribution is derived in Section 2.2.3, where 
+the uncoupled diffusion is handled by time-fractional diffusion equation, and the coupled fractional 
+diffusion is handled by coupled random walk model [27]. The results here give additional insights into 
+the NMR chemical exchange, which could improve the analysis of NMR and magnetic resonance imaging 
+(MRI) experiments, particularly in complicated systems.  
+2. Theory 
+ The chemical exchange occurs when the spin jumps among different sites where the spin precession 
+frequencies are different [1,2,5]. The precession frequencies of the spin moment are proportional to the 
+intensity of the local magnetic field, which is affected by the surrounding electron cloud and nearby spin 
+moments [2]. For simplicity, we consider only the basic exchange between two sites with equal 
+populations [3,5] in this paper, neglecting the relaxation effect. The average precession angular frequencies 
+for these two sites are arbitrarily set as  𝜔1 and 𝜔2 respectively, with 𝜔1  < 𝜔2 and 
+
+3 
+ 
+ ∆𝜔 = 𝜔2 − 𝜔1.  
+ If the angular frequency of the rotating frame reference is set as  
+𝜔1+𝜔2
+2
+;  the two sites have relative 
+angular frequencies -𝜔0 and  𝜔0, respectively, 𝜔0 =
+𝜔2−𝜔1
+2
+.   
+ 
+ The phase of spin undergoes chemical exchange during a time interval 𝜏 changes either by 𝜔0𝜏 or 
+-𝜔0𝜏  depending on the site. From the traditional exchange equations for chemical exchange, the spin 
+always jumps to a site with a different angular frequency after a time interval 𝜏.   Here, the choice for 
+the next sites is assumed to be random; the next site's angular frequency could be either the same or 
+different. This assumption may be more realistic; for instance, after a time interval 𝜏, a spin may 
+successfully jump to another site or return to the original site; or in a heterogeneous system, the spin 
+moves to a similar environment with the same frequency or a different environment with a different 
+frequency.   The random phase jump can be viewed as a random walk process in phase space and 
+analyzed by phase diffusion [16,17].   Both the normal and fractional phase diffusion will be 
+considered in the following. 
+2.1 Simple normal diffusion with a fixed jump time 
+     If the jump time interval 𝜏 is a constant, the random phase jumps with average jump length ∆𝜙 equaling  
+−𝜔0𝜏 or 𝜔0𝜏. The effective phase diffusion constant 𝐷𝜙,𝑠 for such a simple normal diffusion can be obtained 
+by [16] 
+ 
+ 
+ 
+𝐷𝜙,𝑠 =
+〈(∆𝜙)2〉
+2𝜏
+=
+(𝜔0𝜏)2
+2𝜏
+=
+𝜔02
+2 τ, 
+ 
+ 
+ 
+(1) 
+and the normal phase diffusion equation can be described by [16,17] 
+𝑑𝑃(𝜙,𝑡)
+𝑑𝑡
+= 𝐷𝜙,𝑠Δ𝑃(𝜙, 𝑡), 
+ 
+ 
+ 
+ 
+( 2) 
+where 𝜙 is the phase and 𝑃(𝜙, 𝑡) is the probability density function of spin at time t with 𝜙. The solution 
+of Eq. (3) is [16] 
+𝑃(𝜙, 𝑡) =
+1
+√4𝜋𝐷𝜙,𝑠𝑡 𝑒𝑥𝑝 [−
+𝜙2
+4𝐷𝜙,𝑠𝑡].   
+ 
+ 
+(3) 
+The total magnetization 𝑀(𝑡) is obtained by  
+𝑀(𝑡) = ∫
+𝑑𝜙
+∞
+−∞
+𝑒𝑖𝜙𝑃(𝜙, 𝑡) = 𝑒𝑥𝑝(−𝐷𝜙,𝑠𝑡),  
+ 
+ 
+(4) 
+which is a time domain signal. By Fourier transform, we have the frequency domain signal  
+𝑆(𝜔) = 
+𝐷𝜙,𝑠
+𝐷𝜙,𝑠2+𝜔2 =
+𝜔02
+2 τ
+(
+𝜔02
+2 τ)
+2
++𝜔2
+.   
+ 
+ 
+ 
+(5) 
+Note that both 𝜔 and 𝜔0 are the angular frequencies in the rotating frame reference.  
+2.2 Diffusion with waiting time distribution  
+2.2.1 General expressions for phase evolution in chemical exchange 
+     A more realistic exchange time should follow a certain type of time distribution function 𝜑(𝑡), which is 
+often related to the time correlation function 𝐺(𝑡) by 𝜑(𝑡) = −
+𝑑𝐺(𝑡)
+𝑑𝑡 . A commonly used simple time 
+correlation function is the mono exponential distribution; in contrast, in a complicated system, it can be a 
+Mittage Leffler function [20]  or stretched exponential function such as the KWW function [11-14].  
+    For a spin starting jumps at a time 𝑡′ from the site with frequency  𝜔𝑖,0, the probability of acquiring 
+phase  𝜔𝑖,0𝑡′+ 𝜙 at time t is 
+
+4 
+ 
+𝑃𝜔𝑖,0(𝜙, 𝑡) = 𝜑(𝑡′)𝑃(𝜙, 𝑡 − 𝑡′),     
+ 
+ 
+(6) 
+where the phase change 𝜔𝑖,0𝑡′ is obtained from time 0 to time 𝑡′  when the spin stays immobile at the site, 
+and 𝑃(𝜙, 𝑡 − 𝑡′) is the phase PDF resulting from the diffusion, or random walk in the phase space during  
+𝑡 − 𝑡′. Summing all possible magnetization vectors with different phase 𝜔𝑖,0𝑡′ + 𝜙, at time t, the net 
+magnetization 𝑀𝜔𝑖,0(𝑡′, 𝑡) contributed from these spins beginning to jump randomly from time 𝑡′ is 
+𝑀𝜔𝑖,0(𝑡′, 𝑡) = ∫
+𝑑𝜙
+∞
+−∞
+𝑒𝑖𝜔𝑖,0𝑡′+𝜙𝑃𝜔𝑖,0(𝜙, 𝑡 − 𝑡′) = 𝑒𝑖𝜔𝑖,0𝑡′𝜑(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1   , 
+(7) 
+where 
+𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 = ∫
+𝑑𝜙
+∞
+−∞
+𝑒𝑖𝑘𝜙𝑃(𝜙, 𝑡 − 𝑡′).  
+ 
+ 
+(8) 
+The total magnetization from all spins in the systems at time t is   
+𝑀(𝑡) = ∫ 𝑑𝑡′
+𝑡
+0
+ ∑ 𝑝𝑖𝑀𝜔𝑖,0(𝑡′, 𝑡)
+𝑖
+,   
+ 
+ 
+(9) 
+where 𝑝𝑖 is the population of spins at site i. For simplicity, only exchange with two equal population sites 
+will be considered here;  let 𝜔1,0 = −𝜔0, 𝜔2,0 = 𝜔0 and all the subindexes i will be dropped out throughout 
+the rest of the paper.   For a two-site system with equal populations 𝑝1 = 𝑝2 = 
+1
+2, 
+𝑀(𝑡) = ∫ 𝑑𝑡′
+𝑡
+0
+ 1
+2 [𝑀−𝜔0(𝑡′, 𝑡) + 𝑀𝜔0(𝑡′, 𝑡)] 
+               
+        = ∫ 𝑑𝑡′
+𝑡
+0
+ 
+1
+2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′] 𝜑(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 
+         = ∫ 𝑑𝑡′
+𝑡
+0
+  𝐵(𝑡′)𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 ,                                                  (10a) 
+where  
+𝐵(𝑡′) =
+1
+2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′]𝜑(𝑡′).                                           (10b) 
+Eq. (10a) involves the convolution of 𝐵(𝑡′) and 𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1. In Laplace representation [25],  
+𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1;     
+ 
+ 
+ 
+(11) 
+in many cases (the coupled diffusion in the paper), the frequency domain signal can be obtained by the 
+Fourier transform of  𝑀(𝑡): 
+𝑆(𝜔) = ∫
+𝑒𝑖𝜔𝑡𝑀(𝑡)𝑑𝑡
+∞
+0
+= 𝐵(𝜔)𝑝(𝑘, 𝜔)|𝑘=1 
+   
+(12) 
+2.2.2 Normal diffusion with monoexponential distribution function 
+Now, let us consider if the jump time follows a monoexponential distribution  𝜑(𝑡) described by 
+[25] 
+𝜑(𝑡) =
+1
+𝜏 exp (−
+𝑡′
+𝜏 ) ,   
+ 
+ 
+ 
+ 
+(13) 
+whose Laplace representation is  
+𝜑(𝑠) =
+1
+𝑠𝜏+1   
+ 
+ 
+ 
+ 
+ 
+ (14) 
+According to Eq. (10), 
+ 𝐵(𝑡) =
+1
+2 [𝑒−𝑖𝜔0𝑡′ + 𝑒𝑖𝜔0𝑡′]
+1
+𝜏 exp (−
+𝑡′
+𝜏 ),    
+ 
+(15a) 
+whose Laplace representation is [25] 
+
+5 
+ 
+𝐵(𝑠) = 
+1
+2 [
+1
+𝜏(𝑠−𝑖𝜔0)+1 +
+1
+𝜏(𝑠+𝑖𝜔0)+1] ≈
+1
+1+𝜔02𝜏2+𝜏𝑠(1−𝜔02𝜏2) =
+1
+1+𝜔02𝜏2
+1+𝑠
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2
+.   
+(15b) 
+I. 
+Uncoupled normal diffusion 
+The spin angular frequency is often affected by a random fluctuating magnetic field, which is produced 
+by surrounding spins undergoing the thermal motion [**]; additionally, the angular frequency could be 
+affected by the electron cloud change during the exchange process;  further, the chemical exchange may 
+take place because the spin moves among different domains in a heterogeneous system where the 
+frequency fluctuating around positive or negative 𝜔0 . This angular frequency can be denoted as 𝜔, and 
+the  average of its absolute value is 〈|𝜔|〉 = 𝜔0. Because 𝜔 is randomly fluctuating, the individual random 
+phase jump ∆𝜙 = 𝜔��𝑗𝑢𝑚𝑝  randomly fluctuates for each jump time 𝜏𝑗𝑢𝑚𝑝; the space and time uncoupled 
+phase diffusion could be applied to treat the phase random walk, a more complicated coupled diffusion 
+will be considered in Section in subsequence.   For an uncoupled diffusion,   
+〈𝜏𝑗𝑢𝑚𝑝〉 = ∫
+𝑡
+𝜏 exp (−
+𝑡
+𝜏) 𝑑𝑡 = 𝜏
+∞
+0
+,  
+ 
+ 
+ 
+(16) 
+〈𝜏𝑗𝑢𝑚𝑝
+2
+〉 = ∫
+𝑡2
+𝜏 exp (−
+𝑡
+𝜏) 𝑑𝑡 = 2𝜏2
+∞
+0
+.   
+ 
+ 
+ 
+(17) 
+The average phase jump length square 〈(∆𝜙)2〉 is 
+ 
+〈(∆𝜙)2〉 = 〈(|𝜔|𝜏𝑗𝑢𝑚𝑝)2〉 = 〈𝜔2〉〈𝜏𝑗𝑢𝑚𝑝
+2
+〉 = 𝜔0
+22𝜏2 = 2𝜔0
+2𝜏2.  
+ 
+(18) 
+Such an uncoupled random walk has an effective phase diffusion constant 
+ 
+𝐷𝜙 =
+〈(∆𝜙)2〉
+2〈𝜏𝑗𝑢𝑚𝑝〉 =
+〈(𝜔𝜏𝑗𝑢𝑚𝑝)2〉
+2𝜏
+=
+〈𝜔2〉〈𝜏𝑗𝑢𝑚𝑝
+2
+〉
+2𝜏
+=
+𝜔022𝜏2
+2𝜏
+= 𝜔0
+2𝜏.   
+ 
+ 
+(19) 
+With 𝐷𝜙 , the normal phase diffusion equation can be described by [18] 
+𝑑𝑃(𝜙,𝑡)
+𝑑𝑡
+= 𝐷𝜙Δ𝑃(𝜙, 𝑡). 
+ 
+ 
+ 
+ 
+(20) 
+From Eq (20), the probability density function is  
+𝑃(𝜙, 𝑡) =
+1
+√4𝜋𝐷𝜙𝑡 𝑒𝑥𝑝 [−
+𝜙2
+4𝐷𝜙𝑡].     
+ 
+ 
+ (21) 
+Substituting Eq. (21) into Eq. (8), we have  
+𝑝(𝑘, 𝑡 − 𝑡′)|𝑘=1 = ∫
+𝑑𝜙
+∞
+−∞
+𝑃(𝜙, 𝑡 − 𝑡′) = 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)],   
+ 
+ (22) 
+whose Laplace transform representation is 
+𝑝(𝑘, 𝑠)|𝑘=1 =
+1
+𝑠+𝐷𝜙.   
+ 
+ 
+ 
+ 
+ 
+(23) 
+Substituting Eqs. (15b) and (23) into Eq. (11) yields 
+𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1=
+1
+1+𝜔02𝜏2
+1+𝑠
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2
+1
+𝑠+𝐷𝜙.   
+ 
+ 
+(24) 
+From 𝑀(𝑠), the inverse Laplace transform gives 
+𝑀(𝑡) = ∫
+𝑑𝑡 
+∞
+0
+1
+𝜏(1−𝜔02𝜏2) exp (−
+𝑡′
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2
+) 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)].  
+  (25) 
+
+6 
+ 
+Eq. (25) includes the convolution of two parts, the 
+1
+𝜏(1−𝜔02𝜏2) exp (−
+𝑡′
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2
+) comes from the Fourier 
+transform of the 𝜑(𝑡), while 𝑒𝑥𝑝[−𝐷𝜙(𝑡 − 𝑡′)] results from the phase diffusion;  the Frequency domain 
+signal can be obtained from the Fourier Transform of expression (25) as  
+𝑆(𝜔) =
+1
+1+𝜔02𝜏2
+1+[
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2 ]
+2
+𝜔2
+ 
+𝐷𝜙
+𝐷𝜙2+𝜔2.    
+ 
+ 
+(26) 
+II. 
+ Coupled normal diffusion with monoexponential distribution function 
+The coupled random phase walk has a joint probability function 𝜓(𝜙, 𝑡)  expressed by {25} 
+𝜓(𝜙, 𝑡) = 𝜑(𝑡)Φ(𝜙|𝑡), 
+ 
+ 
+ 
+ 
+ (27) 
+where 𝜑(𝑡) is the waiting time function, and  Φ(𝜙|𝑡) is the conditional probability that a phase jump length 
+𝜙 requiring time t. In Fourier-Laplace representation, the probability density function of a coupled random 
+walk has been derived in Ref. [25] as  
+𝑃(𝑘, 𝑠) =
+Ψ(𝑘,𝑠)
+1−𝜓(𝑘,𝑠)  ,  
+ 
+ 
+ 
+ 
+ (28) 
+where 𝜓(𝑘, 𝑠) is the joint probability, and Ψ(𝜙, 𝑡) is the PDF for the phase displacement of the last, 
+incomplete walk, which is  [25] 
+Ψ(𝜙, 𝑡) = 𝛿( 𝜙) ∫
+𝜑(𝑡′)𝑑𝑡′
+∞
+𝑡
+,  
+ 
+ 
+ 
+ (29a) 
+and 
+ Ψ(𝑘, 𝑠) = 
+1−𝜑(𝑠)
+𝑠
+.   
+ 
+ 
+ 
+ 
+(29b) 
+By substituting Eq. (29b) into Eq. (28), it arrives [25] 
+𝑃(𝑘, 𝑠) =
+Ψ(𝑘,𝑠)
+1−𝜓(𝑘,𝑠)  = 
+1−𝜑(𝑠)
+𝑠
+1
+1−𝜓(𝑘,𝑠).   
+ 
+ 
+ 
+(30) 
+In the chemical exchange, the joint probability could be described by  
+Φ(𝜙|𝑡) =
+1
+2 𝛿(|𝜙| − 𝜔0𝑡),  
+ 
+ 
+ 
+(31a) 
+𝜓(𝜙, 𝑡) =
+1
+2 𝜑(𝑡)𝛿(|𝜙| − 𝜔0𝑡),   
+ 
+ 
+ 
+(31b) 
+and 
+𝜓(𝑘, 𝑠)=∫ 𝑒𝑖𝑘𝜙−𝑠𝑡𝜓(𝜙, 𝑡)𝑑𝜙𝑑𝑡= 
+1
+2 [
+1
+𝜏(𝑠−𝑖𝑘𝜔0)+1 +
+1
+𝜏(𝑠+𝑖𝑘𝜔0)+1] ≈
+1+𝜏𝑠
+1+𝑘2𝜔02𝜏2+2𝜏𝑠. 
+ 
+(31c) 
+Substitute Eq. (31c) into Eq. (30), and we get 
+𝑃(𝑘, 𝑠) =
+Ψ(𝑘,𝑠)
+1−𝜓(𝑘,𝑠)  = 
+1−𝜑(𝑠)
+𝑠
+1
+1−𝜓(𝑘,𝑠) =
+𝜏
+1−
+1+𝜏𝑠
+1+𝑘2𝜔02𝜏2+2𝜏𝑠
+.   
+ 
+(32) 
+and 
+  
+ 
+ 
+ 
+ 
+𝑝(𝑘, 𝑠)|𝑘=1 =
+𝜏
+1−
+1+𝜏𝑠
+1+𝜔02𝜏2+2𝜏𝑠
+.   
+ 
+ 
+  
+(33) 
+ 
+Substituting Eqs. (15b) and (33) into Eq. (11) yields 
+
+7 
+ 
+ 
+𝑀(𝑠) =
+1+𝜏𝑠
+1+𝜔02𝜏2+2𝜏𝑠
+𝜏
+1−
+1+𝜏𝑠
+1+12𝜔02𝜏2+2𝜏𝑠
+=
+𝜏(1+𝜏𝑠)
+𝜔02𝜏2+𝜏𝑠 ≈
+𝜏
+𝜏𝑠(1−𝜔02𝜏2)+𝜔02𝜏2 =
+𝜏/𝜔02𝜏2
+𝜏𝑠(1−𝜔02𝜏2)/𝜔02𝜏2+1   
+ 
+(34) 
+From 𝑀(𝑠), the inverse Laplace transform gives 
+𝑀(𝑡) =
+1
+1−𝜔02𝜏2 exp (−
+𝑡
+𝜏(1−𝜔02𝜏2)
+𝜔02𝜏2
+).   
+ 
+ 
+ 
+  (35) 
+The frequency domain NMR signal can be obtained from the Fourier Transform of ���(𝑡) in Eq. (35) as  
+𝑆(𝜔) =
+1
+1−𝜔02𝜏2 
+𝜏(1−𝜔02𝜏2)
+𝜔02𝜏2
+(
+𝜏(1−𝜔02𝜏2)
+𝜔02𝜏2
+)
+2
+𝜔2+1
+= 
+𝜏
+𝜏2(1−𝜔02𝜏2)2𝜔2+𝜔02𝜏2.   
+ 
+  (36) 
+2.2.3  Fractional phase diffusion 
+For a complicated system, the time correlation function may not be a simple monoexponential function, 
+such as the Kohlrausch-Williams-Watts (KWW) function, or Mittag-Leffler function, and the 
+corresponding phase diffusion could be an anomalous diffusion [21-24]. The time fractional phase 
+diffusion will be investigated here, and the time correlation function is assumed as a MLF, 
+G(t) = 𝐸𝛼 (− (
+𝑡
+𝜏)
+𝛼
+),  
+ 
+ 
+ 
+ 
+(37) 
+ and its waiting time distribution function will be a heavy-tailed time distribution [27] 
+𝜑𝑓(𝑡) = −
+𝑑
+𝑑𝑡 𝐸𝛼 (− (
+𝑡
+𝜏)
+𝛼
+) ,   
+ 
+ 
+ 
+ (38) 
+whose Laplace transform is [25,27] 
+𝜑𝑓(𝑠) =
+1
+𝑠𝛼𝜏𝛼+1.  
+ 
+ 
+ 
+ 
+  (39) 
+Based on Eqs. (10b), (38) and (39), we have  
+𝐵(𝑠) =  𝑅𝑒𝜑𝑓(𝑠 + 𝑖𝜔) = 𝑅𝑒 1
+2 [
+1
+𝜏𝛼
+𝜔0
+𝛼 [cos (𝜋
+2 𝛼 − 𝑠𝛼
+𝜔0) + 𝑖 sin (𝜋
+2 𝛼 − 𝑠𝛼
+𝜔0)] + 1
+𝜏𝛼
+] 
+≈
+𝜔0𝛼𝜏𝛼(cos𝜋
+2𝛼+
+1
+𝜔0𝛼𝜏𝛼)
+1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼+𝑠𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
+2𝛼
+1−𝜔02𝛼𝜏2𝛼
+𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼+1 
+=
+𝑐
+1+𝑠𝜏′ ,      
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(40a) 
+where 
+ 𝑐 =
+𝜔0𝛼𝜏𝛼(cos𝜋
+2𝛼+
+1
+𝜔0𝛼𝜏𝛼)
+1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼 ,        
+ 
+ 
+ 
+ 
+ 
+ 
+(40b) 
+and   
+   
+ 
+ 
+𝜏′ =
+𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
+2𝛼
+1−𝜔02𝛼𝜏2𝛼
+𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼+1
+1+𝜔02𝛼𝜏2𝛼+2𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼 .        
+ 
+ 
+ 
+(40c) 
+ 
+I. 
+Uncoupled fractional diffusion  
+ 
+ 
+For such a time distribution function, the phase diffusion constant can be calculated according to Ref. [16, 
+
+8 
+ 
+22,23] as 
+ 
+ 
+ 
+ 
+𝐷𝜙𝑓 =
+〈(∆𝜙)2〉
+2Γ(1+𝛼)𝜏𝛼.  
+ 
+ 
+ 
+(41a) 
+The average phase jump may be assumed as  〈(∆𝜙)2〉 = 𝜔0
+22𝜏2, then 
+ 
+ 
+ 
+ 
+ 
+ 
+ 𝐷𝜙𝑓 =
+𝜔02𝜏2−𝛼
+Γ(1+𝛼)𝜏𝛼  .    
+ 
+ 
+ 
+ (41b) 
+With 𝐷𝜙𝑓 , the fractional phase diffusion equation can be described by [16,17,21,23,24] 
+ 𝑡𝐷∗
+𝛼𝑃𝑓 = 𝐷𝜙𝑓Δ𝑃𝑓(𝜙, 𝑡).  
+ 
+ 
+ 
+  (42) 
+where 0 < 𝛼, 𝛽 ≤ 2, 𝐷𝑓𝑟 is the rotational diffusion coefficient, a is the spherical radius, 𝑡𝐷∗
+𝛼  is the Caputo 
+fractional derivative defined by [22,23] 
+ 𝑡𝐷∗
+𝛼𝑓(𝑡): = {
+1
+𝛤(𝑚−𝛼) ∫
+𝑓(𝑚)(𝜏)𝑑𝜏
+(𝑡−𝜏)𝛼+1−𝑚 , 𝑚 − 1 < 𝛼 < 𝑚,
+𝑡
+0
+𝑑𝑚
+𝑑𝑡𝑚 𝑓(𝑡), 𝛼 = 𝑚,
+   
+ 
+   
+Fourier transform of Eq. (42) give [ 
+                                                                𝑡𝐷∗
+𝛼𝑝(𝑘, 𝑡) = −𝐷𝜙𝑓𝑘2𝑝(𝑘, 𝑡).  
+  
+ 
+ 
+ 
+ (43) 
+The solution of Eq. (43) is  𝑝(𝑘, 𝑡) = 𝐸𝛼[−𝐷𝜙𝑓𝑘2𝑡𝛼] [16,17], whose Laplace representation is [22,23,25] 
+ 
+ 
+ 
+ 
+ 
+𝑝(𝑘, 𝑠) =
+𝑆𝛼−1
+𝑆𝛼+𝐷𝜙𝑓𝑘2,  
+ 
+ 
+ 
+ 
+(44) 
+then 
+ 
+ 
+ 
+ 
+ 
+𝑝(𝑘, 𝑠)|𝑘=1 =
+𝑆𝛼−1
+𝑆𝛼+𝐷𝜙𝑓.   
+ 
+ 
+ 
+ 
+(45) 
+Substituting Eqs. (40) and (45) into Eq. (11), we get 
+ 
+𝑀(𝑠) =
+𝑐
+1+𝑠𝜏′
+𝑆𝛼−1
+𝑆𝛼+𝐷𝜙𝑓,     
+ 
+ 
+ 
+ 
+(46) 
+whose inverse Laplace transform gives  
+M(𝑡) = ∫ 𝑑𝑡′ 
+𝑡
+0
+𝑐
+𝜏′ exp (−
+𝑡′
+𝜏′)𝐸𝛼[−𝐷𝜙𝑓(𝑡 − 𝑡′)𝛼].    
+ 
+ 
+(47) 
+  The NMR signal can be obtained from the Fourier transform of M(𝑡), which is  
+𝑆(𝜔) = 𝐵(𝜔) ∙ 𝐸(𝜔) =
+𝑐
+1+𝜏′2𝜔2 ∙
+𝜔𝛼−1(
+1
+𝐷𝜙𝑓
+) sin(𝜋
+2𝛼)
+𝜔2𝛼(
+1
+𝐷𝜙𝑓
+)
+2
++2𝜔𝛼(
+1
+𝐷𝜙𝑓
+) cos(𝜋
+2𝛼)+1
+.                            (48) 
+II. Coupled fractional diffusion 
+Similarly,  for the coupled normal diffusion, the joint probability could be described by [25] 
+Φ(𝜙|𝑡) =
+1
+2 𝛿(|𝜙| − 𝜔0𝑡),   
+ 
+ 
+ 
+ 
+(49a) 
+𝜓(𝜙, 𝑡) =
+1
+2 𝜑(𝑡)𝛿(|𝜙| − 𝜔0𝑡),      
+ 
+ 
+ 
+(49b) 
+𝜓(𝑘, 𝑠)=∫ 𝑒𝑖𝑘𝜙−𝑠𝑡𝜓(𝜙, 𝑡)𝑑𝜙𝑑𝑡= 
+1
+2 [
+1
+𝜏𝛼(𝑠−𝑖𝑘𝜔0)𝛼+1 +
+1
+𝜏𝛼(𝑠+𝑖𝑘𝜔0)𝛼+1].                        
+ (49c) 
+and 
+
+9 
+ 
+𝜓(1, 𝑠) ≈
+1
+2 [
+1
+𝜏𝛼(𝑠−𝑖𝜔0)𝛼+1 +
+1
+𝜏𝛼(𝑠+𝑖𝜔0)𝛼+1].  
+ 
+ 
+ 
+ (50)  
+Compared to Eq. (40a), it is obvious that 
+𝜓(1, 𝑠) = 𝐵(𝑠) =
+𝑐
+1+𝑠𝜏′ .   
+ 
+ 
+ 
+  (51) 
+Substituted Eq. (51) into Eq. (30), we get 
+𝑝(𝑘, 𝑠)|𝑘=1 =
+Ψ(𝑘,𝑠)
+1−𝜓(1,𝑠)  = 
+1−𝜑(𝑠)
+𝑠
+1
+1−𝜓(1,𝑠)=
+𝜏𝛼𝑠𝛼−1
+1���
+𝑐
+1+𝑠𝜏′
+.    
+ 
+ 
+(52) 
+Eq. (52) can be substituted into Eq. (11) to give 
+𝑀(𝑠) = 𝐵(𝑠)𝑝(𝑘, 𝑠)|𝑘=1 =
+𝑐
+1+𝑠𝜏′
+𝜏𝛼𝑠𝛼−1
+1−
+𝑐
+1+𝑠𝜏′
+=
+𝑐𝜏𝛼𝑠𝛼−1
+1+𝑠𝜏′−𝑐 =
+1
+𝑆𝛼−1
+𝑐𝜏𝛼
+1+𝑠𝜏′−𝑐=
+1
+𝑆𝛼−1
+𝑐𝜏𝛼
+1−𝑐
+1+𝑠 𝜏′
+1−𝑐
+,      
+ 
+(53) 
+whose inverse Laplace transform yields 
+ 𝑀(𝑡) = ∫ 𝑑𝑡′
+1
+Γ(1−𝛼) 𝑡′−𝛼 𝑐
+𝑡
+0
+𝜏𝛼 1
+𝜏′ exp (−
+𝑡−𝑡′
+𝜏′
+1−𝑐
+) . 
+ 
+ 
+(54) 
+The Fourier transform of 𝑀(𝑡) gives NMR frequency domain signal 
+𝑆(𝜔) = sin (
+𝜋
+2 𝛼) |𝜔|𝛼−1 𝑐𝜏𝛼
+𝜏′
+𝜏′
+1−𝑐
+1+( 𝜏′
+1−𝑐)
+2
+𝜔2 = sin (
+𝜋
+2 𝛼) |𝜔|𝛼−1
+𝑐𝜏𝛼
+1−𝑐
+1+( 𝜏′
+1−𝑐)
+2
+𝜔2.  
+(55) 
+3. Results  
+ 
+     A phase diffusion equation method is proposed to describe the effect of chemical exchange on NMR 
+spectrum, based on uncoupled and coupled normal and fractional diffusions. The exchange between two 
+sites with equal populations is considered, and the theoretical expressions are organized in Table 1.    
+Table 1  
+ Comparison of theoretical NMR line shape expressions from phase diffusion method to traditional results for 
+chemical exchange between two sites with equal populations. 
+Frequency domain signal expression from phase diffusion results: 
+Simple phase diffusion with a constant jump time 
+𝑆(𝜔) = 
+𝐷𝜙,𝑠
+𝐷𝜙,𝑠2+𝜔2, 𝐷𝜙,𝑠 =
+𝜔02
+2 τ. 
+Normal phase diffusion with monoexponential function 
+Uncoupled diffusion 
+𝑆(𝜔) =
+1
+1+𝜔02𝜏2
+1+[
+𝜏(1−𝜔02𝜏2)
+1+𝜔02𝜏2 ]
+2
+𝜔2  
+𝐷𝜙
+𝐷𝜙2+𝜔2,  𝐷𝜙 = 𝜔0
+2𝜏. 
+Coupled diffusion 
+𝑆(𝜔) = 
+𝜏
+𝜏2(1−𝜔0
+2𝜏2)2𝜔2+𝜔0
+2𝜏2. 
+Fractional phase diffusion with heavy-tailed time distribution 
+ 𝑐 =
+𝜔0𝛼𝜏𝛼(cos𝜋
+2𝛼+
+1
+𝜔0𝛼𝜏𝛼)
+1+𝜔0
+2𝛼𝜏2𝛼+2𝜔0
+𝛼𝜏𝛼cos𝜋
+2𝛼 , 𝜏′ =
+𝛼𝜔0𝛼−1𝜏𝛼sin𝜋
+2𝛼
+1−𝜔02𝛼𝜏2𝛼
+𝜔0𝛼𝜏𝛼cos𝜋
+2𝛼+1
+1+𝜔0
+2𝛼𝜏2𝛼+2𝜔0
+𝛼𝜏𝛼cos𝜋
+2𝛼  
+Uncoupled diffusion 
+𝑆(𝜔) =
+𝑐
+1+𝜏′2𝜔2 ∙
+𝜔𝛼−1(
+1
+𝐷𝜙𝑓) sin(𝜋
+2𝛼)
+𝜔2𝛼(
+1
+𝐷𝜙𝑓)
+2
++2𝜔𝛼(
+1
+𝐷𝜙𝑓)cos(𝜋
+2𝛼)+1
+,  𝐷𝜙 =
+𝜔02𝜏2
+Γ(1+𝛼)𝜏𝛼 
+Coupled diffusion 
+𝑆(𝜔) = sin (𝜋
+2 𝛼) |𝜔|𝛼−1
+𝑐𝜏𝛼
+1 − 𝑐
+1 + ( 𝜏′
+1 − 𝑐)
+2
+𝜔2
+ 
+Frequency domain signal expression from traditional method: 
+ 
+𝑆(𝜔) =
+𝜔02𝜏
+2
+[𝜏
+2(𝜔0
+2−𝜔2)]
+2
++𝜔2  [3,4,5] 
+
+10 
+ 
+ 
+ 
+phasediff_tau_2overkex_kex30dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+-150
+-100
+-50
+0
+50
+100
+150
+/2  (Hz)
+S()
+a
+  = 2'  
+  traditional model, fixed time diffusion
+'  coupled and uncoupled 
+     norrmal and fractional diffusion
+ = 2'  =(15)
+phasediff_tau_2overkex_kex5dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+-150
+-100
+-50
+0
+50
+100
+150
+S()
+/2  (Hz)
+ = 2'  =(2.5)
+b
+  traditional model, fixed time diffusion
+'  coupled and uncoupled 
+     norrmal and fractional diffusion
+ 
+0
+0.005
+0.01
+0.015
+0.02
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex2dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+S()
+/2  (Hz)
+c
+ = 2' = 1 
+0
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+0.007
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex1dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+/2  (Hz)
+S()
+d
+ = 2'  =1(0.5)
+ 
+0
+0.001
+0.002
+0.003
+0.004
+0.005
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex0.6dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+S()
+/2  (Hz)
+e
+ = 2'  = 1 (0.3)
+tau = 2/kex  = 2/0.6 dw
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex0.1dw_beta_0.75_122722 - Copy
+Traditional
+Fixed_jump_time_diffusion
+Uncoupled_normal_diffusion
+Coupled_Normal_diffusion
+Uncoupled_fractional_diffusion
+Coupled_fractional_diffusion
+S()
+/2  (Hz)
+f
+ = 2'  =(0.05 )
+ 
+Fig. 1  The comparison among the various theoretical results obtained from the phase diffusion models and those 
+obtained by the traditional two-site exchange model, all equations listed in Table 1, with ∆𝜔/2𝜋 = 100 Hz, and 𝛼 = 0.75 
+for fractional phase diffusion. 
+ 
+ 
+ 
+ 
+ 
+ 
+
+11 
+ 
+4. Discussion 
+ 
+      In rotating frame reference, the spin phase in chemical exchange undergoes random phase jumps, 
+which can be intrinsically describe by either uncoupled effective phase diffusion equation or coupled 
+random walk.  
+       Figure 1 shows the comparison among the various theoretical results obtained from the phase diffusion 
+models and those obtained by the traditional two-site exchange model, all equations listed in Table 1. From 
+Figure 1, when the exchange is sufficiently fast, 𝜏 = 2𝜏′ ≤ 1/∆𝜔, the theoretical curves from diffusion with 
+a fixed jump time, uncoupled and coupled normal diffusion almost overlap with that predicted from the 
+traditional model.  However, the exchange time constant 𝜏 for the traditional model and the fixed time 
+diffusion is two times as 𝜏′ for the coupled and uncoupled normal diffusion with the monoexponential 
+distribution. The difference in exchange time could be explained by the following: the effective phase 
+diffusion constant is 
+𝜔02
+2 τ  for diffusion with a fixed jump time τ; in contrast, it is 𝜔0
+2τ for the uncoupled 
+diffusion with a monoexponential time distribution. The two-time difference in diffusion coefficients 
+resulted from the 〈𝜏𝑗𝑢𝑚𝑝
+2
+〉 = ∫
+𝑡2
+𝜏 exp (−
+𝑡
+𝜏) 𝑑𝑡 = 2𝜏2
+∞
+0
+, while in is the fixed time jump 〈𝜏𝑗𝑢𝑚𝑝
+2
+〉 = 𝜏2. The same 
+phase diffusion coefficient 𝜔0
+2τ has been used in Ref. [17] to obtain NMR relaxation expressions, which 
+replicate the traditional NMR relaxation theories;  these NMR relaxation expressions have been verified by 
+numerous experimental results; although this theoretical and experimental confirm is from relaxation 
+NMR, it still provide a strong support to select 𝜔0
+2τ rather than 
+𝜔02
+2 τ  as a phase diffusion coefficient, 
+considering both the exchange and relaxation are random phase walk processes. Therefore, in the analysis 
+of NMR chemical exchange line shape, the exchange time constant could be a two-time difference 
+depending on the employed models.  
+  Additionally, in Figure 1, the exchange line shapes in normal diffusion and fractional diffusion are 
+significantly different. The spectrum line from coupled fractional diffusion is broader than that of 
+uncoupled fractional diffusion, which may be reasonable because there is a more direct effect of heavy-
+tailed time distribution on the phase length in the coupled fractional diffusion than that of uncoupled 
+fractional diffusion.    Meanwhile,   the effect of coupled and uncoupled diffusion on the NMR line shape 
+is different in normal and fractional diffusions; the difference between the coupled and uncoupled 
+diffusion is negligible in normal diffusion but significant in fractional diffusion.  
+      Both the theoretical curves from the coupled and uncoupled fractional phase diffusion become 
+narrower when the fractional derivative parameter 𝛼 decreases.  The overlapped curves in Figure 2 imply 
+the fractional diffusion results reduce to the normal diffusion results when 𝛼 = 1. While, Figure 3 shows 
+the changes in coupled and uncoupled fractional diffusion among different fractional derivative orders, 𝛼 
+= 1, 0.9, 0.75 and 0.5. The smaller the 𝛼  is, the broader the NMR peak is.  Additionally, the middle part and 
+the end part of the fractional diffusion curves have different features: the middle part is a narrow peak, 
+while the end parts are broad shoulders. The narrower peak could come from fast exchange time, while 
+broader end shoulders come from slow exchange. In the view of the traditional model, this could be 
+interpreted as a bimodal exchange. However, both the fast and slow exchange times come from the same 
+heavy-tailed time distribution.  
+     The diffusion method proposed here shows excellent results in the fast exchange range, but it 
+encounters challenges in slow exchange.  This difficulty in slow exchange results from that the diffusion 
+limit is not met because the experimental time window in NMR is not infinite. It requires further effort to 
+overcome the hurdle.   The current method can be combined with other anomalous diffusion models, such 
+as the fractal derivative [28,29,30]. Further research is needed to understand and apply the models, 
+
+12 
+ 
+particularly the fractional diffusion model, and to extend the current method for multiple sites and unequal 
+population exchange.   
+ 
+ 
+ 
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex2dw_beta_1_122722 3:20:13 PM 12/29/2022
+Uncoupled_normal_diffusion
+Uncoupled_fractional_diffusion
+' =  1/  =
+S()
+/2  (Hz)
+a
+ 
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+-150
+-100
+-50
+0
+50
+100
+150
+phasediff_tau_2overkex_kex2dw_beta_1_122722
+Coupled_Normal_diffusion
+Coupled_fractional_diffusion
+S()
+/2  (Hz)
+b
+' =  1/  =
+ 
+ 
+Fig. 2  The fractional diffusion results reduce to the normal diffusion results when 𝛼 = 1, and ∆𝜔/2𝜋 = 100 Hz. 
+ 
+0
+0.005
+0.01
+0.015
+0.02
+-100
+-50
+0
+50
+100
+Uncoupled Fractional Diffusion
+ = 
+ = 
+ = 
+ = 
+S()
+a
+/2  (Hz)
+' =  1   =
+' =  1/
+0
+0.005
+0.01
+0.015
+0.02
+-100
+-50
+0
+50
+100
+Coupled Fractional Diffusion
+ = 
+ = 
+ = 
+ = 
+S()
+b
+/2  (Hz)
+' =  1/
+ 
+ 
+Fig. 3  The changes in coupled and uncoupled fractional diffusion among different fractional derivative orders, 𝛼 = 1, 
+0.9, 0.75 and 0.5, and ∆𝜔/2𝜋 = 100 Hz. 
+ 
+ 
+ 
+ 
+ 
+
+13 
+ 
+5. Conclusion 
+This paper proposes a phase diffusion method to describe the chemical exchange NMR spectrum. The 
+major conclusions are summarized in the following: 
+1. This method directly analyzes the spin system evolution in phase space rather than real space used 
+by most other traditional models.   
+2. The line shape difference between coupled and uncoupled phase diffusion is not obvious in 
+normal diffusion but significant in fractional diffusion.  
+3. There is a significant difference in the line shape between the normal and fractional diffusions. 
+4. Unlike the traditional method, the exchange time constant can follow certain types of distributions. 
+Additionally, the exchange time constant is two times faster based on the monoexponential time 
+distribution than that obtained by the traditional model.  
+5. The method could be extended to multiple sites and unequal population chemical exchange. 
+Furthermore, this phase diffusion method could be combined with other phase diffusion equations 
+in relaxation and PFG diffusion to deal with more complicated scenarios. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+14 
+ 
+References 
+ 
+1. A. Abragam, Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961. 
+2. C. P. Slichter. Principles of magnetic resonance, Springer series in Solid‐State Sciences, Vol. 
+1, Ed by M. Cardoua, P. Fulde and H. J. Queisser, Springer‐Verlag, Berlin (1978). 
+3. A. G. Palmer, H. Koss, Chapter Six - Chemical Exchange, Editor(s): A. J. Wand, Methods in 
+Enzymology, Academic Press, Volume 615, 2019, 177-236. 
+4. J. I. Kaplan, G. Fraenkel, NMR of Chemically Exchanging Systems, Academic Press, New 
+York, 1980. 
+5. C. S. Johnson,  Chemical rate processes and magnetic resonance, Adv. Magn. Reson. 1, 33 
+(1965). 
+6. N. Daffern, C. Nordyke, M. Zhang, A. G. Palmer, J. E. Straub. Dynamical Models of Chemical 
+Exchange in Nuclear Magnetic Resonance Spectroscopy, The Biophysicist 3(1) (2022): 13-34 
+7. J. M. Schurr, B. S. Fujimoto, R. Diaz, B. H. Robinson. 1999. Manifestations of slow site exchange 
+processes in solution NMR: a continuous Gaussian exchange model, J. Magn. Reson. 140 
+(1999) 404–431. 
+8. G. Lin, A. A. Jones. A lattice model for the simulation of one and two dimensional 129Xe 
+exchange spectra produced by translational diffusion, Solid State Nuclear Magnetic 
+Resonance. 26(2) (2004) 87-98. 
+9. R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin. Mittag-Leffler Functions, Related Topics 
+and Applications; Springer: Berlin, 2014. 
+10. T. Sandev, Z. Tomovsky, Fractional equations and models. Theory and applications, Springer 
+Nature Switzerland AG, Cham, Switzerland, 2019. 
+11. R. Kohlrausch, Theorie des elektrischen Rückstandes in der Leidner Flasche, Annalen der 
+Physik und Chemie. 91 (1854) 179–213. 
+12. G. Williams, D. C. Watts, Non-Symmetrical Dielectric Relaxation Behavior Arising from a 
+Simple Empirical Decay Function, Transactions of the Faraday Society 66 (1970) 80–85. 
+13. T. R. Lutz, Y. He, M. D. Ediger, H. Cao, G. Lin, A. A. Jones, Macromolecules 36 (2003) 1724. 
+14. E. Krygier, G. Lin, J. Mendes, G. Mukandela, D. Azar, A.A. Jones, J. A.Pathak, R. H. Colby, S. 
+K. Kumar, G. Floudas, R. Krishnamoorti, R. Faust, Macromolecules 38 (2005), 7721. 
+15. G. Lin, General pulsed-field gradient signal attenuation expression based on a fractional 
+integral modified-Bloch equation, Commu Nonlinear Sci. Numer. Simul. 63 (2018) 404-420.. 
+16. G. Lin, An effective phase shift diffusion equation method for analysis of PFG normal and 
+fractional diffusions, J. Magn. Reson. 259 (2015) 232–240. 
+17. G. Lin, Describing NMR relaxation by effective phase diffusion equation, Communications in 
+Nonlinear Science and Numerical Simulation. 99 (2021) 105825. 
+18. R. L. Magin, Weiguo Li, M. P. Velasco, J. Trujillo, D. A. Reiter, A. Morgenstern, R. G. Spencer, 
+Anomalous NMR relaxation in cartilage matrix components and native cartilage: Fractional-
+order models, J. Magn. Reson. 210 (2011)184-191. 
+
+15 
+ 
+ 
+19. T. Zavada, N. Südland, R. Kimmich, T.F. Nonnenmacher, Propagator representation of 
+anomalous diffusion: The orientational structure factor formalism in NMR, Phys. Rev. 60 
+(1999) 1292-1298. 
+20. G. Lin, Describe NMR relaxation by anomalous rotational or translational diffusion, Commu 
+Nonlinear Sci. Numer. Simul. 72 (2019) 232. 
+21. W. Wyss, J. Math. Phys. (1986) 2782-2785. 
+22. A. I. Saichev, G.M. Zaslavsky, Fractional kinetic equations: Solutions and applications. Chaos 
+7 (1997) 753–764. 
+23. R. Gorenflo, F. Mainardi, Fractional Diffusion Processes: Probability Distributions and 
+Continuous Time Random Walk, in: Lecture Notes in Physics, No 621, Springer-Verlag, Berlin, 
+2003, pp. 148–166. 
+24. F. Mainardi, Yu. Luchko, G. Pagnini, The fundamental solution of the space-time fractional 
+diffusion equation, Fract. Calc. Appl. Anal. 4 (2001) 153–192. 
+25. Y. Povstenko. Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, 
+Birkhäuser, New York, 2015. 
+26. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics 
+approach, Phys. Rep. 339 (2000) 1–77. 
+27. G. Germano, M. Politi, E. Scalas, R. L. Schilling. Phys Rev E 2009;79:066102. 
+28. W. Chen, Time space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006) 
+923. 
+29. W. Chen, H. Sun, X. Zhang, D. Korošak, Anomalous diffusion modeling by fractal and 
+fractional derivatives, Comput. Math. Appl. 5 (5) (2010) 1754–1758. 
+30. H. Sun, M.M. Meerschaert, Y. Zhang, J. Zhu, W. Chen, A fractal Richards’ equation to capture 
+the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resour. 52 
+(2013) 292–295. 
+

B9FQT4oBgHgl3EQfNza6/content/2301.13273v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5c095e2eec9a0acab55a0fce87404221faff6bb1201c383ef634897691828ffd
+size 999109

B9FQT4oBgHgl3EQfNza6/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:23bf556e08c24a06de86c6004ca0fa54745b612266a65149236336be39051469
+size 309174

BdE2T4oBgHgl3EQfRge6/content/tmp_files/2301.03782v1.pdf.txt

@@ -0,0 +1,1366 @@
+Multiple phenotypes in HL60 leukemia cell population
+Yue Wang1,2, Joseph X. Zhou3, Edoardo Pedrini3, Irit Rubin3, May Khalil3, Hong
+Qian2, and Sui Huang3
+1Department of Computational Medicine, University of California, Los Angeles,
+California, United States of America
+2Department of Applied Mathematics, University of Washington, Seattle,
+Washington, United States of America
+3Institute for Systems Biology, Seattle, Washington, United States of America
+Abstract
+Recent studies at individual cell resolution have revealed phenotypic heterogene-
+ity in nominally clonal tumor cell populations. The heterogeneity affects cell growth
+behaviors, which can result in departure from the idealized exponential growth. Here
+we measured the stochastic time courses of growth of an ensemble of populations of
+HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture
+the departure from the exponential growth model in the initial growth phase. De-
+spite being derived from the same cell clone, we observed significant variations in the
+early growth patterns of individual cultures with statistically significant differences in
+growth kinetics and the presence of subpopulations with different growth rates that
+endured for many generations. Based on the hypothesis of existence of multiple inter-
+converting subpopulations, we developed a branching process model that captures the
+experimental observations.
+1
+Introduction
+Cancer has long been considered a genetic disease caused by oncogenic mutations in so-
+matic cells that confer a proliferation advantage. According to the clonal evolution theory,
+accumulation of random genetic mutations produces cell clones with cancerous cell phe-
+notype. Specifically, cells with the novel genotype(s) may display increased proliferative
+fitness and gradually out-grow the normal cells, break down tissue homeostasis and gain
+other cancer hallmarks [15]. In this view, a genetically distinct clone of cells dominates the
+cancer cell population and is presumed to be uniform in terms of the phenotype of indi-
+vidual cells within an isogenic clone. In this traditional paradigm, non-genetic phenotypic
+variation within one clone is not taken into account.
+1
+arXiv:2301.03782v1  [q-bio.PE]  10 Jan 2023
+
+With the advent of systematic single-cell resolution analysis, however, non-genetic cell
+heterogeneity within clonal (cancer) cell populations is found to be universal [33]. This
+feature led to the consideration of the possibility of biologically (qualitatively) distinct
+(meta)stable cell subpopulations due to gene expression noise, representing intra-clonal
+variability of features beyond the rapid random micro-fluctuations.
+Hence, transitions
+between the subpopulations, as well as heterotypic interactions among them may influence
+cell growth, migration, drug resistance, etc. [39, 13, 9]. Thus, an emerging view is that
+cancer is more akin to an evolving ecosystem [11] in which cells form distinct subpopulations
+with persistent characteristic features that determine their mode of interaction, directly
+or indirectly via competition for resources [10, 36]. However, once non-genetic dynamics
+is considered, cell “ecology” differs fundamentally from the classic ecological system in
+macroscopic biology: the subpopulations can reversibly switch between each other whereas
+species in an ecological population do not convert between each other [7]. This affords
+cancer cell populations a remarkable heterogeneity, plasticity and evolvability, which may
+play important roles in their growth and in the development of resistance to treatment
+[30].
+Many new questions arise following the hypothesis that phenotypic heterogeneity and
+transitions between phenotypes within one genetic clone are important factors in cancer.
+Can tumors arise, as theoretical considerations indicate, because of a state conversion
+(within one clone) to a phenotype capable of faster, more autonomous growth as opposed
+to acquisition of a new genetic mutation that confers such a selectable phenotype [55,
+1, 18, 34, 33, 56, 23, 41]? Is the macroscopic, apparently sudden outgrowth of a tumor
+driven by a new fastest-growing clone (or subpopulation) taking off exponentially, or due
+to the cell population reaching a critical mass that permits positive feedback between its
+subpopulations that stimulates outgrowth, akin to a collectively autocatalytic set [17]?
+Should therapy target the fastest growing subpopulations, or target the interactions and
+interconversions of cancer cells?
+At the core of these deliberations is the fundamental question on the mode of tumor
+cell population growth that now must consider the influence of inherent phenotypic hetero-
+geneity of cells and the non-genetic (hence potentially reversible) inter-conversion of cells
+between the phenotypes that manifest various growth behaviors and the interplay between
+these two modalities.
+Traditionally tumor growth has been described as following an exponential growth law,
+motivated by the notion of uniform cell division rate for each cell, i.e. a first order growth
+kinetics [29]. But departure from the exponential model has long been noted. To better fit
+experimental data, two major modifications have been developed, namely the Gompertz
+model and the West law model [53]. While no one specific model can adequately describe
+any one tumor, each model highlights certain aspects of macroscopic tumor kinetics, mainly
+the maximum size and the change in growth rate at different stages. These models however
+are not specifically motivated by cellular heterogeneity. Assuming non-genetic heterogene-
+ity with transitions between the cell states, the population behavior is influenced by many
+2
+
+intrinsic and extrinsic factors that are both variable and unpredictable at the single-cell
+level. Thus, unlike macroscopic population dynamics [43], tumor growth cannot be ad-
+equately captured by a deterministic model, but a stochastic cell and population level
+kinetic model is more realistic.
+Using stochastic processes in modeling cell growth via clonal expansion has a long
+history [54]. An early work is the Luria-Delbr¨uck model, which assumes cells grow deter-
+ministically, with wildtype cells mutating and becoming (due to rare and quasi-irreversible
+mutations) cells with a different phenotype randomly [28]. Since then, there have been
+many further developments that incorporate stochastic elements into the model, such as
+those proposed by Lea and Coulson [25], Koch [22], Moolgavkar and Luebeck [27], and
+Dewanji et al. [8]. We can find various stochastic processes: Poisson processes [2], Markov
+chains [14], and branching processes [19], or even random sums of birth-death processes [8],
+all playing key roles in the mathematical theories of cellular clonal growth and evolution.
+These models have been applied to clinical data on lung cancer [31], breast cancer [37],
+and treatment of cancer [38].
+At single-cell resolution, another cause for departure from exponential growth is the
+presence of positive (growth promoting) cell-cell interactions (Allee effect) in the early
+phase of population growth, such that cell density plays a role in stimulating division,
+giving rise to the critical mass dynamics [20, 24].
+To understand the intrinsic tumor growth behavior (change of tumor volume over time)
+it is therefore essential to study tumor cell populations in culture which affords detailed
+quantitative analysis of cell numbers over time, unaffected by the tumor microenvironment,
+and to measure departure from exponential growth.
+This paper focuses on stochastic
+growth of clonal but phenotypically heterogeneous HL60 leukemia cells with near single-cell
+sensitivities in the early phase of growth, that is, in sparse cultures. We and others have in
+the past years noted that at the level of single cells, each cell behaves akin to an individual,
+differently from another, which can be explained by the slow correlated transcriptome-wide
+fluctuations of gene expression [4, 26]. Given the phenotypic heterogeneity and anticipated
+functional consequences, grouping of cells is necessary. Such classification would require
+molecular cell markers for said functional implication, but such markers are often difficult
+to determine a priori.
+Here, since most pertinent to cancer biology, we directly use a
+functional marker that is of central relevance for cancer: cell division, which maps into cell
+population growth potential — in brief “cell growth”.
+Therefore, we monitored longitudinally the growth of cancer cell populations seeded at
+very small numbers of cells (1, 4, or 10 cells) in statistical ensembles of microcultures (wells
+on a plate of wells). We found evidence that clonal HL60 leukemia cell populations contain
+subpopulations that exhibit diverse growth patterns.
+Based on statistical analysis, we
+propose the existence of three distinctive cell phenotypic states with respect to cell growth.
+We show that a branching process model captures the population growth kinetics of a
+population with distinct cell subpopulations. Our results suggest that the initial phase cell
+growth (“take-off” of a cell culture) in the HL60 leukemic cells is predominantly driven by
+3
+
+the fast-growing cell subpopulation. Reseeding experiments revealed that the fast-growing
+subpopulation could maintain its growth rate over several cell generations, even after the
+placement in a new environment. Our observations underscore the need to not only target
+the fast-growing cells but also the transition to them from the other cell subpopulations.
+2
+Results
+2.1
+Experiment of the cell population growth from distinct initial cell
+numbers.
+To expose the variability of growth kinetics as a function of initial cell density N0 (“initial
+seed number”), HL60 cells were sorted into wells of a 384-well plate (0.084 cm2 area)
+to obtain “statistical ensembles” of replicate microcultures (wells) of the same condition,
+distinct only by N0. Based on prior titration experiments to determine ranges of interest
+for N0 and statistical power, for this experiment we plated 80 wells with N0 = 10 cells
+(N0 = 10-cell group), 80 wells with N0 = 4 cells (N0 = 4-cell group), and 80 wells with
+N0 = 1 cell (N0 = 1-cell group). Cells were grown in the same conditions for 23 days (for
+details of cell culture and sorting, see the Methods section). Digital images were taken
+every 24 hours for each well from Day 4 on, and the area occupied by cells in each well
+was determined using computational image analysis. We had previously determined that
+one area unit equals approximately 500 cells. This is consistent and readily measurable
+because the relatively rigid and uniformly spherical HL60 cells grow as a non-adherent
+“packed” monolayer at the bottom of the well. Note that we are interested in the initial
+exponential growth (and departure from it) and not in the latter phases when the culture
+becomes saturated as has been the historical focus of analysis (see Introduction).
+Wells that have reached at least 5 area units were considered for the characterization
+of early phase (before plateau) growth kinetics by plotting the areas in logarithmic scale as
+a function of time (Fig. 1). All the N0 = 10-cell wells required 3.6-4.6 days to grow from
+5 area units to 50 area units (mean=4.05, standard deviation=0.23). For the N0 = 1-cell
+wells, we observed a diversity of behaviors. While some of the cultures only took 3.5-5
+days to grow from 5 area units to 50 area units, others needed 6-7.2 days (mean=5.02,
+standard deviation=0.75). The N0 = 4-cell wells had a mean=4.50 days and standard
+deviation=0.44 to reach that same population size.
+To examine the exponential growth model, in Fig. 2 (left panel), we plotted the per
+capita growth rate versus cell population size, where each point represents a well (popu-
+lation) at a time point. As expected, as the population became crowded, the growth rate
+decreased toward zero. But in the earlier phase, many populations in the N0 = 1-cell group
+had a lower per capita growth rate than those in the N0 = 10-cell group, even at the same
+population size – thus departing from the expected behavior of exponential growth. The
+weighted Welch’s t-test showed that the difference in these growth rates was significant
+(see the Methods section).
+4
+
+While qualitative differences in the behaviors of cultures with different initial seeding
+cell numbers N0 can be expected for biological reasons (see below), in the elementary
+exponential growth model, the difference of growth rate should disappear when populations
+with distinct seeding numbers are aligned for the same population size that they have
+reached as in Fig. 2. A simple possibility is that the deviations of expected growth rates
+emanate from difference in cell-intrinsic properties.
+Some cells grew faster, with a per
+capita growth rate of 0.6 ∼ 0.9 (all N0 = 10-cell wells and some N0 = 1-cell wells), while
+some cells grew slower, with a per capita growth rate of 0.3 ∼ 0.5 (some of the N0 = 1-
+cell wells). In other words, there is intrinsic heterogeneity in the cell population that is
+not “averaged out” in the culture with low N0, and the sampling process exposes these
+differences between the cells that appear to be relatively stable.
+To illustrate the inherent diversity of initial growth rates, in Fig. 3 (left panel), we
+display the daily cell-occupied areas plotted on a linear scale starting from Day 4. All wells
+with seed of N0 = 10 or N0 = 4 cells grew exponentially. Among the N0 = 1-cell wells,
+14 populations died out. Four wells in the N0 = 1-cell group had more than 10 cells on
+Day 8 but never grew exponentially, and had fewer than 1000 cells after 15 days (on Day
+23). For these non-growing or slow-growing N0 = 1-cell wells, the per capita growth rate
+was 0 ∼ 0.2. In comparison, all the N0 = 10-cell wells needed at most 15 days to reach
+the carrying capacity (around 80 area units, or 40000 cells). See Table 1 for a summary of
+the N0 = 1-cell group’s growth patterns. This behavior is not idiosyncratic to the culture
+system because they recapitulate a pilot experiment performed in the larger scale format
+of 96-well plates (not shown).
+From the above experimental observations, we asserted that there might be at least
+three stable cell growth phenotypes in a population: a fast type, whose growth rate was
+0.6 ∼ 0.9/day for non-crowded conditions; a moderate type, whose growth rate was 0.3 ∼
+0.5/day for non-crowded conditions; and a slow type, whose growth rate was 0 ∼ 0.2/day
+for the non-crowded population.
+The graphs of Fig. 3 also revealed other phenomena of growth kinetics: (1) Most
+N0 = 4-cell wells plateaued by Day 14 to Day 17, but some lagged significantly behind.
+(2) Similarly, four wells in the N0 = 1-cell group exhibited longer lag-times before the
+exponential growth phase, and never reached half-maximal cell numbers by Day 23. These
+outliers reveal intrinsic variability and were taken into account in the parameter scanning
+(see the Methods section).
+2.2
+Reseeding experiments revealing the enduring intrinsic growth pat-
+terns.
+When a well in the N0 = 1-cell group had grown to 10 cells, population behavior was
+still different from those in the N0 = 10-cell group at the outset. In view of the spate of
+recent results revealing phenotypic heterogeneity, we hypothesized that the difference was
+cell-intrinsic as opposed to being a consequence of the environment (e.g., culture medium
+5
+
+Growth pattern
+Well label
+Day 1
+Day 8
+Day 14
+Day 19
+Day 23
+No growth,
+extinction
+162,167,170,176,
+177,179,182,183,
+186,201,234,236,
+239,240
+1
+<10
+<10
+∼0
+Empty
+Slow growth,
+no exponential
+growth
+165
+1
+89
+∼300
+∼350
+∼500
+166
+1
+36
+∼110
+∼120
+∼150
+178
+1
+43
+∼140
+∼170
+∼200
+211
+1
+16
+∼90
+∼200
+∼400
+Delayed
+exponential
+growth
+163
+1
+12
+∼130
+∼300
+∼5000
+181
+1
+44
+∼270
+∼550
+∼5500
+193
+1
+25
+∼200
+∼800
+∼9000
+204
+1
+21
+∼100
+∼600
+∼6000
+Normal
+exponential
+growth
+200 and
+many others
+1
+∼130
+∼20000
+∼40000
+(full)
+∼40000
+(full)
+Table 1: The population of some wells in the N0 = 1-cell group in the growth experiment
+with different initial cell numbers, where ∼ meant approximate cell number. These wells
+illustrated different growth patterns from those wells starting with N0 = 10 or N0 = 4
+cells. Such differences implied that cells from wells with different initial cell numbers were
+essentially different.
+6
+
+Time (days) to
+reach one half area
+11
+12
+13
+14
+15
+16–20
+>20
+Faster wells
+26
+2
+1
+2
+1
+0
+0
+Slower wells
+0
+0
+0
+1
+1
+25
+5
+Table 2: The distribution of time needed for each well to reach the “half area” population
+size in the reseeding experiment. We reseeded equal numbers of cells that grew faster (from
+a full well) and cells that grew slower (from a half-full well), and cultivated them under the
+same new fresh medium environment to compare their intrinsic growth rates. The results
+showed that faster growing cells, even reseeded, still grew faster.
+in N0 = 1 vs N0 = 10 -cell wells).
+To test our hypothesis and exclude differences in the culture environment as determi-
+nants of growth behavior, we reseeded the cells that exhibited the different growth rates
+in fresh cultures. We started with a number of N0 = 1-cell wells. After a period of almost
+3 weeks, again some wells showed rapid proliferation, with cells covering the well, while
+others were half full and yet others wells were almost empty. We collected cells from the full
+and half-full wells and reseeded them into 32 wells each (at about N0 = 78 cells per well).
+These 64 wells were monitored for another 20 days. We found that most wells reseeded
+from the full well took around 11 days to reach the population size of a half-full well, while
+most wells reseeded from the half-full well required around 16 ∼ 20 days to reach the same
+half full well population size. Five wells reseeded from the half-full wells were far from even
+reaching half full well population size by Day 20 (see Table 2). Permutation test showed
+that this difference in growth rate was significant (see the Methods section).
+This reseeding experiment shows that the difference in growth rate was maintained
+over multiple generations, even after slowing down in the plateau phase (full well) and
+was maintained when restarting a microculture at low density in fresh medium devoid of
+secreted cell products. Therefore, it is plausible that there exists endogenous heterogeneity
+of growth phenotypes in the clonal HL60 cell line and that these distinct growth phenotypes
+are stable for at least 15 ∼ 20 cell generations.
+2.3
+Quantitative analysis of experimental results.
+In the experiments with different initial cell numbers N0, we observed at least three patterns
+with different growth rates, and the reseeding showed that these growth patterns were
+endogenous to the cells. Therefore, we propose that each growth pattern discussed above
+corresponded to a cell phenotype that dominated the population: fast, moderate, and slow.
+In the initial seeding of cells that varies N0, the cells were randomly chosen (by FACS);
+thus, their intrinsic growth phenotypes were randomly distributed. During growth, the
+population of a well would be dominated by the fastest type that existed in the seeding
+cells, thus qualitatively, we have following scenarios: (1) A well in the N0 = 10-cell group
+7
+
+almost certainly had at least one initial cell of fast type, and the population would be
+dominated by fast type cells. Different wells had almost the same growth rate, reaching
+saturation at almost the same time. (2) For an N0 = 1-cell well, if the only initial cell is of
+the fast type, then the population has only the fast type, and the growth pattern will be
+close to that of N0 = 10-cell wells. If the only initial cell is of the moderate type, then the
+population could still grow exponentially, but with a slower growth rate. This explains why
+after reaching 5 area units, many but not all N0 = 1-cell wells were slower than N0 = 10-
+cell wells. (3) Moreover, in such an N0 = 1-cell well with a moderate type initial cell, the
+cell might not divide quite often during the first few days due to randomness of entering
+the cell cycle. This would lead to a considerable delay in entering the exponential growth
+phase. (4) By contrast, for an N0 = 1-cell well with a slow type initial cell, the growth rate
+could be too small, and the population might die out or survive without ever entering the
+exponential growth phase in duration of the experiment. (5) Most N0 = 4-cell wells had at
+least one fast type initial cell, and the growth pattern was the same as N0 = 10-cell wells.
+A few N0 = 4-cell wells only had moderate and slow cells, and thus had slower growth
+patterns.
+The above verbal argument is shown in Fig. 4 and entails mathematical modeling with
+the appropriate parameters that relate the relative frequency of these cell types in the
+original population, their associated growth and transition rates to examine whether it
+explains the data.
+2.4
+Branching process model.
+To construct a quantitative dynamical model to recapitulate the growth dynamics differ-
+ences from cell populations with distinct initial seed cell numbers N0, and three intrinsic
+types of proliferation behaviors, we used a multi-type discrete-time branching process.
+The traditional method of population dynamics based on ordinary differential equation
+(ODE), which is deterministic and has continuous variables, is not suited when the cell
+population is small as is the case for the earliest stage of proliferation from a few cells
+being studied in our experiments. Deterministic models are also unfit because with such
+small populations and measurements at single-cell resolution, stochasticity in cell activity
+does not average out. The nuanced differences between individual cells cannot be captured
+by a different deterministic mechanism of each individual cell, and the only information
+available is the initial cell number. Thus, the unobservable nuances between cells are taken
+care of by a stochastic model.
+Given the small populations, our model should be purely stochastic, without determin-
+istic growth. The focus is the concrete population size of a finite number (three) of types,
+thus Poisson processes are not suitable. Markov chains can partially describe the propor-
+tions under some conditions [47], but population sizes are known, not just their ratios,
+therefore Markov chains are not necessary. Even the lifted Markov chains [48] and random
+dynamical systems [52] are not applicable in this situation, since the population should be
+8
+
+non-negative. Branching processes can describe the population size of multiple types with
+symmetric and asymmetric division, transition, and death [19]. Also, the parameters can
+be temporally and spatially inhomogeneous, which is convenient. Therefore, we utilized
+branching processes in our model.
+In the branching process, each cell during each time interval independently and ran-
+domly chooses a behavior: division, death, or stagnation in the quiescent state, whose rates
+depend on the cell growth type. Denoting the growth rate and death rate of the fast type
+by gF and dF respectively, and the population size of fast type cells on Day n by F(n), the
+population at Day n + 1 is:
+F(n + 1) =
+F(n)
+�
+i=1
+Ai,
+where Ai for different i are independent. Ai represents the descendants of a fast type cell i
+after one day. It equals 2 with probability gF, 0 with probability dF, and 1 with probability
+1 − gF − dF. Therefore, given F(n), the distribution of F(n + 1) is:
+P[F(n + 1) = N] =
+�
+2a+b=N
+F(n)!
+a!b![F(n) − a − b]!ga
+Fd[F(n)−a−b]
+F
+(1 − gF − dF)b,
+where the summation is taken for all non-negative integer pairs (a, b) with 2a + b = N.
+Moderate and slow types evolve similarly, with their corresponding growth rates gM, gS,
+and death rates dM, dS.
+As shown in Fig. 2, the growth rates gF, gM, and gS should be decreasing functions of
+the total population. In our model, we adopted a quadratic function.
+We performed a parameter scan to show that our model could reproduce experimental
+phenomena for a wide range of model parameters (see details in Table 3).
+The simulation results are shown on the right panels of Figs. 1–3, in comparison with
+the experimental data in the left. Our model qualitatively captured the growth patterns
+of groups with different initial seeding cell numbers. For example, in Fig. 2, when wells
+were less than half full (cell number < 20000), most wells in the N0 = 10-cell group grew
+faster than the N0 = 1-cell group even when they had the same cell number. In Fig. 3,
+all wells in the N0 = 10-cell group in our model grew quickly until saturation. Similar to
+the experiment, some wells in the N0 = 1-cell group in our model never grew, while some
+began to take off very late.
+In our model, the high extinction rate in the N0 = 1-cell group (14/80) was explained
+as “bad luck” at the early stage, since birth rate and death rate were close, and a cell could
+easily die without division. Another possible explanation for such a difference in growth
+rates was that the population would be 10 small colonies when starting from 10 initial cells,
+while starting from 1 initial cell, the population would be 1 large colony. With the same
+area, 10 small colonies should have a larger total perimeter, thus larger growth space and
+larger growth rate than that of 1 large colony. However, we carefully checked the photos,
+9
+
+Parameters
+Appearance of experimental phenomena
+pF
+pM
+pS
+d
+g0
+r
+Feature 1
+Feature 2
+Feature 3
+Feature 4
+0.4
+0.4
+0.2
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.05
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.1
+0.5
+0.1
+No
+Yes
+Yes
+No
+0.4
+0.4
+0.2
+0.01
+0.45
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.01
+0.6
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.01
+0.4
+0.1
+Yes
+Yes
+Yes
+No
+0.4
+0.4
+0.2
+0.01
+0.5
+0.05
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.01
+0.5
+0
+Yes
+Yes
+Yes
+Yes
+0.4
+0.4
+0.2
+0.01
+0.5
+0.15
+Yes
+Yes
+Yes
+No
+0.4
+0.4
+0.2
+0.01
+0.5
+0.2
+No
+Yes
+Yes
+No
+0.3
+0.5
+0.2
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.5
+0.3
+0.2
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.5
+0.1
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.4
+0.3
+0.3
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.5
+0.4
+0.1
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.3
+0.4
+0.3
+0.01
+0.5
+0.1
+Yes
+Yes
+Yes
+Yes
+0.1
+0.1
+0.8
+0.01
+0.5
+0.1
+No
+Yes
+Yes
+No
+0.5
+0.5
+0
+0.01
+0.5
+0.1
+Yes
+Yes
+No
+Yes
+0
+0.5
+0.5
+0.01
+0.5
+0.1
+No
+Yes
+Yes
+Yes
+0.5
+0
+0.5
+0.01
+0.5
+0.1
+Yes
+No
+Yes
+No
+1
+0
+0
+0.01
+0.5
+0.1
+Yes
+No
+No
+No
+Table 3: Performance of our model with different parameters. Here we adjusted the param-
+eters of our model in a wide range and observed whether the model could still reproduce
+four important “features” in the experiment. This parameter scan showed that our model
+is robust under perturbations on parameters. Here pF, pM, pS are the probabilities that an
+initial cell is of fast, moderate, or slow type; d is the death rate; g0 is the growth factor;
+r is the range of the random modifier. See the Methods section for explanations of these
+parameters.
+Feature 1, all wells in the N0 = 10-cell group were saturated; Feature 2,
+presence of late-growing wells in the N0 = 1-cell group; Feature 3, presence of non-growing
+wells in the N0 = 1-cell group; Feature 4, different growth rates at the same population
+size between the N0 = 10-cell group and the N0 = 1-cell group.
+10
+
+and found that almost all wells produced 1 large colony with nearly the same shape, and
+there was no significant relationship between colony perimeter and growth rate.
+3
+Discussion
+As many recent single-cell level data have shown, a tumor can contain multiple distinct
+subpopulations engaging in interconversions and interactions among them that can in-
+fluence cancer cell proliferation, death, migration, and other features that contribute to
+malignancy [33, 55, 1, 18, 34, 56, 20, 24, 5, 32, 6]. Presence of these two intra-population
+behaviors can be manifest as departure from the elementary model of exponential growth
+[35] (in the early phase of population growth, far away from carrying capacity of the culture
+environment which is trivially non-exponential). The exponential growth model assumes
+uniformity of cell division rates across all cells (hence a population doubling rate that is
+proportional to a given population size N(t)) and the absence of cell-cell interactions that
+affect cell division and death rates. Investigating the “non-genetic heterogeneity” hypoth-
+esis of cancer cells quantitatively is therefore paramount for understanding cancer biology
+but also for elementary principles of cell population growth.
+As an example, here we showed that clonal cell populations of the leukemia HL60
+cell line are heterogeneous with regard to growth behaviors of individual cells that can
+be summarized in subpopulations characterized by a distinct intrinsic growth rates which
+were revealed by analysis of the early population growth starting with microcultures seeded
+with varying (low) cell number N0.
+Since we have noted only very weak effect of cell-cell interactions on cell growth be-
+haviors (Allee effect) in this cell line (as opposed to another cell tumor cell line in which
+we found that departure from exponential growth could be explained by the Allee effect
+[20]), we focused on the very presence among HL60 cells of subpopulations with distinct
+proliferative capacity as a mechanism for the departure of the early population growth
+curve from exponential growth.
+The reseeding experiment demonstrated that the characteristic growth behaviors of
+subpopulations could be inherited across cell generations and after moving to a new envi-
+ronment (fresh culture), consistent with long-enduring endogenous properties of the cells.
+This result might be explained by cells occupying distinct stable cell states (in a multi-
+stable system). Thus, we introduced multiple cell types with different growth rates in our
+stochastic model. Specifically, in a branching process model, we assumed the existence
+of three types: fast, moderate, and slow cells. The model we built could replicate the
+key features in the experimental data, such as different growth rates at the same popula-
+tion size between the N0 = 10-cell group and the N0 = 1-cell group, and the presence of
+late-growing and non-growing wells in the N0 = 1-cell group.
+While we were able to fit the observed behaviors in which the growth rate depended not
+only on N(t) but also on N0, the existence of the three or even more cell types still needs
+11
+
+to be verified experimentally. For instance, statistical cluster analysis of transcriptomes of
+individual cells by single-cell RNA-seq [3] over the population may identify the presence
+of transcriptomically distinct subpopulations that could be isolated (e.g., after association
+with cell surface markers) and evaluated separately for their growth behaviors. We might
+apply inference methods on such sequencing data to determine the gene regulatory relations
+that lead to multiple phenotypes [50, 44], although the causal relationship might not always
+be determined [49]. Besides, since the existence of transposons might affect the growth
+rates, corresponding analysis should be conducted [21, 40].
+The central assumption of coexistence of multiple subpopulations in the cell line stock
+must be accompanied by the second assumption that there are transitions between these
+distinct cell populations. For otherwise, in the stock population the fastest growing cell
+would eventually outgrow the slow growing cells. Furthermore, one has to assume a steady-
+state in which the population of slow growing cells are continuously replenished from the
+population of fast-growing cells. Finally, we must assume that the steady-state proportions
+of the subpopulations are such that at low seeding wells with N0 = 1 cells, there is a sizable
+probability that a microculture receives cells from each of the (three) presumed subtypes of
+cells. The number of wells in the ensemble of replicate microcultures for each N0- condition
+has been sufficiently large for us to make the observations and inform the model, but a
+larger ensemble would be required to determine with satisfactory accuracy the relative
+proportions of the cell types in the parental stock population.
+Transitions might also have been happening during our experiment. For example, those
+late growing wells in the N0 = 1-cell group could be explained by such a transition: Initially,
+only slow type cells were present, but once one of these slow growing cells switched to the
+moderate type, an exponential growth ensued at the same rate that is intrinsic to that of
+moderate cells.
+If there are transitions, what is the transition rate? Our reseeding experiments are
+compatible with a relatively slow rate for interconversion of growth behaviors in that the
+same growth type was maintained across 30 generations. An alternative to the principle
+of transition at a constant intrinsic to each of the types of cells may be that transition
+is extrinsically determined. Specifically, the seeding in the “lone” condition of N0 = 1
+may induce a dormant state, that is a transition to a slower growth mode that is then
+maintained, on average over 30+ generations, with occasional return to the faster types
+that account for the delayed exponential growth. The lack of experimental data might be
+partially made up by inference methods [51].
+This model however would bring back the notion of “environment awareness”, or the
+principle of a “critical density” for growth implemented by cell-cell interaction (Allee effect)
+which we had deliberately not considered (see above) since it was not necessary. We do not
+exclude this possibility which could be experimentally tested as follows: Cultivate N0 = 1-
+cell wells for 20 days when the delayed exponential growth has happened in some wells,
+but then use the cells of those wells with fast-growing population (which should contain of
+the fast type) to restart the experiment, seeded at N0 = 10, 4, 1 cells. If wells with different
+12
+
+seeding numbers exhibit the same growth rates, then the growth difference in the original
+experiment is solely due to preexisting (slow interconverting) cell phenotypes. If now the
+N0 = 1-cell wells resumes the typical slow growth, this would indicate a density induced
+transition to the slow growth type. If cell-cell interaction needs to be taken into account,
+certain results in developmental biology might help, since they study the emergence of
+patterns through strong cell-cell interactions [46, 45, 42].
+In the spirit of Occam’s razor, and given the technical difficulty in separate experiments
+to demonstrate cell-cell interactions in HL60 cells, we were able to model the observed
+behaviors with the simplest assumption of cell-autonomous properties, including existence
+of multiple states (growth behaviors) and slow transitions between them but without cell
+density dependence or interactions.
+Taken together, we showed that one manifestation of the burgeoning awareness of ubiq-
+uitous cell phenotype heterogeneity in an isogenic cell population is the presence of distinct
+intrinsic types of cells that slowly interconvert among them, resulting in a stationary popu-
+lation composition. The differing growth rates of the subtypes and their stable proportions
+may be an elementary characteristic of a given population that by itself can account for the
+departure of early population growth kinetics from the basic exponential growth model.
+4
+Methods
+4.1
+Setup of growth experiment with different initial cell numbers.
+HL60 cells were maintained in IMDM wGln, 20% FBS(heat inactivated), 1% P/S at a
+cell density between 3 × 105 and 2.5 × 106 cells/ml (GIBCO). Cells were always handled
+and maintained under sterile conditions (tissue culture hood; 37◦C, 5% CO2, humidified
+incubator). At the beginning of the experiment, cells were collected, washed two times in
+PBS, and stained for vitality (Trypan blue GIBCO). The population of cells was first gated
+for morphology and then for vitality staining. Only Trypan negative cells were sorted (BD
+FACSAria II). The cells were sorted in a 384 well plate with IMDM wGln, 20% FBS(heat
+inactivated), and 1% P/S (GIBCO).
+Cell population growth was monitored using a Leica microscope (heated environmental
+chamber and CO2 levels control) with a motorized tray. Starting from Day 4, the 384
+well plate was placed inside the environmental chamber every 24 hours. The images were
+acquired in a 3 × 3 grid for each well; after acquisition, the 9 fields were stitched into a
+single image. Software ImageJ was applied to identify and estimate the area occupied by
+“entities” in each image. The area (proportional to cell number) was used to follow the
+cell growth.
+13
+
+4.2
+Setup of reseeding experiment for growth pattern inheritance.
+HL60 cells were cultivated for 3 weeks, and then we chose one full well and one half full
+well. We supposed the full well was dominated by fast type cells, and the half-full well
+was dominated by moderate type cells, which had lower growth rates. We reseeded cells
+from these two wells and cultivated them in two 96-well (rows A-H, columns 1-12) plates.
+In each plate, B2-B11, D2-D11, and F2-F11 wells started with 78 fast cells, while C2-C11,
+E2-E11, and G2-G11 wells started with 78 moderate cells. Rows A, H, columns 1, 12 had
+no cells and no media, and we found that wells in rows B, G, columns 2, 11, which were
+the outmost non-empty wells, evaporated much faster than inner wells. Therefore, the
+growth of cells in those wells was much slower than inner wells. Hence we only considered
+inner wells, where D3-D10 and F3-F10 started with fast cells, C3-C10 and E3-E10 started
+with moderate cells, namely 32 fast wells and 32 moderate wells in total.
+During the
+experiment, no media was added. Each day, we observed those wells to check whether
+their areas exceeded one-half of the whole well. The experiment was terminated after 20
+days.
+4.3
+Weighted Welch’s t-test.
+The weighted Welch’s t-test is used to test the hypothesis that two populations have equal
+mean, while sample values have different weights [12].
+Assume for group i (i = 1, 2),
+the sample size is Ni and the jth sample is the average of cj
+i independent and identically
+distributed variables. Let Xj
+i be the observed average for the jth sample. Set ν1 = N1 − 1,
+ν2 = N2 − 1. Define
+¯
+Xi
+W = (
+Ni
+�
+j=1
+Xj
+i cj)/(
+Ni
+�
+j=1
+)cj,
+s2
+i,W =
+Ni[�Ni
+j=1(Xj
+i )2cj]/(�Ni
+j=1 cj
+i) − Ni( ¯
+Xi
+W )2
+Ni − 1
+,
+t =
+¯
+X1
+W − ¯
+X2
+W
+�
+s2
+1,W
+N1 +
+s2
+2,W
+N2
+,
+ν =
+(
+s2
+1,W
+N1 +
+s2
+2,W
+N2 )2
+s4
+1,W
+N2
+1 ν1 +
+s4
+2,W
+N2
+2 ν2
+.
+If two populations have equal mean, then t satisfies the t-distribution with degree of freedom
+ν.
+The weighted Welch’s t-test was applied to the growth experiment with different initial
+cell numbers, in order to determine whether the growth rates during exponential phase
+14
+
+(5–50 area units) were different between groups. Here Xj
+i corresponded to growth rate,
+and cj
+i corresponded to cell area. The p-value for N0 = 10-cell group vs. N0 = 4-cell group
+was 2.12 × 10−8; the p-value for N0 = 10-cell group vs. N0 = 1-cell group was smaller than
+10−12; the p-value for N0 = 4-cell group vs. N0 = 1-cell group was 5.35 × 10−5. Therefore,
+the growth rate difference between any two groups was statistically significant.
+4.4
+Permutation Test.
+The permutation test is a non-parametric method to test whether two samples are signifi-
+cantly different with respect to a statistic (e.g., sample mean) [16]. It is easy to calculate
+and fits our situation, thus we adopt this test rather than other more complicated tests,
+such as the Mann-Whitney test.
+For two samples {x1, · · · , xm}, {y1, · · · , yn}, consider
+the null hypothesis: the mean of x and y are the same. For these samples, calculate the
+mean of the first sample: µ0 =
+1
+m
+� xi. Then we randomly divide these m + n samples
+into two groups with size m and n: {x′
+1, · · · , x′
+m}, {y′
+1, · · · , y′
+n}, such that each permuta-
+tion has equal probability. For these new samples, calculate the mean of the first sample:
+µ′
+0 = 1
+m
+� x′
+i. Then the two-sided p-value is defined as
+p = 2 min{P(µ0 ≤ µ′
+0), 1 − P(µ0 ≤ µ′
+0)}.
+If µ0 is an extreme value in the distribution of µ′
+0, then the two sample means are different.
+In the reseeding experiment, the mean time of exceeding half well for the fast group
+was 11.4375 days. For all
+�64
+32
+�
+possible result combinations, only 7 combinations had equal
+or less mean time. Thus the p-value was 2 × 7/
+�64
+32
+�
+= 7.6 × 10−18. This indicated that the
+growth rate difference between fast group and moderate group was significant.
+4.5
+Model Details.
+The simulation time interval was half day, but we only utilized the results in full days. For
+each initial cell, the probabilities of being fast, moderate or slow type, pF, pM, pS, were 0.4,
+0.4, 0.2.
+Each half day, a fast type cell had probability d to die, and probability gF to divide.
+The division produced two fast cells, capturing the intrinsic growth behavior that is to
+some extent inheritable. Denote the total cell number of previous day as N, then
+gF = g0(1 − N2/C2) + δ,
+where δ is a random variable that satisfies the uniform distribution on [−r, r], and it is a
+constant for all cells in the same well. If gF < 0, set gF = 0. If gF > 1 − d, set gF = 1 − d.
+In the simulation displayed, death rate d = 0.01, carrying capacity C = 40000, growth
+factor g0 = 0.5, and the range of random modifier r = 0.1.
+Each half day, a moderate type cell had probability d to die, and probability gM to
+divide. The division produced two moderate cells. gM = gF/1.5.
+15
+
+Similarly, each half day, a slow type cell had probability d to die, and probability gS to
+divide. The division produced two slow-growing cells. gS = gF/3.
+4.6
+Parameter scan.
+Since growth is measured by the area covered by cells, we could not experimentally verify
+most assumptions of our model, or determine the values of parameters.
+Therefore, we
+performed a parameter scan by evaluating the performance of our model for different sets
+of parameters.
+We adjusted 6 parameters: initial type probabilities pF, pM, pS, death
+rate d, growth factor g0, and random modifier r. We checked whether these 4 features
+observable in the experiment could be reproduced: growth of all wells in the N0 = 10-cell
+group to saturation; existence of late-growing wells in the N0 = 1-cell group; existence of
+non-growing wells in the N0 = 1-cell group; difference in growth rates in the N0 = 10-cell
+group and the N0 = 1-cell group at the same population size. Table 3 shows the results
+of the performance of simulations with the various parameter sets. Within a wide range
+of parameters, our model is able to replicate the experimental results shown in Figs. 1–3,
+indicating that our model is robust under perturbations.
+Acknowledgements
+We would like to thank Ivana Bozic, Yifei Liu, Georg Luebeck, Weili Wang, Yuting Wei
+and Lingxue Zhu for helpful advice and discussions.
+References
+[1] Angelini, E., Wang, Y., Zhou, J. X., Qian, H., and Huang, S. A model for
+the intrinsic limit of cancer therapy: Duality of treatment-induced cell death and
+treatment-induced stemness. PLOS Comput. Biol. 18, 7 (2022), e1010319.
+[2] Bartoszynski, R., Brown, B. W., McBride, C. M., and Thompson, J. R.
+Some nonparametric techniques for estimating the intensity function of a cancer re-
+lated nonstationary Poisson process. Ann. Stat. (1981), 1050–1060.
+[3] Bhartiya, D., Kausik, A., Singh, P., and Sharma, D. Will single-cell rnaseq
+decipher stem cells biology in normal and cancerous tissues?
+Hum. Reprod. Update
+27, 2 (2021), 421–421.
+[4] Chang, H. H., Hemberg, M., Barahona, M., Ingber, D. E., and Huang,
+S. Transcriptome-wide noise controls lineage choice in mammalian progenitor cells.
+Nature 453, 7194 (2008), 544–547.
+16
+
+[5] Chapman, A., del Ama, L. F., Ferguson, J., Kamarashev, J., Wellbrock,
+C., and Hurlstone, A. Heterogeneous tumor subpopulations cooperate to drive
+invasion. Cell Rep. 8, 3 (2014), 688–695.
+[6] Chen, X., Wang, Y., Feng, T., Yi, M., Zhang, X., and Zhou, D. The overshoot
+and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic
+plasticity. J. Theor. Biol. 390 (2016), 40–49.
+[7] Clark, W. H. Tumour progression and the nature of cancer. Br. J. Cancer 64, 4
+(1991), 631.
+[8] Dewanji, A., Luebeck, E. G., and Moolgavkar, S. H. A generalized Luria–
+Delbr¨uck model. Math. Biosci. 197, 2 (2005), 140–152.
+[9] Durrett, R., Foo, J., Leder, K., Mayberry, J., and Michor, F. Intratumor
+heterogeneity in evolutionary models of tumor progression. Genetics 188, 2 (2011),
+461–477.
+[10] Egeblad, M., Nakasone, E. S., and Werb, Z.
+Tumors as organs: Complex
+tissues that interface with the entire organism. Dev. Cell 18, 6 (2010), 884–901.
+[11] Gatenby, R. A., Cunningham, J. J., and Brown, J. S. Evolutionary triage gov-
+erns fitness in driver and passenger mutations and suggests targeting never mutations.
+Nat. Commun. 5 (2014), 5499.
+[12] Goldberg, L. R., Kercheval, A. N., and Lee, K.
+T-statistics for weighted
+means in credit risk modeling. J. Risk Finance 6, 4 (2005), 349–365.
+[13] Gunnarsson, E. B., De, S., Leder, K., and Foo, J. Understanding the role of
+phenotypic switching in cancer drug resistance. J. Theor. Biol 490 (2020), 110162.
+[14] Gupta, P. B., Fillmore, C. M., Jiang, G., Shapira, S. D., Tao, K., Kuper-
+wasser, C., and Lander, E. S. Stochastic state transitions give rise to phenotypic
+equilibrium in populations of cancer cells. Cell 146, 4 (2011), 633–644.
+[15] Hanahan, D., and Weinberg, R. A. Hallmarks of cancer: The next generation.
+Cell 144, 5 (2011), 646–674.
+[16] Hastie, T., Tibshirani, R., and Friedman, J. The Elements of Statistical Learn-
+ing, 2nd ed. Springer, New York, 2016.
+[17] Hordijk, W., Steel, M., and Dittrich, P.
+Autocatalytic sets and chemical
+organizations: modeling self-sustaining reaction networks at the origin of life. New J.
+Phys. 20, 1 (2018), 015011.
+17
+
+[18] Howard, G. R., Johnson, K. E., Rodriguez Ayala, A., Yankeelov, T. E.,
+and Brock, A. A multi-state model of chemoresistance to characterize phenotypic
+dynamics in breast cancer. Sci. Rep. 8, 1 (2018), 1–11.
+[19] Jiang, D.-Q., Wang, Y., and Zhou, D. Phenotypic equilibrium as probabilistic
+convergence in multi-phenotype cell population dynamics. PLOS ONE 12, 2 (2017),
+e0170916.
+[20] Johnson, K. E., Howard, G., Mo, W., Strasser, M. K., Lima, E. A., Huang,
+S., and Brock, A. Cancer cell population growth kinetics at low densities deviate
+from the exponential growth model and suggest an Allee effect. PLOS Biol. 17, 8
+(2019), e3000399.
+[21] Kang, Y., Gu, C., Yuan, L., Wang, Y., Zhu, Y., Li, X., Luo, Q., Xiao, J.,
+Jiang, D., Qian, M., et al. Flexibility and symmetry of prokaryotic genome re-
+arrangement reveal lineage-associated core-gene-defined genome organizational frame-
+works. mBio 5, 6 (2014), e01867–14.
+[22] Koch, A. L. Mutation and growth rates from Luria-Delbr¨uck fluctuation tests. Mutat.
+Res. -Fund. Mol. M. 95, 2-3 (1982), 129–143.
+[23] Kochanowski, K., Sander, T., Link, H., Chang, J., Altschuler, S. J., and
+Wu, L. F. Systematic alteration of in vitro metabolic environments reveals empirical
+growth relationships in cancer cell phenotypes. Cell Rep. 34, 3 (2021), 108647.
+[24] Korolev, K. S., Xavier, J. B., and Gore, J. Turning ecology and evolution
+against cancer. Nat. Rev. Cancer 14, 5 (2014), 371–380.
+[25] Lea, D. E., and Coulson, C. A. The distribution of the numbers of mutants in
+bacterial populations. J. Genet. 49, 3 (1949), 264–285.
+[26] Li, Q., Wennborg, A., Aurell, E., Dekel, E., Zou, J.-Z., Xu, Y., Huang, S.,
+and Ernberg, I. Dynamics inside the cancer cell attractor reveal cell heterogeneity,
+limits of stability, and escape. Proc. Natl. Acad. Sci. U.S.A. 113, 10 (2016), 2672–2677.
+[27] Luebeck, G., and Moolgavkar, S. H. Multistage carcinogenesis and the incidence
+of colorectal cancer. Proc. Natl. Acad. Sci. 99, 23 (2002), 15095–15100.
+[28] Luria, S. E., and Delbr¨uck, M. Mutations of bacteria from virus sensitivity to
+virus resistance. Genetics 28, 6 (1943), 491.
+[29] Mackillop, W. J. The growth kinetics of human tumours. Clin. Phys. Physiol. M.
+11, 4A (1990), 121.
+[30] Meacham, C. E., and Morrison, S. J.
+Tumour heterogeneity and cancer cell
+plasticity. Nature 501, 7467 (2013), 328–337.
+18
+
+[31] Newton, P. K., Mason, J., Bethel, K., Bazhenova, L. A., Nieva, J., and
+Kuhn, P.
+A stochastic markov chain model to describe lung cancer growth and
+metastasis. PLOS ONE 7, 4 (2012), e34637.
+[32] Niu, Y., Wang, Y., and Zhou, D. The phenotypic equilibrium of cancer cells:
+From average-level stability to path-wise convergence. J. Theor. Biol. 386 (2015),
+7–17.
+[33] Pisco, A. O., and Huang, S. Non-genetic cancer cell plasticity and therapy-induced
+stemness in tumour relapse: ‘What does not kill me strengthens me’. Br. J. Cancer
+112, 11 (2015), 1725–1732.
+[34] Sahoo, S., Mishra, A., Kaur, H., Hari, K., Muralidharan, S., Mandal, S.,
+and Jolly, M. K. A mechanistic model captures the emergence and implications of
+non-genetic heterogeneity and reversible drug resistance in ER+ breast cancer cells.
+NAR Cancer 3, 3 (2021), zcab027.
+[35] Skehan, P., and Friedman, S. J. Non-exponential growth by mammalian cells in
+culture. Cell Prolif. 17, 4 (1984), 335–343.
+[36] Sonnenschein, C., and Soto, A. M. Somatic mutation theory of carcinogenesis:
+Why it should be dropped and replaced. Mol. Carcinog. 29, 4 (2000), 205–211.
+[37] Speer, J. F., Petrosky, V. E., Retsky, M. W., and Wardwell, R. H. A
+stochastic numerical model of breast cancer growth that simulates clinical data. Cancer
+Res. 44, 9 (1984), 4124–4130.
+[38] Spina, S., Giorno, V., Rom´an-Rom´an, P., and Torres-Ruiz, F. A stochastic
+model of cancer growth subject to an intermittent treatment with combined effects:
+Reduction in tumor size and rise in growth rate. Bull. Math. Biol. 76, 11 (2014),
+2711–2736.
+[39] Tabassum, D. P., and Polyak, K. Tumorigenesis: It takes a village. Nat. Rev.
+Cancer 15, 8 (2015), 473–483.
+[40] Wang, Y.
+Algorithms for finding transposons in gene sequences.
+arXiv preprint
+arXiv:1506.02424 (2015).
+[41] Wang, Y. Some Problems in Stochastic Dynamics and Statistical Analysis of Single-
+Cell Biology of Cancer. Ph.D. thesis, University of Washington, 2018.
+[42] Wang, Y. Two metrics on rooted unordered trees with labels. Algorithms for Molec-
+ular Biology 17, 1 (2022), 1–17.
+19
+
+[43] Wang, Y., Dessalles, R., and Chou, T. Modelling the impact of birth control
+policies on China’s population and age: effects of delayed births and minimum birth
+age constraints. Royal Society Open Science 9, 6 (2022), 211619.
+[44] Wang, Y., and He, S. Inference on autoregulation in gene expression. arXiv preprint
+arXiv:2201.03164 (2022).
+[45] Wang, Y., Kropp, J., and Morozova, N. Biological notion of positional informa-
+tion/value in morphogenesis theory. International Journal of Developmental Biology
+64, 10-11-12 (2020), 453–463.
+[46] Wang, Y., Minarsky, A., Penner, R., Soul´e, C., and Morozova, N. Model
+of morphogenesis. Journal of Computational Biology 27, 9 (2020), 1373–1383.
+[47] Wang, Y., Mistry, B. A., and Chou, T. Discrete stochastic models of selex:
+Aptamer capture probabilities and protocol optimization. The Journal of Chemical
+Physics 156, 24 (2022), 244103.
+[48] Wang, Y., and Qian, H. Mathematical representation of Clausius’ and Kelvin’s
+statements of the second law and irreversibility. Journal of Statistical Physics 179, 3
+(2020), 808–837.
+[49] Wang, Y., and Wang, L. Causal inference in degenerate systems: An impossibility
+result. In International Conference on Artificial Intelligence and Statistics (2020),
+PMLR, pp. 3383–3392.
+[50] Wang, Y., and Wang, Z. Inference on the structure of gene regulatory networks.
+Journal of Theoretical Biology 539 (2022), 111055.
+[51] Wang, Y., Zhang, B., Kropp, J., and Morozova, N. Inference on tissue trans-
+plantation experiments. Journal of Theoretical Biology 520 (2021), 110645.
+[52] Ye, F. X.-F., Wang, Y., and Qian, H. Stochastic dynamics: Markov chains and
+random transformations. Discrete & Continuous Dynamical Systems-B 21, 7 (2016),
+2337.
+[53] Yorke, E. D., Fuks, Z., Norton, L., Whitmore, W., and Ling, C. C. Model-
+ing the development of metastases from primary and locally recurrent tumors: Com-
+parison with a clinical data base for prostatic cancer.
+Cancer Res. 53, 13 (1993),
+2987–2993.
+[54] Zheng, Q. Progress of a half century in the study of the Luria–Delbr¨uck distribution.
+Math. Biosci. 162, 1 (1999), 1–32.
+[55] Zhou, D., Wang, Y., and Wu, B.
+A multi-phenotypic cancer model with cell
+plasticity. J. Theor. Biol. 357 (2014), 35–45.
+20
+
+[56] Zhou, J. X., Pisco, A. O., Qian, H., and Huang, S. Nonequilibrium population
+dynamics of phenotype conversion of cancer cells. PLOS ONE 9, 12 (2014), e110714.
+21
+
+Figure 1: Growth curves of the experiment (left) and simulation (right), starting from
+the time of reaching 5 area units (experiment) or having 2500 cells (simulation), with a
+logarithm scale for the y-axis. The time required for reaching 5 area units was determined
+by exponential extrapolation, as reliable imaging started at > 5 area units. The x-axis is
+the time from reaching 5 area units (experiment) or 2500 cells (simulation). Red, green,
+or blue curves correspond to 10, 4, or 1 initial cell(s). Although starting from the same
+population level, patterns are different for distinct initial cell numbers. The N0 = 1-cell
+group has higher diversity.
+22
+
+experimental
+80
+cell area
+40
+20
+10-cell group
+4-cell group
+10
+1-cell group
+5
+0
+5
+10
+15
+time (day)
+80
+cell area
+40
+20simulation
+40000
+cell number
+20000
+10000
+5000
+2500
+0
+5
+10
+15
+time (day)
+40000
+ell number
+20000
+100005
+0
+5
+10
+15
+time (day)
+80
+cell area
+40
+20
+10
+5
+0
+5
+10
+15
+time (day)8
+QQQ
+2500
+0
+5
+10
+15
+time (day)
+40000
+cell number
+20000
+10000
+5000
+2500
+0
+5
+10
+15
+time (day)20
+40
+60
+80
+cell area
+0
+0.5
+1
+1.5
+growth rate
+experimental
+10-cell group
+4-cell group
+1-cell group
+0
+20
+40
+60
+80
+cell area
+0
+0.5
+1
+1.5
+growth rate
+1
+2
+3
+4
+cell number
+104
+0
+0.5
+1
+1.5
+growth rate
+simulation
+0
+1
+2
+3
+4
+cell number
+104
+0
+0.5
+1
+1.5
+growth rate
+Figure 2: Per capita growth rate (averaged within one day) vs. cell population for the
+experiment (left) and simulation (right). Each point represents one well in one day. Red,
+green, or blue points correspond to 10, 4, or 1 initial cell(s).
+23
+
+0
+5
+10
+15
+20
+time (day)
+0
+20
+40
+60
+80
+cell area
+experimental
+10-cell group
+4-cell group
+1-cell group
+0
+5
+10
+15
+20
+time (day)
+5
+10
+20
+40
+80
+cell area
+0
+5
+10
+15
+20
+time (day)
+0
+1
+2
+3
+4
+cell number
+104
+simulation
+0
+5
+10
+15
+20
+time (day)
+2500
+5000
+10000
+20000
+40000
+cel number
+Figure 3: Growth curves of the experiments with different initial cell numbers N0 (left)
+and growth curves of corresponding simulation (right). Each curve describes the change in
+the cell population (measured by area or number) over a well along time. Red, green, or
+blue curves correspond to N0 = 10, 4, or 1 initial cell(s).
+24
+
+Figure 4: Schematic illustration of the qualitative argument: Three cell types and growth
+patterns (three colors) with different seeding numbers. One N0 = 10-cell well will have
+at least one fast type cell with high probability, which will dominate the population. One
+N0 = 1-cell well can only have one cell type, thus in the microculture ensemble of replicate
+wells, three possible growth patterns for wells can be observed.
+25
+
+fast
+moderate
+slow
+fast
+fast
+moderate
+slow

BdE2T4oBgHgl3EQfRge6/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:92a209d77bde2f71d9801d94de1e39f975e17a8eaeb69d4ad8362b722afe07a1
+size 151696

BdE4T4oBgHgl3EQf5Q5A/content/2301.05321v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6d96a590ac96bc21cd11ad454bc2ad0b2970aec2742165ec1bf0ef55d16bce30
+size 983040

BdE4T4oBgHgl3EQf5Q5A/content/tmp_files/2301.05321v1.pdf.txt

@@ -0,0 +1,2198 @@
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell
+populations via crowding control: stability,
+robustness and quasi-dedifferentiation
+Cristina Parigini1,2,3 and Philip Greulich1,2*
+1*School of Mathematical Sciences, University of Southampton,
+Southampton, United Kingdom.
+2Institute for Life Sciences, University of Southampton,
+Southampton, United Kingdom.
+3Te P¯unaha ¯Atea - Space Institute, University of Auckland,
+Auckland, New Zealand.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: [email protected];
+Abstract
+To maintain renewing epithelial tissues in a healthy, homeostatic state,
+(stem) cell divisions and differentiation need to be tightly regulated.
+Mechanisms of homeostatic control often rely on crowding control: cells
+are able to sense the cell density in their environment (via various
+molecular and mechanosensing pathways) and respond by adjusting
+division, differentiation, and cell state transitions appropriately. Here
+we determine, via a mathematically rigorous framework, which general
+conditions for the crowding feedback regulation (i) must be minimally
+met, and (ii) are sufficient, to allow the maintenance of homeosta-
+sis in renewing tissues. We show that those conditions naturally allow
+for a degree of robustness toward disruption of regulation. Further-
+more, intrinsic to this feedback regulation is that stem cell identity is
+established collectively by the cell population, not by individual cells,
+which implies the possibility of ‘quasi-dedifferentiation’, in which cells
+committed to differentiation may reacquire stem cell properties upon
+depletion of the stem cell pool. These findings can guide future exper-
+imental campaigns to identify specific crowding feedback mechanisms.
+Keywords: keyword1, Keyword2, Keyword3, Keyword4
+1
+arXiv:2301.05321v1  [q-bio.TO]  12 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Homeostatic regulation of renewing tissue cell populations via crowding control
+1 Introduction
+Many adult tissues are renewing, that is, terminally differentiated cells are
+steadily removed and replaced by new cells produced by the division of cycling
+cells (stem cells and progenitor cells), which then differentiate. In order to
+maintain those tissues in a healthy, homeostatic state, (stem) cell divisions
+and differentiation must be tightly balanced. Adult stem cells are the key
+players in maintaining and renewing such tissues due to their ability to produce
+cells through cell division and differentiation persistently [1]. However, the
+underlying cell-intrinsic and extrinsic factors that regulate a homeostatic state
+are complex and not always well understood.
+Several experimental studies have identified mechanisms and pathways that
+regulate homeostasis. For example, cell crowding can trigger delamination and
+thus loss of cells in Drosophila back [2], and differentiation in cultured human
+colon, various zebrafish epiderimises, and canine kidney cells [3, 4]. On the
+other hand, cell crowding can affect cell proliferation: overcrowding can inhibit
+proliferation [5], whereas a reduction in the cell density, obtained, for example,
+by stretching a tissue [6] causes an increase in proliferative activity (both
+shown in cultured canine kidney cells). Although the mechanisms to mediate
+this regulation are not always clear, experimental studies on mechanosensing
+showed that cell overcrowding reduces cell motility and consequently produces
+a compression on cells that inhibits cell proliferation [5, 7]. Another mechanism
+utilising crowding feedback is the competition for limited growth signalling
+factors [8]. More specifically, in the mouse germ line, cells in the niche respond
+to a growth factor (FGF5) that promotes proliferation over differentiation,
+which they deplete upon being exposed to it. Therefore, the more cells are
+in the niche, the less FGF5 is available per cell, and the less proliferation (or
+more differentiation) occurs.
+Despite differing in the involved molecular pathways and many other
+details, all these regulatory mechanisms are, in essence, sensing the cell den-
+sity in their environment and responding by adjusting their propensities to
+divide, differentiate, die, or emigrate from the tissue. This class of mechanisms,
+for which cell fate propensities depend on the cell density, can be classified as
+crowding feedback regulation: the cell density determines the cells’ prolifera-
+tion and differentiation, which affects their population dynamics and thus the
+cell density. However, the crowding response to changes in cell density cannot
+be arbitrary in order to maintain homeostasis. It must provide a (negative)
+feedback, in the sense that cells sense the cell density and adjust proliferation,
+differentiation, and cell loss, such that the cell density is decreased if it is too
+high and increased if it is too low. For simple tissues consisting of a single
+cell type with a unique cell state, it is relatively straightforward to give the
+conditions for crowding feedback to maintain homeostasis successfully. In this
+case, when the cell division rate decreases with cell density and differentiation
+and or death rate increase with cell density, a homeostatic state is maintained.
+However, such conclusions are not as simple to make when a tissue consists of
+a complex lineage hierarchy and a multitude of underlying cellular states. In
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+3
+the latter, more realistic case, conditions for successful homeostatic regulation
+– in which case we speak of crowding control – may take more complex forms.
+Previous studies based on mathematical modelling have shed some light
+on quantitative mechanisms for homeostatic control [9–13]. In particular, in
+[13], a mathematical assessment of crowding feedback modelling shows that
+a (dynamic) homeostatic state exists under reasonable biological conditions.
+Nevertheless, the case of dynamic homeostasis considered there may not nec-
+essarily be a steady state but could also exhibit oscillations in cell numbers
+(as does realistically happen in the uterus during the menstrual cycle). While
+the criterion presented in [13] provides a valid sufficient condition for dynamic
+homeostasis, it relies on a rather abstract mathematical quantity – the domi-
+nant eigenvalue of the dynamical matrix – that is difficult, if not impossible,
+to measure in reality.
+Here, we wish to generalise previous findings and seek to identify general
+conditions for successful homeostatic control if propensities for cell division,
+differentiation, and loss are responsive to variations in cell density. More
+precisely, we derive conditions that must be minimally fulfilled (necessary
+conditions) and conditions which are sufficient, to ensure that homeostasis pre-
+vails. To identify and formulate those conditions, we note that homeostasis is
+a property of the tissue cell population dynamics, which can be mathemati-
+cally expressed as a dynamical system. Even if a numerically exact formulation
+of the dynamics may not be possible, one can formulate generic yet mathe-
+matically rigorous conditions by referring to the criteria for the existence of
+stable steady states in the cell population dynamics of renewing tissues. We
+will derive those conditions by mathematical, analytical means, augmented by
+a numerical analysis testing the limits of those conditions.
+We will also show that homeostatic control by crowding feedback possesses
+inherent robustness to failures and perturbations of the regulatory pathways,
+which may occur through external influences (e.g. wide-spread biochemical fac-
+tors) and genetic mutations. Finally, we will assess the response of cells when
+the pool of stem cells is depleted. Crucially, we find that inherent to crowd-
+ing feedback control is that formerly committed progenitor cells reacquire
+self-renewal capacity without substantial changes in their internal states. Ded-
+ifferentiation has been widely reported under conditions of tissue regeneration
+[14, 15] or when stem cells are depleted [16–19], which is usually thought to
+involve a substantial reprogramming of the cell-intrinsic states towards a stem
+cell type. On the other hand, our analysis suggests the possibility of “quasi”-
+dedifferentiation, the reversion from a committed cell to a stem cell by a
+mere quantitative adjustment of the pacing of proliferation and differentiation,
+without a substantial qualitative change in its expression profiles.
+
+Springer Nature 2021 LATEX template
+4
+Homeostatic regulation of renewing tissue cell populations via crowding control
+2 Modelling of tissue cell dynamics under
+crowding feedback
+We seek to assess the conditions for homeostasis in renewing tissue cell pop-
+ulations, that is, either a steady state of the tissue cell population (strict
+homeostasis) or long-term, bounded oscillations or fluctuations (dynamic
+homeostasis), which represent well-defined constraints on the dynamics of the
+tissue cell population. To this end, we will here derive a formal, mathematical
+representation of the tissue cell dynamics under crowding feedback.
+The cell population is fully defined by (i) the number of cells, (ii) the
+internal (biochemical and mechanical) states of each cell, and (iii) the spatial
+position of cells. We assume that a cell’s behaviour can depend on the cell
+density and the states of cells in its close cellular environment. As we examine
+a situation close to a homeostatic state, we assume that the cell density is
+homogeneous over the range of interaction between cells, which expands over
+a volume V . Hence, the cell density ρ is proportional to the average number
+of cells, ¯n, in that volume, ρ =
+¯n
+V . Similarly, we define the number of cells
+in internal state i as ni, and the cell density of cells in internal state i as
+ρi = ¯ni
+V , where ¯ni is the expected value of ni. As we consider only the crowding
+feedback response of cells, which only accounts for the cell densities ρi but
+not the explicit position of cells, the spatial configuration (iii) is not relevant
+to our considerations. Thus, the configuration of the cell population and its
+time evolution is entirely determined by the average number of cells in each
+state i, as a function of time t, ¯ni(t). The configuration of cell numbers ni
+can change only through three processes: (1) cell division, whereby it must be
+distinguished between the cell state of daughter cells, (2) the transition from
+one cell state to another, (3) loss of a cell, through cell death or emigration
+out of the tissue. Following the lines of Refs. [13, 20] and denoting as Xi,j,k a
+cell in internal states i, j, k, respectively, we can formalise these events as:
+cell division: Xi
+λirjk
+i
+−−−→ Xj + Xk
+(1)
+cell state transition: Xi
+ωij
+−−→ Xj
+(2)
+cell loss: Xi
+γi
+−→ ∅ ,
+(3)
+where the symbols above the arrows denote the dynamical rates of the transi-
+tions, i.e. the average frequency at which such events occur. In particular, γi
+is the rate at which a cell in state i is lost, ωij the rate at which a cell changes
+its state from i to j and λirjk
+i
+denotes the rate at which a cell i divides to pro-
+duce two daughter cells, one in state j and one in state k (i = j, j = k, k = i
+are possible). For later convenience, we distinguish here the overall rate of cell
+division in state i, λi and the probability rjk
+i
+that such a division produces
+daughter cells in states j and k.
+Since we consider a situation where cells can respond to the cell densities
+ρi via crowding feedback, all the rates and probabilities (λi, γi, ωij, rjk
+i ) may
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+5
+depend on the cell densities of either state j, ρj. For convenience, we discretise
+the number of states in case the state space is a continuum and only distinguish
+states which have substantially different propensities (λi, γi, ωij, rjk
+i ). Without
+loss of generality, we assume that there are m states, that is, i, j, k = 1, ..., m
+(for a rigorous argument for the discretisation of the state space, see [13]).
+The rates given above denote the average number of events happening per
+time unit. Thus, we can express the total rate of change of the average (i.e.
+expected) number of cells ¯ni(t), that is, the derivative ˙¯ni = d¯ni
+dt , in terms of
+the rates of those events. This defines a set of ordinary differential equations.
+Following the lines of Refs. [13, 20], we can write ˙ni as,
+˙¯ni =
+��
+j
+ωji¯nj + λj
+��
+k
+rik
+j + rki
+j
+�
+¯nj
+�
+− ¯ni
+�
+λi + γi +
+�
+j
+ωij
+�
+,
+(4)
+where for convenience, we did not write the time dependence explicitly, i.e.
+ni = ni(t), and all parameters may depend on the cell densities ρj. Since V is
+constant, we can divide by V to equivalently express this in terms of the cell
+state densities, ρi = ¯ni
+V , and then write Eq. (4) compactly as,
+d
+dtρ(t) = A(ρ(t)) ρ(t)
+(5)
+where ρ = (ρ1, ρ2, ...) is the vector of cell state densities and A(ρ) is the matrix,
+A =
+�
+�
+λ1 − �
+j̸=1 κ1j − γ1
+κ21
+κ31
+· · ·
+κ12
+λ2 − �
+j̸=2 κ2j − γ2 κ32
+· · ·
+κ1m
+κ2m
+· · · λm − �
+j̸=m κmj − γm
+�
+� ,
+(6)
+in which κij = λi2rj
+i + ωij, with rj
+i = �
+k(rjk
+i
++ rkj
+i )/2, is the total transition
+rate, that combines all transitions from Xi to Xj by cell divisions and direct
+state transitions (again, all parameters may depend on ρ, as therefore also
+does A). We can thus generally write the elements of the matrix A, aij with
+i, j = 1, ..., m as
+aij =
+� λi − γi − �
+k̸=i κik
+for i = j
+κji
+for i ̸= j
+(7)
+We now make the mild assumption that divisions of the form Xi → Xj+Xk
+are effectively three events, namely, cell duplication, Xi → Xi + Xi coupled to
+cell state changes, Xi → Xj and Xi → Xk, if j ̸= i or k ̸= i. In this view, the
+parameters relevant for crowding feedback are the total cell state transition
+propensities κij and the cell division rate λi, as in (6), instead of ωij and rjk
+i .
+These equations describe a dynamical system which, for given initial con-
+ditions, determines the time evolution of the cell densities, ρi(t). Crucially,
+
+Springer Nature 2021 LATEX template
+6
+Homeostatic regulation of renewing tissue cell populations via crowding control
+this description allows for a rigorous mathematical definition of what a home-
+ostatic state is, and to apply tools of dynamical systems analysis to determine
+the circumstances under which a homeostatic state prevails. In particular, we
+define a strict homeostatic state as a steady state of the system, (5), when the
+cell numbers – and thus cell densities, given that V is fixed – in each state
+do not change, mathematically expressed as dρ
+dt = 0 (a fixed point of the sys-
+tem). A dynamic homeostatic state is when cell densities may also oscillate
+or fluctuate but remain bounded and thus possess a finite long-term average
+cell population (in which case the system either approaches a steady state or
+limit cycles – that is, oscillations – or chaotic but bounded behaviour). Based
+on these definitions, we can now analyse under which circumstances crowd-
+ing feedback can maintain those states, which in the case of strict homeostasis
+requires, in addition, that the corresponding steady state is stable.
+2.1 Cell types and lineage hierarchies
+According to [13], cell population dynamics of the type (5) can be associated
+with a cell state network, in which each state is a node, and the nodes are
+connected through cell state transition (direct transitions and cell divisions).
+Furthermore, by decomposing this network in strongly connected components
+(SCCs), the cell fate model can be viewed as a directed acyclic network [21],
+generally called the condensed network. Here, we follow the definitions of [13]
+and define a cell type as an SCC of the cell state network, so that any cell states
+connected via cyclic cell state trajectories (sequences of cell state transitions)
+are of the same type, and the condensed network of cell types represents the
+cell lineage hierarchy. This definition ensures that cells of the same type have
+the same lineage potential (outgoing cell state trajectories) and that the stages
+of the cell cycle are associated with the same cell type. In this context, we
+will in the following also speak of differentiation when a cell state transition
+between different cell types occurs.
+Each cell type can be classified as self-renewing, declining or hyper-
+proliferating, depending on the dominant eigenvalue µ (called growth parame-
+ter) of the dynamical matrix A (from Eq. (5) ff.) reduced to that SCC. This
+is µ = 0 for self-renewing cell types, when cell numbers of that type remain
+constant over time, µ < 0 (µ > 0) for the declining (hyperproliferating) types
+when cell numbers decline (increase) in the long term [13]. Importantly, for the
+population dynamics to be strictly homeostatic, which means that a steady
+state of model (5) exists, the cell type network must fulfil strict rules. These
+are: (i) at least one self-renewing cell type (with µ = 0) must exist; (ii) self-
+renewing cell types must stay at an apex of the condensed network; (iii) all
+the other cells must be of declining types. This means that the critical task of
+homeostatic control is to ensure that the kinetic parameters of the cell type at
+the apex of the cell lineage hierarchy are fine-tuned to maintain exactly µ = 0.
+Therefore, we can restrict our analysis to find conditions for the cell type
+at the lineage hierarchy’s apex to be self-renewing, which we will do in the
+following. Other cell types simply need that differentiation (transition towards
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+7
+another cell type) or loss is faster than proliferation, so that they become
+declining cell types, µ < 0, but those rates do not require fine-tuning and thus
+trivially regulated. We note that when we consider only cell states of the type
+at the apex of the cell lineage hierarchy, any differentiation event is – according
+to this restricted model – a cell loss event and included as event occurring with
+rates γi. Given that cell loss from a cell type at the lineage apex is rare, we
+will therefore in the following also denote the rates γi simply as differentiation
+rates.
+3 Results
+We will now determine necessary and sufficient conditions for the establish-
+ment of strict and dynamical homeostasis when subject to crowding feedback,
+which we here define through the derivatives of the dynamical parameters
+λi, rjk
+i , ωij, γi as a function of the cell densities. As argued before, we only need
+to consider cell types at an apex of the cell type network, which, for home-
+ostasis to prevail, must have a growth parameter (i.e. dominant eigenvalue of
+matrix A in Eq. (6)) µ = 0. Furthermore, we assume that the apex cell type
+resides in a separate stem cell niche. Therefore, the parameters only depend
+on cell densities ρi of states associated with that cell type, i.e. we can write
+A = A(ρ), where ρ = �
+i∈S ρi comprises only cell states of the apex cell type
+S. Provided that, the matrix elements are functions of ρ, and therefore also µ
+is a function of ρ. Thus, self-renewal corresponds to a non-trivial fixed point,
+ρ∗, of Eq. (5), restricted to cell type S, for which the dominant eigenvalue of
+A is zero, that is µ(ρ∗) = 0 (ρ∗ = �
+i∈S ρ∗
+i ).
+For convenience, we will often generally refer to parameters as αi, i =
+1, ..., 2m + m2, where αi stands for any of the parameters, {λi, γi, κij|i, j =
+1, ..., m}, respectively1. Hence, we study which conditions the functions αi(ρ)
+must meet to maintain homeostasis. In particular, we study how those param-
+eters qualitatively change with the cell density – increase or decrease – that
+is, we study how the sign and magnitude of derivatives α′
+i :=
+dαi
+dρ
+affects
+homeostasis.
+A crucial property of the matrix A(ρ) is that it is always a Metzler matrix,
+since all its off-diagonal elements, κij ≥ 0. Since the cell state network of a
+cell type is strongly connected, we can further state that A(ρ) is irreducible.
+Notably, for irreducible Metzler matrices holds the Perron-Frobenius theorem
+[22], and thus A(ρ) possesses a simple, real dominant eigenvalue µ. Besides,
+it as left and right eigenvectors associated with µ, respectively indicated as v
+and w, which are strictly positive, that is, all their entries are vi > 0, wi > 0.
+From this follows that the partial derivative of the dominant eigenvalue µ by
+1More
+precisely,
+αi|i=1,..,m
+:=
+λi, αi|i=m+1,..,2m
+:=
+γi−m, αi|i=2m+1,..,2m+m2
+:=
+κ⌊(i−2m)/m⌋,i−⌊(i−2m)/m⌋m
+
+Springer Nature 2021 LATEX template
+8
+Homeostatic regulation of renewing tissue cell populations via crowding control
+the i, j-th element of A, aij = [A]ij is always positive:
+∂µ
+∂aij
+= viwj
+vw > 0
+(8)
+where the left equality is according to [23] and is generally valid for simple
+eigenvalues. Here, v is assumed to be in row form, and vw thus corresponds
+to a scalar product.
+3.1 Sufficient condition for dynamic homeostasis
+In [13], it was shown that a dynamic homeostatic state, where cell numbers
+may change over time but stay bounded, is assured if, 2
+µ′(ρ) < 0 for all ρ > 0.
+(9)
+This sufficient condition requires that the dominant eigenvalue of A as a func-
+tion of the cell density, µ(ρ), is a strictly decreasing function of cell density.
+Also, the range of this function must be sufficiently large so that it has a root,
+i.e. a value ρ∗ with µ(ρ∗) = 0 must exist for the function µ(ρ).
+Assuming that a non-trivial steady state, ρ∗ > 0, exists, we now translate
+the sufficient condition for a dynamic homeostatic state, Eq. (9), into condi-
+tions on the parameters as a function of the cell density, αi(ρ). In particular,
+we can write,
+µ′(ρ) =
+�
+ij
+∂µ
+∂aij
+∂aij
+∂ρ =
+�
+ij
+viwj
+vw a′
+ij =
+�
+i
+viwi
+vw a′
+ii +
+�
+i,j̸=i
+viwj
+vw a′
+ij
+=
+�
+i
+viwi
+vw
+�
+�λ′
+i − γ′
+i −
+�
+j̸=i
+κ′
+ij
+�
+� +
+�
+i,j̸=i
+vjwi
+vw κ′
+ij ,
+(10)
+where we used Eq. (8) and the explicit forms of aij, the elements of the matrix
+A according to Eq. (7). Provided that all the parameters depend on ρ, condition
+(9) results in:
+0 > µ′ =⇒ 0 >
+�
+i
+viwi (λ′
+i − γ′
+i) + wi
+�
+j̸=i
+(vj − vi)κ′
+ij
+for all ρ > 0 ,
+(11)
+While we cannot give an explicit general expression for the dominant eigen-
+vectors v, w, this condition is sufficiently fulfilled if each term of the sum on
+the right-hand side of Eq. (11) is negative. More restrictively, we have Eq. (11)
+2In [13], this condition, defined through dependency on cell number, can be directly translated
+into a condition on the cell density derivative if the volume is assumed as a constant.
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+9
+sufficiently fulfilled if
+�
+�
+�
+�
+�
+λ′
+i ≤ 0, γ′
+i ≥ 0 for all i
+λ′
+i < 0 or γ′
+i > 0 at for least one i
+κ′
+ij = 0 for all i, j
+for ρ > 0
+(12)
+This means that, excluding rates that are zero, which are biologically mean-
+ingless, if no direct state transitions within a cell type are subject to crowding
+feedback (κ′
+ij = 0), while all (non-zero) cell division rates depend negatively
+on ρ (λ′
+i < 0), and differentiation rates depend positively (γ′
+i > 0), for all
+attainable levels of ρ, then dynamical homeostasis is ensured.
+Alternatively, we can rewrite Eq. (11) as
+0 >
+�
+i
+viwi
+vw
+�
+�λ′
+i − γ′
+i −
+�
+j̸=i
+κ′
+ij +
+�
+j̸=i
+vj
+vi
+κ′
+ij
+�
+�
+for all ρ > 0 ,
+(13)
+which, due to
+vj
+vi
+> 0, implies another sufficient condition for dynamic
+homeostasis:
+�
+�
+�
+�
+�
+λ′
+i ≤ 0, γ′
+i ≥ 0 for all i
+λ′
+i < 0 or γ′
+i > 0 at for least one i
+κ′
+ij ≤ 0 with |�
+j κ′
+ij| ≤ γ′
+i − λ′
+i for all i, j
+(14)
+The above condition is less restrictive than Eq. (12), allowing for some non-
+zero crowding feedback dependency of state transition rates κij, as long as the
+crowding feedback strength of the total outgoing transition rate of each state
+does not outweigh the feedback on proliferation and differentiation rate of that
+state (if there is).
+3.2 Necessary condition for strict homeostasis
+We now consider the circumstances under which a strict homeostatic is main-
+tained, that is, when a steady state of the cell population exists and is
+asymptotically stable.
+A necessary condition for the existence of a steady state ρ∗ (irrespective
+of stability) has been given in [13], namely, that the cell type at the apex of
+the lineage hierarchy is self-renewing, i.e. its dynamical matrix A has µ = 0.
+µ depends on the cell density ρ of the apex cell type, since the dynamical
+parameters αi and thus A depend on ρ. As before, it is required that µ(ρ∗)
+has sufficient range so that a value ρ∗ with µ(ρ∗) = 0 exists. This condition is
+fulfilled if the range of the feedback parameters αi(ρ) is sufficiently large. In
+that case there exists an eigenvector ρ∗ with A(ρ∗)ρ∗ = 0, which can be chosen
+by normalisation to fulfil �
+i∈S ρ∗
+i = ρ∗. Thus, ρ∗ is a fixed point (steady state)
+
+Springer Nature 2021 LATEX template
+10
+Homeostatic regulation of renewing tissue cell populations via crowding control
+of the cell population system (5). Hence, we need to establish what is required
+for this state to be asymptotically stable.
+To start with, we give the Jacobian matrix of the system (5) at the fixed
+point ρ∗ :
+[J]ij = ∂[A(ρ)ρ]i
+∂ρj
+����
+ρ=ρ∗
+= a∗
+ij + ηi ,
+(15)
+where
+ηi =
+�
+k
+a′
+ikρ∗
+k .
+(16)
+Here and in the following, we assume the derivatives to be taken at the steady
+state, i.e. a′
+ij :=
+daij
+dρ |ρ=ρ∗. The eigenvalues of the Jacobian matrix J at ρ∗
+determine the stability of the steady state ρ∗: it is asymptotically stable if and
+only if the real part of all eigenvalues of J(ρ∗) is negative.
+The Routh-Hurwitz theorem [24] states that for a polynomial to have only
+roots with negative real part, all its coefficients must necessarily be positive.
+Given that the eigenvalues of the Jacobian matrix J are the roots of its char-
+acteristic polynomial, a necessary condition for ρ∗ to be asymptotically stable
+is that the coefficients of the characteristic polynomial of J are all positive.
+Let us start by considering a self-renewing cell type with exactly two cell
+states being at the apex of a lineage hierarchy. This system has a 2 × 2 dynam-
+ical matrix A and Jacobian J, whereby A is irreducible and has dominant
+eigenvalue µA = 0. The characteristic polynomial of a generic 2×2 matrix, M,
+is
+P M(s) = s2 + pM
+1 s + pM
+0 .
+(17)
+with pM
+1 = −tr(M) and pM
+0 = det(M). In particular, since A has an eigenvalue
+zero,
+pA
+0 = det(A) = a11a22 − a12a21 = 0 .
+(18)
+From this follows that the right and left eigenvectors to the matrix A
+associated with the dominant eigenvalue µA = 0, w and v, are:
+w =
+�
+−a22
+a21
+�
+and v =
+�
+−a22 a12
+�
+.
+(19)
+From the Jacobian matrix J, we get equivalently,
+pJ
+0 = det(J) = (a21 − a22)(−a22η1 + a12η2)
+a22
+= vη |w|
+a22
+,
+(20)
+with the L1-norm |w| = w1 + w2 = −a22 + a213. Here we used the form of J
+in Eq. (15) with η = (η1, η2) from (16), as well as the relations (18) and (19),
+and we factorised the determinant.
+3Note that aii is always negative or zero
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+11
+From Eq. (10), we can further establish:
+µ′ =
+�
+ij
+viwj
+vw a′
+ij =
+�
+ij
+|w|
+ρ∗
+viρ∗
+j
+vw a′
+ij = |w|
+ρ∗
+vη
+vw
+(21)
+= − a22pJ
+0
+ρ∗pJ
+1 a22
+.
+(22)
+Here, we used that ρ∗ is a dominant right eigenvector, and thus ρ∗ =
+ρ∗
+|w|w,
+and furthermore we used the definition of ηi = �
+j a′
+ijρ∗
+j, we substituted Eq.
+(20), and used that vw = a2
+22 + a12a21 = −pA
+1 a22. Finally, we get:
+pJ
+0 = −µ′ρ∗pA
+1 .
+(23)
+Notably, we can show that this relation also holds for higher dimensions by
+explicitly computing the coefficients of characteristic polynomials pA,J
+i
+, the
+eigenvalues and eigenvectors, and then evaluating both sides of the equation.
+For systems with three states, this can be done analytically. For systems with
+4,5, and 6 states we tested relation (23) numerically by generating N =1000
+random matrices with entries chosen from a uniform distribution4. In each
+case, this relation was fulfilled. Hence we are confident that this relation holds
+up to 6 states, and it is reasonable to expect this to hold also for larger systems.
+Since A has a simple dominant eigenvalue µA = 0, we can factorise one term
+from the characteristic polynomial, P(s) = sQ(s) knowing that all roots of
+Q(s) are negative. Applying the Routh-Hurwitz necessary condition to Q(s), it
+follows that the coefficients of the polynomial Q, pQ
+i > 0, where i = 1, 2, ..., n−
+1. Thus, pA
+1 > 0 and considering that ρ∗ > 0 by definition, then for having
+pJ
+0 > 0 we must require µ′ < 0. Therefore, a necessary condition for a stable,
+strict homeostatic state is
+0 > µ′ =⇒ 0 >
+�
+i
+viwi (λ′
+i − γ′
+i) + wi
+�
+j̸=i
+(vj − vi)κ′
+ij
+������
+ρ=ρ∗
+,
+(24)
+where on the right-hand side, we used Eq. (11). This condition is bound to the
+validity of Eq. (23), that is, we can show it analytically for up to three states
+and numerically up to 6 states. Nonetheless, we also expect this to be true for
+larger systems.
+One way to satisfy this necessary condition is if at ρ = ρ∗
+�
+�
+�
+�
+�
+�
+i ≤ 0, γ′
+i ≥ 0 for all i
+λ′
+i < 0 or γ′
+i > 0 at for least one i
+κ′
+ij = 0
+.
+(25)
+4The diagonal elements of the random matrix are tuned using a local optimiser (fmincon
+function of Matab) so that the matrix has a zero dominant eigenvalue.
+
+Springer Nature 2021 LATEX template
+12
+Homeostatic regulation of renewing tissue cell populations via crowding control
+Notably, the necessary conditions (24) and (25) only differ from the suffi-
+cient conditions for dynamic heterogeneity, Eqs. (11) and (12), by needing to
+be fulfilled only at the steady-state cell density ρ∗, whereas to ensure dynamic
+homeostasis, those should be valid for a sufficiently large range of ρ.
+3.3 Sufficient condition for strict homeostasis
+Now we assess under which circumstances a strict homeostatic state is assured
+to prevail.
+First of all, the necessary conditions from above need to be fulfilled. In
+particular, the parameter functions αi(ρ) must have a sufficient range so that
+µ(ρ) has a root, i.e. ρ∗ with µ(ρ∗) = 0 exists, from which the existence of
+a steady state follows. The question now is whether we can find sufficient
+conditions assuring that the fixed point ρ∗ with �
+i ρ∗
+i = ρ∗ is (asymptotically)
+stable.
+Let us define a matrix B(x), x = (x1, ..., xm) with bij(x) = [B]ij(x) =
+a∗
+ij +xi. Hence, B(xi = 0) = A(ρ∗) and B(xi = ηi) = J, where J, the Jacobian
+matrix, and ηi are defined as in (15) and (16), respectively. We consider now the
+dominant eigenvalue as function of the entries of B, µ[B] := µ({bij}|i,j=1,...,m)
+(the square brackets are chosen to denote the difference from the function
+µ(ρ)). For sufficiently small ηi, we can then express the dominant eigenvalue
+of the Jacobian matrix J, µ[J], relative to the dominant eigenvalues of A∗ :=
+A(ρ∗) as,
+µ[J] = µ[A∗] +
+�
+i
+∂µ
+∂xi
+|xi=0 ηi + O(η2) ,
+(26)
+with,
+∂µ
+∂xi
+|xi=0 =
+�
+ij
+∂µ
+∂bij
+∂bij
+∂xi
+|xi=0 =
+�
+ij
+∂µ
+∂aij
+|B=A∗ ,
+(27)
+since for x = 0, bij = aij for all i, j. It follows that for sufficiently small5 ηi,
+and if all ηi < 0, we have
+µJ = µ[A∗](ρ∗) +
+�
+i
+∂µB
+∂xi
+|xi=0ηi + O(η2
+i ) ≈
+�
+i
+∂µA
+∂aij
+ηi < 0
+(28)
+since all ∂µA
+∂aij > 0 (according to Eq. (8)) and µA(ρ∗) = 0. Hence, since µJ < 0,
+the steady state ρ∗ is asymptotically stable if all ηi < 0. Thus, we get a
+sufficient condition for asymptotic stability of the steady state ρ∗:
+0 > ηi = ρ∗
+i (λ′
+i − γ′
+i) +
+�
+k̸=i
+(κ′
+kiρ∗
+k − κ′
+ikρ∗
+i ) > −ϵi for all i
+(29)
+5That is, there exist ϵi > 0 so that this is valid for any |ηi| < ϵi
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+13
+where ϵi > 0 is sufficiently small. As this is an asymptotically stable steady
+state, it corresponds to a controlled strict homeostatic state. In this case,
+even if the cell numbers are disturbed (to some degree), the cell population is
+regulated to return to the strict homeostatic state.
+Notably, condition (29) is fulfilled if,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+λ′
+i ≤ 0, γ′
+i ≥ 0 for all i
+λ′
+i < 0 or γ′
+i > 0 at for least one i
+κ′
+ij = 0
+and |λ′
+i|, |γ′
+i|, < ϵi
+(30)
+Furthermore, we may soften the condition on κij to
+κ′
+ij
+κ′
+ji <
+ρ∗
+j
+ρ∗
+i to allow also
+some degree of feedback for the κij.
+The conditions (30) are very similar to the ones for dynamic homeostasis,
+(12), but here these conditions only need to be fulfilled at ρ = ρ∗, whereas
+for dynamic homeostasis they need to be fulfilled for a sufficient range of ρ.
+Moreover, in addition to the qualitative nature of the feedback (related to
+the signs of λ′
+i, γ′
+i), the ‘strength’ of the crowding feedback, i.e. the absolute
+values of λ′
+i, γ′
+i must not be ‘too large’, that is, smaller than ϵi. We cannot, in
+general and for all system sizes, give a definite value for the feedback strength
+bound ϵi below which strict homeostasis is assured. Nevertheless, by using the
+sufficient stability criterion based on the Routh-Hurwitz criterion [24] we can
+identify those bounds for systems with up to three cell states, which guides
+expectations for larger systems. The details of this criterion and the necessary
+derivations are shown in Appendix A. There, we show that for systems with one
+or two cell states, ϵi = ∞, which means that asymptotic stability is ensured for
+arbitrary feedback strengths. For systems with three cell states, we can assure
+that ϵi = ∞ if certain further conditions are met (see Eq. (A13)). Otherwise,
+ϵi can be determined implicitly from the roots of a quadratic form (Eq. (A14)),
+and thus stability may depend on the magnitude of the feedback. In principle,
+such bounds can be found for larger systems too, but the algebraic complexity
+of this process renders it unfeasible to do this in practical terms.
+3.4 Robustness to perturbations and failures
+Now, we wish to assess the robustness of the above crowding control mecha-
+nism, i.e. what occurs if it is disrupted, for example, by the action of toxins,
+other environmental cues, or by cell mutations. More precisely, we will study
+what happens if one or more feedback pathways, here characterised as a param-
+eter αi with α′
+i ̸= 0 fulfilling the conditions for (dynamic or strict) homeostatic
+control, is failing, that is, it becomes α′
+i = 0. We will first address the case
+of tissue-extrinsic factors, i.e. those affecting all the cells in the tissue, and
+then the case of single-cell mutations. In the latter case, only a single cell
+would initially show a dysregulated behaviour, yet, if this confers a proliferative
+advantage, it can lead to hyperplasia and possibly cancer [25–27].
+
+Springer Nature 2021 LATEX template
+14
+Homeostatic regulation of renewing tissue cell populations via crowding control
+First, we note that the sufficient condition for strict homeostasis, given
+by Eq. (30), is overly restrictive. In a tissue cell type under crowding feed-
+back control with λ′
+i < 0 and γ′
+i > 0 for more than one i, there is a degree
+of redundancy. That is, if the feedback is removed for one or more of these
+parameters (changing to λ′
+i = 0 and, or γ′
+i = 0), then the sufficient condition
+for a strict homeostatic state remains fulfilled as long as at least one λ′
+i or
+γ′
+i remains non-zero. This possible redundancy confers a degree of robustness,
+meaning that feedback pathways can be removed – setting α′
+i = 0 – without
+losing homeostatic control. Since the necessary conditions, Eqs. (24), are even
+less restrictive, tissue homeostasis may even tolerate more severe disruptions
+that reverse some feedback pathways, e.g. switching from λ′
+i < 0 to λ′
+i > 0,
+as long as other terms in the sum on the right-hand side of (24) compensate
+for this changed sign, ensuring that the sum as a whole is negative. In any
+case, it is important to remind the underlying assumption for which a non-
+trivial steady state exists. In case the variability of the kinetic parameters is
+not enough to assure the condition µ(ρ∗ = 0), then the tissue will degenerate,
+either shrinking and eventually disappearing or indefinitely growing.
+From the above considerations, we conclude that if crowding control applies
+to more than one parameter αi, that is, α′
+i ̸= 0 with appropriate sign and
+magnitude, homeostasis is potentially robust to feedback disruption. This may
+include a simple variation of the feedback function α′
+i but also perturbation in
+the feedback functions shape and complete feedback failure, α′
+i = 0.
+An illustrative example of this situation is shown in Figure 1. Here, the time
+evolution of the cell density is shown for a three-state cell fate model, which has
+been computed by integration of the dynamical system (5) (the details of this
+model are given in Appendix B as Eq. (B15) and illustrated in Figure B1). Four
+kinetic parameters are regulated via crowding control satisfying the sufficient
+condition for strict homeostasis, (30). Then, starting from this homeostatic
+configuration, feedback disruption is introduced at a time equal to zero. In one
+case (“Single failure”), a single kinetic parameter suffers a complete failure of
+the type α′
+i = 0. In this case, the remaining feedback functions compensate
+for this failure, and a new homeostatic condition is achieved. Instead, in the
+second case (“Multiple failures”), failures are applied so that three of the four
+kinetic parameters initially regulated do not adjust with cell density6. Notably,
+the only feedback function left satisfies the condition for asymptotic stability,
+(30). Nevertheless, the variability of this kinetic parameter is not enough to
+assure the existence of a steady state, since in this case, the function µ(ρ) does
+not possess any root. Hence µ > 0 for all ρ, leading to an indefinite growth of
+the cell population. Additional test cases are presented in Appendix B.2.
+So far, we modelled the feedback dysregulation as acting on a global scale,
+thus changing the whole tissue’s dynamics behaviour. This situation represents
+a feedback mechanism affected by cell-extrinsic signals, in which any dysregu-
+lation applies to all the cells in the same way. However, dysregulation can also
+6Only in this example, feedback control fails upon multiple failures, while in general, multiple
+failures may still be compensated to maintain homeostatic control.
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+15
+-10
+0
+10
+20
+30
+40
+0
+2
+4
+6
+8
+10
+Homeostasis
+Single failure
+Multiple failures
+-10
+0
+10
+20
+30
+40
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+Homeostasis
+Single failure
+Multiple failures
+Fig. 1
+Cell dynamics in terms of cell density, scaled by the steady state in the homeostatic
+case, as a function of time (left) and the corresponding variation of the dominant eigenvalue µ
+(right). Time is scaled by the inverse of ¯α = mini α∗
+i . The homeostatic model is perturbed at
+a time equal to zero to include feedback failure of the type α′
+i = 0. In the case where only one
+feedback function fails (“Single failure”), the system is able to achieve and maintain a new
+homeostatic state, characterised by a constant cell density and a zero dominant eigenvalue.
+In case more than one feedback fails (“Multiple failures”), the cell dynamics are unstable
+since a steady state does not exist and µ > 0 for all ρ. The cell fate model corresponds to
+model (B15) with parameters given in Table B1 and Table B2.
+act at the single-cell level, for example, when DNA mutations occur. In this
+case, the impact of the dysregulation is slightly different, as explained in the
+following.
+Suppose, upon disruption of crowding control in a single cell, for example,
+by DNA mutations, a sufficient number of crowding feedback pathways remain
+so that there is a steady state and the sufficient condition (30) is still fulfilled.
+In that case, homeostasis is retained, just as when this occurs in a tissue-wide
+disruption. However, if the homeostatic control of that single cell fails such that
+the cell becomes hyperproliferative, µ > 0, or declining, µ < 0, the tissue may
+still remain homeostatic. If µ < 0, the single mutated cell will be lost, upon
+which only a population of crowding controlled cells remain, which remain in
+homeostasis. If µ > 0 in a single cell, hyper-proliferation is not ensured either:
+while the probability for mutated cells to grow in numbers is larger than to
+decline, due to the low numbers, mere randomness can lead to the loss of the
+mutated cell with a non-zero probability, which results in the extinction of
+the dysregulated mutant7. In that case, the mutant cells go extinct and the
+tissue remains homeostatic despite the disruption of homeostatic control in
+the mutated cells; a stark contrast to disruption on the tissue level. Otherwise,
+if the mutant clone (randomly) survives, it will continue to hyper-proliferate
+and eventually dominate the tissue, which is thus rendered non-homeostatic.
+However, the tissue divergence time scale may be much longer than the case
+where the same dysregulation occurs in all cells.
+The deterministic cell population model (5) is suitable for describing the
+average cell numbers. Nevertheless, it fails to describe the stochastic nature of
+7For example, in the case of a single state with cell division rate λ and loss rate γ – a simple
+branching process – the probability for a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ,
+which is less than certainty.
+
+Springer Nature 2021 LATEX template
+16
+Homeostatic regulation of renewing tissue cell populations via crowding control
+single-cell fate choice. Thus, assessing a single cell’s impact on tissue dynamics
+requires stochastic modelling. To that end, we implemented this situation as
+a Markov process with the same rates as the tissue cell population dynamics
+model8 (see Appendix B.3 for more details).
+In Figure 2, we show numerical simulation results of a stochastic version
+of the model used for previous results in Figure 1, depicted in terms of tissue
+cell density as a function of time. Here, two possible realisations of the same
+stochastic process are presented. We note that the initially homeostatic tissue
+results in stochastic fluctuations of the cell density, which remain, on average,
+constant. At a time equal to zero, a single cell in this tissue switches behaviour,
+presenting multiple failures which, if applied to all the cells, would determine
+the growth of the tissue (corresponding to Multiple failures curve in Figure 1).
+In one instance of the stochastic simulation, however, the mutated clone goes
+extinct after some time, leaving a tissue globally unaffected by the mutation.
+On the other hand, in another instance, the mutated clone prevails, leading to
+the growth of the tissue cell population. The fact that vastly different outcomes
+can occur with the same parameters and starting conditions demonstrates the
+impact of stochasticity in the case of a single-cell mutation.
+-10
+0
+10
+20
+30
+40
+50
+1
+1.2
+1.4
+Homeostasis
+Multiple Failure (instance #1)
+Multiple Failure (instance #2)
+Fig. 2
+Numerical simulation results of a stochastic version of the model used in Figure 1
+upon disruption of crowding control in a single cell, mimicking a DNA mutation. At a
+time equal to 0, the initially homeostatic model is disrupted with a single cell presenting
+multiple failures in the feedback control, as in Figure 1. Two instances of simulations run
+with identical parameters are presented. The rescaled cell density ρ/ρ∗ is shown as a function
+of the time, scaled by the inverse of ¯α = mini α∗
+i . Whilst the mutated cell and its progeny go
+extinct in one instance (#1), in the other (#2), mutated cells prevail and hyper-proliferate
+so that tissue homeostasis is lost. The simulation stops when the clone goes extinct or when
+instability is detected. Full details of the simulation are provided in Appendix B.3.
+3.5 Quasi-dedifferentiation
+In the previous section, we addressed the case where external or cell-intrinsic
+factors disrupt homeostatic control in self-renewing cells of a tissue. However,
+8While a Markov process is an approximation which not necessarily reflects the probability
+distribution of subsequent event times realistically, it is often sufficient to assess the qualitative
+behaviour of a system with low numbers, subject to random influences from the environment.
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+17
+situations such as injury, poisoning, or cell radiation might also affect home-
+ostasis in other ways. An example is when stem cells are completely depleted
+from the tissue. In this context, many studies about tissue regeneration after
+injury report evidence of cell plasticity [17, 18], when committed cells regain
+the potential of the previously depleted stem cells. Cell dedifferentiation is just
+an example where differentiated cells return to an undifferentiated state as a
+response to tissue damage. Lineage tracing experiments confirmed this feature
+in vivo in several cases [16, 28–30].
+In the following, we assess how committed progenitor cells respond to the
+depletion of the stem cell pool if they are under crowding feedback control.
+Without loss of generality, let us consider an initially homeostatic scenario
+where there is a self-renewing (i.e. stem) cell type (S) – with growth param-
+eter µ = 0 – at the apex of a lineage hierarchy, and a committed progenitor
+cell type (C) – with µ < 0, but with at least one state that has a non-zero
+cell division rate – below type S in the hierarchy, as depicted in Figure 3.
+Based on this cell fate model, S-cells proliferate and differentiate into C-cells
+while maintaining the S-cell population. The C-cells also proliferate and dif-
+ferentiate into other downstream cell types which we do not explicitly consider
+here. C-cells do not maintain their own population; only the steady influx of
+new cells of that type via differentiation of S-cells into C-cells maintains the
+latter population (see [13]). We further assume that both S- and C-cells are
+under appropriate crowding control, fulfilling both the sufficient conditions for
+dynamic homeostasis, (12), and for stable, strict homeostasis, (30).
+Based on the above modelling, we can write the dynamics of the cell
+densities belonging to the committed progenitor type as,
+d
+dtρc = Ac(ρc)ρc + u ,
+(31)
+where ρc = (ρms+1, ρms+2, .., ρms+mc) are the cell densities in the committed
+C-type, with ms being the number of states of the self-renewing S-type. Ac is
+the dynamical matrix restricted to states in the C-type and ui = �ms
+j=1 κjiρj
+is a constant vector quantifying the influx of cells into the C-type.
+First, we note that the Jacobian matrix of a committed cell type, described
+by (31), J =
+�
+∂A(ρc)ρc
+∂ρj
+�
+j=ms+1,...,ms+mc
+, has the same form as a cell type at the
+apex of the hierarchy, since u does not depend on the densities ρms+1,...,ms+mc.
+From this follows that if C-cells are regulated by crowding control, fulfilling the
+conditions (30), then also the population of C-cells is stable around a steady
+state ρ∗
+c, albeit with a growth parameter µc(ρ∗
+c) < 09.
+We now consider the scenario where all stem cells are depleted at some
+point, as was experimentally done in [16, 18]. This would stop any replen-
+ishment of C-cells through differentiation of S-cells, corresponding to setting
+9This can be seen when multiplying the steady state condition for (31), Ac(ρs, ρc)ρc + u = 0
+with a positive left dominant eigenvector v, giving, µcvρ∗
+c + vu = 0. Since ρ∗ and v have all
+positive entries and u is non-negative, this equation can only be fulfilled for µc < 0.
+
+Springer Nature 2021 LATEX template
+18
+Homeostatic regulation of renewing tissue cell populations via crowding control
+Fig. 3 Sketch representative of the quasi-dedifferentiation scenario. A homeostatic system
+enclosed in the black box comprises two cell types: a stem cell type, S, (blue) and a com-
+mitted cell type, C, (green). In the unperturbed homeostatic scenario, S is self-renewing,
+characterised by a growth parameter at the steady state µ∗ = 0, and C is transient, with
+a growth parameter at the steady state µ∗ < 0. Both cell types are subject to crowding
+control, fulfilling both conditions (12), and (30). By removing the stem cell type XS, the
+committed cell type acquires self-renewing property through crowding control, effectively
+becoming a stem cell type (see Figure 4).
+u = 0 in (31). Hence we end up with the dynamics ˙ρc = A(ρc)ρc. Now, assum-
+ing that the function µ(ρ) has sufficient range, so that µ(ρ∗∗
+c ) = 0 for some
+ρ∗∗
+c , and provided that A(ρc) is under crowding control fulfilling the sufficient
+conditions for asymptotic stability of a steady state, then, following our argu-
+ments from section 3.3, the population of C-cells will attain a stable steady
+state. In other words, those previously committed cells become self-renewing
+cells. Also, since they now reside at the apex of the lineage hierarchy (given
+that S-cells are absent), they effectively become stem cells.
+Hence, under crowding control, previously committed progenitor cells
+(committed cells that can divide) will automatically become stem cells if the
+original stem cells are depleted. Commonly, such a reversion of a committed cell
+type to a stem cell type would be called ‘dedifferentiation’ or ‘reprogramming’.
+However, in this case, no genuine reversion of cell states occurs; previously
+committed cells do not transition back to states associated with the stem cell
+type. Instead, they respond by crowding feedback and adjust their dynamical
+rates so that µ becomes zero, hence attaining a self-renewing cell type. Cru-
+cially, this new stem cell type is fundamentally different to the original one
+and still most similar to the original committed type. We call this process
+quasi-dedifferentiation. The quasi-dedifferentiation follows the same reversion
+of proliferative potential as in ‘genuine’ dedifferentiation but without explicit
+reversion in the cell state trajectories.
+The following numerical example illustrates this situation. We focus on the
+cell dynamics of a single C-type regulated via crowding feedback (detail of
+the model are provided in Appendix B.4). The cell density as a function of
+the time, shown in Figure 4, is obtained by integrating the corresponding cell
+population model according to Eq. (5). The system is initially in a homeostatic
+condition, meaning that there is a constant influx of cells from some upstream
+self-renewing types. Such upstream types are assumed to be properly regulated
+such that this cell influx is constant over time. At a time equal to zero, the cell
+influx becomes suddenly zero, representing an instantaneous removal of all the
+
+Xo-
+HomeostasisSpringer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+19
+-10
+0
+10
+20
+30
+40
+0
+0.2
+0.4
+0.6
+0.8
+1
+Homeostasis
+Quasi-dedifferentiation
+-10
+0
+10
+20
+30
+40
+-0.2
+-0.1
+0
+Homeostasis
+Quasi-dedifferentiation
+Fig. 4 Cell dynamics of an initially committed cell type C (µ < 0) upon removal of all stem
+cells. (Left) Cell density scaled by the steady-state density as a function of time. (Right)
+Corresponding variation of the dominant eigenvalue µc. Time is scaled by the inverse of
+¯α = mini α∗
+i . It is assumed that a stem cell type, S, initially resides in the lineage hierarchy
+above the committed cell type (as in Figure 3). S cells differentiate into C cells, which is
+modelled as a constant cell influx of C-cells (S is not explicitly simulated). At a time equal
+to zero, a sudden depletion of S cells is modelled by stopping the cell influx. After some
+transitory phase, the cell population stabilises around a new steady state and becomes self-
+renewing with µc = 0. The full description of the dynamical model, which corresponds to
+model (B15) with parameters given in Table B1, is reported in Appendix B.4.
+self-renewing cells from the tissue. A new homeostatic condition is achieved
+after a transitory phase thanks to the crowding feedback acting on the C-
+type. This example demonstrates how an initially committed cell type, i.e.
+with µc < 0, regulated via crowding feedback, might be able to switch, upon
+disruption, to a self-renewing behaviour µc = 0.
+4 Discussion
+For maintaining healthy adult tissue, the tissue cell population must be
+maintained in a homeostatic state. Here, we assessed one of the most com-
+mon generalised regulation mechanisms of homeostasis, which we refer to as
+crowding feedback. Based on this, progenitor cells (stem cells and committed
+progenitors) adjust their propensities to divide, differentiate, and die, accord-
+ing to the surrounding density of cells, which they sense via biochemical or
+mechanical signals. For this purpose, we used a generic mathematical model
+introduced before in Refs. [13, 20], which describes tissue cell population
+dynamics in the most generic way, including cell divisions, cell state transi-
+tions, and cell loss / differentiation. Based on this model, we rigorously define
+what is meant when speaking of a ‘homeostatic state’, introducing two notions:
+a strict homeostasis is a steady state of the tissue cell population dynamics,
+while dynamical homeostasis allows, in addition to strict homeostasis, for oscil-
+lations and fluctuations, as long as a finite long-term average cell population
+is maintained (such as the endometrium during the menstrual cycle).
+By analysing this dynamical system, we find several sufficient and necessary
+conditions for homeostasis. These conditions are formulated in terms of how
+the propensities of cell division, differentiation, and cell state changes, of cells
+
+Springer Nature 2021 LATEX template
+20
+Homeostatic regulation of renewing tissue cell populations via crowding control
+whose type is at the apex of an adult cell lineage hierarchy, may depend on
+their cell density. We find that when, for a wide range of cell density values,
+the cell division propensity of at least one state decreases with cell density or
+the differentiation propensity increases with it, while other propensities (e.g.
+of cell state transitions) are not affected by the cell density, then dynamic
+homeostasis prevails (12). For strict homeostasis to prevail, this only needs
+to be fulfilled at the steady state itself, but in addition, the magnitude of
+the feedback strength may not be too large (30). We can derive explicit and
+implicit expressions for the bound on feedback strength for systems of two
+and three-cell states but cannot do so for arbitrary systems. Furthermore, we
+find that a necessary condition for strict homeostasis is that the conditions for
+dynamic homeostasis are met at least at the steady state cell density.
+A direct consequence of the conditions we found is that they allow for a
+considerable degree of redundancy when more than one propensity depends
+appropriately on the cell density. Hence feedback pathways, that is, cell dynam-
+ics parameters depending on the cell density, may serve as ‘back-ups’ to each
+other if one fails. We demonstrate that this confers robustness to the home-
+ostatic system in that one or more crowding feedback pathways may fail, yet
+the tissue remains in homeostasis.
+Finally, we assess how crowding feedback regulation affects the response of
+committed progenitor cells to a complete depletion of all stem cells. We showed
+that committed cells which can divide and are under appropriate crowding
+feedback control (that is, meeting the sufficient conditions (12) and (30)), will
+necessarily, without additional mechanisms or assumptions, reacquire stem cell
+identity, that is, become self-renewing and are at the apex of the lineage hierar-
+chy. Notably, while this process resembles that of dedifferentiation, it does not
+involve explicit reprogramming, in that the cell state transitions are reversed.
+Instead, only the cell fate propensities adjust to the changing environment by
+balancing proliferation and differentiation as is required for self-renewal. While
+these are purely theoretical considerations, and such a process has not yet
+been experimentally found, we predict that it must necessarily occur under the
+appropriate conditions. This can be measured by assessing the gene expression
+profiles (e.g. via single-cell RNA sequencing) of cells that ‘dedifferentiate’, i.e.
+reacquire stemness after depletion of stem cells. Moreover, those considerations
+yield further, more general insights:
+• Stem cell identity is neither the property of individual cells nor is it strictly
+associated with particular cell types or states. Any cell that can divide and
+differentiate, committed or not, may become a stem cell under appropriate
+environmental control.
+• From the latter follows that stemness is a property determined by the
+environment, not the cell itself.
+• ‘Cell plasticity’ might need to be seen in a wider context. Usually, cell
+plasticity is associated with a change of a cell’s type when subjected to
+environmental cues, which involves a substantial remodelling of the cell’s
+morphology and biochemical state. However, we see that a committed cell
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+21
+may turn into a stem cell simply by adjusting the pace of the cell cycle
+and differentiation processes to the environment. This may not require
+substantial changes in the cell’s state.
+This exemplifies that homeostatic control through crowding feedback is not
+only a way to render homeostasis stable and robust, but also to create stem
+cell identities as a collective property of the tissue cell population.
+Acknowledgments.
+We thank Ben MacArthur and Ruben Sanchez-Garcia
+for valuable discussions.
+Declarations
+PG is supported by an MRC New Investigator Award, Grant number
+MR/R026610/1. The code generated for numerical computations in the cur-
+rent study is available on Github, https://github.com/cp4u17/Feedback. No
+other data was generated for this work.
+Contributions are as follows: C.P. and P.G. conceptualised the paper, C.P.
+and P.G. did the mathematical analysis, C.P. did the numerical analysis, P.G.
+supervised the work.
+The authors have no competing interests to declare that are relevant to the
+content of this article.
+Appendix A
+Asymptotic stability assessment
+based on Routh-Hurwitz
+A.1
+Background
+In control system theory, a commonly used method for assessing the stability
+of a linear system is the Routh-Hurtwiz (RH) criterion [24]. It is an algebraic
+criterion providing a necessary and sufficient condition on the parameters of a
+dynamic system of arbitrary order to ensure the dynamics are asymptotically
+stable. In particular, the criterion defines a set of conditions on the coefficients,
+pi, of the characteristic polynomial, P(s), written as
+P(s) = sn +
+n
+�
+i=1
+pisn−i ,
+(A1)
+in which n corresponds to the dimension of the system. Note that the notation
+used in this section, based on that from [24], is different from that of the main
+text, where pi is the polynomial coefficient of ith order.
+A first result of the RH criterion is that a necessary condition for the
+dynamical system to be asymptotically stable is that all the coefficients must
+be positive, that is,
+pi > 0, for all i .
+(A2)
+
+Springer Nature 2021 LATEX template
+22
+Homeostatic regulation of renewing tissue cell populations via crowding control
+Additional conditions on the polynomial coefficients are added for a necessary
+and sufficient condition. These conditions are based on Routh’s array, written
+as
+�
+�����
+1 p2 p4 ... 0
+p1 p3 ...
+b1 b2 ...
+c1
+...
+�
+�����
+,
+(A3)
+in which the first two rows contain all the coefficients of the characteristic
+polynomial, and the following ones are recursively computed as
+bi = −
+det
+�
+1
+p2i
+p1 p2i+1
+�
+p1
+,
+(A4)
+ci = −
+det
+�
+p1 p2i+1
+b1
+bi
+�
+b1
+,
+(A5)
+and so on until a zero is encountered. The RH criterion states that the system is
+asymptotically stable if and only if the elements in the first column of Routh’s
+array are positive.
+Based on that, it can be easily shown that for a second-order polynomial,
+the necessary condition (A2) is also sufficient for asymptotic stability (a.s.)
+since b1 = p1p2, which means that
+The system is a. s.
+⇐⇒ pi > 0, for i = 1, 2 .
+(A6)
+Instead, the necessary and sufficient condition for a polynomial of order three
+results in
+The system is a. s.
+⇐⇒ pi > 0, for i = 1, 2, 3 and p1p2 − p3 > 0 .
+(A7)
+The same reasoning can be applied to higher-order dynamics to derive
+additional conditions on the coefficients pi.
+A.2
+Verification of the necessary condition for
+asymptotic stability
+The Matlab code for verifying (23) is provided in https://github.com/
+cp4u17/Feedback.git.
+The strategy used is to evaluate each term in Eq. (23) and simply compare
+the left and right-hand sides of the equation. We followed a symbolic approach
+(based on the Matlab symbolic toolbox) for an arbitrary three-state model. A
+numerical approach was used instead for higher-order dynamics, specifically
+4, 5 and 6 state cell fate models. To do so, we randomly defined the cell
+dynamical matrix at the steady state, A(ρ∗), and its derivative with respect
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+23
+to ρ. Entries were chosen from a uniform distribution and, for assuring a zero
+dominant eigenvalue for A(ρ∗), a local optimiser (fmincon function of Matlab)
+was used to find appropriate diagonal elements. For each dimension of the cell
+fate model, we tested up to 1000 random cases.
+A.3
+Sufficient condition for asymptotic stability
+In this section, we will indicate with the superscripts A and J the coefficients of
+the characteristic polynomial expressed as Eq. (A1) respectively of the matrix
+of the dynamical system, Eq. (6), and those of the Jacobian matrix, Eq. (15).
+For a two and three-state system, the following relations can be alge-
+braically derived
+pJ
+1 = pA
+1 −
+�
+i
+ηi .
+(A8)
+where ηi is according to Eq. (16). Again, considering that pA
+1 > 0, if all ηi ≤ 0
+then pJ
+1 > 0.
+Hence, the above relation implies that in a two-state system, the RH cri-
+terion given by Eq. (A6) is fulfilled when η ≤ 0, with at least one negative
+component (otherwise J = A) and therefore the system is asymptotically sta-
+ble. We recall that asking ηi ≤ 0 without further constraints is equivalent to
+the previously derived condition (30) with ϵi = ∞.
+For applying the RH criterion to a three-state cell dynamic system, given
+by Eq. (A7), we need to evaluate the sign of pJ
+2 and then that of pJ
+1 pJ
+2 − pJ
+3 .
+To do so, we first write
+pJ
+2 = pA
+2 −
+�
+i
+fiηi ,
+(A9)
+in which fi = �
+j aji − Tr(A) for i = 1, 2, 3. Since the off-diagonal elements
+are non-negative, and the trace of A is negative, then fi > 0 for i = 1, 2, 3.
+That means that if all ηi ≤ 0 then pJ
+2 > 0. Concerning the term pJ
+1 pJ
+2 − pJ
+3 ,
+this can be written as a quadratic form in η =
+�
+η1, η2, η3
+�
+as
+pJ
+1 pJ
+2 − pJ
+3 = Q(η) = ηT AQη + bT
+Qη + cQ ,
+(A10)
+in which
+AQ =
+�
+�
+f1 f1 f1
+f2 f2 f2
+f3 f3 f3
+�
+� ,
+(A11)
+bQ = −pA
+1
+�
+�
+f1
+f2
+f3
+�
+� − pA
+2
+vw
+�
+�
+v3(w3 − w1) + v2(w2 − w1)
+v3(w3 − w2) + v1(w1 − w2)
+v2(w2 − w3) + v1(w1 − w3)
+�
+� ,
+(A12)
+and cQ = pA
+1 pA
+2 . Here, v = (v1, v2, v3) is a left dominant eigenvector and
+w = (w1, w2, w3) a right dominant eigenvector.
+We now note that the matrix AQ is semidefinite positive since two eigen-
+values are zero (the rows are two-fold degenerate) and one is positive, equal
+to Tr(AQ) = �
+i fi, and cQ > 0. We now distinguish two cases, depending on
+
+Springer Nature 2021 LATEX template
+24
+Homeostatic regulation of renewing tissue cell populations via crowding control
+the sign of bQ elements. First, if bQ ≤ 0, then Q(η) > 0 for any η ≤ 0. Since
+fi, pA
+1 , pA
+2 , vw > 0, we get a sufficient condition for bQ ≤ 0, namely,
+0 ≤ v3(w3 − w1) + v2(w2 − w1)
+(A13)
+0 ≤ v3(w3 − w2) + v1(w1 − w2)
+0 ≤ v2(w2 − w3) + v1(w1 − w3)
+In that case, asymptotic stability and thus crowding feedback control is assured
+for any η < 0, and thus the bound for feedback strength is ϵi = ∞ for i =
+1, 2, 3.
+Otherwise, if there is at least one positive element in bQ, then Q(η) > 0
+only if |ηi| < ϵi, where ϵ = (ϵ1, ϵ2, ϵ3) are the absolute values of the solutions
+to the equation Q(η) = 0, that is – given that ηi are negative – the solution to,
+0 = ϵT AQϵ − bT
+Qϵ + cQ .
+(A14)
+Importantly, we note that the elements of bQ depend uniquely on the proper-
+ties of the dynamical system and therefore, they can be determined without
+requiring the knowledge of the parameter derivatives, i.e. the specific crowding
+feedback dependencies.
+The Matlab code for verifying (A8), (A9) and (A10) is provided in
+https://github.com/cp4u17/Feedback.git.
+Appendix B
+Test case
+B.1
+Asymptotic stability
+This section reports the details of the model used for numerical examples
+presented in the main text. The cell dynamics correspond to the following
+three-state cell fate model
+X1
+λ1
+−→ X1 + X1,
+X1
+ω13
+−−→ X3,
+X1
+γ1
+−→ ∅
+X2
+ω21
+−−→ X1,
+X2
+ω23
+−−→ X3,
+X2
+γ2
+−→ ∅
+X3
+λ3
+−→ X3 + X3,
+X3
+ω31
+−−→ X1,
+X3
+ω32
+−−→ X2,
+(B15)
+whose network is shown in Figure B1. In such a model, for simplicity, we only
+consider symmetric self-renewing divisions so that κij = ωij. Also, we apply
+the crowding feedback to division rates, λi, and differentiation rates γi. In this
+way, it is straightforward to apply the sufficient condition (30) for asymptotic
+stability since κ′
+ij = 0 for all i, j.
+Hence, each kinetic parameter of the type αi ∈ {λj, γj}j=1,...,3 is expressed
+as a function of ρ, whilst those of the type αi ∈ {κjk}j,k=1,...,3 are constant. In
+particular, we chose a Hill function [31] where αi(ρ) = ci + kiρni/(Kni
+i
++ ρni)
+in case αi is a differentiation rate, so that α′
+i = ∂αi/∂ρ > 0, and αi(ρ) =
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+25
+ci +ki/(Kni
+i +ρ/ni) in case it is a proliferation rate, so that α′
+i < 0. According
+to (30) this choice assures that, if there is a value ρ = ρ∗ for which µ(ρ∗) = 0,
+this corresponds to an asymptotically stable steady state.
+The parameter values used in our example are reported in Table B1, and
+the profiles of the proliferation and differentiation rates as a function of ρ are
+shown in Figure B2. Based on these values, the steady state corresponds to
+ρ∗ = 1 (arbitrary unit). As expected, the dominant eigenvalue of the Jacobian
+at the steady state is negative (µJ = −1.21).
+To test the dynamical behaviour of the tissue cell population, we numer-
+ically solved the system of ODEs (5) for different initial conditions based on
+the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).
+The results are shown in Figure B3 as the time evolution of ρ, normalised
+by the steady-state ρ∗, (left panels), and of the dominant eigenvalue, µ (right
+panels). The label H indicates an initial condition corresponding to the self-
+renewing state ρ∗, that is, the system is initially in homeostasis. In the
+simulations labelled as P− and P+, we applied perturbation in the initial
+state ρ∗ = (ρ∗
+1, ρ∗
+2, ρ∗
+3), which are, respectively,
+�
+0.8ρ∗
+1, 0.75ρ∗
+2, 0.85ρ∗
+3
+�
+and
+�
+1.5ρ∗
+1 1.1ρ∗
+2 1.2ρ∗
+3
+�
+. As expected, in all these cases, the feedback’s effect is sta-
+bilising the system so that it returns to the steady state upon perturbation,
+ρ → ρ∗, (asymptotic stability) and thus regains self-renewal property, µ → 0,
+over time.
+Fig. B1
+Cell state network representing a cell type composed of three states. The links
+represent direct transitions, ωij; symmetric divisions occur with rates λi and differentiation
+with rate γi, where subscripts i, j = 1, 2, 3 indicate the corresponding cell state, as per model
+(B15).
+B.2
+Failure of feedback function
+Based on the cell fate model regulated via crowding feedback described in
+the previous section, we assess the impact of failure in one or more feedback
+functions. In particular, the failure of the crowding regulation is modelled,
+assuming one or more kinetic parameters as a constant. Five different failure
+test cases are assessed. For doing so, we chose αi = (1 + C)α∗
+i being constant
+instead of depending on ρ, in which α∗ is the value at the steady state when
+
+M
+Y1
+XI
+23
+1
+013
+021
+1
+031
+-
+X
+X2
+023
+032Springer Nature 2021 LATEX template
+26
+Homeostatic regulation of renewing tissue cell populations via crowding control
+k
+K
+n
+α∗
+α′
+λ1
+0.74
+0.57
+2.00
+0.61
+-0.84
+λ3
+7.79
+2.07
+2.00
+1.53
+-0.56
+γ1
+3.07
+1.22
+2.00
+1.28
+1.48
+γ2
+2.28
+0.43
+2.00
+1.97
+0.61
+κ13
+–
+0.95
+0.00
+κ21
+–
+1.44
+0.00
+κ23
+–
+1.71
+0.00
+κ31
+–
+2.03
+0.00
+κ32
+–
+1.35
+0.00
+Table B1
+Values of the Hill function parameters describing the kinetic parameters in
+case of homeostasis regulation via crowding feedback for the cell fate model (B15). The
+generic kinetic parameters (represented as αi in the right columns of the table) are a
+function of the total cell density, ρ, and are given by γi(ρ) = c + kρn/(Kn + ρn) and
+λi(ρ) = c + k/(Kn + ρn) with i = 1, 2, 3. A common value c = 0.05 is assumed. State
+transition rates ωij, are constant and equal to κij. For such a cell fate dynamics, the steady
+state is ρ∗ = 1. The unit of the kinetic parameter is arbitrary and therefore omitted. Unless
+specified otherwise, these values apply to all the numerical examples presented in this work.
+0
+0.5
+1
+1.5
+2
+0
+0.5
+1
+1.5
+2
+2.5
+0
+0.5
+1
+1.5
+2
+-4
+-2
+0
+2
+4
+Fig. B2
+Proliferation and differentiation rates (left panels, with α as a generic placeholder
+for parameters), and their derivative with respect to ρ (right panels) as functions of cell
+density normalised by the steady-state ρ∗ for the cell fate model (B15) schematised in
+Figure B1. The profiles in the left panel correspond to Hill functions defined in Table B1.
+there are no failures (reported in Table B1) and C is a constant (reported in
+Table B2). Five test cases, indicated as F1−5, are assessed.
+In test case F1, only one feedback fails. Three of the four kinetic parameters
+fail in cases F2−4. Finally, F5 represents a case where all the feedback functions
+fail. The corresponding variability of the dominant eigenvalue, µ, as a function
+of the cell density is shown in Figure B4. It is clear that whilst F1−4 cases
+satisfy the sufficient condition for strict homeostasis, (30), in test cases F5,
+the dominant eigenvalue being constant means that there is no homeostatic
+regulation. Importantly, there is no steady state in test cases F2,4 since the
+dominant eigenvalue is always positive in one case or negative in the other.
+Based on these assumptions, we numerically solved the system of ODEs
+(5) using the explicit Runge-Kutta Dormand-Prince method (Matlab ode45
+function). The failure test cases start at time 0 from an initially homeostatic
+condition, H. The results are shown in Figure B5 as the time evolution of
+ρ, normalised by the homeostatic steady-state, ρ∗, (left panels), and of the
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+27
+0
+5
+10
+15
+0.8
+1
+1.2
+1.4
+H
+P-
+P+
+0
+5
+10
+15
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+H
+P-
+P+
+Fig. B3
+Effect of perturbation of homeostasis under crowding control, when feedback
+parameters are according to Table B1. (Left) Cell density ρ, scaled by the steady-state ρ∗,
+as a function of time. (Right) Corresponding variation of the dominant eigenvalue µ. Time
+is scaled by the inverse of ¯α = mini α∗
+i . Three different initial condition are tested: H,
+corresponds to the steady state ρ∗ = (ρ∗
+1, ρ∗
+2, ρ∗
+3), P− to
+�0.8ρ∗
+1, 0.75ρ∗
+2, 0.85ρ∗
+3
+�
+and P+
+to
+�1.5ρ∗
+1, 1.1ρ∗
+1, 1.2ρ∗
+1
+�
+. Since the steady state is asymptotically stable, thanks to crowding
+control, the cell population remain in, or return to, a homeostatic state characterised by
+µ = 0.
+Parameter
+F1
+F2
+F3
+F4
+F5
+λ1
++5%
++5%
++5%
+-20%
+-5%
+λ3
+-
++5%
++5%
+-20%
+-5%
+γ1
+-
+-5%
+-
++20%
+-5%
+γ2
+-
+-
+-5%
+-
+-5%
+Table B2
+Value of the constant C in the feedback failure test cases. Whenever a failure
+in the feedback of one kinetic parameter α occurs, that parameter is modelled as a
+constant, α = (1 + C)α∗, in which the steady-state value, α∗, is reported in Table B1. Test
+cases F1 and F2 correspond to those presented in the main text (Figure 1).
+dominant eigenvalue, µ, (right panels). Note that the cases F1,2 correspond
+respectively to the Single failure and Multiple failures reported in the
+main text (Figure 1).
+In two cases, F1,3, despite a single or multiple feedback functions failing, a
+new homeostatic condition is reached after some time, where µ = 0. However,
+suppose a different set of feedback fails, like in F2,4, such that the dominant
+eigenvalue is respectively positive or negative for any ρ. In that case, no steady
+state can be attained, and the tissue cell population will hyper-proliferate or
+decline in the long term. Hence, even if the condition for asymptotic stability
+is met, there is no steady state. Finally, if homeostasis is not regulated at
+all, as in F5, then the population dynamics only depend on the value of the
+dominant eigenvalue (the cell dynamical model (5) turns linear). In the case
+shown, µ > 0 and therefore, the cell population diverges.
+B.3
+Single cell mutation scenario
+To assess the tissue dynamics with a single-cell mutation, as presented in the
+main text, we modelled the clonal dynamics, namely, the dynamics of single
+cells and their progeny. For doing so, we considered the model (B15) as a
+
+Springer Nature 2021 LATEX template
+28
+Homeostatic regulation of renewing tissue cell populations via crowding control
+0
+0.5
+1
+1.5
+2
+-2
+0
+2
+4
+H
+F1
+F2
+F3
+F4
+F5
+Fig. B4
+Variation of the dominant eigenvalue µ as a function of the cell density, ρ,
+normalised by the reference homeostatic state value, ρ∗. The curve H corresponds to the
+reference homeostatic model presented in Appendix B.1. The other curves, F1−5, represent
+different sets of feedback failure, as reported in Table B2.
+-10
+0
+10
+20
+30
+40
+0
+0.5
+1
+1.5
+2
+2.5
+H
+F1
+F2
+F3
+F4
+F5
+-10
+0
+10
+20
+30
+40
+-0.6
+-0.4
+-0.2
+0
+0.2
+H
+F1
+F2
+F3
+F4
+F5
+Fig. B5
+Failure of feedback control. (Left) Cell density, scaled by the steady state in the
+homeostatic case, as a function of time. (Right) Corresponding variation of the dominant
+eigenvalue µ. Time is scaled by the inverse of ¯α = mini α∗
+i . The homeostatic model, H, is
+perturbed at a time equal to zero to include the feedback failure reported in Table B2. Whilst
+in F1,3, the regulation is able to achieve and maintain a new homeostatic state (µ = 0),
+the remaining case fails to regulate the cell population, leading to an indefinite growth or
+shrinking of the tissue.
+Markov process with the same numerical rates as before, but now events are
+treated as stochastic. Then, we run numerical simulations using the Gillespie
+algorithm [32] to evaluate this model. In particular, the results presented in
+this work are based on 100 independent instances, where each instance is a
+possible realisation of the stochastic process. We chose a total cell number
+N0 = 5000 as the initial condition (cell density is based on unitary volume).
+In real tissues, the number of cells could be a few orders of magnitude larger.
+However, this number is sufficiently large to avoid the extinction of the process
+in the time scale analysed, so once rescaled, these dynamics are representative
+of those in the tissue. All the simulations are stopped when the mutated clone
+goes extinct or divergence of the dynamics is detected, defined as reaching
+N = 5N0.
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+29
+From an implementation point of view, we consider a cell fate model
+represented by two disconnected cell state networks to model the tissue dynam-
+ics, including the mutated cell. One network corresponds to the unperturbed
+test case H, and the other to the dysregulated one, F2 (both described in
+Appendix B.2). The simulation starts with N0 cells in the H network, dis-
+tributed in each state proportionally to the expected steady-state distribution
+in the tissue, and no cells in the F2 network. Thus, since the two networks
+are disconnected, F2 remains empty, and the simulation represents the tissue
+dynamics before the dysregulation. At a time equal to zero, we moved one
+cell from a random state in the H network to the corresponding state in the
+F2 one. This simulation represents the tissue dynamics, including the single
+mutated cell.
+In Figure B6 (left), all the trajectories where the mutated clones go extinct
+are shown. In these cases, the tissue dynamics remain globally unaffected by
+the mutation. Due to the stochastic nature of the process, mutant clones can
+go extinct even if the growth parameter is positive. That is, even in cases where
+divergence would be observed for the tissue-wide disruption. However, this does
+not occur in all the instances. The right panel of the same figure shows those
+instances where the mutated clone does not go extinct and eventually prevails,
+resulting in diverging cell population dynamics. For the chosen parameters,
+this divergence of the mutated clone is detected in 6% of all cases. Surprisingly,
+only a few clones survive despite a proliferative advantage, but this is plausible
+for a small fitness advantage (For example, in the case of a single state with
+cell division rate λ and loss rate γ – a simple branching process [33] – the
+probability for the a mutant with µ > 0, that is, λ > γ, to establish is 1−γ/λ,
+which can be very low for λ ≈ γ).
+In the main text (Figure 2), only one profile for each scenario is shown,
+respectively. They correspond to instance #24 for the homeostatic case and
+instance #43 for the diverging case.
+B.4
+Quasi-dedifferentiation
+The numerical example presented in the main text is based on the same cell
+fate model described in Appendix B.1. To model the dynamics of a committed
+cell type, we choose a constant non-negative u =
+�
+0.02 0.07 0.06
+�T to model
+for the cell influx. For such a model, the steady state, ρ∗
+c, is asymptotically
+stable.
+The figures presented in the main text are based on the numerical integra-
+tion of the system of ordinary differential equation (31). In particular, we used
+the explicit Runge-Kutta Dormand-Prince method (Matlab ode45 function).
+References
+[1] National Institute of Health: Stem Cell Basics (2016). https://stemcells.
+nih.gov/info/basics
+
+Springer Nature 2021 LATEX template
+30
+Homeostatic regulation of renewing tissue cell populations via crowding control
+-5
+0
+5
+10
+15
+20
+0.8
+0.9
+1
+1.1
+1.2
+H
+F2
+0
+20
+40
+60
+80
+0.8
+0.9
+1
+1.1
+1.2
+H
+F2
+Fig. B6
+Results of numerical simulations of the stochastic process representing the cell
+dynamics, according to section B.3. The cell density, scaled by the steady state in the
+homeostatic case, as a function of the time is shown for 100 random instances. Each shown
+trajectory is the result of a different instance of the stochastic process. At a time equal to
+zero, the cell mutation is modelled as a switch of a single random cell from the homeostatic
+H cell dynamics to the F2 model assessed in Appendix B.2. On the left panel, only the
+trajectories for which the mutated clone goes extinct are shown. The right panel shows the
+trajectories in which the mutated clone prevails. Dynamics are scaled by ¯α = mini{α∗
+i }.
+[2] Marinari, E., Mehonic, A., Curran, S., Gale, J., Duke, T., Baum, B.:
+Live-cell delamination counterbalances epithelial growth to limit tis-
+sue overcrowding. Nature 484(7395), 542–545 (2012). https://doi.org/10.
+1038/nature10984
+[3] Eisenhoffer, G.T., Loftus, P.D., Yoshigi, M., Otsuna, H., Chien, C.-B.,
+Morcos, P.A., Rosenblatt, J.: Crowding induces live cell extrusion to main-
+tain homeostatic cell numbers in epithelia. Nature 484(7395), 546–549
+(2012). https://doi.org/10.1038/nature10999
+[4] Eisenhoffer, G.T., Rosenblatt, J.: Bringing balance by force: Live cell
+extrusion controls epithelial cell numbers. Trends in Cell Biology 23(4),
+185–192 (2013). https://doi.org/10.1016/j.tcb.2012.11.006
+[5] Puliafito, A., Hufnagel, L., Neveu, P., Streichan, S., Sigal, A., Fygenson,
+D.K., Shraiman, B.I.: Collective and single cell behavior in epithelial con-
+tact inhibition. Proceedings of the National Academy of Sciences 109(3),
+739–744 (2012). https://doi.org/10.1016/j.juro.2012.06.073
+[6] Gudipaty, S.A., Lindblom, J., Loftus, P.D., Redd, M.J., Edes, K., Davey,
+C.F., Krishnegowda, V., Rosenblatt, J.: Mechanical stretch triggers rapid
+epithelial cell division through Piezo1. Nature 543(7643), 118–121 (2017).
+https://doi.org/10.1038/nature21407
+[7] Shraiman, B.I.: Mechanical feedback as a possible regulator of tis-
+sue growth. Proceedings of the National Academy of Sciences 102(9),
+3318–3323 (2005). https://doi.org/10.1073/pnas.0404782102
+[8] Kitadate, Y., J¨org, D.J., Tokue, M., Maruyama, A., Ichikawa, R.,
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+31
+Tsuchiya, S., Segi-Nishida, E., Nakagawa, T., Uchida, A., Kimura-
+Yoshida, C., Mizuno, S., Sugiyama, F., Azami, T., Ema, M., Noda, C.,
+Kobayashi, S., Matsuo, I., Kanai, Y., Nagasawa, T., Sugimoto, Y., Taka-
+hashi, S., Simons, B.D., Yoshida, S.: Competition for Mitogens Regulates
+Spermatogenic Stem Cell Homeostasis in an Open Niche. Cell Stem Cell
+24(1), 79–92 (2019). https://doi.org/10.1016/j.stem.2018.11.013
+[9] Johnston, M.D., Edwards, C.M., Bodmer, W.F., Maini, P.K., Chapman,
+S.J.: Mathematical modeling of cell population dynamics in the colonic
+crypt and in colorectal cancer. Proceedings of the National Academy
+of Sciences 104(10), 4008–4013 (2007). https://doi.org/10.1073/pnas.
+0611179104
+[10] Sun, Z., Komarova, N.L.: Stochastic modeling of stem-cell dynamics with
+control Zheng. Math Bioscience 240(2), 231–240 (2012). https://doi.org/
+10.1016/j.mbs.2012.08.004
+[11] Bocharov, G., Quiel, J., Luzyanina, T., Alon, H., Chiglintsev, E., Cheresh-
+nev, V., Meier-Schellersheim, M., Paul, W.E., Grossman, Z.: Feedback
+regulation of proliferation vs. differentiation rates explains the depen-
+dence of CD4 T-cell expansion on precursor number. Proceedings of the
+National Academy of Sciences of the United States of America 108(8),
+3318–3323 (2011). https://doi.org/10.1073/pnas.1019706108
+[12] Greulich, P., Simons, B.D.: Dynamic heterogeneity as a strategy of
+stem cell self-renewal. Proceedings of the National Academy of Sciences
+113(27), 7509–7514 (2016). https://doi.org/10.1073/pnas.1602779113
+[13] Greulich, P., MacArthur, B.D., Parigini, C., S´anchez-Garc´ıa, R.J.: Uni-
+versal principles of lineage architecture and stem cell identity in renewing
+tissues. Development 148, 194399 (2021)
+[14] Donati, G., Rognoni, E., Hiratsuka, T., Liakath-Ali, K., Hoste, E., Kar,
+G., Kayikci, M., Russell, R., Kretzschmar, K., Mulder, K.W., Teich-
+mann, S.A., Watt, F.M.: Wounding induces dedifferentiation of epidermal
+Gata6+ cells and acquisition of stem cell properties. Nature Cell Biology
+19, 603–613 (2017). https://doi.org/10.1038/ncb3532
+[15] Jopling, C., Boue, S., Belmonte, J.C.I.: Dedifferentiation, transdifferenti-
+ation and reprogramming: three routes to regeneration. Nature Reviews
+Molecular Cell Biology 12, 79 (2011)
+[16] Tata, P.R., Mou, H., Pardo-Saganta, A., Zhao, R., Prabhu, M., Law, B.M.,
+Vinarsky, V., Cho, J.L., Breton, S., Sahay, A., Medoff, B.D., Rajagopal,
+J.: Dedifferentiation of committed epithelial cells into stem cells in vivo.
+Nature 503(7475), 218–223 (2013). https://doi.org/10.1038/nature12777
+
+Springer Nature 2021 LATEX template
+32
+Homeostatic regulation of renewing tissue cell populations via crowding control
+[17] Tetteh, P.W., Farin, H.F., Clevers, H.: Plasticity within stem cell hier-
+archies in mammalian epithelia. Trends in Cell Biology 25(2), 100–108
+(2015). https://doi.org/10.1016/j.tcb.2014.09.003
+[18] Tetteh, P.W., Basak, O., Farin, H.F., Wiebrands, K., Kretzschmar, K.,
+Begthel, H., van den Born, M., Korving, J., De Sauvage, F.J., van Es, J.H.,
+Van Oudenaarden, A., Clevers, H.: Replacement of Lost Lgr5-Positive
+Stem Cells through Plasticity of Their Enterocyte-Lineage Daughters.
+Cell Stem Cell 18(2), 203–213 (2016). https://doi.org/10.1016/j.stem.
+2016.01.001
+[19] Murata, K., Jadhav, U., Madha, S., van Es, J., Dean, J., Cavazza, A.,
+Wucherpfennig, K., Michor, F., Clevers, H., Shivdasani, R.A.: Ascl2-
+Dependent Cell Dedifferentiation Drives Regeneration of Ablated Intesti-
+nal Stem Cells. Cell Stem Cell 26(3), 377–3906 (2020). https://doi.org/
+10.1016/j.stem.2019.12.011
+[20] Parigini, C., Greulich, P.: Universality of clonal dynamics poses funda-
+mental limits to identify stem cell self-renewal strategies. eLife 9, 1–44
+(2020). https://doi.org/10.7554/eLife.56532
+[21] Bollob´as, B.: Modern Graph Theory, 1st edn. Graduate Texts in Mathe-
+matics 184. Springer, New York (1998)
+[22] MacCluer, C.R.: The Many Proofs and Applications of Perron’s Theorem.
+SIAM Review 42(3), 487–498 (2000)
+[23] Horn,
+R.A.,
+Johnson,
+C.R.:
+Matrix
+Analysis,
+2nd
+edn.
+Cam-
+bridge University Press, Cambridge (1985). https://doi.org/10.1017/
+cbo9780511810817
+[24] Franklin, G.F., Powell, J.D., Emami-Naeini, A.: Feedback Control of
+Dynamic Systems, 7th edn. Prentice Hall Press, USA (2014)
+[25] Tomasetti, C., Vogelstein, B., Parmigiani, G.: Half or more of the somatic
+mutations in cancers of self-renewing tissues originate prior to tumor
+initiation. Proceedings of the National Academy of Sciences 110(6),
+1999–2004 (2013). https://doi.org/10.1073/pnas.1221068110
+[26] Colom, B., Jones, P.H.: Clonal analysis of stem cells in differentiation
+and disease. Current Opinion in Cell Biology 43, 14–21 (2016). https:
+//doi.org/10.1016/j.ceb.2016.07.002
+[27] Rodilla, V., Fre, S.: Cellular plasticity of mammary epithelial cells under-
+lies heterogeneity of breast cancer. Biomedicines 6(4), 9–12 (2018). https:
+//doi.org/10.3390/biomedicines6040103
+
+Springer Nature 2021 LATEX template
+Homeostatic regulation of renewing tissue cell populations via crowding control
+33
+[28] Tata, P.R., Rajagopal, J.: Cellular plasticity: 1712 to the present day.
+Current Opinion in Cell Biology 43, 46–54 (2016). https://doi.org/10.
+1016/j.ceb.2016.07.005
+[29] Merrell,
+A.J.,
+Stanger,
+B.Z.:
+Adult
+cell
+plasticity
+in
+vivo:
+De-
+differentiation and transdifferentiation are back in style. Nature Reviews
+Molecular Cell Biology 17(7), 413–425 (2016). https://doi.org/10.1038/
+nrm.2016.24
+[30] Puri, S., Folias, A.E., Hebrok, M.: Plasticity and dedifferentiation within
+the pancreas: Development, homeostasis, and disease. Cell Stem Cell
+16(1), 18–31 (2015). https://doi.org/10.1016/j.stem.2014.11.001
+[31] Lei, J., Levin, S.A., Nie, Q.: Mathematical model of adult stem cell
+regeneration with cross-talk between genetic and epigenetic regulation.
+Proceedings of the National Academy of Sciences 111(10) (2014). https:
+//doi.org/10.1073/pnas.1324267111
+[32] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions.
+The Journal of Physical Chemistry 81(25), 2340–2361 (1977). https://
+doi.org/10.1021/j100540a008
+[33] Haccou, P., Jagers, P., Vatutin, V.A.: Branching Processes: Varia-
+tion, Growth, and Extinction of Populations. Cambridge University
+Press, Cambridge (2005). https://doi.org/10.2277/0521832209. http://
+pure.iiasa.ac.at/7598/
+

BdE4T4oBgHgl3EQf5Q5A/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d32a4ea40e522757e60d681544db9a465613159dea5e610cf94df6b37ad2c1d1
+size 194487

BtE5T4oBgHgl3EQfTQ-D/content/tmp_files/2301.05535v1.pdf.txt

@@ -0,0 +1,1008 @@
+ 
+ 
+ 
+Using the profile of publishers to predict 
+barriers across news articles 
+ 
+Abdul Sittar1,2[0000−0003−0280−9594]  and Dunja Mladeni´c1,2[0000−0002−0360−6505] 
+1 Joˇzef  Stefan  Institute,  Slovenia, 
+2 Joˇzef  Stefan  International  Postgraduate  School,  Slovenia, 
+Jamova cesta 39 
+{abdul.sittar, dunja.mladenic}@ijs.si 
+ 
+Abstract. Detection of news propagation barriers, being economical, 
+cultural, political, time zonal, or geographical, is still an open research 
+issue. We present an approach to barrier detection in news spreading 
+by utilizing Wikipedia-concepts and metadata associated with each bar- 
+rier. Solving this problem can not only convey the information about the 
+coverage of an event but it can also show whether an event has been 
+able to cross a specific barrier or not. Experimental results on IPoNews 
+dataset (dataset for information spreading over the news) reveals that 
+simple classification models are able to detect barriers with high accu- 
+racy. We believe that our approach can serve to provide useful insights 
+which pave the way for the future development of a system for predicting 
+information spreading barriers over the news. 
+ 
+Keywords: news propagation · news spreading  barriers  · cultural  bar- 
+rier · economical barriers · geographical barrier · political barrier · time 
+zone barrier · classification methods 
+ 
+1 Introduction 
+The phenomenon of event-centric news spreading due to globalization has been 
+exposed internationally [8]. International events capture attention from all cor- 
+ners of the world. News agencies play their part to bring our attentions on some 
+events and not on others. Varying nature of living styles, cultures, economic con- 
+ditions, time zone, and geographical juxtaposition of countries present a signifi- 
+cant role in process of publishing news related to different events [3, 6, 13, 19–21]. 
+For example, publishing about sports events could be dependent on culture, epi- 
+demic events can reach firstly to neighboring countries due to geographic prox- 
+imity and, news on a luxury product may be relevant for economically strong 
+countries due to demand of wealthy people. We represent this differentiation 
+along with different barriers. These barriers include but are not limited to 1) 
+Economic Barrier, 2) Cultural Barrier, 3) Political Barrier, 4) Geographical Bar- 
+rier, and 5) Time Zone Barrier. Detection of the overpass of these barriers does 
+Copyright © 2021 for this paper by its authors. Use permitted under Creative 
+Commons License Attribution 4.0 International (CC BY 4.0). 
+
+2 
+A. Sittar et al. 
+ 
+ 
+not only tell us the area where the broadcasting of an event reached, but it also 
+shows us events-location relation as countries have different culture, economic 
+conditions, geographical placement on the globe, political point of view, and 
+time zone. Following are the definitions of news crossing these barriers: 
+Cultural Barrier. If we identify the coverage of specific event-centric news by 
+publishers that are surrounded by different cultures, then we can say that the 
+news related to the event crossed cultural barriers. 
+Political Barrier. If news about a specific event is disseminated from publishers 
+having different political alignment, we can say that the news related to that 
+event crossed the political barrier. 
+Geographical Barrier. We say that some news related to a specific event 
+overpasses geographical barriers if that event gets attention by publishers of 
+countries located in different geographical regions. 
+Time Zone Barrier. We can claim  that  event-centric  news  has  crossed  the 
+time zone barrier if it has been published by publishers located in different time 
+zones. 
+Economic Barrier. It can be asserted that a piece of event-centric news has 
+crossed economic barriers if it is published in countries having different economic 
+conditions. 
+In this paper, we propose a methodology for detection of different barriers 
+during information propagation in form of news that utilize data (IPoNews) [18] 
+related to three contrasting events (earthquake, Global warming, and FIFA world 
+cup) in different domains (natural disasters, climate changes, and sports) in 5 
+different languages: English, Slovene, Portuguese, German, and Spanish. 
+ 
+1.1 Contributions 
+Following are the main scientific contributions of this paper: 
+ 
+– A novel methodology for barrier detection in news spreading. 
+– Experimental comparison of several simple classification models that can 
+serve as a baseline. 
+ 
+1.2 Problem Statement 
+Observing the spreading of news on a particular event over time, we want to 
+predict whether a barrier (cultural, political, geographical, time zone, economi- 
+cal) is likely to hamper information while information propagates over the news 
+(binary classification). 
+ 
+2 RELATED WORK 
+ 
+Multiple barriers come across event-centric news specifically when the news is 
+concerned about international or national events. According to news flow theo- 
+ries, multiple determinants impact international news spreading. The economic 
+
+Using the profile of publishers to predict barriers across news articles 
+3 
+ 
+ 
+power of a country is one of the factors that influence news spreading. Moreover, 
+economic variations has different influence for different events (e.g. protests, con- 
+flicts, disasters) [15]. The magnitude of economic interactivity between countries 
+can also impact the news flow [21]. Economic growth/income level shows the eco- 
+nomic condition of a country. Multiple organizations are working on generating 
+prosperity and welfare index on yearly basis. Among them, “The Legatum Pros- 
+perity Index” and “Human Development Index” are popular 1, 2. Geographical 
+representation of entities and events has been utilized extensively in the past 
+to detect local, global, and critical events [3, 13, 19, 20]. It has been said that 
+countries with close distance share culture and language up to a certain extent 
+which can further unfold interesting facts about shared tendencies in informa- 
+tion spreading [15, 16]. 
+ 
+News agencies tend to follow the national context in which journalists op- 
+erate. One of the related examples is the SARS epidemic study which found 
+that cross-national contextual values such as political and economic situations 
+impact the news selection [5]. It will be true to say that fake news is produced 
+based on many factors and it is surrounded by a paramount factor that is polit- 
+ical effect [11]. A great amount of work regarding fake news dwells on different 
+strategies and few studies considered political alignment to have a compelling 
+effect on news spreading [4, 12]. [12] strongly proved it to be a major strategy 
+in news agencies to control the news and change accordingly due to the involve- 
+ment of journalists and political actors. Countries that share common culture 
+are expected to have heavier news flow about between them reporting on similar 
+events [21]. Many quantitative studies found demographic, psychological, socio- 
+cultural, source, system, and content-related aspects [1]. Many models have tried 
+to explain cultural differences between societies. Hofstede’s national culture di- 
+mensions (HNCD) has been widely used and cited in different disciplines [7, 9]. 
+ 
+News classification for different kinds of problems is a well-known topic since 
+the past and features used to classify varies depending upon the problem. [17] 
+used news content and user profile to classify the news whether it is fake or 
+not. [2] calculated TF-IDF score and Word2Vec score of most frequent words 
+and used them as features to classify into one of the five categories (state, econ- 
+omy, entertainment, international, and sports). Similarly, [14] performed part- 
+of-speech (POS) tagging at sentences level and used them as features, and built 
+supervised learning classifiers to classify news articles based on their location. 
+Mostly classifier trained to utilize popular supervised learning methods such as 
+Random Forest, Support Vector Machine (SVM), Naive Bayes, k-Nearest Neigh- 
+bour (kNN), and Decision Tree. In this work, we used the profile of each barrier 
+for each news publisher (see section 3.5) and most frequent 300 Wikipedia con- 
+cepts from the dataset that appeared in the list of news articles related to three 
+contrasting events (earthquake, Global Warming, and FIFA world cup). We also 
+ 
+1 http://hdr.undp.org/en/content/human-development-index-hdi 
+2 https://www.prosperity.com/ 
+
+4 
+A. Sittar et al. 
+ 
+≥ 
+ 
+compared the results of popular classifiers such as SVM, Random Forest, Deci- 
+sion Tree, Naive Bayes, and kNN (see Section 5.4). 
+ 
+3 DATA DESCRIPTION 
+ 
+3.1 Dataset 
+ 
+We utilized dataset ”A dataset for information spreading over the news (IPoNews)” 
+that consists of pairs of news articles that were labeled based on the level of their 
+similarity, as described in [18]. This dataset was collected from Event Registry, 
+a platform that identifies events by collecting related articles written in differ- 
+ent languages from tens of thousands of news sources [10]. The similarity score 
+among cross-lingual news articles was calculated using concept-based similar- 
+ity employing Wikifier service3. [18] describes the criteria when information is 
+considered to be propagated. Statistics of the data set are shown in table 3. 
+ 
+Table 1. Statistics about dataset 
+ 
+Dataset Domain 
+Event type 
+Articles per Language Total Articles 
+ 
+1 
+ 
+Sports 
+ 
+FIFA World Cup 
+Eng Spa Ger Slv Por 
+ 
+2682 
+983 762 711 10 216 
+2 
+Natural Disaster Earthquake 
+941 999 937 19 251 
+3147 
+3 
+Climate Changes Global Warming 996 298 545 8 
+97 
+1944 
+ 
+ 
+The dataset contains a list of pairs of news articles annotated with one of 
+the labels such as ”information-Propagated”, ”Unsure”, or ”Information-Not- 
+Propagated” (see Table 2). The information is considered to be propagated if the 
+cosine similarity score of the two articles in the pair is above a predefined thresh- 
+old ( 0.7 for Information-Propagated, < 0.4 for Information-not-Propagated, 
+otherwise Unsure). We restructured the original dataset to include only exam- 
+ples labeled as spreading information. In this way, we have pair of news articles 
+where we observe information spreading from one to the other. Furthermore, for 
+each example, instead of having a pair of articles, we kept only the article that 
+was published earlier. In this way, each example contains an article that spreads 
+information. 
+ 
+Table 2. Articles with metadata 
+ 
+from 
+to 
+weight Class 
+from-publisher to-publisher from-pub-uri 
+to-pub-uri 
+Por44 
+Por43 
+0.627 
+Unsure 
+ClicRBS 
+SAPO 24 
+jornald.clicrbs.com.br 24.sapo.pt 
+English881 English880 1 
+Information-Propagated 
+Sky News 
+247 Wall St. 
+news.sky.com 
+247wallst.com 
+English258 English329 0.313 
+Information-Not-Propagated Sify 
+4-traders 
+sify.com 
+4-traders.com 
+English793 English787 0.238 
+Information-Not-Propagated Bioengineer.org 7NEWS Sydney scienmag.com 
+7news.com.au 
+German237 German236 0.979 
+Information-Propagated 
+watson 
+watson 
+aargauerzeitung.ch 
+aargauerzeitung.ch 
+ 
+ 
+3 http://wikifier.org/info.html, https://github.com/abdulsittar/IPoNews 
+
+Using the profile of publishers to predict barriers across news articles 
+5 
+ 
+ 
+3.2 Statistics after restructuring the data 
+ 
+The original dataset describes in Section 3 contains pairs of articles along with 
+the information on whether there was the propagation of information related to a 
+specific event or not. We used only examples labeled as propagating information 
+4. Based on the available metadata for articles, we ignored articles that do not 
+have metadata information in our database (see Section 3.4). Table 3 shows the 
+statistics for each barrier after filtering the original dataset. 
+ 
+Table 3. Statistics about barrier 
+ 
+Dataset Domain 
+Event type 
+Articles for each barrier 
+ 
+1 
+ 
+Sports 
+ 
+FIFA World Cup 
+Time-Zone Cultural Political Geographical Economical 
+724 
+699 
+143 
+726 
+634 
+2 
+Natural Disaster Earthquake 
+1102 
+1113 
+227 
+1113 
+1010 
+3 
+Climate Changes Global Warming 586 
+445 
+108 
+487 
+463 
+ 
+ 
+ 
+ 
+3.3 Wikipedia Concepts as Features 
+ 
+As our dataset already mention (see Section 3) if information in news is spread- 
+ing from an article to another based on Wikipedia-concepts, we utilized the 
+most frequent (top 300) Wikipedia-concepts as features. Figure 1 portrays these 
+Wikipedia-concepts for all three events in form of word clouds. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 1. Word clouds of most frequent words related to earthquake, FIFA 
+World Cup and Global Warming events respectively. 
+ 
+ 
+ 
+ 
+3.4 Barriers Knowledge 
+ 
+Barriers knowledge refers to a database that contains metadata about each bar- 
+rier. Figure 3 shows schema of database and Table 4 presents barriers along with 
+their characteristics. Each barrier depends on one main information that is the 
+country name of the headquarter of the news publishers. Since the utilized data 
+4 https://doi.org/10.5281/zenodo.3950064 
+
+DEBpresident
+United
+Yor
+Wart
+overnmen
+States
+nameGermanname
+football
+SWar
+ummerWorld
+assoclation
+Unitednationa
+CUDFIFAASSO
+ation
+FIFAWorldYorKname
+War
+States
+IInchEarthFranceUnited
+United
+New
+Globa
+disambiguation6 
+A. Sittar et al. 
+ 
+ 
+set already contains headquarter of publishers therefore we fetched the coun- 
+try associated with headquarters. For economical barrier, we fetched economical 
+profile  for  each  country  using  “”The  Legatum  Prosperity  Index””  5.  Cultural 
+differences among different regions were collected using Hofstede’s national cul- 
+ture dimensions (HNCD). For time zone and geographical barrier, we stored 
+general UTC-offset, latitude, and longitude. For political barrier we are using 
+the political alignment of the newspaper/magazine that we determined based on 
+Wikipedia infobox at their Wikipedia page. For instance, for Austrian newspa- 
+per ”Der Standard” we find social liberalism as political alignment (See Figure 
+2), for British newspaper ”Daily Mail” we find right-wing as political alignment, 
+for German ”Stern” magazine there is no information in its Wikipedia infobox 
+on the political alignment thus we label political alignment as unknown. 
+ 
+ 
+ 
+Fig. 2. Three Wikipedia infobox for three different newspapers/magazines 
+with political alignment 
+ 
+ 
+ 
+5 https://www.prosperity.com/ 
+
+Der Standard
+DERSTANDARD
+Type
+Daily newspaper
+Owner(s)
+Oscar Bronner
+Publisher
+Oscar Bronner
+Martin Kotynek
+Founded
+19 October 1988: 32 years
+ago
+Political
+Social liberalism
+alignment
+Headguarters
+Vienna
+Circulation
+86,000 (2013)
+Website
+www.derstandard.de
+www.derstandard.at DailyMail
+DailumlailFREE
+MICHELIN
+SO MUCH
+FOR THE
+BONFIRE
+OF THE
+QUANGOS!
+aplasticheart
+DailyMail frontpageon 4August 2010
+Type
+Dailynewspaper
+Format
+Tabloid
+Owner(s)
+DailyMail and General Trust
+Founder(s)
+AlfredHarmsworthandHarold
+Harmsworth
+Publisher
+DMGMedia
+Editor
+GeordieGreig
+Founded
+4 May1896:124 years ago
+Political
+Right-wing[1]2][3]
+alignment
+Language
+English
+Headquarters Northcliffe House
+2 Derry Street
+LondonW85TT
+Circulation
+1.134.184(asofFebruary
+2020)[4]
+ISSN
+0307-7578
+OCLC
+16310567
+number
+Website
+www.dailymail.co.ukStern
+?
+stern
+?
+stern
+KRERSMID
+Alein in turopa
+IXABE
+HRSECIEENE
+Sternmagazinecoveron18February2016
+Editor
+FlorianGless,Anna-Beeke
+Gretemeier
+Categories Newsmagazine
+FrequencyWeekly
+Circulation390,000(2020)
+Year
+1948
+founded
+Firstissue
+1August1948,72yearsago
+Company
+Gruner+Jahr
+Country
+Gemany
+Basedin
+Hamburg
+Language
+Geman
+Website
+www.stern.de
+ISSN
+0039-1239Using the profile of publishers to predict barriers across news articles 
+7 
+ 
+ 
+3.5 Features for Individual Barrier 
+We represented each barrier with a specific profile containing a list of features. 
+Table 4 depicts the list of features for each barrier. Economic and cultural bar- 
+riers consist of a vector of length 11 and 6 features whereas geographical, time 
+zone, and political only contain 1 or 2 features such as latitude-longitude, UTC- 
+offset, and political alignment. 
+ 
+ 
+ 
+Fig. 3. Database Schema for Barriers 
+ 
+ 
+ 
+ 
+3.6 Dataset Annotation 
+We queried the metadata information for each article and generated a CSV file 
+for each barrier. We annotated each article based on that meta information to be 
+used for model training and classification. For economic and cultural barriers, we 
+calculated cosine similarity between vectors of economical values and vectors of 
+cultural values. Score greater than the threshold value of 0.9 labeled as FALSE 
+otherwise TRUE. We set the lowest value as a threshold based on the fact that 
+if two countries have a little gap concerning culture or economical values then 
+there exists a barrier. For geographical barriers, we compared the latitude and 
+longitude of the country of each publisher. If a country name or lat/lat appeared 
+to be the same then we annotated it with FALSE otherwise TRUE. Lastly, for 
+
+Enterprise Conditions
+Social Capital
+Education
+EconomicQuality
+Marketaccessand
+Individualistic cuiture
+Living Conditions
+Economic
+infrastructure
+Power distance
+Profile
+Governance
+Natural Environment
+afety
+Fam
+Health
+nty
+Cultural Value's
+Has
+Has
+induigence vs
+restraint
+Headquarter
+Has
+Country
+Has
+Geographical
+Values
+(Lat/Lon)
+Has
+name
+Has
+Longitude
+Latitude
+Political barrier
+Time-Zone
+Political
+UTC-offset
+Alignment8 
+A. Sittar et al. 
+ 
+ 
+Table 4. Features of each barrier 
+ 
+Barrier 
+Features 
+ 
+Economic 
+Rank, Safety-Security, 
+Personal-Freedom, Governance, Social-Capital, Investment-Environment, 
+Enterprise-Conditions, Market-Infrastructure, Economic-Quality, 
+Living-Conditions, Health, Education, Natural-Environment 
+ 
+Cultural 
+Power-Distance, 
+Uncertainty-Avoidance-By-Individuals, Individualistic-Cultures, 
+Masculinity-Femininity, Long-Term-Orientation, Indulgence-Restraint 
+Geographical Latitude, Longitude 
+Time Zone 
+UTC-offset 
+Political 
+Political-Alignment 
+ 
+ 
+ 
+ 
+time-zone and political barriers, we followed the same process that was for the 
+geographical barrier. if political alignment or UTC-offset appeared to be  the 
+same for a pair then it is annotated with FALSE otherwise TRUE. Figure  4 
+depicts the class distribution for each barrier. We can notice unbalanced class 
+distribution with majority of the examples being False. This is especially true 
+for Cultural and Political barrier with 91 percent of example being False. Thus 
+in our evaluation we rely more on F1 measure than classification accuracy. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4. Class Distribution for Each Barrier 
+
+2000
+True
+2014
+False
+1500
+1588
+1599
+1324
+1000
+948
+670
+500
+478
+408
+203
+42
+0
+I Barrier
+nical
+olitical Barrier
+Econom
+TimeUsing the profile of publishers to predict barriers across news articles 
+9 
+ 
+ 
+4 MATERIALS AND METHODS 
+4.1 Problem Modeling 
+For each barrier, we have a list of news articles where each article is associated 
+with 300 Wikipedia-concepts and features related to that barrier. The task is to 
+predict the status S of each barrier B. 
+S = f (C, B) 
+f is the learning function for barrier detection, C is donating here Wikipedia- 
+concepts related to an article and B is the list of features related to a specific 
+barrier (see Table 4). 
+ 
+4.2 Methodology 
+We utilized dataset IPoNews [18] and built a database on top of this dataset 
+that includes barrier knowledge. Figure 5 explains the overall process of model 
+construction from news articles to results generation. We created a list of in- 
+stances using the most frequent Wikipedia-concepts based on news articles and 
+joined them along with barrier knowledge. After performing the annotation (see 
+Section 3.6), we trained popular classification models and generated the results 
+on test data (see Section 5.4). 
+ 
+ 
+ 
+Fig. 5. Steps for Model Construction 
+ 
+ 
+ 
+5 EXPERIMENTAL EVALUATION 
+5.1 Baselines 
+We used the following methods as baselines for all our models. 
+– Uniform: Generates predictions uniformly at random. 
+– Stratified: Generates predictions by respecting the training set’s class dis- 
+tribution. 
+– Most Frequent: Always predicts the most frequent label in the training 
+set. 
+
+Barrier's
+Results
+knowledge
+Testset
+Newsarticles (IPoNews)
+Metadata
+Barriers'Annotation
+Wikipediaconcepts
+Model Construction
+Trainset10 
+A. Sittar et al. 
+ 
+sum 
+sum 
+sum 
+sum 
+ 
+5.2 Classification Methods 
+ 
+We trained popular classification models for each barrier such as SVM, kNN, 
+Decision Tree, Random Forest, and Naive  Bayes  using Scikit-Learn.  We  applied 
+a stratified 10-fold cross-validator to split the dataset for training and testing. 
+For Random Forest, kNN, and Decision Tree, we varied the size of n-estimator, 
+value of k, and max-leafs and chosen the one with the best score on test data 
+respectively. Implementation of this methodology to barrier detection can be 
+found on GitHub 6. 
+ 
+5.3 Evaluation Metric 
+ 
+Due to imbalance in the class distribution for all barriers, we used micro averaged 
+precision and recall to evaluate our models. 7 
+– Micro-Precision: The precision of average contributions from each class is 
+calculated in micro-precision whereas the following question is answered by 
+precision: What proportion of positive predictions was correct? It is defined 
+as: 
+  
+TruePositivesum 
+ 
+Micro − Precision =  TruePositive 
++ FalsePositive 
+– Micro-Recall: Recall of average contributions from each class is calculated 
+in micro-recall whereas the following question is answered by recall: What 
+proportion of actual positives was predicted correctly? It is defined as: 
+  
+TruePositivesum 
+ 
+Micro − Recall =  TruePositive 
++ FalseNegative 
+ 
+5.4 Results and Analysis 
+ 
+Table 5 shows the results of all the classifiers for each barrier along with baselines. 
+Analysis of the experimental results show that overall all the machine learning 
+models outperform the three baselines. For all the barriers, we can notice Micro- 
+Recall is equal to Micro-Precision. The best performing baseline is the ”Most- 
+frequent” with Micro-F1 for economic, cultural, geographical, time zone, and 
+political barrier equal to 0.70, 0.90, 0.58, 0.70, and 0.90 respectively. The best 
+performing models on all the barriers are Decision Tree, Random Forest, and 
+kNN. Looking at Micro-F1, we can see that on  the  Economic  and  Cultural 
+barrier kNN achieved the best performance of 0.75 and 0.95 respectively. On 
+Geographical barriers, kNN and Decision Tree performed the best achieving 0.81. 
+On Time-Zone, the best performing classifier is Random Forest with Micro-F1 
+6 https://github.com/cleopatra-itn/BarrierDetection-Classification 
+7 https://peltarion.com/knowledge-center/documentation/evaluation- 
+view/classification-loss-metrics/micro-recall 
+
+Using the profile of publishers to predict barriers across news articles 
+11 
+ 
+ 
+0.83. On Political barriers, SVM, kNN, and Random Forest achieve the best 
+Micro-F1 score of 0.97. 
+In terms of classification accuracy, we can see that Random Forest outper- 
+forms the baselines as well as the other four classifiers for the first four barriers. 
+Notice that Random forest performs better than decision tree but takes more 
+time. Naive-Bayes achieves a little bit lower classification accuracy than the Deci- 
+sion Tree for the first four barriers. On the political barrier Naive-Bayes achieves 
+the best classification accuracy (0.98) but lower Micro-F1 (0.66). 
+ 
+6 CONCLUSIONS AND FUTURE WORK 
+ 
+It is highly important to detect the barriers while information propagates specif- 
+ically through the news. For journalists, marketers, and social scientists, the phe- 
+nomenon of knowing which barrier appeared most frequently for what type of 
+events, is significantly helpful to solve business and marketing problems. In this 
+regard, we proposed a simple methodology. Though its results are good enough 
+for three types of events, we would like to enhance features as well as events. We 
+used only Wikipedia-concepts and meta information to detect barriers. In the 
+future, we would like to use DMoz categories provided by Event Registry [10], 
+and transformation of the text of news articles as a feature for barrier detection. 
+Currently geographical and time zone barriers are calculated in a binary way ei- 
+ther the same or different. In the future, we would like to introduce the distance 
+between countries and between time zones as labels instead of the currently used 
+binary labeling. 
+ 
+7 ACKNOWLEDGMENTS 
+ 
+The research described in this paper was supported by the Slovenian research 
+agency under the project J2-1736 Causalify and co-financed by the Republic 
+of Slovenia and the European Union’s Horizon 2020 research and innovation 
+program under the Marie Sk-lodowska-Curie grant agreement No 812997. 
+
+12 
+A. Sittar et al. 
+ 
+ 
+ 
+Table 5. Classifiers’ comparison with baselines 
+ 
+Barrier 
+Model 
+CA Mic-Pre Mic-Rec Mic-F1 
+Economic 
+Uniform 
+0.50 0.50 
+0.49 
+0.49 
+ 
+Stratified 
+0.58 0.59 
+0.57 
+0.59 
+ 
+Most Frequent 0.70 0.70 
+0.70 
+0.70 
+ 
+SVM 
+0.66 0.69 
+0.69 
+0.69 
+ 
+kNN 
+0.70 0.75 
+0.75 
+0.75 
+ 
+Decision Tree 
+0.69 0.73 
+0.73 
+0.73 
+ 
+Random Forest 0.74 0.74 
+0.74 
+0.74 
+ 
+Naive Bayes 
+0.61 0.63 
+0.63 
+0.63 
+ 
+ 
+Cultural 
+Uniform 
+0.50 0.50 
+0.49 
+0.50 
+ 
+Stratified 
+0.83 0.83 
+0.83 
+0.83 
+ 
+Most Frequent 0.90 0.90 
+0.90 
+0.90 
+ 
+SVM 
+0.84 0.93 
+0.93 
+0.93 
+ 
+kNN 
+0.55 0.95 
+0.95 
+0.95 
+ 
+Decision Tree 
+0.90 0.94 
+0.94 
+0.94 
+ 
+Random Forest 0.93 0.93 
+0.93 
+0.93 
+ 
+Naive Bayes 
+0.83 0.51 
+0.51 
+0.51 
+ 
+ 
+Geographical Uniform 
+0.49 0.50 
+0.50 
+0.50 
+ 
+Stratified 
+0.50 0.51 
+0.51 
+0.51 
+ 
+Most Frequent 0.58 0.58 
+0.58 
+0.58 
+ 
+SVM 
+0.81 0.76 
+0.76 
+0.76 
+ 
+kNN 
+0.79 0.81 
+0.81 
+0.81 
+ 
+Decision Tree 
+0.78 0.81 
+0.81 
+0.81 
+ 
+Random Forest 0.79 0.79 
+0.79 
+0.79 
+ 
+Naive Bayes 
+0.76 0.79 
+0.79 
+0.79 
+ 
+ 
+Time Zone 
+Uniform 
+0.49 0.49 
+0.49 
+0.49 
+ 
+Stratified 
+0.59 0.58 
+0.58 
+0.58 
+ 
+Most Frequent 0.70 0.70 
+0.70 
+0.70 
+ 
+SVM 
+0.78 0.77 
+0.77 
+0.77 
+ 
+kNN 
+0.70 0.78 
+0.78 
+0.78 
+ 
+Decision Tree 
+0.80 0.81 
+0.81 
+0.81 
+ 
+Random Forest 0.83 0.83 
+0.83 
+0.83 
+ 
+Naive Bayes 
+0.72 0.64 
+0.64 
+0.64 
+ 
+ 
+Political 
+Uniform 
+0.51 0.52 
+0.50 
+0.50 
+ 
+Stratified 
+0.84 0.83 
+0.81 
+0.82 
+ 
+Most Frequent 0.90 0.90 
+0.90 
+0.90 
+ 
+SVM 
+0.79 0.97 
+0.97 
+0.97 
+ 
+kNN 
+0.62 0.97 
+0.97 
+0.97 
+ 
+Decision Tree 
+0.79 0.91 
+0.91 
+0.91 
+ 
+Random Forest 0.97 0.97 
+0.97 
+0.97 
+ 
+Naive Bayes 
+0.98 0.66 
+0.66 
+0.66 
+
+Using the profile of publishers to predict barriers across news articles 
+13 
+ 
+ 
+References 
+1. Al-Samarraie, H., Eldenfria, A., Dawoud, H.: The impact of personality traits on 
+users’ information-seeking behavior. Information Processing & Management 53(1), 
+237–247 (2017) 
+2. Alam, M.T., Islam, M.M.: Bard: Bangla article classification using a new compre- 
+hensive dataset. In: 2018 International Conference on Bangla Speech and Language 
+Processing (ICBSLP). pp. 1–5. IEEE (2018) 
+3. Andrews, S., Gibson, H., Domdouzis, K., Akhgar, B.: Creating corroborated cri- 
+sis reports from social media data through formal concept analysis. Journal of 
+Intelligent Information Systems 47(2), 287–312 (2016) 
+4. Bakshy, E., Messing, S., Adamic, L.A.: Exposure to ideologically diverse news and 
+opinion on facebook. Science 348(6239), 1130–1132 (2015) 
+5. Camaj, L.: Media framing through stages of a political discourse: International 
+news agencies’ coverage of kosovo’s status negotiations. International Communica- 
+tion Gazette 72(7), 635–653 (2010) 
+6. Dagon, D., Zou, C.C., Lee, W.: Modeling botnet propagation using time zones. In: 
+NDSS. vol. 6, pp. 2–13 (2006) 
+7. He, M., Lee, J.: Social culture and innovation diffusion: a theoretically founded 
+agent-based model. Journal of Evolutionary Economics pp. 1–41 (2020) 
+8. Hong, X., Yu, Z., Tang, M., Xian, Y.: Cross-lingual event-centered news clustering 
+based on elements semantic correlations of different news. Multimedia Tools and 
+Applications 76(23), 25129–25143 (2017) 
+9. Khosrowjerdi,  M.,  Sundqvist,  A.,  Bystr¨om,  K.:  Cultural  patterns  of  information 
+source use: A global study of 47 countries. Journal of the Association for Informa- 
+tion Science and Technology 71(6), 711–724 (2020) 
+10. Leban, G., Fortuna, B., Brank, J., Grobelnik, M.: Event registry: learning about 
+world events from news. In: Proceedings of the 23rd International Conference on 
+World Wide Web. pp. 107–110 (2014) 
+11. Martens, B., Aguiar, L., Gomez-Herrera, E., Mueller-Langer, F.: The digital trans- 
+formation of news media and the rise of disinformation and fake news (2018) 
+12. Maurer, P., Beiler, M.: Networking and political alignment as strategies to con- 
+trol the news: Interaction between journalists and politicians. Journalism Studies 
+19(14), 2024–2041 (2018) 
+13. Quezada,  M.,  Pen˜a-Araya,  V.,  Poblete,  B.:  Location-aware  model  for  news  events 
+in social media. In: Proceedings of the 38th International ACM SIGIR Conference 
+on Research and Development in Information Retrieval. pp. 935–938 (2015) 
+14. Rao, V., Sachdev, J.: A machine learning approach to classify news articles based 
+on location. In: 2017 International Conference on Intelligent Sustainable Systems 
+(ICISS). pp. 863–867. IEEE (2017) 
+15. Segev, E.: Visible and invisible countries: News flow theory revised. Journalism 
+16(3), 412–428 (2015) 
+16. Segev, E., Hills, T.: When news and memory come apart: A cross-national com- 
+parison of countries’ mentions. International Communication Gazette 76(1), 67–85 
+(2014) 
+17. Shu, K., Zhou, X., Wang, S., Zafarani, R., Liu, H.: The role of user profiles for fake 
+news detection. In: Proceedings of the 2019 IEEE/ACM international conference 
+on advances in social networks analysis and mining. pp. 436–439 (2019) 
+18. Sittar, A., Mladeni´c, D., Erjavec, T.: A dataset for information spreading over the 
+news. In: Proc. of Slovenian KDD Conf. on Data Mining and Data Warehouses 
+(SiKDD) (2020) 
+
+14 
+A. Sittar et al. 
+ 
+ 
+19. Watanabe, K., Ochi, M., Okabe, M., Onai, R.: Jasmine: a real-time local-event 
+detection system based on geolocation information propagated to microblogs. In: 
+Proceedings of the 20th ACM international conference on Information and knowl- 
+edge management. pp. 2541–2544 (2011) 
+20. Wei, H., Sankaranarayanan, J., Samet, H.: Enhancing local live tweet stream to 
+detect news. GeoInformatica pp. 1–31 (2020) 
+21. Wu, H.D.: A brave new world for international news? exploring the determinants of 
+the coverage of foreign news on us websites. International Communication Gazette 
+69(6), 539–551 (2007) 
+

BtFKT4oBgHgl3EQfXS78/content/tmp_files/2301.11794v1.pdf.txt

@@ -0,0 +1,1249 @@
+Thermodynamic features of the 1D dilute Ising model
+in the external magnetic field
+A.V. Shadrina,∗, Yu.D. Panova
+aInstitute of Natural Sciences and Mathematics, Ural Federal University, 620002, 19 Mira
+street, Ekaterinburg, Russia
+Abstract
+We consider the effects of the magnetic field on the frustrated phase states of
+the dilute Ising chain, especially, the behavior of the magnetic entropy change
+and the isentropic dependence of the temperature on the magnetic field, which
+are the key parameters of the magnetocaloric effect. The found temperature
+dependences of entropy demonstrate the nonequivalence of frustrated phases in
+the antiferromagnetic and ferromagnetic cases. In the antiferromagnetic case,
+the nonzero magnetic field at certain parameters causes a charge ordering for
+nonmagnetic impurities at a half-filling, while in the ferromagnetic case, the
+magnetic field reduces the frustration of the ground state only partially. It is
+also shown, that impurities radically change the magnetic Gr¨uneisen parameter
+in comparison with the case of a pure Ising chain.
+Keywords:
+dilute Ising chain, frustrated magnets, magnetic entropy change
+1. Introduction
+One of the remarkable features of low-dimensional systems, such as deco-
+rated Ising models [1–7], the anisotropic Potts chain [8], the diamond Hubbard
+chain [9], is the presence, under certain parameters, of a frustrated ground state
+for which the residual entropy is nonzero. Despite the absence of a real phase
+transition at finite temperatures according to the Perron–Frobenius theorem for
+square real matrices [10] the thermodynamic behavior near the boundaries be-
+tween different phases of the ground state for these systems can exhibit striking
+features. As shown in [11], if one of the phases has a nonzero residual entropy
+that preserves continuity at the boundary with the other phase, then the ther-
+modynamic characteristics of the system will demonstrate pseudo-transitions at
+a finite temperature. Entropy, heat capacity, magnetization, and susceptibility
+have similar features to the behavior of these properties at conventional phase
+transitions, including the presence of quasicritical exponents [12].
+∗Corresponding author
+Email address: [email protected] (A.V. Shadrin)
+Preprint submitted to Journal of Magnetism and Magnetic Materials
+January 30, 2023
+arXiv:2301.11794v1  [cond-mat.stat-mech]  27 Jan 2023
+
+Recently, frustrated magnetic systems have also attracted the attention of
+researchers due to the enhanced magnetocaloric effect in the vicinity of finite-
+field transitions [13, 14]. Besides to geometric factors, the impurities are also
+the reason for the existence of frustrations in the magnetic system. The simplest
+example of a magnetic system that is frustrated by impurities is a diluted Ising
+chain. The Hamiltonian of 1D diluted Ising model can be written in the following
+form
+H = −J
+�
+i
+Sz,iSz,i+1 + V
+�
+i
+P0,iP0,i+1 − h
+�
+i
+Sz,i − µ
+�
+i
+P0,i.
+(1)
+Here we use the S = 1 pseudospin operators. The states for a given lattice
+site with the pseudospin projections Sz = ±1 correspond to the two magnetic
+states with the conventional spin projections sz = ±1/2, while the state with
+Sz = 0 corresponds to the charged nonmagnetic state. Sz,i is a z-projection of
+the on-site pseudospin operator, P0,i = 1 − S2
+z,i is the projection operator onto
+the Sz = 0 state, J is the exchange constant, V > 0 is the inter-site correlation
+parameter for impurities, h is an external magnetic field, and µ is a chemical
+potential for impurities. Further we will assume that nonmagnetic impurities
+are mobile, which corresponds to the annealed system.
+As well known [15],
+V = V0 + V1 − 2V01 describes the interaction for a more general case:
+V0
+�
+i
+P0,iP0,i+1 + V1
+�
+i
+P1,iP1,i+1 + V01
+�
+i
+�
+P0,iP1,i+1 + P1,iP0,i+1
+�
+,
+(2)
+where P1 = S2
+z, is the projection operator onto magnetic states. The solutions
+and various thermodynamic properties of the 1D dilute Ising model at zero
+external magnetic field was found in [16–20]. If h ̸= 0, then there are no explicit
+analytical expressions for various thermodynamic functions of the model (1). It
+is known the account of magnetic field for the S = 1 Ising chain significantly
+expands the list of possible phase states of the system and leads to various
+features of thermodynamic behavior [21, 22].
+In the present paper, we consider the effects of the magnetic field on the
+frustrated phase states of the model (1). We focused on the behavior of entropy
+and, in particular, on the magnetic entropy change, which is the key parameter
+of the magnetocaloric effect.
+Also, we explore the isentropic dependence of
+the temperature on the magnetic field. The paper is organized as follows. We
+briefly describe the methods in section 2, and section 3 the results, including
+the ground state phase diagram, and their discussions are given. Conclusions
+are presented in section 4.
+2. Methods
+We define the transfer matrix for the model (1) as
+τ =
+�
+�
+xz
+z1/2 t1/2
+x−1
+z1/2 t1/2
+y−1t
+z−1/2 t1/2
+x−1
+z−1/2 t1/2
+xz−1
+�
+� ,
+(3)
+2
+
+where x = eβJ, y = eβV , z = eβh, t = eβµ and β = 1/T, and we assume kB = 1.
+From (3), we found the characteristic equation for the eigenvalues λi:
+λ3 − λ2 �
+ty−1 + x(z + z−1)
+�
+− λ
+�
+x2 − x−2 + t
+�
+xy−1 − 1
+�
+(z + z−1)
+�
+− 2t(x − x−1) − tx−2y−1 = 0.
+(4)
+The eigenvalues in a general case are cumbersome functions, but at h = 0 they
+could be reduced to the known expressions [20]:
+λ1,2
+=
+1
+2
+�
+x + x−1 + y−1t
+�
+±
+�
+2t + 1
+4
+�
+x + x−1 − y−1t
+�2�1/2
+,
+λ3
+=
+x − x−1.
+(5)
+According to the Perron–Frobenius theorem [10], there is only one maximum
+eigenvalue, λ1, and in the thermodynamic limit we obtain the grand potential
+and the entropy in the following form:
+Ω = Nω = −NT ln λ1,
+S = −
+� ∂ω
+∂T
+�
+h,µ
+= ln λ1 + T
+λ1
+�∂λ1
+∂T
+�
+h,µ
+.
+(6)
+The grand potential and entropy found depend on parameters J, V , h, µ, and
+T. But in the present problem, it is more convenient to use the concentration n
+of impurities as an external parameter. The dependence n(µ) can be obtained
+from the equation
+n = −
+�∂ω
+∂µ
+�
+T,h
+= T
+λ1
+�∂λ1
+∂µ
+�
+T,h
+.
+(7)
+In a general case h ̸= 0, we used numerical methods to get the inverse
+dependence µ(n) and fix the concentration of impurities n at all temperatures.
+If h = 0, we obtain the explicit expressions [20]:
+µ = ln
+�
+y
+�
+x + x−1� g + m
+g − m
+�
+,
+(8)
+S = 1
+2 ln 2 (1 + 2g)2
+1 − 4m2
++ g + 2m2
+1 + 2g
+ln y − m ln
+��
+x + x−1� g + m
+g − m
+�
+− (1 − 2m) (g − m)
+�
+x − x−1�
+(1 + 2g) (x + x−1) ln x,
+(9)
+where
+g =
+�
+m2 + 1
+2
+�1
+4 − m2
+�
+y−1 �
+x + x−1��1/2
+,
+(10)
+and we introduced the deviation of the concentration of impurities from half-
+filling, m = n − 1/2.
+3
+
+The knowledge of the entropy from Eqs. (6,7) gives an opportunity to explore
+magnetocaloric properties of the dilute Ising chain for a given n. We explore the
+magnetic entropy change, the isentropic dependencies of the temperature on the
+magnetic field and the magnetic Gr¨uneisen parameter, which can be calculated
+from the relation
+Γmag = 1
+T
+�∂T
+∂h
+�
+S,n
+= − 1
+T
+(∂S/∂h)T,n
+(∂S/∂T)h,n
+.
+(11)
+The explicit expression that we use to calculate Γmag for a given n in variables
+(T, h, µ) has the following form:
+Γmag = − 1
+T
+�
+(∂S/∂h)T,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂h)T,µ
+(∂S/∂T)h,µ (∂n/∂µ)T,h − (∂S/∂µ)T,h (∂n/∂T)h,µ
+�
+.
+(12)
+3. Results
+3.1. Phase diagram at zero temperature.
+The phase diagram of the dilute Ising chain in longitudinal magnetic field at
+zero temperature is shown in Fig. 1 for the J − h plane. The limiting case for
+the Ising chain without impurities, m = −1/2, is given in Fig. 1(a). Two ground
+states, the ferromagnetic (FM) state with magnetic moment oriented towards
+the field, and the antiferromagnetic (AFM) state with zero magnetic moment,
+are separated by the critical value of magnetic field |hc| = −2J at which the spin-
+flip transition occurs. Figures 1(b) and 1(c) show the cases of a weakly diluted
+chain, −1/2 < m < 0, and a strongly diluted chain, 0 ≤ m < 1/2, respectively.
+Dilution with impurities leads to the appearance of two new boundary lines on
+the J − h plane, J = V and |h| = −J − V . If J > V , the ground state is
+represented by macroscopic FM domains (or drops) separated by macroscopic
+impurity domains.
+Similarly, the AFM domains arise when J < −V − |h|.
+Schematically, this is shown in Fig. 1(b,c), where the arrows correspond to the
+spins, and the circles correspond to the impurities. For both FM and AFM
+phases, the entropy is zero.
+Analysis of the ground state of the model (1) in zero magnetic field shows [19,
+20] that phases with the nonzero residual entropy exist at |J| < V and at
+|J| = V . For the weak exchange, when |J| < V , the spin correlation length is
+always finite, but the impurity correlation length with the temperature lowering
+tends to infinity at the half-filling concentration, m = 0, due to the formation of
+charge ordering [20]. If |J| = V and h = 0, the spin correlation length and the
+impurity correlation length are finite for all values of the impurity concentration
+and temperature.
+In magnetic field, if −V − |h| < J < V , the residual entropy is also not zero,
+so the ground state is frustrated. If −1/2 < m < 0, the ground state of the
+chain is a set of finite AFM or FM spin clusters separated by impurities. This
+state we call frustrated ferromagnetic (FR-FM) or frustrated antiferromagnetic
+(FR-AFM) respectively.
+The FR-FM and FR-AFM states separated by the
+4
+
+J
+J
+J
+h
+h
+h
+V
+V
+-V
+-V
+|h|=-V-J
+|h|=-V-J
+|h|=-2J
+|h|=-2J
+0 Ј m < 0.5
+-0.5 < m < 0
+m = -0.5
+(b)
+(a)
+(c)
+AFM
+AFM
+AFM
+FR-PM
+FR-AFM
+FR-FM
+FM
+FM
+FM
+0
+0
+0
+Figure 1:
+Phase diagram at zero temperature for a dilute Ising chain in a longitudinal
+magnetic field: (a) the Ising chain without impurities, n = 0, (b) the case of a weakly diluted
+chain, (c) the case of a strongly diluted chain.
+5
+
+spin-flip line, |h| = −2J, as it is shown in Fig. 1(b). In the strongly diluted
+case, 0 ≤ m < 1/2, there are single spins separated by impurity clusters, and
+the system exhibits a paramagnetic response, which is uniform over the entire
+range −V −|h| < J < V . This frustrated paramagnetic (FR-PM) state is shown
+in Fig. 1(c).
+3.2. The magnetic entropy change.
+Temperature dependences of the entropy S and the magnetic entropy change,
+∆S = S(h = 0)−S(h ̸= 0), are shown in Fig. 2 for the antiferromagnetic (AFM)
+sign of the exchange constant, J < 0, and in Fig. 3 for the ferromagnetic (FM)
+sign, J > 0. The correlation parameter for impurities V accepted and used as
+a positive scaling factor.
+Fig. 2 shows the temperature dependences of the entropy for J/V = −1,
+h = 0 in panel (a), and for J/V = −1, h/V = 0.5 in panel (b). If at h = 0 the
+entropy monotonically depends on |m| and has a maximum at m = 0, at h ̸= 0
+the dependence on |m| has a local minimum at m = 0. The magnetic entropy
+change for J/V = −1 is shown in panel (c). The maximum ∆S for all m is
+achieved at T = 0 and also has a minimum with ∆S < 0 at finite temperature
+for small values of impurity concentrations.
+It is worth noting that in the AFM chain, for any value of the applied
+magnetic field, we get zero entropy at m = 0, because there is only one way to
+minimize the energy: alternating spins and charges, and all spins are oriented
+along the magnetic field. In a certain sense, in this case we get a kind of magneto-
+electric effect: an external magnetic field causes a charge ordering. This also
+gives us the maximum change in entropy at half-filling.
+The temperature dependences of the entropy for J/V = −0, 5, h = 0 are
+shown in Fig. 2(d), and for J/V = −0.5, h/V = 0.5 in Fig. 2(e).
+The de-
+pendences of S on |m| have a local minimum at m = 0 both at h = 0 and at
+h/V = 0.5. The magnetic entropy change for J/V = −0.5 is shown in Fig. 2(f).
+In a contrast to the previous case, the ∆S dependences show a maximum at
+finite temperature for some m ≥ 0, and also show a minimum with ∆S < 0 at
+finite temperature in some range for m < 0.
+Fig. 3 shows the temperature dependences of the entropy for J/V = 1, h = 0
+in panel (a), and for J/V = 1, h/V = 0.5 in panel (b). The magnetic entropy
+change for J/V = 1 is shown in panel (c). Both at h = 0 and h ̸= 0 the entropy
+monotonically depends on |m| and has a maximum at m = 0. The magnetic
+entropy change has a maximum at finite temperature for some m < 0, and near
+the m = 0 it also has a local minimum at a finite temperature. In the case
+of FM, we will not get the same effect at h > 0 as for J/V = −1, because it
+makes no sense to split the spin clusters into more than one spin in order to
+minimize the energy. But the entropy is still slightly reduced, because there is
+no ordering chaos for different the spin clusters: they will all be oriented by a
+magnetic field.
+The temperature dependences of the entropy for J/V = 0.5, h = 0 are
+shown Fig. 3 in panel (d), and for J/V = 0.5, h/V = 0.5 in panel (e). The
+6
+
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+S
+S
+S
+S
+T/V
+T/V
+T/V
+T/V
+T/V
+T/V
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+DS
+DS
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+0.5
+0.4
+0.3
+0.2
+0.1
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0.
+0.
+-0.4
+-0.5
+-0.3
+0.5
+0.4
+0.3
+0.2
+0.1
+-0.1
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+0.1
+-0.5
+-0.4
+0.
+0.
+-0.3
+-0.2
+-0.1
+-0.2
+-0.5
+-0.1
+-0.2
+-0.3
+-0.4
+0.
+-0.5
+0.
+-0.4
+-0.3
+-0.2
+-0.1
+0.1
+0.2
+0.3
+0.4
+0.5
+-0.1
+-0.2
+-0.3
+-0.4
+-0.5
+0.1
+0.2
+0.3
+0.4 0.5
+Figure 2: (color online) Temperature dependences of the entropy S and the magnetic entropy
+change ∆S in the AFM case (J < 0). Panels (a), (b), and (c) correspond to J/V = −1;
+(d), (e), (f) – to J/V = −0.5. Panels (a) and (d) show the entropy S at h = 0, (b) and (e)
+– at h/V = 0.5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.5).
+The numbers near lines correspond to the deviation of the impurity concentration n from
+half-filling, m = n − 1/2. Solid (dashed) lines correspond to m ≤ 0 (m > 0).
+7
+
+S
+S
+S
+S
+T/V
+T/V
+T/V
+T/V
+T/V
+T/V
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.1
+0.2
+0.3
+DS
+DS
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (d)
+(a)
+(b)
+(c)
+(e)
+(f)
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+0.1
+-0.1
+0.
+0.
+-0.5
+-0.5
+-0.4
+-0.4
+-0.3
+-0.3
+-0.2
+-0.2
+-0.1
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+-0.5
+-0.5
+-0.4
+-0.4
+-0.3
+-0.3
+0.
+0.
+0.2
+0.2
+0.1
+0.1
+-0.1
+-0.1
+-0.2
+0.5
+-0.5
+0.4
+-0.4
+-0.3
+-0.2
+-0.1
+0.
+0.3
+0.2
+0.1
+-0.5
+-0.4
+-0.3
+0.
+-0.2
+-0.1
+0.5
+0.4
+0.3
+0.2
+0.1
+-0.2
+Figure 3: (color online) Temperature dependences of the entropy S and the magnetic entropy
+change ∆S in the FM case (J > 0). Panels (a), (b), and (c) correspond to J/V = 1; (d),
+(e), (f) – to J/V = 0.5.
+Panels (a) and (d) show the entropy S at h = 0, (b) and (e) –
+at h/V = 0.5, (c) and (f) – the magnetic entropy change ∆S = S(h = 0) − S(h = 0.5).
+The numbers near lines correspond to the deviation of the impurity concentration n from
+half-filling, m = n − 1/2. Solid (dashed) lines correspond to m ≤ 0 (m > 0).
+magnetic entropy change for J/V = 0.5 is shown in panel (f). Qualitatively,
+the behavior of entropy differs from J/V = 1 case at some region near |m| = 0,
+where the tendency to the charge ordering causes the decreasing of S. The ∆S
+dependences also show local maxima at finite temperature in some range for
+m < 0, and a monotonic behavior with maximal value at T = 0 for m ≥ 0. The
+magnetic entropy change for FM case is always positive.
+The concentration dependences of entropy at T/V = 0.05 shown in Fig. 4
+allow estimating approximately the features of the residual entropy S0. The
+dependences of S0 on m have the following form [20]:
+S0 = ln
+�1
+2 + g0
+�
++ 1
+2 ln
+2
+1
+4 − m2 − m ln g0 + m
+g0 − m,
+|J|/V = 1,
+(13)
+S0 = 1
+2 ln
+1
+2 + |m|
+1
+2 − |m| + |m| ln
+1
+4 − m2
+8m2
++ 1
+2 ln 2,
+|J|/V < 1,
+(14)
+where
+g0 =
+1
+√
+2
+�1
+4 + m2
+�1/2
+.
+(15)
+These expressions depend only on |m| and are identical for the AFM and FM
+cases. The curves of S(h = 0) in Fig. 4(a) and (c), and in Fig. 4(b) and (d)
+8
+
+(d)
+S
+0.8
+0.0
+0.2
+0.4
+0.6
+S
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+m
+m
+m
+m
+S
+0.5
+0.5
+0.5
+0.5
+0.8
+0.0
+0.2
+0.4
+0.6
+0.25
+0.25
+0.25
+0.25
+-0.25
+-0.25
+-0.25
+-0.25
+-0.5
+-0.5
+-0.5
+-0.5
+S
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+(a)
+(c)
+(b)
+DS
+DS
+DS
+DS
+S(h=0)
+S(h=0)
+S(h=0)
+S(h=0)
+S(h№0)
+S(h№0)
+S(h№0)
+S(h№0)
+0.0
+0.0
+0.0
+0.0
+Figure 4: (color online) The dependence of the entropy S (solid lines) and the magnetic entropy
+change ∆S (dashed lines) on the deviation of the impurity concentration n from half-filling,
+m = n − 1/2, at T/V = 0.05. Panel (a) corresponds to J/V = −1, (b) – to J/V = −0.5, (c)
+– to J/V = 1, and (d) – to J/V = 0.5.
+confirm this property. For h ̸= 0 the concentration dependences of the residual
+entropy become asymmetric with respect to m = 0 for AFM case, but save
+the symmetry in FM case. The same dependence for the AFM and FM cases
+holds only at m > 0 for |J| < V , when the ground state consists of single spins
+separated by nonmagnetic impurities, and the sign of the exchange constant has
+no effect.
+3.3. The isentropic dependence of the temperature on the magnetic field.
+Fig. 5 shows the isentropic lines in the h − T parameter plane for the fer-
+romagnetic sign of exchange constant, J > 0.
+For the Ising chain without
+impurities, the isentropes slope near the critical field hc = 0 is almost vertical
+that leads to extremely high and narrow peak of the Gr¨uneisen parameter [23],
+which is proportional to e2J/T at h ∝ T e−2J/T . The impurities change this
+9
+
+h/V
+T/ V
+T/ V
+T/ V
+0.01
+0.1
+0.3
+0.55
+0.65
+0.75
+0.85
+0.95
+0.5
+0.5
+0.55
+0.55
+0.6
+0.6
+0.65
+0.7
+-3
+10
+-6
+10
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.1
+0.3
+0.5
+0.7
+0.1
+0.3
+0.5
+0.7
+0.1
+0.3
+0.5
+0.7
+n=0
+n=0.25
+n=0.75
+0.1
+0.135
+0.15
+0.2
+0.25
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.2
+0.5
+0.65
+0.75
+0.85
+h/V
+0.1
+0.3
+0.5
+0.7
+T/ V
+0.1
+0.3
+0.5
+0.7
+T/ V
+0.1
+0.3
+0.5
+0.7
+T/ V
+-5
+10
+0.01
+0.05
+-8
+10
+-3
+10
+n=0
+n=0.03
+n=0.5
+(a)
+(b)
+Figure 5:
+The isentropic lines in the h − T parameter plane (a) for J/V = 1.3 and (b) for
+J/V = 0.5. The value of the impurity concentration n is given in the frame. The numbers
+next to the lines show the entropy values.
+picture drastically: the entropy value increases by several orders of magnitude,
+and the isentropes slope near hc = 0 remains finite.
+The magnetic Gr¨uneisen parameter can be rewritten [24] in the scaling form
+as
+Γmag = −Gr
+1
+h − hc
+,
+(16)
+where −Gr is a prefactor, and hc is a critical magnetic field. Fig. 6 shows the
+value −Gr for hc = 0 as a function of T and h for J/V = 1.3, n = 0.03 in
+panel (a), and for J/V = 0.5, n = 0.25 in panel (b). As can be seen, in both
+cases, impurities lead to suppression of the singular behavior of the magnetic
+Gr¨uneisen parameter which is observed for the Ising chain without impurities.
+At low temperatures, in the FR-FM state, the system behaves like an ideal
+paramagnet near h = 0 with a prefactor value −Gr = 1 [24].
+Fig. 7 shows the isentropic lines in the h − T parameter plane for the anti-
+ferromagnetic sign of exchange constant, J < 0. The case of a moderate value
+of the exchange constant, J/V = −1.3, is given in panel (a). In the absence of
+impurities, there is practically no dependence of entropy on the magnetic field.
+Impurities lead to an increase in the entropy of the system by several orders
+of magnitude and the appearance of two critical values of the magnetic field,
+10
+
+(a)
+(b)
+Figure 6: (color online) The prefactor of the Gr¨uneisen parameter −Gr (a) for J/V = 1.3,
+n = 0.03, (b) for J/V = 0.5, n = 0.25.
+|hc| = −J − V , which correspond to the transition lines from AFM to FR-AFM
+or FR-PM states. It is worth to note, that for comparable values of |J| and
+V , the critical field |hc| = −J − V can be much smaller than the spin-flip field
+|hc| = −2J. The case of a small exchange constant, J/V = −0.15, is given
+in panel (b). Without impurities, the system has two critical spin-flip fields,
+|hc| = −2J. Impurities lead to the appearance of a critical field hc = 0. As
+a result, there are three critical fields for a weakly diluted case, and only one
+critical field hc = 0 for a strongly diluted case.
+Fig. 8 shows the prefactor of the magnetic Gr¨uneisen parameter as a function
+of T and h near the corresponding critical fields: for J/V = −1.3, n = 0.25,
+hc/V = −1 − J/V = 0.3 in panel (a), for J/V = −0.15, n = 0.25, hc = 0 in
+panel (b), and for J/V = −0.15, n = 0.25, hc/V = −2J/V = 0.3 in panel (c).
+In all cases, the prefactor tends to −Gr = 1 at sufficiently low temperatures.
+4. Conclusion
+We examined the effects of the magnetic field on the frustrated phase states
+of the dilute Ising chain.
+The temperature dependences of entropy and the
+magnetic entropy change show the nonequivalence of frustrated phases in AFM
+and FM cases. The largest effect is achieved when |J|/V = 1. In the AFM case,
+J/V = −1, the nonzero magnetic field causes a charge ordering for nonmagnetic
+impurities and leads to the maximal value of the magnetic entropy change at a
+half-filling. In the FM case, J/V = 1, the magnetic field reduces the frustration
+of the ground state only partially. Impurities radically change the magnetic
+Gr¨uneisen parameter in comparison with the case of a pure Ising chain. They
+suppress the singular behavior of Γmag near h = 0 in the FM case and produce
+the paramagnetic behavior in the FR-FM case. In the AFM case, additional
+values of the critical magnetic field for Γmag appear, which are associated with
+the transition line from AFM to frustrated ground state. In the FR-AFM state,
+Γmag exhibits paramagnetic behavior at h = 0.
+11
+
+0.6
+0.4
+0.8
+0.2
+0.6
+0.0
+-0.5
+0.4 T/ V
+0.0
+0.2
+h/V
+0.51.0
+0.8
+0.5
+0.6
+0.0
+-0.5
+0.4
+T/V
+0.0
+0.2
+h/ V
+0.5h/V
+T/ V
+T/ V
+T/ V
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.1
+0.1
+0.1
+0.2
+0.2
+0.2
+0.3
+0.3
+0.3
+0.4
+0.4
+0.4
+0.5
+0.1
+0.2
+0.3
+0.1
+0.2
+0.3
+0.1
+0.2
+0.3
+0.4
+0.4
+0.5
+0.5
+0.5
+0.5
+0.6
+0.6
+0.6
+0.6
+0.7
+0.8
+0.48
+0.48
+0.5
+0.5
+0.55
+0.55
+0.6
+0.6
+0.65
+n=0
+n=0.25
+n=0.75
+h/V
+T/ V
+T/ V
+T/ V
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.05
+0.05
+0.05
+0.1
+0.1
+0.1
+0.15
+0.15
+0.15
+0.2
+0.2
+0.2
+0.1
+0.3
+0.4
+0.4
+0.4
+0.5
+0.6
+0.1
+0.25
+0.4
+0.5
+0.5
+0.5
+0.6
+-13
+10
+-10
+10
+-8
+10
+-6
+10
+-5
+10
+n=0
+n=0.25
+n=0.75
+(a)
+(b)
+Figure 7:
+The isentropic lines in the h − T parameter plane (a) for J/V = −1.3 and (b) for
+J/V = −0.15. The value of the impurity concentration n is given in the frame. The numbers
+next to the lines show the entropy values.
+(a)
+(b)
+(c) 
+Figure 8: (color online) The prefactor of the Gr¨uneisen parameter −Gr near the critical field
+(a) for J/V = −1.3, n = 0.25, hc/V = 0.3, (b) for J/V = −0.15, n = 0.25, hc = 0, (c) for
+J/V = −0.15, n = 0.25, hc = 0.3.
+12
+
+1.0
+0.5
+0.20
+0.0
+0.15
+0.0
+T/V
+0.10
+0.2
+h/ V
+0.4
+0.051.0
+0.3
+0.5
+0.2
+0.0
+T/V
+-0.1
+0.1
+0.0
+h/ V
+0.11.0
+0.5
+0.3
+0.0
+0.2
+T/V
+0.2
+0.1
+0.3
+h/ V
+0.4
+0.5Acknowledgments
+This work was supported by the Ministry of Education and Science of the
+Russian Federation, project FEUZ-2020-0054.
+References
+[1] L. ˇCanov´a, J. Streˇcka, M. Jaˇsˇcur, Geometric frustration in the class of ex-
+actly solvable Ising–Heisenberg diamond chains, Journal of Physics: Con-
+densed Matter 18 (20) (2006) 4967–4984. doi:10.1088/0953-8984/18/
+20/020.
+[2] J. Torrico, M. Rojas, S. M. de Souza, O. Rojas, N. S. Ananikian, Pair-
+wise thermal entanglement in the ising-xyz diamond chain structure in an
+external magnetic field, EPL (Europhysics Letters) 108 (5) (2014) 50007.
+doi:10.1209/0295-5075/108/50007.
+[3] J. Streˇcka, Anomalous Thermodynamic Response in the Vicinity of a
+Pseudo-Transition of a Spin-1/2 Ising Diamond Chain, Acta Physica
+Polonica A 137 (5) (2020) 610–612. doi:10.12693/APhysPolA.137.610.
+[4] L. G´alisov´a,
+J. Streˇcka,
+Vigorous thermal excitations in a double-
+tetrahedral chain of localized Ising spins and mobile electrons mimic a
+temperature-driven first-order phase transition, Physical Review E 91 (2)
+(2015) 022134. doi:10.1103/PhysRevE.91.022134.
+[5] O. Rojas, J. Streˇcka, S. de Souza, Thermal entanglement and sharp specific-
+heat peak in an exactly solved spin-1/2 Ising-Heisenberg ladder with alter-
+nating Ising and Heisenberg inter–leg couplings, Solid State Communica-
+tions 246 (2016) 68–75. doi:10.1016/j.ssc.2016.08.002.
+[6] J. Streˇcka, R. C. Al´ecio, M. L. Lyra, O. Rojas, Spin frustration of a spin-
+1/2 Ising–Heisenberg three-leg tube as an indispensable ground for thermal
+entanglement, Journal of Magnetism and Magnetic Materials 409 (2016)
+124–133. doi:10.1016/j.jmmm.2016.02.095.
+[7] S. de Souza, O. Rojas, Quasi-phases and pseudo-transitions in one-
+dimensional models with nearest neighbor interactions, Solid State Com-
+munications 269 (2018) 131–134. doi:10.1016/j.ssc.2017.10.006.
+[8] Y. Panov, O. Rojas, Unconventional low-temperature features in the one-
+dimensional frustrated q-state potts model, Physical Review E 103 (6)
+(2021) 062107. doi:10.1103/PhysRevE.103.062107.
+[9] O. Rojas, S. M. de Souza, J. Torrico, L. M. Ver´ıssimo, M. S. S. Pereira,
+M. L. Lyra, Low-temperature pseudo-phase-transition in an extended hub-
+bard diamond chain, Physical Review E 103 (4) (2021) 042123.
+doi:
+10.1103/PhysRevE.103.042123.
+13
+
+[10] F. Ninio, A simple proof of the Perron-Frobenius theorem for positive sym-
+metric matrices, Journal of Physics A: Mathematical and General 9 (8)
+(1976) 1281–1282. doi:10.1088/0305-4470/9/8/017.
+[11] O. Rojas, A Conjecture on the Relationship Between Critical Residual
+Entropy and Finite Temperature Pseudo-transitions of One-dimensional
+Models, Brazilian Journal of Physics 50 (6) (2020) 675–686. doi:10.1007/
+s13538-020-00773-8.
+[12] O. Rojas, J. Streˇcka, M. L. Lyra, S. M. de Souza, Universality and qua-
+sicritical exponents of one-dimensional models displaying a quasitransi-
+tion at finite temperatures, Physical Review E 99 (4) (2019) 042117.
+doi:10.1103/PhysRevE.99.042117.
+[13] M. E. Zhitomirsky, Enhanced magnetocaloric effect in frustrated magnets,
+Physical Review B 67 (10) (2003) 104421.
+doi:10.1103/PhysRevB.67.
+104421.
+[14] M. E. Zhitomirsky, A. Honecker, Magnetocaloric effect in one-dimensional
+antiferromagnets, Journal of Statistical Mechanics:
+Theory and Ex-
+periment 2004 (07) (2004) P07012.
+doi:10.1088/1742-5468/2004/07/
+P07012.
+[15] T. L. Hill, Statistical Mechanics: Principles and Selected Applications,
+Dover, New York, 1956.
+[16] F. Rys, A. Hintermann, Gittermodell eines ungeordneten Ferromagneten
+II. Exakte Losung des eindimensionalen Dodells, Helv. Phys. Acta 42 (4)
+(1969) 608.
+[17] F. Matsubara, K. Yoshimura, S. Katsura, Magnetic Properties of One-
+Dimensional Dilute Ising Systems. I, Canadian Journal of Physics 51 (10)
+(1973) 1053–1063. doi:10.1139/p73-140.
+[18] Y. Termonia, J. Deltour, Thermodynamic properties of one-dimensional
+dilute Ising systems with interacting impurities, Journal of Physics C: Solid
+State Physics 7 (24) (1974) 4441–4451. doi:10.1088/0022-3719/7/24/
+007.
+[19] B. Balagurov, V. Vaks, R. Zaitsev, Statistics of a One-Dimensional Model
+of a Solid Solution, Sov Phys Solid State 16 (8) (1974) 1498–1502.
+[20] Y. Panov, Local distributions of the 1D dilute Ising model, Journal of
+Magnetism and Magnetic Materials 514 (July) (2020) 167224.
+doi:10.
+1016/j.jmmm.2020.167224.
+[21] F. A. Kassan-ogly,
+One-dimensional ising model with next-nearest-
+neighbour interaction in magnetic field, Phase Transitions 74 (4) (2001)
+353–365. doi:10.1080/01411590108227581.
+14
+
+[22] A. V. Zarubin, F. A. Kassan-Ogly, A. I. Proshkin, A. E. Shestakov, Frustra-
+tion Properties of the 1D Ising Model, Journal of Experimental and Theo-
+retical Physics 128 (5) (2019) 778–807. doi:10.1134/S106377611904006X.
+[23] G. O. Gomes, L. Squillante, A. C. Seridonio, A. Ney, R. E. Lagos,
+M. de Souza, Magnetic Gr¨uneisen parameter for model systems, Physical
+Review B 100 (5) (2019) 054446. doi:10.1103/PhysRevB.100.054446.
+[24] B. Wolf, A. Honecker, W. Hofstetter, U. Tutsch, M. Lang, Cooling through
+quantum criticality and many-body effects in condensed matter and cold
+gases, International Journal of Modern Physics B 28 (26) (2014) 1–35.
+doi:10.1142/S0217979214300175.
+15
+

CNFAT4oBgHgl3EQfsx5b/content/2301.08660v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:27af67c791568940162b8257793e5675179fb69e7418c7739df9b9a401b1b3b6
+size 2004952

CNFAT4oBgHgl3EQfsx5b/content/tmp_files/2301.08660v1.pdf.txt

@@ -0,0 +1,1189 @@
+1 
+ 
+A BIG-DATA DRIVEN FRAMEWORK TO ESTIMATING VEHICLE VOLUME BASED 
+ON MOBILE DEVICE LOCATION DATA 
+ 
+Mofeng Yang1, Weiyu Luo2, Mohammad Ashoori3, Jina Mahmoudi4, Chenfeng Xiong5*, Jiawei 
+Lu6, Guangchen Zhao7, Saeed Saleh Namadi8, Songhua Hu9 and Aliakbar Kabiri10 
+ 
+ 
+1. Ph.D. ([email protected]) 
+2. Graduate Research Assistant ([email protected]) 
+3. Graduate Research Assistant ([email protected]) 
+4. Ph.D., P.E., Research Scientist ([email protected])  
+5. Assistant Professor ([email protected]), *Corresponding Author  
+6. Graduate Research Assistant ([email protected])  
+7. Graduate Research Assistant ([email protected]) 
+8. Graduate Research Assistant ([email protected])  
+9. Graduate Research Assistant ([email protected])  
+10. Graduate Research Assistant ([email protected])  
+ 
+ 
+1-4, 7-10: Maryland Transportation Institute (MTI), Department of Civil and Environmental 
+Engineering, 1173 Glenn Martin Hall, University of Maryland, College Park MD 20742, USA. 
+5: Department of Civil and Environmental Engineering, College of Engineering, Villanova 
+University, Villanova, PA 19085, USA 
+6. School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, 
+AZ 85281, USA 
+ 
+ 
+Words Count: 4,623 + 2 Tables (250*2) = 5,123 
+  
+Submission Date: 07/31/2022 
+ 
+ 
+ 
+
+2 
+ 
+ABSTRACT 
+  
+Vehicle volume serves as a critical metric and the fundamental basis for traffic signal control, 
+transportation project prioritization, road maintenance plans and more. Traditional methods of 
+quantifying vehicle volume rely on manual counting, video cameras, and loop detectors at a limited 
+number of locations. These efforts require significant labor and cost for expansions. Researchers 
+and private sector companies have also explored alternative solutions such as probe vehicle data, 
+while still suffering from a low penetration rate. In recent years, along with the technological 
+advancement in mobile sensors and mobile networks, Mobile Device Location Data (MDLD) have 
+been growing dramatically in terms of the spatiotemporal coverage of the population and its 
+mobility. This paper presents a big-data driven framework that can ingest terabytes of MDLD and 
+estimate vehicle volume at a larger geographical area with a larger sample size. The proposed 
+framework first employs a series of cloud-based computational algorithms to extract multimodal 
+trajectories and trip rosters. A scalable map matching and routing algorithm is then applied to snap 
+and route vehicle trajectories to the roadway network. The observed vehicle counts on each 
+roadway segment are weighted and calibrated against ground truth control totals, i.e., Annual 
+Vehicle-Miles of Travel (AVMT), and Annual Average Daily Traffic (AADT). The proposed 
+framework is implemented on the all-street network in the state of Maryland using MDLD for the 
+entire year of 2019. Results indicate that our proposed framework produces reliable vehicle 
+volume estimates and also demonstrate its transferability and the generalization ability. 
+ 
+Keywords: mobile device location data; big data analytics; vehicle volume; cloud computing; map 
+matching and routing. 
+ 
+ 
+ 
+ 
+
+3 
+ 
+1. INTRODUCTION 
+  
+Vehicle volume measures the amount of traffic traveling through a roadway segment given a 
+specific period of time. It serves as a critical metric and the fundamental basis for various 
+transportation applications including traffic signal control, transportation project prioritization and 
+road maintenance plan. Traditional methods to quantify vehicle volume rely on manual counting, 
+video cameras, and loop detectors at a limited number of locations, a practice that requires 
+significant human labor and a high cost for expansions (1-5). Researchers and private sector 
+companies have also explored alternative solutions such as probe vehicle data, while still suffering 
+from the low penetration rate issue (6-10). 
+ 
+In the past two decades, along with the technological advancement in mobile sensors and mobile 
+networks, mobile device location data (MDLD) have been growing dramatically in terms of 
+coverage and size, with broader spatiotemporal coverage of the population and its mobility. A 
+series of research studies have demonstrated the usefulness of MDLD for enhancing the traditional 
+travel survey and have revealed its potential to substitute surveys (11, 12). At the same time, 
+obtaining travel statistics solely based on MDLD is also worth investigating to reduce human labor 
+and cost. However, MDLD do not include any ground truth information such as trip origins and 
+destinations, travel modes, and trip purposes, which requires computational algorithms to be 
+developed and validated against the existing travel surveys. More importantly, unlike travel 
+surveys which collect information from representative samples to obtain population-representative 
+statistics, MDLD contain all available mobile devices with uneven data quality. 
+ 
+This study was conducted as part of the Vulnerable Road User Density Exposure Dashboard 
+project (https://mti.umd.edu/sdi) - an interactive dashboard that utilizes MDLD to provide data 
+and insights on multimodal volume and safety risk exposure of vulnerable road users (e.g., 
+pedestrians, bicycles) at intersections and roadway segments within Maryland. In this study, we 
+present a big-data driven framework that ingests terabytes of MDLD and estimates vehicle volume 
+for all roadway segments. First, a series of cloud-based computational algorithms are applied—
+including but not limited to—a trip and tour identification algorithm to mine travel behavior 
+information and a travel mode imputation model that impute multimodal trajectories from MDLD. 
+A map matching and routing algorithm is then applied to snap and route vehicle trajectories to the 
+roadway network. The observed vehicle counts on each roadway segment are weighted to match 
+the Annual Vehicle Miles of Travel (AVMT) by county, urban/rural status, and functional classes. 
+Further, a random forest regression model is used to calibrate the weighted vehicle volume against 
+the Annual Average Daily Traffic (AADT) acquired from loop detectors. The proposed framework 
+is implemented on the all-street network in the state of Maryland using MDLD data for the entire 
+year of 2019. 
+ 
+2. LITERATURE REVIEW 
+  
+2.1. Application of Mobile Device Location Data in Transportation Research 
+ 
+The appearance of MDLD in the transportation industry started in the 1990s. Since the mid-1990s, 
+researchers began installing Global Positioning System (GPS) data loggers in vehicles to 
+supplement travel surveys (13-15). With high-frequency in-vehicle GPS data, this approach can 
+
+4 
+ 
+significantly improve the accuracy of travel surveys by recording the exact origin and destination 
+as well as the departure and arrival times. However, only a small number of vehicles can be 
+sampled with this technique, a drawback limiting its capability. Similarly, the wearable GPS, 
+which was introduced in the early 2000s, allowed respondents to report non-vehicle travel modes 
+while still suffering from small sample size issues (16, 17). In the past decade, private sector 
+entities such as INRIX and RITIS also started to incorporate the probe vehicle data into their 
+commercial products (18-21). Nonetheless, the low penetration rate (i.e., 2%-10%) of the 
+commercial probe vehicle data remains the core challenge with respect to drawing the whole 
+picture of travel patterns. 
+  
+As mentioned above, despite having high precision, traditional MDLD usually suffer from small-
+sample-size issues, which significantly limits the usefulness of the data. Since mobile devices, 
+such as smartphones and tablets, have become more popular, MDLD generated from these devices 
+have a greater potential for being used in transportation applications. These new types of MDLD, 
+namely cellular data and Location-Based Service (LBS) data, offer a more extensive 
+spatiotemporal coverage and a larger sample size. The cellular data are generated through 
+communication between cellphones and cell towers (22) and can be further categorized into Call 
+Detail Record (CDR) and sightings (11). The CDR data can only capture the cell tower location, 
+whereas the sightings provide the exact latitude and longitude values. Both types of cellular data 
+have been widely applied to research topics such as travel behavior, human mobility, and social 
+networks in the past two decades (23-31). Despite the large volume of data, cellular data are limited 
+by their spatial and temporal resolution, which is determined by the density of cell towers and 
+users’ cellphone usage levels (32). On a positive note, however, cellular data require less advanced 
+phones and can raise fewer user privacy concerns. The LBS data provide the exact locations 
+generated when a mobile application updates the device’s location with the most accurate sources, 
+based on the existing location sensors such as Wi-Fi, Bluetooth, cellular tower, and GPS (11, 23-
+25, 33, 34). Many applications have been developed using the LBS data. For instance, a recent 
+smartphone-enhanced travel survey conducted in the U.S. used a mobile application, rMove 
+developed by Resource Systems Group (RSG), to collect high-frequency location data and allow 
+the respondents to recall their trips by showing the trajectories in rMove (35-38). Additionally, 
+Airsage leveraged LBS data to develop a traffic platform that can estimate traffic flow, speed, 
+congestion and road user sociodemographic for every road and time of day (39). Further, the 
+Maryland Transportation Institute (MTI) at the University of Maryland (UMD) developed the 
+COVID-19 Impact Analysis Platform (https://data.covid.umd.edu) to provide insight on COVID-
+19’s impact on mobility, health, economy, and society across the U.S. (40-43). 
+ 
+2.2. Vehicle Volume Estimation Methods 
+ 
+2.2.1. Estimating Vehicle Volume with Loop Detectors 
+ 
+Loop detectors are widely used to record traffic volumes and occupancy levels. These sensors are 
+usually buried under the pavements to detect the induction change from the presence of a vehicle. 
+Kwon et al. 2003 developed an algorithm using data from single loop detectors to estimate truck 
+traffic volumes (1). The results showed a 5.7% error compared with the ground truth highway data. 
+Loop detector data were also applied together with probe vehicle data to estimate queue length (44) 
+and vehicle volume at a city-wide scale (45). Although proven to be efficient in estimating vehicle 
+
+5 
+ 
+volume, the high installation and maintenance cost of loop detectors limit their capability of being 
+scaled up to cover the entire transportation network. Therefore, loop detector datasets are often 
+incomplete and mostly unavailable at minor arterials and local streets. 
+ 
+2.2.2. Estimating Vehicle Volume with Probe Vehicle Data 
+ 
+In the past two decades, MDLD have gained significant attention and have been utilized for 
+estimating various traffic characteristics including vehicle volume. With the development of 
+MDLD, estimating vehicle volume at the city scale became a reality. Probe vehicles can record 
+their trajectory data with high granularity (i.e., 1Hz). Based on the trajectory data obtained from 
+probe vehicles, a wide range of methods can be used by researchers to solve transportation 
+problems. Zhao et al. proposed novel methods to estimate queue length and vehicle volume based 
+on the probability theory without prior information about the penetration rate or queue length 
+distribution (6). Guo et al. estimated vehicle volume and queue length at signalized intersections 
+and proposed a new framework to optimize traffic signal control operations (7). Sekuła et al. 
+applied several machine learning and neural networks to estimate historical hourly vehicle volume 
+between sparsely located sensors based on the probe vehicle data (8). Shockwave theories were 
+also applied to probe vehicle data by a few studies (9, 10).      
+ 
+2.2.3. Estimating Vehicle Volume with Mobile Device Location Data 
+Many studies have been conducted focusing on estimating traffic flow and detecting congestion 
+using cellular data (46, 47). Xing et al. 2019 utilized CDR with Time Difference of Arrival (TDOA) 
+positioning technique in order to estimate multimodal traffic volumes on different types of urban 
+roadways by identifying three modes of travel – namely, drive alone, carpooling, and bus (48). 
+The results showed that compared with the ground truth vehicle volume obtained from License 
+Plate Recognition (LPR) cameras, the mean relative error was in the range of 17.1% to 25.7%, 
+depending on the roadway type. Despite significant advances in positioning techniques, cellular 
+data still suffers from low accuracy issues, whereas LBS data have a noticeable advantage due to 
+utilizing different sources to accurately locate the user – a feature that has resulted in an increased 
+usage of this type of data by researchers and the private sector for estimating vehicle volume. Fan 
+et al. 2019 developed a computing framework alongside a heuristic map matching algorithm to 
+estimate Vehicle Miles of Travel (VMT) and AADT for the state of Maryland using INRIX data. 
+The results showed an R2 of 0.878 when fitting the estimated AADT with the ground truth AADT 
+(49). Moreover, a number of state agencies conducted rigorous evaluations of vehicle volume 
+obtained through traditional methods as well as from MDLD obtained by private sector companies. 
+They found the latter to be a promising source for supplementing current surveys and traditional 
+methods (50).  
+3. THE BIG-DATA DRIVEN VEHICLE VOLUME ESTIMATION FRAMEWORK 
+  
+3.1. Overview of the Framework 
+ 
+In this study, we propose a big-data driven vehicle volume estimation framework, which offers the 
+capability of efficiently estimating vehicle volume ingested from terabytes of MDLD. Figure 1 
+shows the proposed framework. The proposed framework is built on Amazon Web Services 
+(AWS). MDLD and all supporting data are stored in Simple Cloud Storage (S3). All algorithms 
+
+6 
+ 
+are developed based on Apache Spark, which uses Resilient Distributed Datasets (RDD), and are 
+coded in PySpark using the Elastic MapReduce (EMR) services. In the cloud environment, MDLD 
+are spliced into RDDs given the number of executors (43, 49). At the same time, all external data 
+sources (i.e., K-D Tree, network, routing engine) are broadcasted into all executors for master and 
+core nodes. The same algorithms are applied to each RDD along with the broadcasted variables, 
+and the results are aggregated and outputted into S3.  
+ 
+ 
+Figure 1. The Big-Data Driven Vehicle Volume Estimation Framework 
+ 
+3.2. Trip End Identification and Travel Mode Imputation 
+ 
+Trip is the basic unit of analysis for almost all transportation applications. However, MDLD 
+usually do not contain any trip-related information. Therefore, in this study, a trip end 
+identification algorithm is used to extract trip-level information from the MDLD, including trip 
+start location, trip end location, departure time, and arrival time. Then, a travel mode imputation 
+model is further applied to infer four travel modes–namely, the air, drive, rail, and nonmotorized 
+modes based on heuristic rules and a random forest model. Detailed descriptions of the trip end 
+identification algorithm and the travel mode imputation model can be found in the following 
+references (12, 51). 
+ 
+3.3. Map Matching and Routing 
+  
+To ensure flexibility and scalability of our map matching and routing method across the entire 
+U.S., we extract the drivable network from OpenStreetMap (OSM) using the latest open-source 
+Python package osm2gmns. The osm2gmns package can parse roadway network data from OSM 
+and output networks to csv files in the General Modeling Network Specification (GMNS) format. 
+It provides customized and practical functions to facilitate traffic modeling. Functions include 
+complex intersection consolidation, movement generation, traffic zone creation, short link 
+
+Cloud Computing
+aws
+Data Source
+Local Server Backup
+MobileDevice
+5
+DATA
+Location Data
+S3Online Bucket
+Geospatial Maps
+spark
+Smart Location
+AnnualAverage
+Database
+DailyTraffic
+Amazon EMR
+AmazonEC2
+DailyUpdate:1176.52to3401.80million
+Annual VehicleMiles
+OpenStreetMap
+PySpark
+points; 15.05 to 17.36 million devices.
+ofTravel
+Computation
+Roadway Network
+osm2gmns
+WeightingandCalibration
+ALGORITHM
+Algorithms
+1.Network parsing
+RandomForestModel
+Data Preprocessing
+2. Missing value
+3.ScalableacrossU.S.
+8
+TripEnd Identification
+County
+networkx
+Urban/Rural Status
+Travel ModeImputation
+1.Routing engine
+#ofLanes,Speed Limit
+MapMatchingandRouting
+2. Short path algorithm
+Weighting and Calibration
+Built Environments
+APPLICATION
+VulnerableUserExposureRiskDashboard
+Decision Support
+Mobility Tracking
+Safety Improvement7 
+ 
+combination, and network visualization. More details about osm2gmns can be found here: 
+https://osm2gmns.readthedocs.io/en/latest/ 
+ 
+To match each location sighting to our OSM network, the OSM network is firstly parsed and 
+converted into the routable formats, where roadway segments are represented by links and nodes. 
+With the network topology, we use the networkX package to build a shortest path-based routing 
+engine. We then transform the latitude and longitude of the start node and end node for each link 
+to the plane coordinate (in meters), and then calculate link direction (degree) using the arctan value 
+between the two nodes (see Figure 3 for details). The travel direction between consecutive 
+sightings is also calculated. Similar to the method for link direction calculation, the coordinates of 
+each sighting are converted to plane coordinates, then the degree is calculated using the arctan 
+value between consecutive sightings. A spatial index structure, K-Dimensional Tree (K-D Tree), 
+is built using the link geometric nodes (i.e., link nodes). Then, for each sighting, we search all link 
+nodes that are within 100 meters. The 100-meter threshold is selected to balance the algorithm 
+efficacy and the computing speed. If we increase the value, more candidate links will be considered 
+but this will require more computing resources. If we decrease the value, we might not be able to 
+find a candidate link when the observation is sparse. To validate, we calculate the distance between 
+consecutive link nodes using the Maryland OSM network as an example. Results indicate that 
+more than 95% of the link nodes are within 100 meters of their neighbors, as shown in Figure 2. 
+Therefore, using the 100-meter value as the radius for searching candidate nodes is reasonable. 
+ 
+ 
+Figure 2. Distribution of Distance between Link Nodes in the OSM Network 
+ 
+As the next step, for each sighting, we compare its travel direction to all candidate links. The 
+closest link with an absolute travel direction difference smaller than 30 degrees will be selected as 
+a valid matched link for the sighting. This 30-degree threshold is selected mainly to avoid the 
+sighting being matched to the link in the opposite direction. In common cases, the degree 
+difference between the travel direction and the link direction should be approximately 0. Here, we 
+use a 30-degree threshold to consider the uncertainty of location accuracy in MDLD. After the 
+matched link for each sighting is found, given the observed link sequence, the routing engine can 
+fill the gap between consecutively observed links and retrieve the complete route. Another layer 
+of reasonable checks is conducted at the routing stage. For each pair of consecutive sightings that 
+
+Distribution of Distance between Link Nodes
+40%
+35%
+30%
+25%
+20%
+15%
+10%
+5%
+0%8 
+ 
+are snapped to links, the routed distance is calculated by summing the link length of all the links 
+traveled between the two sightings. Two reasonableness checks are carried out:  
+ 
+(1) If the routed distance is greater than the cumulative distance between the two sightings 
+snapped to links by 2,000 meters or more, we consider the route invalid.  
+(2) The travel time on these links will be calculated based on the timestamp difference between 
+the two sightings. With the routed distance and travel time, the average travel speed on 
+these links can be calculated. If the speed exceeds 50 m/s (i.e., 112 mph or 180 km/h), we 
+assume that one of the two sightings is matched to the wrong link. 
+ 
+If either of these two violations is observed, we apply a trial-and-error process by removing the 
+latter sighting and performing the routing using the next sighting snapped to the network until it 
+does not violate the 2,000-meter threshold or the 50 m/s threshold (52). A simple example of the 
+map matching and routing method is illustrated in Figure 3. 
+ 
+ 
+Figure 3. Example of Map Matching and Routing. 
+ 
+3.4. Weighting 
+  
+After map matching and routing, we collect routes for all vehicle trips and aggregate them by links 
+to obtain the observed vehicle volume for each link. Afterward, we develop a link-based weighting 
+method to match the AVMT in the region. We classify each link by county, urban/rural status, and 
+functional classes and calculate the link weight using the formula below: 
+ 
+𝑤𝐶,𝑢,𝑓 = 𝐴𝑉𝑀𝑇𝐶,𝑢,𝑓
+∑
+𝑂𝐶,𝑢,𝑓,𝑖
+𝑁𝐶
+ 
+ 
+where 𝑤𝐶,𝑢,𝑓 represents the weight for links in county C, with urban/rural status of u, and with 
+functional class f; 𝐴𝑉𝑀𝑇𝑐,𝑢,𝑓 represent the AVMT; and 𝑂𝐶,𝑢,𝑓,𝑖 represents the observed vehicle 
+volume on link i; 𝑁𝐶 represents the total number of links in county C. For instance, if the study 
+area has 20 counties, 2 urban/rural status and 6 functional classes, then a total of 240 link-based 
+
+>TravelDirection
+o Link Centroid
+Observation
+Node
+Matched Link
+Degree
+CandidateLink
+Routed Link9 
+ 
+weights will be generated. Subsequently, the weighted vehicle volume for each link can be 
+calculated as: 
+ 
+𝑉𝑐,𝑢,𝑓,𝑖 = 𝑤𝐶,𝑢,𝑓 × 𝑂𝑐,𝑢,𝑓,𝑖 
+ 
+where 𝑉𝑐,𝑢,𝑓,𝑖 represents the weighted vehicle volume on link i. 
+ 
+3.5. Volume Calibration 
+  
+The weighted vehicle volume is further calibrated to match the ground truth AADT collected from 
+loop detectors at a limited number of locations. In this study, we use the random forest regression 
+to calibrate the weighted vehicle volume against the AADT to obtain the final vehicle volume. 
+During the calibration process, a 10-fold cross-validation (CV) process is used to fine-tune the 
+random forest regression hyperparameters with 90% training data. The fine-tuned models are then 
+applied to the 10% testing data. 
+ 
+4. CASE STUDY: THE STATE OF MARYLAND 
+  
+4.1. Data 
+ 
+4.1.1. Mobile Device Location Data and the Study Area 
+  
+This study used MDLD data obtained from Maryland Transportation Institute (MTI). MTI 
+integrated and cleaned the raw MDLD from multiple data vendors and built a national MDLD data 
+panel that consists of more than 270,000,000 Monthly Active Users (MAU) and represents 
+movements across the nation.  (40-43, 51). Figure 4 shows the density of location sightings 
+covering locations within and outside of the boundaries of the state of Maryland. In this study, we 
+used all MDLD data that are observed in the state of Maryland for the entire year of 2019. The 
+MDLD is processed on a daily basis and the results are aggregated to produce an annual total result. 
+ 
+ 
+Figure 4. Mobile Device Location Data around the State of Maryland. 
+  
+
+10 
+ 
+4.1.2. OpenStreetMap Network  
+ 
+Using the osm2gmns package, we extracted a total of 634,516 drivable roadway segments within 
+the state of Maryland. Information about the number of lanes and speed limits was recorded for 
+only 111,835 roadway segments (17.6%) and 84,728 roadway segments (13.4%), respectively. As 
+shown on the left-hand side in Figure 5, the missing values for the number of lanes and speed 
+limits were estimated based on the corresponding values on nearby roadways in the same county, 
+and with the same urban/rural status, and road functional classes. These two variables are further 
+used as features in the vehicle volume calibration model.  
+ 
+ 
+Figure 5. Number of Lanes and Speed Limits in OSM 
+ 
+4.1.3. Annual Vehicle Miles of Travel Data 
+  
+We use the vehicle miles traveled data from the Maryland Department of Transportation State 
+Highway Administration (MDOT SHA) as a control total number to weight observed vehicle 
+volume. Every year, MDOT SHA publishes an annual vehicle miles of travel (AVMT) report by 
+county and functional classification for the state, county, and municipal highway systems. This 
+AVMT report features the current FHWA Functional Classification Codes (1-7) and provides 
+additional classifications (i.e., Urban, Rural, Principal Arterial and Other Freeways and 
+Expressways, and Minor Collector). As discussed in the methodology section, the weights are 
+generated based on county, urban/rural status and functional classes. Here, 23 Maryland counties 
+plus Baltimore City, urban or rural, and two function classes (highway and non-highway) are 
+considered. We map the OSM link type to the FHWA Functional Classification Codes and 
+generated the highway and non-highway classes. More specifically, “motorway”, “trunk” and 
+“ramp” are classified as highway (i.e., 1, 2 in FHWA class), and the other types are classified as 
+non-highway (i.e., 3,4,5,6,7 in FHWA class). More details about the AVMT data can be found 
+here: https://www.roads.maryland.gov/mdotsha/Pages/index.aspx?PageId=302  
+ 
+
+EstimatedNumberof
+Lanes (lane)
+1Lane
+2Lanes
+3or4Lanes
+5or6Lanes
+Morethan6Lanes
+(a)
+(b)
+EstimatedSpeedLimits
+(mph)
+5-15
+15-25
+25-45
+45-60
+60-70
+(c)
+(d)11 
+ 
+4.1.4. Annual Average Daily Traffic Data 
+  
+We use the AADT also from MDOT SHA to calibrate weighted vehicle volume against the ground 
+truth at a limited number of locations. The AADT data consists of linear and point geometric 
+features which represent the geographic locations and segments of roadway throughout the state 
+of Maryland that include traffic volume metrics such as AADT. More details about the AADT can 
+be found here:https://data.imap.maryland.gov/maps/77010abe7558425997b4fcdab02e2b64/about  
+ 
+4.1.5. Smart Location Database and Features for Volume Calibration 
+ 
+The Smart Location Database (SLD) is a nationwide geographic data resource for measuring 
+location efficiency. The SLD is produced by the U.S. Environmental Protection Agency (EPA)’s 
+Smart Growth Program. It provides more than 90 variables on land use and built environment 
+characteristics such as population and employment densities, land use diversity, urban design 
+attributes, destination accessibility, transit accessibility, and socioeconomic/sociodemographic 
+characteristics at the census block group level. Most attributes are available for every census block 
+group in the United States. In this study, we use SLD variables as features in the random forest 
+regression to calibrate weighted vehicle volume to account for the effects of the built environment. 
+The SLD variables used in this study include “TotEMP”, “Pct_AO0”, “D1A”, “D1C”, “D3AAO”, 
+“D3B”, and “D5AR”: 
+• TotEMP = total employment; 
+• Pct_AO0 = percent of zero-car households; 
+• D1A = gross residential density (housing units per acre) on unprotected land; 
+• D1C = gross employment density (jobs per acre) on unprotected land; 
+• D3AAO = network density in terms of facility miles of auto-oriented links per square 
+miles; 
+• D3B = street intersection density (weighted, auto-oriented intersections eliminated); 
+• D5AR = jobs within 45 minutes auto travel time, time decay (network travel time) 
+weighted 
+We also include urban/rural status, county code, link type, number of lanes, and speed limits as 
+features in the calibration process.  
+ 
+4.2. Results 
+ 
+4.2.1. Overall Comparison 
+ 
+Figure 6 shows the weighting and calibration results for both training and testing sets. The blue 
+dots represent weighted volume comparisons and the green dots represent calibrated vehicle 
+volume comparisons with MDOT SHA AADT. Figure 6 (a) and (b) compares the weighted vehicle 
+volume and calibrated vehicle volume with the MDOT SHA AADT in the training set respectively; 
+Figure 6 (c) and (d) compares the weighted vehicle volume and calibrated vehicle volume with the 
+MDOT SHA AADT in the testing set respectively. As it can be seen from Figure 6 (a), for the 
+training set, the Pearson correlation value and the Root Mean Square Error (RMSE) between the 
+weighted vehicle volume and the ground truth AADT are 0.746 and 7,912, respectively. These 
+values are improved to 0.966 and 2,996 after calibration, as shown in Figure 6 (b). Similarly, for 
+
+12 
+ 
+the testing set, the Pearson correlation and RMSE are improved from 0.764 and 7,548, to 0.854 
+and 5,701 respectively after calibration. 
+ 
+ 
+Figure 6. (a) Weighted Vehicle Volume in Training Set; (b) Calibrated Vehicle Volume in Training Set; 
+(c) Weighted Vehicle Volume in Testing Set; (d) Calibrated Vehicle Volume in Testing Set. 
+ 
+4.2.2. Vehicle Volume Validation by Link Types and Urban/Rural Status 
+ 
+Figure 7 and Table 1 show the calibrated vehicle volume by link types for both the training and 
+testing sets. For all link types, a good correlation (i.e., over 0.80) can be observed between the 
+calibrated vehicle volume and the ground truth AADT, except for Local Roads and Highway 
+Ramps in the testing set. The results indicate that our proposed framework can accurately estimate 
+vehicle volume on higher-level roadways (i.e., Interstate Highways and Highways, Primary Roads, 
+Secondary Roads), while concurrently maintaining high correlations for lower-level roadways (i.e., 
+Tertiary Roads, Local Roads, Highway Ramps). The relatively weaker performance for the case 
+of lower-level roadways can be attributed to limitations in technology. The MDLD only capture 
+part of the daily trips of a device within the area with mobile network connections and higher-level 
+roadways usually have a better coverage compared to lower-level ones. This variability might also 
+result in capturing more travelers on highways and major arterials. In addition, the LBS data 
+sample is more likely to include the active travelers that make more trips and/or longer-duration 
+
+140000
+140000
+Corr.=0.746
+MDOT SHA AADT (veh/day)
+Corr.=0.966
+120000
+RMSE=7912
+120000
+RMSE=2996
+100000
+100000
+80000
+80000
+.
+60000
+60000
+40000
+:
+40000
+20000
+20000
+0
+0
+0
+20000
+40000
+60000
+80000100000120000
+140000
+0
+20000
+40000
+60000
+80000100000120000
+140000
+WeightedVehicleVolume(veh/day)
+CalibratedVehicleVolume (veh/day)
+140000
+140000
+Corr.= 0.764
+ (veh/day)
+Corr.=0.854
+MDOT SHA AADT (veh/day)
+120000
+RMSE=7548
+120000
+RMSE=5701
+100000
+100000
+MDOT SHA AADT
+80000
+80000
+60000
+60000
+40000
+40000
+20000
+20000
+0
+0
+0
+20000
+40000
+60000
+80000
+100000120000140000
+0
+20000
+40000
+60000
+80000
+100000
+120000
+140000
+Weighted Vehicle Volume (veh/day)
+CalibratedVehicleVolume(veh/day)13 
+ 
+trips, such as long-distance travel for leisure or business purposes or long-distance commute which 
+usually happen on interstate highways. 
+ 
+ 
+Figure 7. Volume Calibration Results Comparison by Link Type. 
+ 
+Figure 8 and Table 2 show the calibration of vehicle volume by urban/rural status for both the 
+training and testing sets. In summary, for both urban and rural roads, a good correlation (i.e., over 
+
+100000
+100000
+100000
+100000
+50000
+50000
+50000
+50000
++0
+-0
+50000100000
+0
+50000100000
+0
+50000100000
+0
+50000100000
+Ro
+Road
+60000
+60000
+60000
+60000
+40000
+40000
+40000
+40000
+20000
+20000
+20000
+20000
+0
+0-
+0
+0-
+0
+2500050000
+0
+2500050000
+0
+2500050000
+0
+2500050000
+yR
+Roa
+40000
+40000
+40000
+40000
+(veh/day)
+(veh/day)
+20000
+20000
+20000
+20000
+1
+0
+20000 40000
+2000040000
+2000040000
+0
+2000040000
+MDOT SHA AADT
+AADT
+Ro
+load
+SHA
+60000
+60000
+60000
+60000
+40000
+40000
+MDOT :
+40000
+40000
+20000
+20000
+20000
+20000
+0.
+0
+50000
+0
+50000
+0
+50000
+50000
+Roa
+ads
+30000
+30000
+30000
+30000
+20000
+20000
+20000
+20000
+10000
+10000
+10000
+10000
+1
++0
+20000
+0
+20000
+0
+20000
+20000
+Ral
+amj
+80000
+80000
+80000
+80000
+60000
+60000
+60000
+60000
+40000
+40000
+40000
+40000
+20000
+20000
+20000-
+20000
+：
+0
+0
+0
+50000
+0
+50000
+0
+50000
+0
+5000014 
+ 
+0.80) can be observed between the calibrated vehicle volume and the ground truth AADT, whereas 
+a higher correlation can be observed for urban roads.  The relatively weaker performance in rural 
+roadways can also be attributed to the technology limitation mentioned above.  
+ 
+Figure 8. Volume Calibration Results Comparison by Urban/Rural Status. 
+ 
+Table 1. Volume Calibration Results Comparison by Link Type 
+Link Type 
+Training Set 
+Testing Set 
+Corr. 
+RMSE 
+Corr. 
+RMSE 
+Before 
+After 
+Before 
+After 
+Before 
+After 
+Before 
+After 
+All 
+0.746 
+0.966 
+7912 
+2996 
+0.764 
+0.854 
+7548 
+5701 
+Interstate Highways 
+and Highways 
+0.752 
+0.975 
+20081 
+6559 
+0.712 
+0.775 
+19633 
+15246 
+Primary Roads 
+0.699 
+0.971 
+7909 
+2695 
+0.721 
+0.846 
+8665 
+6509 
+Secondary Roads 
+0.627 
+0.960 
+4899 
+1776 
+0.617 
+0.813 
+3667 
+2667 
+Tertiary Roads 
+0.414 
+0.959 
+3486 
+994 
+0.511 
+0.869 
+3090 
+1877 
+Local Roads 
+0.374 
+0.944 
+2474 
+853 
+0.426 
+0.742 
+1701 
+1083 
+Highway Ramps 
+0.242 
+0.866 
+10426 
+4722 
+0.182 
+0.402 
+9119 
+6846 
+ 
+Table 2. Volume Calibration Results by Urban/Rural Status. 
+Link Type 
+Training Set 
+Testing Set 
+Corr. 
+RMSE 
+Corr. 
+RMSE 
+Before 
+After 
+Before 
+After 
+Before 
+After 
+Before 
+After 
+All 
+0.746 
+0.966 
+7912 
+2996 
+0.764 
+0.854 
+7548 
+5701 
+Rural 
+0.769 
+0.967 
+3583 
+1442 
+0.727 
+0.826 
+4810 
+4075 
+
+-
+.
+60000
+60000
+60000
+60000
+40000
+(veh/day)
+40000
+20000
+20000
+20000
+20000
+E
+200004000060000
+200004000060000
+200004000060000
+200004000060000
+MDOT SHA
+125000
+125000
+2
+100000
+100000
+100000
+75000
+75000
+75000
+50000
+50000
+50000
+C
+25000
+25000
+25000
+0
+0
+50000
+100000
+50000
+100000
+0
+50000
+100000
+50000
+10000015 
+ 
+Urban 
+0.738 
+0.964 
+8913 
+3363 
+0.764 
+0.853 
+8311 
+6179 
+ 
+Figure 9 visualizes the calibrated vehicle volume averaged from the entire year of 2019 
+(represented as AADT) on the all-street network in the state of Maryland. It can be seen that the 
+interstate highway and the highway skeletons can be clearly identified from the map. Major 
+arterials also stand out from the map. Figure 9 (b) zooms into the Washington D.C. area, where I-
+495, I-270, I-95 and the Baltimore/Washington Parkway are clearly seen. Figure 9(c) zooms into 
+the Baltimore area, where I-395, I-695, I-795, I-95, and I-70 are all captured. Figure 9(d) zooms 
+into Hagerstown, MD, which is a city in Washington County, MD near the border of Pennsylvania. 
+The I-70, I-81, and MD-40 are all captured, demonstrating the ability of our proposed framework 
+to produce reliable results in rural areas.  
+ 
+ 
+Figure 9. Visualization of Calibrated Vehicle Volume. (a) the State of Maryland; (b) Washington D.C.; 
+(c) Baltimore City; (d) Hagerstown, MD. 
+ 
+5. CONCLUSIONS AND DISCUSSIONS  
+ 
+This paper presents a big-data driven framework that is able to ingest terabytes of MDLD and 
+estimate vehicle volume based on MDLD. The proposed framework first employs a series of 
+cloud-based computational algorithms to extract vehicle trajectories. A map-matching and routing 
+algorithm is then applied to snap and route vehicle trajectories to the road network. The observed 
+vehicle counts on each road segment are weighted and calibrated against the control total, i.e., 
+annual vehicle miles traveled (VMT), and data collected from real-world loop detectors. The 
+proposed framework is implemented and validated on the all-street network in the state of 
+
+(a)
+(b)
+Calibrated VehicleVolume
+(AADT) (veh/day)
+<=5,000
+5,000-10,000
+10,00025,000
+25,000-50,000
+50,000-120,000
+(c)
+(d)16 
+ 
+Maryland using MDLD data from 2019. After weighting and calibration processes, high 
+correlation and low RMSE values are observed between our vehicle volume estimates and the 
+ground truth data. 
+ 
+The framework proposed in this study and the study findings have practical implications. For 
+instance, estimated vehicle volume based on MDLD can be leveraged in safety risk exposure 
+analysis. In particular, the proposed estimation method can particularly be beneficial for safety 
+risk exposure and crash analysis with respect to vulnerable road users (e.g., pedestrians and 
+bicyclists). Pedestrian and bicyclist exposure data have traditionally been collected through 
+surveys or count collections at sample locations (53, 54). In addition to being costly and labor-
+intensive, these conventional data collection methods are susceptible to subjectivity and may yield 
+inaccurate data. Consequently, high-quality and readily-available pedestrian and bicyclist 
+exposure data are considered as a limitation in safety analysis (55). As exposure data are crucial 
+for contextualization of crash analysis and prioritization of safety countermeasures (53), utilization 
+of high-quality and consistent exposure data is imperative. When it comes to safety analysis, using 
+MDLD for volume estimation—as performed in this study—provides a tremendous advantage 
+over using data obtained from traditional volume estimation methods. This is due to the potential 
+of the MDLD to produce more reliable exposure data. Employment of such high-fidelity exposure 
+data (i.e., MDLD-estimated volumes) as input for safety and crash analyses can lead to more 
+accurate results and guide data-driven, evidence-based policy decision-making to improve the 
+safety of all road users including the most vulnerable ones. 
+ 
+ACKNOWLEDGEMENTS 
+This study was conducted as part of a collaboration among the Maryland Department of 
+Transportation State Highway Administration (MDOT SHA), Maryland Transportation Institute 
+(MTI) at the University of Maryland College Park, and Shock, Trauma and Anesthesiology 
+Research (STAR) Center at the University of Maryland Baltimore through the sponsorship from 
+the Safety Data Initiative from the U.S. Department of Transportation (USDOT).  
+ 
+CONFLICT OF INTEREST 
+The authors declare that they have no conflict of interest. 
+ 
+AUTHOR CONTRIBUTION STATEMENT 
+The authors confirm contribution to the paper as follows: study conception and design: M.Y., W.L., 
+C.X.; data collection: M.Y., M.A., J.M., G.C., S.S.N.; analysis and interpretation of results: M.Y., 
+J.M., W.L.; methodology support (osm2gmns): J.L.; draft manuscript preparation: M.Y., W.L., 
+M.A., J.M., G.C., S.S.N., A.K.. 
+ 
+REFERENCE 
+ 
+1. Kwon, J., Varaiya, P. and Skabardonis, A.. Estimation of truck traffic volume from single loop 
+detectors with lane-to-lane speed correlation. Transportation Research Record, 1856(1), 
+pp.106-117 (2003). 
+2. Muñoz, L., Sun, X., Horowitz, R., and Alvarez, L.. Traffic density estimation with the cell 
+transmission model. In American Control Conference. Proceedings of the 2003, Vol. 5. IEEE, 
+3750–3755 (2003). 
+
+17 
+ 
+3. Okutani, I. and Stephanedes, Y. J.. Dynamic prediction of traffic volume through Kalman 
+filtering theory. Transportation Research Part B: Methodological 18, 1 (1984), 1–11 
+4. Wilkie, D., Sewall, J., and Lin, M.. Flow reconstruction for data driven traffic animation. ACM 
+Transactions on Graphics (TOG) 32, 4 (2013), 89 
+5. Jia, Z., C. Chen, B. Coifman, and P. Varaiya. The PeMS Algorithms for Accurate, Real-Time 
+Estimates of g-Factors and Speeds from Single Loop Detectors. Proc., IEEE Intelligent 
+Transportation Systems Council Annual Meeting, IEEE, Piscataway, N.J., pp. 536–541 (2001). 
+6. Zhao, Y., Zheng, J., Wong, W., Wang, X., Meng, Y., & Liu, H. X.. Various methods for queue 
+length and traffic volume estimation using probe vehicle trajectories. Transportation Research 
+Part C: Emerging Technologies, 107, 70–91 (2019). https://doi.org/10.1016/j.trc.2019.07.008 
+7. Guo, Q., Li, L., & (Jeff) Ban, X.. Urban traffic signal control with connected and automated 
+vehicles: A survey. In Transportation Research Part C: Emerging Technologies (Vol. 101, pp. 
+313–334) (2019). Elsevier Ltd. https://doi.org/10.1016/j.trc.2019.01.026 
+8. Sekuła, P., Marković, N., vander Laan, Z., & Sadabadi, K. F.. Estimating historical hourly 
+traffic volumes via machine learning and vehicle probe data: A Maryland case study. 
+Transportation Research Part C: Emerging Technologies, 97, 147–158 (2018). 
+https://doi.org/10.1016/j.trc.2018.10.012 
+9. Anuar, K., & Cetin, M.. Estimating Freeway Traffic Volume Using Shockwaves and Probe 
+Vehicle Trajectory Data. Transportation Research Procedia, 22, 183–192 (2017). 
+https://doi.org/10.1016/j.trpro.2017.03.025 
+10. Li, F., Tang, K., Yao, J., & Li, K.. Real-Time Queue Length Estimation for Signalized 
+Intersections Using Vehicle Trajectory Data. Transportation Research Record, 2623(1), 49–59 
+(2017). https://doi.org/10.3141/2623-06 
+11. Chen, C., Ma, J., Susilo, Y., Liu, Y., & Wang, M.. The promises of big data and small data for 
+travel behavior (aka human mobility) analysis. Transportation Research Part C: Emerging 
+Technologies. 68, 285-299, (2016). 
+12. Yang, M., Pan, Y., Darzi, A., Ghader, S., Xiong, C. and Zhang, L.. A data-driven travel mode 
+share estimation framework based on mobile device location data. Transportation, pp.1-45 
+(2021). 
+13. Battelle. Global Positioning Systems for Personal Travel Surveys: Lexington Area Travel Data 
+Collection Test. Final Report. FHWA, U.S. Department of Transportation, (1997). 
+14. 2000–2001 California Statewide Household Travel Survey. Final Report. NuStats, Austin, Tex 
+(2002). 
+15. Kansas City Regional Travel Survey. Final Report. NuStats, Austin, Tex, (2004). 
+16. Mid-Region Council of Governments 2013 Household Travel Survey. Final Report. Westat, 
+Rockville, Md, (2014). 
+17. 2014 Southern Nevada Household Travel Survey. Final Report. Westat, Rockville, Md, (2015). 
+18. INRIX Traffic. http://www.inrix.com/, (2020). 
+19. Haghani, Ali, Masoud Hamedi, and Kaveh Farokhi Sadabadi. I-95 Corridor coalition vehicle 
+probe project: Validation of INRIX data. I-95 Corridor Coalition 9, (2009). 
+20. Schrank, D., Eisele, B., & Lomax, T.. 2014 Urban mobility report: powered by Inrix Traffic 
+Data (No. SWUTC/15/161302-1), (2015). 
+21. Cui, Z., Ke, R., Pu, Z., & Wang, Y.. Deep bidirectional and unidirectional LSTM recurrent 
+neural network for network-wide traffic speed prediction. arXiv preprint arXiv:1801.02143, 
+(2018). 
+
+18 
+ 
+22. Horak, Ray. Telecommunications and data communications handbook. John Wiley & Sons, 
+(2007). 
+23. Gonzalez, M. C., Hidalgo, C. A., & Barabasi, A. L.. Understanding individual human mobility 
+patterns. Nature, 453(7196), 779-782, (2008). 
+24. Kang, C., Liu, Y., Ma, X., & Wu, L.. Towards estimating urban population distributions from 
+mobile call data. Journal of Urban Technology, 19(4), 3-21, (2012). 
+25. Kang, C., Ma, X., Tong, D., & Liu, Y.. Intra-urban human mobility patterns: An urban 
+morphology perspective. Physica A: Statistical Mechanics and its Applications, 391(4), 1702-
+1717, (2012). 
+26. Pappalardo, L., F. Simini, S. Rinzivillo, D. Pedreschi, F. Giannotti and A.-L. Barabási. 
+Returners and Explorers Dichotomy in Human Mobility. Nature communications. Vol. 6,pp. 
+8166. (2015). 
+27. Song, C., T. Koren, P. Wang and A.-L. Barabási. Modelling the Scaling Properties of Human 
+Mobility. Nature Physics. Vol. 6, No. 10, pp. 818. (2010). 
+28. Song, C., Z. Qu, N. Blumm and A.-L. Barabási. Limits of Predictability in Human Mobility. 
+Science. Vol. 327, No. 5968, pp. 1018-102. (2010). 
+29. Çolak, S., A. Lima and M. C. González. Understanding Congested Travel in Urban Areas. 
+Nature communications. Vol. 7, pp. 10793. (2016). 
+30. Bachir, D., Khodabandelou, G., Gauthier, V., El Yacoubi, M. and Puchinger, J. Inferring 
+dynamic origin-destination flows by transport mode using mobile phone data. Transportation 
+Research Part C: Emerging Technologies, 101, pp.254-275. (2019). 
+31. Fekih, M., Bellemans, T., Smoreda, Z., Bonnel, P., Furno, A. and Galland, S.. A data-driven 
+approach for origin–destination matrix construction from cellular network signalling data: a 
+case study of Lyon region (France). Transportation, pp.1-32. (2020). 
+32. Landmark, A.D., Arnesen, P., Södersten, C.J. and Hjelkrem, O.A.. Mobile phone data in 
+transportation research: methods for benchmarking against other data sources. Transportation, 
+pp.1-23. (2021). 
+33. Wang, F., & Chen, C.. On data processing required to derive mobility patterns from passively-
+generated mobile phone data. Transportation Research Part C: Emerging Technologies. 87, 
+58-74, (2018). 
+34. Wang, F., Wang, J., Cao, J., Chen, C., & Ban, X. J.. Extracting trips from multi-sourced data 
+for mobility pattern analysis: An app-based data example. Transportation Research Part C: 
+Emerging Technologies. 105, 183-202, (2019). 
+35. Puget Sound Regional Travel Study. Report: Spring 2014 Household Travel Survey. RSG, 
+(2014). 
+36. In-The-Moment Travel Study. Revised Report. RSG, (2015). 
+37. Puget Sound Regional Travel Study. Report: 2015 Household Travel Survey. RSG, (2015). 
+38. 2017 Puget Sound Regional Travel Study. Draft Final Report. RSG, (2017). 
+39. Airsage. https://www.airsage.com/, (2020). 
+40. Zhang, L., Darzi, A., Ghader, S., Pack, M.L., Xiong, C., Yang, M., Sun, Q., Kabiri, A. and Hu, 
+S.. 
+Interactive 
+covid-19 
+mobility 
+impact 
+and 
+social 
+distancing 
+analysis 
+platform. Transportation Research Record, p.03611981211043813 (2020). 
+41. Xiong, C., Hu, S., Yang, M., Luo, W., and Zhang, L.. Mobile device data reveal the dynamics 
+in a positive relationship between human mobility and COVID-19 infections. Proceedings of 
+the National Academy of Sciences, 117(44), 27087-27089 (2020a). 
+
+19 
+ 
+42. Xiong, C., Hu, S., Yang, M., Younes, H., Luo, W., Ghader, S. and Zhang, L.. Mobile device 
+location data reveal human mobility response to state-level stay-at-home orders during the 
+COVID-19 pandemic in the USA. Journal of the Royal Society Interface, 17(173), p.20200344. 
+(2020b). 
+43. Hu, S., Xiong, C., Yang, M., Younes, H., Luo, W. and Zhang, L., 2021. A big-data driven 
+approach to analyzing and modeling human mobility trend under non-pharmaceutical 
+interventions during COVID-19 pandemic. Transportation Research Part C: Emerging 
+Technologies, 124, p.102955. 
+44. Li, J.Q., Zhou, K., Shladover, S.E. and Skabardonis, A.. Estimating queue length under 
+connected vehicle technology: Using probe vehicle, loop detector, and fused 
+data. Transportation research record, 2356(1), pp.17-22 (2013). 
+45. Meng, C., Yi, X., Su, L., Gao, J. and Zheng, Y.. City-wide traffic volume inference with loop 
+detector data and taxi trajectories. In Proceedings of the 25th ACM SIGSPATIAL International 
+Conference on Advances in Geographic Information Systems (pp. 1-10) (2017). 
+46. Caceres, N., Romero, L. M., Benitez, F. G. and del Castillo, J. M.. Traffic Flow Estimation 
+Models Using Cellular Phone Data. IEEE Transactions on Intelligent Transportation Systems, 
+vol. 13, no. 3, pp. 1430-1441 (2012). doi: 10.1109/TITS.2012.2189006 
+47. Janecek, A., Valerio, D., Hummel, K. A., Ricciato, F. and Hlavacs, H.. The Cellular Network 
+as a Sensor: From Mobile Phone Data to Real-Time Road Traffic Monitoring. IEEE 
+Transactions on Intelligent Transportation Systems, vol. 16, no. 5, pp. 2551-2572, doi: 
+10.1109/TITS.2015.2413215 (2015). 
+48. Xing, J., Liu, Z., Wu, C., & Chen, S.. Traffic volume estimation in multimodal urban networks 
+using cell phone location data. IEEE Intelligent Transportation Systems Magazine, 11(3), 93-
+104 (2019). 
+49. Fan, J., Fu, C., Stewart, K., and Zhang, L.. Using big GPS trajectory data analytics for vehicle 
+miles traveled estimation. Transportation research part C: emerging technologies 103 (2019): 
+298-307. 
+50. Codjoe, Julius, Grace Ashley, and William Saunders. Evaluating cell phone data for AADT 
+estimation. No. FHWA/LA. 18/591, LTRC Project Number: 16-3SA, State Project Number: 
+DOTLT1000110. Louisiana Transportation Research Center, (2018). 
+51. Zhang, L., Ghader, S., Darzi, A., Pan, Y., Yang, M., Sun, Q., Kabiri, A. and Zhao, G.. Data 
+analytics and modeling methods for tracking and predicting origin-destination travel trends 
+based on mobile device data. Federal Highway Administration Exploratory Advanced 
+Research Program (2020). 
+52. Newson, P. and Krumm, J.. Hidden Markov map matching through noise and sparseness. 
+In Proceedings of the 17th ACM SIGSPATIAL international conference on advances in 
+geographic information systems (pp. 336-343) (2009). 
+53. Sanders, R.L., Frackelton, A., Gardner, S., Schneider, R., Hintze, M. Ballpark Method for 
+Estimating Pedestrian and Bicyclist Exposure in Seattle, Washington: Potential Option for 
+Resource-constrained Cities in an Age of Big Data. Transportation Research Record, 2605(1), 
+pp.32–44 (2017). 
+54. Lee, K., and Sener, I.N. Emerging Data Mining for Pedestrian and Bicyclist Monitoring: A 
+Literature Review Report. Safety through Disruption, National University Transportation 
+Center (UTC) Program. (2017). https://safed.vtti.vt.edu/wp-content/uploads/2020/07/UTC-
+Safe-D_Emerging-Data-Mining-for-PedBike_TTI-Report_26Sep17_final.pdf. Accessed July 
+10, 2021. 
+
+20 
+ 
+55. Synthesis of Methods of Estimating Pedestrian and Bicyclist Exposure to Risk at Areawide 
+Levels and on Specific Transportation Facilities. FHWA-SA-17-041. FHWA, U.S. 
+Department of Transportation, (2017). 
+

CNFAT4oBgHgl3EQfsx5b/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:96d0bce6dfc1b7672d9a66e83a94acadd6b37857a79fe2d91a32599af60599d6
+size 4325421

CNFAT4oBgHgl3EQfsx5b/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:be96a0404281fd1420de0277849ab9072e3c08199b1709c75513f56b67a207a0
+size 145839

F9FKT4oBgHgl3EQfbS5P/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d9cff610be896d59369b5be5ff5c2b3fad8b788bba9d25d2c816c082ecb0daed
+size 175551

H9AyT4oBgHgl3EQfrvmy/content/tmp_files/2301.00567v1.pdf.txt

@@ -0,0 +1,1551 @@
+Strange Stars within Bosonic and Fermionic Admixed Dark Matter
+Luiz L. Lopes1∗ and H. C. Das2,3†
+1Centro Federal de Educac¸˜ao Tecnol´ogica de Minas Gerais Campus VIII; CEP 37.022-560, Varginha - MG - Brasil
+2Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India and
+3Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
+(Dated: January 3, 2023)
+In this work, we study dark matter (DM) admixed strange quark stars exploring the different possibilities
+about the nature of the DM and their effects on the macroscopic properties of strange stars, such as maximum
+masses, radii, as well the dimensionless tidal parameter. We observe that the DM significantly affects the
+macroscopic properties that depend on the DM mass, type, and fraction inside the star.
+I.
+INTRODUCTION
+Recently, it was suggested that quarks probable appear in-
+side the core of a massive neutron star (NS) due to a very high
+density, where hadronic matter undergoes phase transitions to
+a new phase of quarks and gluons [1]. Furthermore, several
+exotic particles, such as hyperons production, dark matter ac-
+cretions, kaon condensations, and so on, appeared primarily in
+the core of the NS. As a result, in those regimes, the equation
+of state (EoS) is the fundamental component that can describe
+both micro/macroscopic properties. Another possibility is that
+at least some of the observed pulsars are indeed stable quark
+stars, or strange stars. Several theoretical works have pro-
+posed the existence of strange quark stars (SQSs) [2], which
+are made up of u, d, and s quarks in equilibrium in terms
+of weak interactions. The Bodmen-Witten conjecture states
+that strange quark matter (SQM) can have a lower energy per
+baryon than pure nucleons because the exclusion principle
+may be dominant at absolute zero pressure and temperature
+[3, 4]. Hence, the SQM might be the true ground state of the
+hadronic matter. Hence, it stands to reason that the SQS must
+be more stable than the ordinary NS.
+Various phenomenological models have been proposed to
+explore the SQSs properties.
+Among them, the MIT bag
+model and Nambu-Jona-Lasinio (NJL) have been widely
+used. In this study, we use the vector MIT (vMIT) bag model
+to describe the quark matter interactions [5, 6]. In the MIT
+bag model, it has been assumed that the quarks are bound
+in a bag of finite dimensions. In contrast to their absolute
+mass, which is very high, it is hypothesized that quarks in-
+side the bag have a very low mass. The system is given a
+bag constant B as a constant energy density to balance the
+bag’s behavior and determine its size. The inward pressure
+at the bag’s surface counterbalances the outward pressure the
+quarks cause, which means the pressure between the true and
+perturbative vacuum. Consequently, as B increases, the quark
+pressure lowers, which impacts the star’s structure. The value
+of B relies on the mass of the strange quark when u and d
+quarks have very low masses. The values of B still need to
+be established and are fully model-dependent. One can con-
+strain its values with the help of observational results. For
+∗ [email protected]
+† [email protected]
+example, in the observational limit of GW170817, the pre-
+dicted values of B1/4 = 134.1 − 141.4 MeV with low-spin
+prior and B1/4 = 126.1 − 141.4 MeV with high spin prior
+for SQSs [7]. In Ref. [8], they have predicted the range of
+B1/4 = 133.68 − 222.5 MeV for SQSs. However, in the
+vMIT bag model [5, 6], the value of B can be obtained by
+including the stability window, as mentioned in Refs. [3, 4].
+Moreover, there are different phenomenological and macro-
+scopic studies suggesting that the quark phases inside the
+compact stars can undergo a phase transition into a color su-
+perconducting state of 2-flavour superconducting (2SC), and
+color-flavor locked (CFL) [9, 10]. They form Cooper pairs
+at high density and low-temperature [11]. The gap parameter
+(∆) determines the pairing strength of Cooper pairs influence
+the formation of pure CFL stars [12–17] and CFL magnetars
+[18, 19]. Recently, it has been suggested that with the proper
+choice of ∆ and bag pressure B, the CFL stars and their EoS
+can successfully reproduce various observational constraints
+such as GW and NICER results [20–22]. In this study, we
+want to explore the dark matter (DM) effects on the strange
+stars with with and without CFL phases and try to constrain
+the macroscopic properties with various observational data.
+The compact objects such as NS, white dwarfs captures
+some amount of DM inside it in their evolving time. The
+amount of DM particles acrreted inside the star due to its im-
+mense gravitational potential. Various theoretical predictions
+provide us with the unknown nature of DM. Still, numerous
+work has been fully dedicated to explaining its properties by
+applying it to different systems such as white dwarf [23], NS
+[24–28], and even our earth [29]. In the present study, we as-
+sume that the SQSs might contain a certain amount of DM
+in their life time. The types of DM particles may be either
+bosonic or fermionic, and also the percentage of DM depends
+on the (i) evolution time and (ii) types of accretions. How-
+ever, the accreted DM particles interact directly or indirectly
+with hadrons by exchanging other bosonic particles, mainly
+depending on the model used. Here, we take different types
+of possible scenarios for DM admixed SQS.
+The direct detection experiments have already been estab-
+lished, such as XENON100 [30], XENON1T[31], CDMS
+[32], LUX [33], PANDAX-II [34] etc. to measure the scat-
+tering cross-section of the DM and nucleons.
+Although,
+they provided some exclusions bound to the scattering cross-
+section. Still, the null results provided by the experiments
+alluded to an inconclusive nature of DM. However, the exclu-
+sion bounds prescribed by such direct detection experiments
+arXiv:2301.00567v1  [astro-ph.HE]  2 Jan 2023
+
+2
+depend on the local DM density around the solar neighbor-
+hood, which does not affect the density of DM in the NS/SQS
+environment. After the accretion of DM inside NS/SQS, it
+collides with nucleons or quarks by losing its kinetic energy,
+and eventually, it is bound inside the star. When the accre-
+tion ends, the DM particles finally reach thermal equilibrium
+with one another due to their internal interactions. This ex-
+plains why NSs with admixed DM have essentially constant
+DM particle densities [25, 27, 28, 35]. Therefore, the accreted
+DM particles are restricted to a narrow radius area inside the
+star. In this study, we choose two types of DM and see their
+effects on the SQS properties with the vMIT bag model and a
+model with superconducting phases.
+Recently, the fastest and heaviest Galactic NS named PSR
+J0952-0607 (black widow) in the disk of the Milky Way has
+been detected to have mass M = 2.35 ± 0.17 M⊙ in continu-
+ation of the pulsars PSR J0740+6620 (M = 2.08 ± 0.07 M⊙
+[36, 37]). The simultaneous measurements of the M and R
+for NS are done by neutron star interior composition explorer
+(NICER) [38, 39] while the limit on the dimensionless tidal
+deformability of Λ1.4 = 190+390
+−120 was provided in GW170817
+event [40]. We calculate the mass, radius, and tidal deforma-
+bility for the DM admixed SQS and put constraints using
+the observational results obtained from different x-ray/pulsars
+data, GW170817 data.
+II.
+FORMALISM
+A.
+Vector MIT bag model
+We use the thermodynamic consistent vector MIT bag
+model introduced in Ref. [5, 6] to describe the quark matter.
+In this model, the quark interaction is mediated by the vec-
+tor channel V µ, analogous to the ω meson in QHD [41]. Its
+Lagrangian reads:
+LvMIT =
+�
+¯ψq
+�
+γµ(i∂µ − gqV Vµ) − mq
+�
+ψq
+−B + 1
+2m2
+V V µVµ
+�
+Θ( ¯ψqψq),
+(1)
+where mq is the mass of the quark q of flavor u, d or s, ψq
+is the Dirac quark field, B is the constant vacuum pressure,
+and Θ( ¯ψqψq) is the Heaviside step function to assure that
+the quarks exist only confined to the bag. Applying Euler-
+Lagrange, we obtain the energy eigenvalue, which at T = 0
+K, is also the chemical potential:
+Eq = µq =
+�
+m2q + k2 + gqV Vµ,
+(2)
+now, using Fermi-Dirac statistics, we can obtain the EoS in
+mean field approximation. The energy density of the quarks
+is:
+ϵq = Nc
+π2
+� kf
+0
+Eqk2d3k,
+(3)
+where Nc = 3 is the number of colors and kf is the Fermi
+momentum. The contribution of the bag and the mesonic mass
+term is obtained with the Hamiltonian: H = −⟨L⟩. The total
+quark energy density now reads:
+ϵ =
+�
+q
+ϵq + B − 1
+2m2
+vV 2
+0 .
+(4)
+To construct an electrically neutral, beta-stable matter, leptons
+are added as a free Fermi gas. The pressure is obtained via the
+relation: p = � µn − ϵ, where the sum runs over all the
+fermions.
+The parameters utilized in this work are the same as pre-
+sented in Ref. [5]. We use mu = md = 4 MeV, and ms = 95
+MeV. We also assume a universal coupling of quarks with the
+vector meson, i.e., guV = gdV = gsV = gV , and use a value
+of GV = 0.3 fm2 as defined below:
+GV =
+� gV
+mV
+�2
+= 0.3 fm2.
+(5)
+Now, the value of GV is somewhat arbitrary. To reproduce
+stable strange matter, the value of GV combined with the bag
+must lie in the range known as the stability window. The sta-
+bility window is related to the so-called Bodmer-Witten con-
+jecture [3, 4], which states that the true ground state of the
+strongly interacting matter is not protons and neutrons but
+consists of strange quark matter, which in turn is composed
+of deconfined up, down, and strange quarks. For the SQM
+hypothesis to be accurate, the energy per baryon of the decon-
+fined phase (for p = 0 and T = 0) is lower than the nonstrange
+infinite baryonic matter [3–5].
+Euds/A < 930 MeV,
+(6)
+at the same time, the nonstrange matter still needs to have
+an energy per baryon higher than nonstrange infinite baryonic
+one; otherwise, protons and neutrons would decay into u and
+d quarks:
+Eud/A > 930 MeV.
+(7)
+Therefore, both, Eqs. 6 and 7 must simultaneously satisfied.
+For GV = 0.3 fm2 used in this work, the stability window
+lies between 139 MeV < B1/4 < 146 MeV [5]. Here, we
+assume the maximum allowed value: B1/4 = 146 MeV, as it
+will produce the lower radius for the canonical star, as well the
+lower value of the dimensionless tidal parameter Λ, while still
+producing very massive strange quark stars, M > 2.40 M⊙.
+B.
+Superconducting CFL quark matter via analytical
+approximation
+Due to the low temperature and high densities reached in
+the strange star interiors, the quark matter may be a color
+superconductor, which is a degenerate Fermi gas of quarks
+with a condensate of Cooper pairs near the Fermi surface
+that induces color Meissner effects [11]. Among the vari-
+ous possible configurations of superconducting matter, we can
+cite two possibilities: The two-flavor color-superconducting
+
+3
+phase, where quarks with two out of three colors and two out
+of three flavors pair in the standard BCS fashion. The flavors
+with the most phase space near their Fermi surfaces, namely,
+u and d, are the ones that pair, leaving the strange quark and
+the remaining color unpaired. Such phase is expected at den-
+sities around 2 < n/n0 < 4 [42]. Another one is the color-
+flavor locked phase, where the up, down, and strange quarks
+can be treated on an equal footing, and the disruptive effects
+of the strange quark mass can be neglected. In this phase,
+quarks of all three colors and all three flavors form conven-
+tional spinless Cooper pairs. The CFL phase is expected at
+n > 4n0 [42]. For additional discussion about 2SC, CFL,
+and other color superconducting phases, see Ref. [11] and the
+references therein.
+The 2SC and the CFL phases were explored within the NJL
+model in Ref. [43], while in Ref. [42], the authors show that
+the color superconducting NJL EoS is very well fitted by an
+analytical approximation, called constant-sound-speed (CSS)
+parameterization, whose EoS reads [42, 44, 45]:
+p = a(ϵ − ϵ∗),
+n = n∗[(1 + a)p/(aϵ∗)]1/(1+a).
+(8)
+We have, therefore, three free parameters, the square of the
+speed of sound (v2
+s = a), the energy density at p = 0 (ϵ∗),
+which plays a role similar to the bag in the MIT base models,
+and the number density at p = 0 (n∗). In Ref. [42], the authors
+freely vary the value of a in the range 0.2 < a < 0.8 and
+found that - depending on the NJL parametrization - the 2SC
+phase is well described by a < 0.33 while the CFL phase is
+described by a > 0.35. On the other hand, Ref. [44] uses the
+extreme case a = 1. Here we consider that the quark matter
+is in the CFL phase and use an intermediate value, a = 0.6
+(see the text and Fig. 4 from Ref. [42], as well Ref. [45]).
+The value of ϵ∗ is chosen as 203 MeV/fm3 to match the value
+coming from the vector MIT bag model. Finally, n∗ has to
+be constrained, as we still need to reproduce strange quark
+stars in accordance with the Bodmer-Witten conjecture. We
+choose n∗ = 0.24 fm−3, which is very close to n0 = 0.23 fm−3
+coming from the vector MIT. Within this value, we have E/A
+= 906 MeV, with implies that the analytical approximation
+of the CFL satisfies Eq. 6 and, therefore, the Bodmer-Witten
+conjecture.
+III.
+RESULTS AND DISCUSSIONS
+A.
+Bosonic DM
+This section briefly reviews the formalism of a bosonic DM
+model initially proposed in Refs. [46, 47]. At very low tem-
+peratures, all particles in a dilute Bose gas condense to the
+same quantum ground state, forming a Bose-Einstein Con-
+densate (BEC). Particles become correlated when their wave-
+lengths overlap; that means the thermal wavelength is greater
+than the mean inter-particle distance. Assuming T = 0 K
+approximation, almost all the DM particles are in the con-
+densate. Only binary collisions at low energy are relevant in
+a dilute and cold gas. These collisions are characterized by
+a single parameter, the s-wave scattering length la, indepen-
+dently of the details of the two-body potential. Therefore, one
+can replace the interaction potential with an effective repul-
+sive interaction [48]:
+V (⃗r − ⃗r′) = 4πla
+mx
+δ(⃗r − ⃗r′),
+(9)
+where mx is the mass of the bosonic DM.
+The ground state properties of the DM are described by the
+mean-field Gross-Pitaevskii (GP) equation, and the equation
+of the state (EoS) has the form [27, 46, 47, 49]:
+px = 2πla
+m3x
+ϵ2
+x..
+(10)
+The scattering length la is assumed equal to 1 fm, as in the
+Ref. [27, 46, 47, 49]. Moreover, the pressure strongly de-
+pends on the bosonic DM’s mass due to the cubic dependence.
+Therefore this parameter must be taken with care. Based on
+the self-interaction cross-section of the DM constraint (see
+Refs. [27, 49], the DM mass in the range 50 MeV < mx <
+160 MeV. However, the original works from Ref. [46, 47] sug-
+gest a mass of around 1 GeV. It is worth emphasizing that a
+mass ten times larger imply in pressure 1000 times lower! In
+Ref. [50], the authors use a slightly different model of bosonic
+DM, where the self-interaction is based on a scalar quartic
+term in the potential.
+They use the same constraint based
+on the self-interaction cross-section of the DM and suggest
+a mass of 400 MeV. To explore the ambiguity relative to the
+mass of the bosonic DM, we use here two values: 100 MeV,
+which agrees with Ref. [27, 49] and 400 MeV, which is in
+agreement with Ref. [50], and it is not so far from 1 GeV as
+suggested in Ref. [46, 47]. With these settings, the pressure
+for mx = 400 MeV is 64 times lower than for mx = 100 MeV.
+The total EoS of the strange star is, therefore, the sum of
+the contribution of the ordinary quark matter and the DM:
+p = pq + px,
+and
+ϵ = ϵq + ϵx.
+(11)
+Another important quantity is the fraction of the DM. To solve
+the TOV equations [51], we need to specify the central values
+both for normal matter and for DM: pq(0), px(0) respectively.
+Here, we follow Ref. [27, 49] and define:
+fx =
+px(0)
+pq(0) + px(0),
+(12)
+and use three different values for fx = 0.05, 0.075 and 0.10.
+As pointed out in Ref. [27, 49], these values agree with the
+current DM constraints obtained from stars like the Sun.
+1. Bosonic DM within vector MIT bag model
+In Fig. 1, we plot the TOV solution for bosonic DM ad-
+mixed strange stars with the mass of 100 MeV and 400 MeV.
+As can be seen, for a bosonic DM mass of 100 MeV, we
+have an increase in the maximum mass with the increase of the
+
+4
+ 1
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 10
+ 10.5
+ 11
+ 11.5
+ 12
+ 12.5
+ 13
+mx = 100 MeV
+M/M0
+R  (km)
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+ 1
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 10
+ 10.5
+ 11
+ 11.5
+ 12
+ 12.5
+ 13
+mx = 400 MeV
+M/M0
+R  (km)
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+FIG. 1. Mass-radius relation for bosonic DM admixed strange stars
+with mx =100 MeV (left) and mx = 400 MeV (right).
+fraction of DM. This result is coherent with those presented in
+Ref. [27, 49] for the original, massless MIT. Moreover, as in
+the case of the original massless MIT, with the massive vector
+MIT, we also see that the presence of DM affects only massive
+stars. Strange stars with M < 1.5 M⊙ reproduced essentially
+the same radii. The maximum masses vary from 2.41 M⊙ for
+pure strange stars to 2.51M⊙ for bosonic DM admixed with
+a fraction of 0.10. This indicates that the PSR J0740+6620
+with a gravitational mass of 2.08 ± 0.07 M⊙ [37] can indeed
+be a stable strange star with or without admixed bosonic DM.
+Even the possible mass of 2.35 ± 0.17 M⊙ of the black widow
+pulsar PSR J0952-0607 [52] can be explained as bosonic DM
+matter admixed strange star. On the other hand, the radius
+of the canonical star is in the narrow range of 11.37 km to
+11.40 km. In the literature, there is no consensus about the
+true value of the radius of the canonical star. For instance, in
+ref. [53], the constraint on the radius of the canonical star is
+10.1 − 11.1 km, which provides a very narrow range. If this
+is true, neither of our results can fulfill such tight constraints.
+In Ref. [54], an upper limit of 11.9 km was provided. In this
+case, our results are in full agreement. However, recent results
+from the NICER x-ray telescope point that the radius of the
+canonical star is between 11.52 km and 13.85 km [39] and
+between 11.96 km and 14.26 km as given in Ref. [38]. In
+these cases, our radii are too small.
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 100 MeV
+Λ
+M/M0
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 400 MeV
+Λ
+M/M0
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+FIG. 2. Dimensionless tidal parameter Λ for bosonic DM admixed
+strange stars with mx = 100 MeV (top) and mx = 400 MeV (bot-
+tom).
+Now, we have opposite results for a mass mx = 400 MeV!
+First, the maximum mass decrease with the increase of DM
+fraction, dropping from 2.41 M⊙ to 2.29 M⊙ for a fraction fx
+of 0.10. However, all values agree with the mass of the PSR
+J0740+6620 [37] and the PSR J0952-0607 [52]. Secondly,
+we see that even low-mass strange stars are already affected
+by the DM and are significantly more compact. The radius
+of the 1.4 M⊙ strange star can reach a value as low as 11.08
+km. Therefore, this result is in agreement with both Refs. [53,
+54]. The polytropic EoS of Eq. 10 can easily explain these
+results. A four times higher DM matter mass produces sixty-
+four times smaller pressure! The reduction of the pressure
+causes the reduction of the maximum mass and increases the
+star compression.
+Another essential quantity and constraint is the so-called
+dimensionless tidal deformability parameter Λ. If we put an
+extended body in an inhomogeneous external field, it will ex-
+perience different forces throughout its surface. The result is a
+tidal interaction. The tidal deformability of a compact object
+is a single parameter λ that quantifies how easily the object
+is deformed when subjected to an external tidal field. Larger
+tidal deformability indicates that the object is easily deformed.
+Conversely, a compact object with a small tidal deformability
+parameter is more compact and more difficult to deform. The
+
+5
+TABLE I. Macroscopic properties of bosonic DM admixed strange
+stars
+mx (MeV)
+fx
+M/M⊙ R (km) R1.4 (km) Λ1.4
+100
+0.000
+2.41
+11.86
+11.37
+644
+100
+0.050
+2.46
+12.01
+11.37
+638
+100
+0.075
+2.48
+12.06
+11.38
+645
+100
+0.100
+2.51
+12.08
+11.40
+652
+400
+0.000
+2.41
+11.86
+11.37
+644
+400
+0.050
+2.31
+11.42
+11.16
+526
+400
+0.075
+2.30
+11.38
+11.12
+497
+400
+0.100
+2.29
+11.31
+11.08
+480
+tidal deformability is defined as:
+Λ ≡
+λ
+M 5 ≡ 2k2
+3C5 ,
+(13)
+where M is the compact object mass and C = GM/R is
+its compactness. The parameter k2 is called the second (or-
+der) Love number. Additional discussion about the theory of
+tidal deformability and the tidal Love numbers are beyond the
+scope of this work and can be found in Refs. [22, 40, 55–
+59] and references therein. Nevertheless, as pointed out in
+Refs. [22, 58], the value of yR must be corrected since strange
+stars are self-bound and present a discontinuity at the surface.
+Therefore we must have
+yR → yR − 4πR3∆ϵS
+M
+,
+(14)
+where R and M are the star radius and mass, respectively,
+and ∆ϵS is the difference between the energy density at the
+surface (p = 0) and the star’s exterior (which implies ϵ = 0).
+The results for the dimensionless tidal parameter are displayed
+in Fig. 2.
+As we can be seen, some features present in the mass-radius
+relation are also present here. For instance, for a mass mx =
+100 MeV, the low masses of strange stars have similar tidal
+parameters, despite their DM fraction. The tidal parameter for
+the canonical mass lies between 638 and 644. These values
+are in agreement with the constraint Λ < 800 [55], but fail to
+fulfill the constraint 70 < Λ < 580 [40].
+In the case of mx = 400 MeV, the strange stars’ huge com-
+pression due to an increase in the DM fraction reduces the
+tidal parameter Λ. The tidal parameter now lies around 500.
+This indicates that for mx = 400 MeV, we are able to explain
+very massive neutron stars as the PSR J0952-0607 [52], and
+simultaneously fulfills the constraints of Λ < 800 [55] and 70
+< Λ < 580 [40]. We summarize the results of this section in
+Tab I.
+2. Bosonic DM within CFL quark matter
+In order to better understand the effects of the DM in
+strange stars, we now assume that the quark matters are in
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 10
+ 11
+ 12
+ 13
+ 14
+mx = 100 MeV
+M/M0
+R  (km)
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 10
+ 11
+ 12
+ 13
+ 14
+mx = 400 MeV
+M/M0
+R  (km)
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+FIG. 3. Mass-radius relation for bosonic DM admixed CFL strange
+stars with mx =100 MeV (left) and mx = 400 MeV (right).
+the CFL superconducting phase via the analytical approxima-
+tion EoS in Eq. 8. The mass-radius relations are presented in
+Fig. 3.
+As can be seen, for a bosonic DM mass of 100 MeV, we
+have an increase in the maximum mass with the increase of the
+fraction of DM. The qualitative results for CFL superconduct-
+ing quark stars are analogous to both the original, massless
+MIT as showed in Ref. [27, 49], as well for the massive vector
+MIT bag model as presented in the last section. This indicates
+a possible model-independent behavior about the effect of the
+bosonic DM. Moreover, as in the case of the original mass-
+less MIT and the massive vector MIT, in the CFL phase, we
+also see that the presence of DM affects only massive stars.
+CFL strange stars with M < 1.8 M⊙ reproduced essentially
+the same radii. The maximum masses vary from 2.81 M⊙ for
+pure strange stars to 2.88M⊙ for bosonic DM admixed with a
+fraction of 0.10. This indicates that the PSR J0740+6620 with
+M = 2.08 ± 0.07 M⊙ [37] can be a stable CFL strange star
+with or without admixed bosonic DM. Even the possible mass
+of 2.35 ± 0.17 M⊙ of the pulsar PSR J0952-0607 [52] can
+be explained as bosonic DM matter admixed strange star. On
+the other hand, the radius of the canonical star presents almost
+no variation and is fixed at around 11.57 km. Such a value is
+too low to reproduce the constraint range of 10.1 − 11.1 km,
+shown in Ref. [53] while agreeing with Ref. [54], whose upper
+
+6
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 100 MeV
+Λ
+M/M0
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 400 MeV
+Λ
+M/M0
+fx = 0.000
+fx = 0.050
+fx = 0.075
+fx = 0.100
+FIG. 4. Dimensionless tidal parameter Λ for bosonic DM admixed
+CFL superconducting strange stars with mx = 100 MeV (top) and
+mx = 400 MeV (bottom).
+limit is 11.9 km. About the NICER x-ray telescope, the con-
+straint between 11.52 km and 13.85 km pointed in Ref. [39]
+is fulfilled, but the bound in the range between 11.96 km and
+14.26 km (Ref. [38]) is not.
+For a mass mx = 400 MeV, the results for CLF super-
+conducting strange stars are analogous to the massive MIT
+bag model discussed in the last section. The maximum mass
+decrease with the increase of DM fraction, dropping from
+2.81 M⊙ to 2.61 M⊙ for a fraction fx of 0.10. However, all
+values agree with the mass of the PSR J0740+6620 [37] and
+the black widow pulsar PSR J0952-0607 [52]. Secondly, we
+see that even low-mass strange stars are already affected by
+the DM and are significantly more compact. The radius of
+the 1.4 M⊙ for fx = 0.10 is about 11.29 km. Such a low ra-
+dius fails to fulfill both NICER constraints [38, 39], but is in
+agreement with Capano et al. [54]. The reduction of the CFL
+strange star and its compression can again be explained by the
+polytropic EoS of Eq. 10. A four times higher DM matter
+mass produces sixty-four times smaller pressure! The reduc-
+tion of the pressure causes the reduction of the maximum mass
+and increases the star compression.
+We also calculate the dimensionless tidal parameter Λ for
+the CFL superconducting strange stars. The results are pre-
+sented in Fig. 4. As we can be seen, the results are analogous
+TABLE II. Macroscopic properties of bosonic DM admixed CFL su-
+perconducting strange stars
+mx (MeV)
+fx
+M/M⊙ R (km) R1.4 (km) Λ1.4
+100
+0.000
+2.81
+12.89
+11.57
+721
+100
+0.050
+2.83
+12.84
+11.57
+709
+100
+0.075
+2.86
+12.96
+11.58
+717
+100
+0.100
+2.88
+13.00
+11.58
+717
+400
+0.000
+2.81
+12.89
+11.57
+721
+400
+0.050
+2.63
+12.30
+11.37
+570
+400
+0.075
+2.62
+12.22
+11.32
+545
+400
+0.100
+2.61
+12.13
+11.29
+531
+to the vector MIT bag model. As in the case of the mass-radius
+relation, for low mass stars there is very low variation in the
+Λ. For instance, for a mass mx = 100 MeV, the low masses
+strange stars have similar tidal parameters, despite their DM
+fraction. The tidal parameter for the canonical mass lies be-
+tween 709 and 721. These values are in agreement with the
+constraint Λ < 800 [55], but fail to fulfill the constraint 70
+< Λ < 580 [40].
+In the case of mx = 400 MeV, the results for CFL super-
+conducting strange stars are again analogous to vector MIT
+strange stars. The stars’ huge compression as the DM fraction
+increases reduce the tidal parameter Λ. The tidal parameter
+now lies around 550. This indicates that for mx = 400 MeV,
+we are able to explain very massive neutron stars as the PSR
+J0952-0607 [52], and simultaneously fulfills the constraints
+of Λ < 800 [55] and 70 < Λ < 580 [40]. We summarize the
+results of this section in Tab II.
+B.
+Fermionic DM
+The Lagrangian of the fermionic DM reads [22, 25, 35]:
+LDM = ¯χ(iγµ∂µ − (mx − gHh))χ
++1
+2(∂µh∂µh − m2
+Hh2).
+(15)
+Here, we assume a dark fermion represented by the Dirac
+field χ that self-interacts through the exchange of the Higgs
+boson, whose mass is mH = 125 GeV. The coupling con-
+stant is assumed to be gH = 0.1, which agrees with the con-
+straints in Refs. [25, 27]. Within this prescription, the DM
+self-interaction is very feeble and behaves as a free Fermi gas.
+More explicitly, the strength of the interaction is:
+GH =
+� gH
+mH
+�2
+= 2.492 × 10−8
+fm2.
+(16)
+The EoS is easily obtained in mean field approximation,
+completely analogous to the QHD model [41]. The fermionic
+DM is assumed to be the lightest neutralino, with mx = 200
+GeV, as done in Ref. [25, 35]. However, as pointed out in
+Ref. [60], the lower limit for weakly interacting massive par-
+ticles (WIMP) is 60 GeV. Therefore we also use mx = 60 GeV
+
+7
+ 1
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+mx = 200 GeV
+M/M0
+R  (km)
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+ 1
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+mx = 60 GeV
+M/M0
+R  (km)
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+FIG. 5. Mass-radius relation for fermionic DM admixed strange stars
+with mx = 200 GeV (top) and mx = 60 GeV (bottom).
+to better study the influence of the DM mass. As in the case
+of the bosonic DM, we must fix the DM fraction. As we are
+dealing here with fermionic DM, we follow ref. [25, 28, 35]
+and use the Fermi momentum to fix the DM fraction, using
+three different values: kDM
+F
+= 0.02 GeV, 0.04 GeV, and 0.06
+GeV.
+1. Fermionic DM within vector MIT bag model
+We display in Fig. 5 the TOV solution for a fermionic DM
+with a mass of 200 GeV and 60 GeV within the vector MIT
+bag model. As can be seen, the results for fermionic DM are
+significantly different when compared with bosonic DM. The
+maximum masses are always reduced, and the star compres-
+sion always increases, even for very low masses. Also, differ-
+ent DM fractions always produce different mass-radius rela-
+tions, affecting all the strange star families, unlike the bosonic
+case, where we have very similar stars for different DM frac-
+tions, which is easily understood by the different criteria of
+the DM fraction. In the case of bosonic DM, the DM frac-
+tion is dependent on the quark EoS via Eq. 12. In the case of
+fermionic DM, the Fermi momentum is fixed and independent
+of the quark EoS.
+Qualitatively, the results for mx = 200 GeV and 60 GeV
+TABLE III. Macroscopic properties of fermionic DM admixed
+strange stars
+mx (GeV) kDM
+F
+(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4
+200
+0.000
+2.41
+11.86
+11.37
+644
+200
+0.02
+2.37
+11.75
+11.22
+586
+200
+0.04
+2.16
+10.70
+10.39
+346
+200
+0.06
+1.80
+8.72
+8.95
+108
+60
+0.000
+2.41
+11.86
+11.37
+644
+60
+0.02
+2.40
+11.84
+11.30
+625
+60
+0.04
+2.33
+11.46
+11.05
+524
+60
+0.06
+2.16
+11.31
+10.42
+351
+are the same. Increasing the DM fraction compress the star
+and reduces the maximum mass. Quantitatively, we see that
+a higher DM mass has a strong influence once it has a higher
+increase in the energy density, and at the same time, that pro-
+duces a lower contribution to the pressure. The maximum
+mass drops from 2.41 M⊙ for kDM
+F
+= 0.00 to only 1.80 M⊙
+for kDM
+F
+= 0.06 GeV in the case of mx = 200 GeV and to 2.16
+M⊙ for mx = 60 GeV. In the same sense, the radius of the
+canonical star drops from 11.37 km for kDM
+F
+= 0.00 to only
+8.95 km for kDM
+F
+= 0.06 GeV in the case of mx = 200 GeV,
+and to 10.42 km for mx = 60 GeV. As can be seen, the results
+for kDM
+F
+= 0.06 GeV with mx = 200 GeV can be ruled out
+once it has a very low maximum mass in disagreement with
+the NICER result of the PSR J0740+6620 with a gravitational
+mass of 2.08 ± 0.07 M⊙ [37], and also a very low radius for
+the canonical star, in disagreement even with the low limit of
+10.1 km presented in Ref. [53].
+We plot in Fig. 6 the dimensionless parameter Λ for
+fermionic DM admixed strange stars with mx = 200 GeV and
+mx = 60 GeV within the vector MIT bag model. As we can
+see, the strong compression due to the fermionic DM contri-
+bution reduces the tidal parameter significantly. In the case
+with mx = 200 GeV and kDM
+F
+= 0.06 GeV, the tidal parame-
+ter drops to only 108, which is six times lower than for kDM
+F
+=
+0.00, although, as we pointed out before, such parametrization
+must be ruled out.
+As can be seen, most of the parametrizations are able to
+fulfill the main constraints for pulsar observations, i.e., M >
+2.01M⊙ and 70
+<
+Λ
+< 580. Indeed, the presence of
+DM improves the theoretical prediction and the observational
+constraints, although it can be some debate about the radius of
+the canonical star. They do not fulfill NICER results [38, 39]
+but agree with Ref. [54].
+It is also worth to point the existence of almost degenerate
+results. As can be seen, for mx = 200 GeV with kDM
+f
+= 0.04
+GeV, the macroscopic are essentially the same for the mx = 60
+GeV and kDM
+f
+= 0.06 GeV. The main results are summarized
+in Tab. III.
+
+8
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 200 GeV
+Λ
+M/M0
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 60 GeV
+Λ
+M/M0
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+FIG. 6. Dimensionless tidal parameter Λ for fermionic DM admixed
+strange stars with mx = 200 GeV (top) and mx = 60 GeV (bottom).
+2. Fermionic DM within CFL quark matter
+We now study the effect of Fermionic DM in CFL super-
+conducting matter described by the analytical approximation
+of Eq. 8.
+We display in Fig. 7 the TOV solution for a fermionic DM
+with a mass of 200 GeV and 60 GeV. As in the case of the vec-
+tor MIT bag model, for CFL superconducting quark matter,
+the results for fermionic DM are significantly different when
+compared with bosonic DM. And again, the qualitative effect
+of fermionic DM is the same for CFL as it is for the vector
+MIT bag model. The maximum masses are always reduced,
+and the star compression always increases, even for very low
+masses. Again, different DM fractions always produce differ-
+ent mass-radius relations, affecting all the strange star fami-
+lies.
+From the quantitative point of view, the maximum mass
+drops from 2.81 M⊙ for kDM
+F
+= 0.00 to 2.04 M⊙ for kDM
+F
+= 0.06 GeV in the case of mx = 200 GeV and to 2.49 M⊙
+for mx = 60 GeV. In the same sense, the radius of the canon-
+ical star drops from 11.57 km for kDM
+F
+= 0.00 to 9.21 km for
+kDM
+F
+= 0.06 GeV in the case of mx = 200 GeV, and 10.66
+km for mx = 60 GeV. Now, unlike the case of the vector MIT,
+none of the CFL superconducting strange stars can be ruled
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+ 14
+mx = 200 GeV
+M/M0
+R  (km)
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+ 14
+mx = 60 GeV
+M/M0
+R  (km)
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+FIG. 7. Mass-radius relation for fermionic DM admixed CFL super-
+conducting strange stars with mx = 200 GeV (top) and mx = 60
+GeV (bottom).
+out in the light of the PSR J0740+6620, M
+= 2.08 ± 0.07
+M⊙ [37], although for kDM
+F
+= 0.06 GeV and mx = 200 GeV
+the radius of the canonical star is below the lower limit of 10.1
+km presented in Ref. [53].
+We plot in Fig. 8 the dimensionless parameter Λ for
+fermionic DM admixed superconducting strange stars with
+mx = 200 GeV and mx = 60 GeV. The results are completely
+analogous to the case of the vector MIT bag model; however,
+the value of Λ here is always higher. The compression due to
+the fermionic DM contribution reduces the tidal parameter. In
+the case with mx = 200 GeV and kDM
+F
+= 0.06 GeV, the tidal
+parameter drops from 721 to 151.
+It is also worth noting that some parametrizations can ful-
+fill the main constraints for pulsar observations, 70 < Λ <
+580, and yet produce a very high maximum mass, sometimes
+reaching 2.50 M⊙. The presence of DM again improves the
+theoretical prediction and the observational constraints, al-
+though it can be some debate about the radius of the canonical
+star. They do not fulfill NICER results [38, 39], but agree with
+Ref. [54]. Moreover, most parametrizations can explain even
+the black widow pulsar PSR J0952-0607 [52].
+Finally, even when we use a different model for the quark
+matter, the existence of almost degenerate results is still
+present: for mx = 200 GeV with kDM
+f
+= 0.04 GeV and mx =
+
+9
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 200 GeV
+Λ
+M/M0
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+ 0
+ 200
+ 400
+ 600
+ 800
+ 1000
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+mx = 60 GeV
+Λ
+M/M0
+kf = 0.00 GeV
+kf = 0.02 GeV
+kf = 0.04 GeV
+kf = 0.06 GeV
+FIG. 8. Dimensionless tidal parameter Λ for fermionic DM admixed
+CFL strange stars with mx = 200 GeV (top) and mx = 60 GeV
+(bottom).
+TABLE IV. Macroscopic properties of fermionic DM admixed color
+superconducting quark stars
+mx (GeV) kDM
+F
+(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4
+200
+0.000
+2.81
+12.89
+11.57
+721
+200
+0.02
+2.75
+12.65
+11.43
+653
+200
+0.04
+2.50
+11.50
+10.67
+421
+200
+0.06
+2.04
+9.38
+9.21
+151
+60
+0.000
+2.81
+12.89
+11.57
+721
+60
+0.02
+2.78
+12.75
+11.53
+694
+60
+0.04
+2.69
+12.45
+11.28
+610
+60
+0.06
+2.49
+11.53
+10.66
+422
+60 GeV with kDM
+f
+= 0.06 GeV. The main results are summa-
+rized in Tab. IV.
+3. Fermionic DM with a vector channel
+Now we study if the presence of a dark, repulsive vector
+channel affects the macroscopic properties of the fermionic
+DM admixed strange stars. The new Lagrangian is the La-
+grangian in Eq. 15 plus the repulsive channel and the respec-
+tive meson mass, and reads [28]:
+LVDM = gξ ¯χ(γµξµ)χ + 1
+2m2
+ξξµξµ − 1
+4V µνVµν.
+(17)
+The Lagrangian of Eq. 17 is analogous to the ω contribution
+to the QHD Lagrangian [5, 41]. Indeed, the junction of Eq. 15
+and Eq. 17 makes this model of DM fully analogous to the
+original σ −ω model of the QHD [41]. The coupling constant
+gξ = 0.1 is fixed, and it is equal to the gH, while the mass of
+the vector of the dark meson is assumed to be 34 MeV, follow-
+ing Ref. [28]. As the mass of the dark vector meson is 3000
+times smaller than the mass of the Higgs boson, the repulsive
+channel is much stronger than the attractive one. Indeed, we
+have
+Gξ =
+� gξ
+mξ
+�2
+= 0.337
+fm2,
+(18)
+which is stronger than the quark repulsion and millions of
+times higher than the DM scalar coupling (Eq. 16). Never-
+theless, despite the strong self-repulsion of the fermionic DM,
+the numerical results are barely affected by the repulsive chan-
+nel. For the vector MIT bag model, the only noticeable dif-
+ference appears for mx = 200 GeV and kDM
+F
+= 0.04 GeV. In
+this case, the maximum mass increase from 2.16 M⊙ to 2.17
+M⊙. The radius of the canonical star also grows from 11.39
+km to 11.46 km. The tidal parameter Λ1.4 also increases from
+346 to 358. It is worth noticing that all these variations are far
+beyond the precision with which experimental measurements
+are made. All the other parametrizations present even lower
+(or none) differences. Herefore, we do not provide any figures
+in this section since they would be visually indistinguishable
+from those in the last paragraph. We only display the main
+results in Tab. V. In the case of CFL superconducting quark
+matter, the differences are even smaller!
+The nature of the vector coupling can explain why the
+differences are so small. The vector mesons couple to the
+number density, and we are dealing with a very low-density
+regime. Indeed, even kDM
+F
+= 0.06 GeV implies a number den-
+sity is around 9.6 × 10−4 fm−3. Of course, we could increase
+the repulsion of the dark vector boson, but we believe this
+would be very unrealistic since DM was proposed to explain
+higher attraction in galaxy curves [61].
+IV.
+CONCLUSIONS
+In this work, we calculate the properties for the DM ad-
+mixed for strange quark stars. We use two different mod-
+els for the quark model: the vector MIT bag model, as pre-
+sented in Refs. [5, 6] and the CFL color superconducting
+quark matter via an analytical approximation, as discussed
+in Refs. [42, 44, 45]; and two different kinds of dark mat-
+ter: a bosonic as discussed in Refs. [27, 46, 47, 49] and for
+fermionic [22, 25, 35]. For each kind of DM, we use two
+different mass values, and the strange stars always agree with
+the Bodmer-Witten conjecture [3, 4]. Our main conclusions
+can be summarized as follows:
+
+10
+TABLE V. Macroscopic properties of dark vector boson fermionic
+DM admixed strange stars within the vector MIT bag model. The
+only significant differences are for kDM
+F
+= 0.04 GeV
+mx (GeV) kDM
+F
+(GeV) M/M⊙ R (km) R1.4 (km) Λ1.4
+200
+0.000
+2.41
+11.86
+11.37
+644
+200
+0.02
+2.37
+11.75
+11.22
+586
+200
+0.04
+2.17
+10.70
+10.46
+358
+200
+0.06
+1.80
+8.72
+9.01
+112
+60
+0.000
+2.41
+11.86
+11.37
+644
+60
+0.02
+2.40
+11.84
+11.30
+625
+60
+0.04
+2.33
+11.46
+11.10
+532
+60
+0.06
+2.16
+11.31
+10.43
+353
+• The qualitative results for DM admixed strange stars are
+independent of the quark model utilized. This is true for
+both bosonic and fermionic, as well it is independent of
+the DM mass.
+• For a bosonic DM with a mass of mx = 100 MeV, we
+have an increase of the maximum mass, while the prop-
+erties of low-mass strange stars are not significantly af-
+fected. This is the only case in that we have an in-
+crease in the star’s mass. Such a situation happens for
+the vector MIT, the CFL superconducting quark mat-
+ter, and also for the massless MIT, as pointed out in
+Refs. [27, 49].
+• For a bosonic DM with a mass of mx = 400 MeV, we
+have a decrease of the maximum mass, whilst the radii
+of the low-mass strange stars, in this case, are also af-
+fected.
+• For a fermionic DM, the maximum mass always de-
+creases. The higher the DM fraction, the lower the max-
+imum mass, and the smaller the radii. Also, the higher
+the DM mass, the higher the stellar compression and the
+lower the maximum mass.
+• Although we introduce a repulsive dark vector field
+with a mass 3000 times smaller than the attractive scalar
+field, we do not find significant variation in the stellar
+macroscopic properties.
+• There are almost degenerate results both for mx = 200
+GeV with kDM
+f
+= 0.04 GeV and mx = 60 GeV with
+kDM
+f
+= 0.06 GeV, the maximum mass, as well the prop-
+erties of the canonical star are essentially the same.
+• About the observational constraints, we can see that the
+mass of the PSR J0740+6620 pulsar, M = 2.08 ± 0.07
+M⊙ [37] is easily obtained. Even the mass range of
+2.35 ± 0.17 M⊙ of the black widow pulsar PSR J0952-
+0607 [52] can be reached for some parametrization.
+• The radius of the canonical star is still a matter of de-
+bate. Most of our results point to a radius between 11.0
+km to 11.5 km. In general, our results are in agreement
+with Ref. [54] but are too low to reproduce the NICER
+results [38, 39] whilst at the same time are too high to
+agree with Ref. [53].
+• Except for bosonic DM with a mass of mx = 100 MeV,
+in all other cases, the presence of the DM reduces the
+dimensionless tidal parameter Λ. In most of these cases,
+the constraint 70 < Λ < 580 [40] is easily fulfilled.
+[1] E. Annala, T. Gorda, A. Kurkela, J. N¨attil¨a, and A. Vuorinen,
+Nature Physics 16, 907 (2020).
+[2] R. X. Xu, Symposium - Intern. Astron. U. 214, 191 (2003).
+[3] A. R. Bodmer, Phys. Rev. D 4, 1601 (1971).
+[4] E. Witten, Phys. Rev. D 30, 272 (1984).
+[5] L. L. Lopes, C. Biesdorf, and D. P. Menezes, Physica Scripta
+96, 065303 (2021).
+[6] L. L. Lopes, C. Biesdorf, and D. P. Menezes, Monthly Notices
+of the Royal Astronomical Society 512, 5110 (2022).
+[7] E.-P. Zhou, X. Zhou, and A. Li, Phys. Rev. D 97, 083015
+(2018).
+[8] A. Aziz, S. Ray, F. Rahaman, M. Khlopov, and B. K. Guha,
+International Journal of Modern Physics D 28, 1941006 (2019).
+[9] M. Alford, Journal of Physics G: Nuclear and Particle Physics
+30, S441 (2003).
+[10] I. A. Shovkovy, Foundations of Physics 35, 1309 (2005).
+[11] M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨afer, Rev.
+Mod. Phys. 80, 1455 (2008).
+[12] M. Alford and S. Reddy, Phys. Rev. D 67, 074024 (2003).
+[13] A. Drago, A. Lavagno, and G. Pagliara, Phys. Rev. D 69,
+057505 (2004).
+[14] C. V. Flores and G. Lugones, Phys. Rev. C 95, 025808 (2017).
+[15] I. Lopes and J. Silk, Phys. Rev. D 99, 023008 (2019).
+[16] R. S. Bogadi, M. Govender, and S. Moyo, Phys. Rev. D 102,
+043026 (2020).
+[17] A. Li, Z.-Q. Miao, J.-L. Jiang, S.-P. Tang, and R.-X. Xu,
+Monthly Notices of the Royal Astronomical Society 506, 5916
+(2021).
+[18] X.-J. Wen, Phys. Rev. D 88, 034031 (2013).
+[19] H. Liang and X.-J. Wen, International Journal of Modern
+Physics A 34, 1950170 (2019).
+[20] Z. Roupas, G. Panotopoulos, and I. Lopes, Phys. Rev. D 103,
+083015 (2021).
+[21] Z. Miao, J.-L. Jiang, A. Li, and L.-W. Chen, The Astrophysical
+Journal Letters 917, L22 (2021).
+[22] O. Lourenc¸o, C. H. Lenzi, M. Dutra, E. J. Ferrer, V. de la In-
+cera, L. Paulucci, and J. E. Horvath, Phys. Rev. D 103, 103010
+(2021).
+[23] N. F. Bell, G. Busoni, M. E. Ramirez-Quezada, S. Robles, and
+M. Virgato, Journal of Cosmology and Astroparticle Physics
+2021 (10), 083.
+[24] H. C. Das, A. Kumar, B. Kumar, S. Kumar Biswal, T. Nakat-
+sukasa, A. Li, and S. K. Patra, Mon. Not. Roy. Astron. Soc. 495,
+4893 (2020), arXiv:2002.00594 [nucl-th].
+[25] H. Das, A. Kumar, B. Kumar, S. Biswal, and S. Patra, Journal
+of Cosmology and Astroparticle Physics 2021 (01), 007.
+
+11
+[26] H. C. Das, A. Kumar, and S. K. Patra, Phys. Rev. D 104, 063028
+(2021), arXiv:2109.01853 [astro-ph.HE].
+[27] G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 023002 (2017).
+[28] A. Guha and D. Sen, Journal of Cosmology and Astroparticle
+Physics 2021 (09), 027.
+[29] A. Green and P. Tanedo, Computer Physics Communications
+242, 120 (2019).
+[30] E. Aprile, M. Alfonsi, K. Arisaka, et al. (XENON100 Collabo-
+ration), Phys. Rev. Lett. 109, 181301 (2012).
+[31] E. Aprile, J. Aalbers, F. Agostini, et al. (XENON Collaboration
+7), Phys. Rev. Lett. 121, 111302 (2018).
+[32] R. Agnese, T. Aralis, T. Aramaki, et al., Phys. Rev. Lett. 121,
+051301 (2018).
+[33] D. Akerib, X. Bai, E. Bernard, et al., Astroparticle Physics 45,
+34 (2013).
+[34] Q. Wang, A. Abdukerim, W. Chen, and others (PandaX-II Col-
+laboration), Chinese Physics C 44, 125001 (2020).
+[35] O. Lourenc¸o, T. Frederico, and M. Dutra, Phys. Rev. D 105,
+023008 (2022).
+[36] H. T. Cromartie, E. Fonseca, S. M. Ransom, et al., Nature As-
+tronomy 4, 72 (2020).
+[37] E. Fonseca, H. T. Cromartie, T. T. Pennucci, et al., The Astro-
+physical Journal Letters 915, L12 (2021).
+[38] M. C. Miller, F. K. Lamb, A. J. Dittmann, et al., The Astrophys-
+ical Journal 887, L24 (2019).
+[39] T. E. Riley, A. L. Watts, S. Bogdanov, et al., The Astrophysical
+Journal 887, L21 (2019).
+[40] B. P. Abbott, R. Abbott, T. D. Abbott, et al. (The LIGO Sci-
+entific Collaboration and the Virgo Collaboration), Phys. Rev.
+Lett. 121, 161101 (2018).
+[41] B. D. Serot, Reports on Progress in Physics 55, 1855 (1992).
+[42] J. L. Zdunik and P. Haensel, Astronomy and Astrophysics 551,
+A61 (2013).
+[43] B. K. Agrawal, Phys. Rev. D 81, 023009 (2010).
+[44] S. Han and A. W. Steiner, Phys. Rev. D 99, 083014 (2019).
+[45] M. G. Alford, S. Han, and M. Prakash, Phys. Rev. D 88, 083013
+(2013).
+[46] X. Li, T. Harko, and K. Cheng, Journal of Cosmology and As-
+troparticle Physics 2012 (06), 001.
+[47] X. Li, F. Wang, and K. Cheng, Journal of Cosmology and As-
+troparticle Physics 2012 (10), 031.
+[48] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev.
+Mod. Phys. 71, 463 (1999).
+[49] I. Lopes and G. Panotopoulos, Phys. Rev. D 97, 024030 (2018).
+[50] D. Rafiei Karkevandi, S. Shakeri, V. Sagun, and O. Ivanytskyi,
+Phys. Rev. D 105, 023001 (2022).
+[51] J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374
+(1939).
+[52] R. W. Romani, D. Kandel, A. V. Filippenko, T. G. Brink, and
+W. Zheng, The Astrophysical Journal Letters 934, L17 (2022).
+[53] F. ¨Ozel, D. Psaltis, T. G¨uver, G. Baym, C. Heinke, and S. Guil-
+lot, Astrophys. J. 820, 28 (2016).
+[54] C. D. Capano, I. Tews, S. M. Brown, B. Margalit, S. De, S. Ku-
+mar, D. A. Brown, B. Krishnan, and S. Reddy, Nature Astron-
+omy 4, 625 (2020).
+[55] B. P. Abbott, R. Abbott, T. D. Abbott, et al. (LIGO Scientific
+Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119,
+161101 (2017).
+[56] Flores, Cesar V., Lopes, Luiz L., Castro, Luis B., and Menezes,
+D´ebora P., Eur. Phys. J. C 80, 1142 (2020).
+[57] K.
+Chatziioannou,
+Gen.
+Rel.
+Grav.
+52,
+109
+(2020),
+arXiv:2006.03168 [gr-qc].
+[58] S. Postnikov, M. Prakash, and J. M. Lattimer, Phys. Rev. D 82,
+024016 (2010).
+[59] L. L. Lopes and D. P. Menezes, The Astrophysical Journal 936,
+41 (2022).
+[60] I. P. Lopes and J. Silk, Phys. Rev. Lett. 88, 151303 (2002).
+[61] G. Bertone and D. Hooper, Rev. Mod. Phys. 90, 045002 (2018).
+

I9FAT4oBgHgl3EQfux7Z/content/tmp_files/2301.08672v1.pdf.txt

@@ -0,0 +1,1521 @@
+arXiv:2301.08672v1  [math.CT]  20 Jan 2023
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+OLIVIA MONJON, J´ERˆOME SCHERER, AND FLORENCE STERCK
+Abstract. The correspondence between the concept of conditional flatness and admissibil-
+ity in the sense of Galois appears in the context of localization functors in any semi-abelian
+category admitting a fiberwise localization. It is then natural to wonder what happens in
+the category of crossed modules where fiberwise localization is not always available. In this
+article, we establish an equivalence between conditional flatness and admissibility in the
+sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors.
+We use this equivalence to prove that nullification functors are admissible for the class of
+regular epimorphisms, even if the kernels of their localization morphisms are not acyclic.
+Introduction
+It is a natural question to ask whether the pullback of a nice extension inherits these
+nice properties. When working with localization functors or reflections one particularly nice
+feature for an extension is flatness. We say that an extension is L-flat, for a localization
+functor L, if applying L to the extension yields another extension, see Definition 2.1. The
+question is thus to understand when the pullback of an L-flat extension is again L-flat.
+Such questions have been studied first in a homotopical context by Berrick and Farjoun,
+[1]. For homotopical localization functors in the category of topological spaces (in the sense
+of Bousfield, [5], see also Farjoun’s book [13]), preservation of L-flatness (for fiber sequences)
+under pullbacks was shown to be equivalent for L to be a so-called nullification functor. The
+situation is surprisingly more delicate in the category of groups. Farjoun and the second
+author proved for example that all nilpotent quotient functors have this nice property, which
+they called conditional flatness, see [14].
+The standard strategy to establish conditional flatness for a localization functor consists
+in a few reduction steps culminating in a simpler form, which Gran identified as admissibility
+in the sense of Galois for the class of regular epimorphisms [17, Proposition 3.3]. This shifted
+the study of conditional flatness in homotopy theory to that of admissibility in semi-abelian
+categories, see [15]. Admissibility had been introduced by Janelidze and Kelly in [17] and
+has since then played a central role in the categorical study of extensions, let us mention for
+example Everaert, Gran, and Van der Linden’s work in [12].
+In this article we study admissibility for localization functors in the category of crossed
+modules (of groups), a category of interest to both topologists due to Whitehead’s work on
+connected 2-types, [25], and algebraists since Brown and Spencer [7] proved the equivalence
+between crossed modules and internal groupoids in the category of groups (a result that
+they credit to Verdier). This equivalence relates two interesting notions and allows one to
+deal with the concept of internal groupoid in an alternative way, that is useful for compu-
+tations. Moreover, crossed modules form a semi-abelian category in the sense of Janelidze,
+2020 Mathematics Subject Classification. 18G45, 55P60, 18E50, 55R70, 18E13.
+Key words and phrases. Crossed modules, Localization functors, Admissibility, Regular epimorphisms,
+Conditional flatness, Nullifications.
+1
+
+2
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+M´arki and Tholen, [18]. We adopt the algebraic point of view here and continue our work
+started in [22]. Indeed, among the reduction steps we have mentioned above, the first one
+calls on fiberwise localization techniques. For group theoretical localization and homotopy
+localization functors, it allows one to reduce the study to extensions with local kernel (fiber).
+Fiberwise localization techniques are available in the category of groups thanks to work of
+Casacuberta and Descheemaeker, [10], but we proved in [22] that they are not at hand in
+general for crossed modules. Our aim in this article is thus to modify the strategy to be able
+to study admissibility in this setting.
+We focus on localization functors such that the co-augmentation morphism ℓT: T → LT is
+a regular epimorphism for all crossed modules T. We call them regular-epi localization and
+notice that many examples of interest are provided by nullification functors, as defined in
+Definition 1.10. Any crossed module A determines a nullification functor PA that “kills” all
+morphisms from A and there are other regular-epi localization functors such as abelianization.
+One first important observation which makes the reduction strategy viable is that, even
+though fiberwise localization does not exist in general, even for nullification functors, we can
+use this tool for certain extensions.
+Lemma 2.5. Let L be a regular-epi localization. Let
+(1)
+T
+Q
+N
+1
+1
+κ
+α
+be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules.
+Then, we can construct the fiberwise localization of the pullback of (1) along g:
+N
+N
+T′
+T
+Q′
+Q
+1
+1
+1
+1
+κ
+πT
+κ′
+g
+πQ′
+α
+This allows us to relate conditional flatness with admissibility, in the same spirit as what
+was done in the category of groups, [14], or in the wider context of semi-abelian categories
+where fiberwise localization exists, [15]. A localization functor L is said to be admissible for
+the class of regular epimorphisms if it preserves any pullback of the form
+LT
+T′
+Q
+LQ
+πLT
+ℓQ
+πQ
+α
+where α is a regular epimorphism between L-local objects.
+Theorem 3.4.
+Let L be a regular-epi localization functor. Then the following statements
+are equivalent
+(1) L is conditionally flat;
+(2) L is admissible for the class of regular epimorphisms.
+One difference between groups and crossed modules and maybe the main source of com-
+plication is highlighted by the behavior of kernels. This was already the reason why one
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+3
+cannot always construct fiberwise localization and we were also surprised to find examples of
+nullification functors for which the kernel of the nullification morphism ℓT : T → PAT is not
+always PA-acyclic, see [22, Proposition 4.6]. For groups and spaces, this property actually
+characterizes nullification functors.
+Still we prove here that acyclic kernels implies admissibility and in Proposition 4.3, that if
+the kernels of the localization morphisms are Lf-acyclic, then Lf is a nullification functor. Well
+behaved nullification functors are therefore admissible, but what about arbitrary nullification
+functors, for which fiberwise localization does not necessarily exist and for which the kernel of
+the nullification is not necessarily acyclic? By carefully looking at the inductive construction
+of PAT we show our main result, namely that all nullification functors are admissible.
+Theorem 5.5. Let A be any crossed module. The nullification functor PA is admissible for
+the class of regular epimorphisms.
+We end this introduction with a short outline. The first section consists of preliminaries
+that we use in the rest of the article. Then in Section 2 we introduce L-flat exact sequences
+and conditionally flat localization functors in the context of crossed modules. We show how to
+construct fiberwise localization of L-flat exact sequences. The third section is essential in the
+development of a simpler characterisation of conditional flatness: It provides an equivalence
+with the notion of admissibility in the specific context of regular-epi localization functors. In
+Section 4 the link between L-acyclicity and admissibility is established and the last section
+is devoted to the proof that every nullification functor is admissible.
+Acknowledgments. We would like to thank Marino Gran for sharing his insight about
+admissibility.
+1. Preliminaries
+1.1. The semi-abelian category of crossed modules. In this subsection, following Norrie
+[23] and Brown-Higgins [6], we provide the basic definitions and notation concerning crossed
+modules.
+Definition 1.1. [25] A crossed module of groups is a pair of groups T1 and T2, an action by
+group automorphisms of T2 on T1, denoted by T2 × T1 → T1 : (b, t) �→
+bt, together with a
+group homomorphism ∂T : T1 → T2 such that for any b in T2 and any t, s in T1,
+(2)
+∂T( bt) = b∂T(t)b−1,
+(3)
+∂T(t)s = tst−1.
+Hence we often write a crossed module as a triple (T1, T2, ∂T), or simply T for short, and
+we refer sometimes to ∂T as the connecting morphism.
+Definition 1.2. Let N := (N1, N2, ∂N) and M := (M1, M2, ∂M) be two crossed modules. A
+morphism of crossed modules α: N → M is a pair of group homomorphisms α1: N1 → M1
+and α2 : N2 → M2 such that the two following diagrams commute
+N2
+N1
+M1
+M2
+∂N
+∂M
+α1
+α2
+M2 × M1
+N2 × N1
+N1
+M1.
+(α2, α1)
+α1
+
+4
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+where the horizontal arrows in the diagram on the right are the respective group actions of
+the two crossed modules.
+We write XMod for the category of crossed modules of groups.
+Remark 1.3. There is an embedding of the category of groups in this category via two
+functors which are respectively left and right adjoint to the truncation functor Tr: XMod →
+Grp that sends a crossed module T := (T1, T2, ∂T) to T2. The functor X: Grp → XMod which
+sends a group G to the crossed module XG = (1, G, 1) reduced to the group G at level 2 is
+the left adjoint functor and the functor R: Grp → XMod: G �→ (G, G, IdG) is the right ajoint
+functor. This will help us to import group theoretical results into XMod.
+There is an obvious notion of subcrossed module, see [23]. One simply requires the sub-
+object to be made levelwise of subgroups, the connecting homomorphism and the action are
+induced by the given connecting homomorphism and action. The notion of normality is less
+obvious.
+Definition 1.4. A subcrossed module N := (N1, N2, ∂N) of T := (T1, T2, ∂T) is normal if the
+following three conditions hold
+(1) N2 is a normal subgroup of T2;
+(2) for any t2 ∈ T2 and n1 ∈ N1, we have t2n1 ∈ N1;
+(3) [N2, T1] := ⟨ n2t1t−1
+1
+| t1 ∈ T1, n2 ∈ N2⟩ ⊆ N1.
+In contrast to limits, which are built component-wise, colimits are generally more delicate
+to construct. In particular, the construction of cokernels is not straightforward, but when
+N is a normal subcrossed module of T the cokernel is simply the levelwise quotient by the
+normal subgroups N1 ⊳ T1 and N2 ⊳ T2.
+The category of crossed modules shares many nice properties with the category of groups.
+The traditional homological lemmas, [2], the Split Short Five Lemma, [3], and the Noether
+Isomorphism Theorems, [2], hold.
+One can recognize pullbacks by looking at kernels or
+cokernels, [2, Lemmas 4.2.4 and 4.2.5], and in fact Xmod is a semi-abelian category, as
+introduced by Janelidze, M´arki, and Tholen in [18]. This is shown in [18]. There is one result
+we will use several times in this article, namely [2, Lemma 4.2.4], which we recall now.
+Proposition 1.5. Let C be a semi-abelian (or homological) category. Consider the following
+diagram of exact rows:
+T ′
+Q′
+N′
+T
+Q
+N
+1
+1
+1
+(2)
+w
+u
+v
+κ
+α
+κ′
+α′
+Then the following statements hold.
+(1) If u is an isomorphism then (2) is a pullback.
+(2) If u and w are regular epimorphisms then v is also a regular epimorphism.
+1.2. Localization functors. In this subsection we recall the definition of localization func-
+tors in the category of crossed modules. We also recall some important properties of such
+functor as well as some examples.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+5
+Definition 1.6. A localization functor in the category of crossed modules is a coaugmented
+idempotent functor L: XMod → XMod. The coaugmentation ℓ: Id → L is a natural transfor-
+mation such that ℓLX and LℓX are isomorphisms.
+In particular we have ℓLX = LℓX, see [9, Proposition 1.1].
+Definition 1.7. Let L be a localization functor. A crossed module T is L-local if ℓT : T → LT
+is an isomorphism. A morphism f : N → M is an L-equivalence if Lf is an isomorphism.
+We recall a few basic and useful closure properties of L-equivalences.
+Lemma 1.8.
+(1) The pushout of an L-equivalence is an L-equivalence.
+(2) The composition of L-equivalences is an L-equivalence.
+(3) A κ-filtered colimit of a diagram Tβ of L-equivalences Tβ → Tβ+1 for all successor
+ordinals β + 1 < κ yields an L-equivalence T0 → Tκ = colimβ<κTβ.
+(4) Let F be an I-indexed diagram of L-equivalences in the category of morphisms of
+crossed modules. Then the colimit colimIF is an L-equivalence.
+Sometimes a localization functor L is associated to a full reflexive subcategory L of XMod.
+The pair of adjoint functors U: L ⇆ XMod: F provides a localization functor L = FU, as
+Cassidy, H´ebert, and Kelly do in [11]. Some other times there is a morphism f one wishes
+to invert so as to construct a localization functor often written Lf.
+Definition 1.9. Let f be a morphism of crossed modules. A crossed module T is Lf-local if
+Hom(f, T) is an isomorphism. A morphism g in XMod is an Lf-equivalence if Hom(g, T) is
+an isomorphism for any Lf-local crossed module T.
+Such localization functors exist in XMod, see for example Bousfield’s foundational work
+[4]. Local objects and local equivalences coincide then with the notions introduced in Defini-
+tion 1.7. Proposition 1.8 is the analogue of Hirschhorn’s [16, Proposition 1.2.20 and Propo-
+sition 1.2.21].
+If the codomain of the morphism f is the trivial crossed module, the functor Lf is of
+particular interest.
+Definition 1.10. Let A be a crossed module and f be the morphism A → 1. The localization
+functor Lf is then written PA and is called a nullification functor. An f-local object is called
+A-null, or A-local and a crossed module T is A-acyclic if PAT = 1. The localization morphism
+ℓT : T → PAT is written pT.
+Proposition 1.11. Let A and T be crossed modules. Then there exists an ordinal λ depending
+on A such that PAT is constructed as a transfinite filtered colimit of a diagram of the form
+T = T0 → T1 → · · · → Tβ → . . . for β < λ where all morphisms are PA-equivalences and
+regular epimorphims.
+This inductive construction has been carefully described in [22, Proposition 2.8]. The rea-
+son why each step is a PA-equivalence and a regular epimorphism is that Tβ+1 is constructed
+from Tβ by taking the cokernel of all morphisms A → Tβ. We recall the details and use
+them in Section 5. There is a larger class of localization functors we investigate in this se-
+quel to [22]. They share with PA the property that the localization morphism is a regular
+epimorphism.
+Definition 1.12. A localization functor L is a regular-epi localization if for any crossed
+module T the coaugmentation ℓT: T → LT is a regular epimorphism.
+
+6
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+Remark 1.13. In the category of crossed modules, a morphism α = (α1, α2) is a regular
+epimorphism (a coequalizer of a pair of parallel arrows) if and only if both α1 and α2 are sur-
+jective group homomorphisms [20, Proposition 2.2]. A surjective homomorphism of crossed
+modules is an epimorphism but there exist epimorphisms that are not surjective. In a pointed
+protomodular category such as XMod, regular epimorphisms and normal epimorphisms (the
+cokernel of some morphism) coincide.
+We present now some interesting examples of localization functors that will illustrate our
+results in the rest of the article, see also the end of [22, Section 2].
+Example 1.14. The nullification functor PXZ with respect to the crossed module XZ is given
+by:
+PXZ
+
+
+
+
+
+N1
+N2
+∂
+
+
+
+
+ =
+N1/[N2, N1]
+1
+Example 1.15. The abelianization functor Ab: XMod → XMod is already described in [24].
+It is defined by:
+Ab
+
+
+
+
+
+N1
+N2
+∂
+
+
+
+
+ =
+N1/[N2, N1]
+N2/[N2, N2]
+˜∂
+Example 1.16. Our third and last example of localization functor of crossed modules is
+I: XMod → XMod, see [22, Example 2.15]:
+I
+
+
+
+
+
+N1
+N2
+∂N
+
+
+
+
+ =
+N2
+N2
+IdN2
+This functor is induced by the adjunction between the truncation functor Tr: XMod → Grp,
+defined by Tr(T1, T2, ∂T) = T2, see Remark 1.3, and its right adjoint R: Grp → XMod that
+sends a group T to (T, T, IdT).
+Remark 1.17. The functor considered in Example 1.14 is a regular-epi localization, since all
+nullification functors are so. However regular-epi localizations are not nullification functors in
+general as illustrated by the functor Ab in Example 1.15. Indeed, if Ab were a nullification PA,
+then A = (A1, A2, ∂A) would be a perfect crossed module, i.e. one such that Ab(A) = (1, 1, Id).
+In particular, the group A2 would be a perfect group. But then PA(XS3) = XS3 since there
+are no non-trivial homomorphisms from a perfect group to the symmetric group S3. But we
+know that Ab(XS3) = XC2, where C2 is the cyclic group of order two, so abelianization is not
+a nullification.
+We finally note that a localization functor Lf is a regular-epi localization functor if f itself
+is a regular epimorphism, an analogous observation appears in [8] for groups.
+To conclude these preliminaries, let us recall the notion of fiberwise localization.
+We
+introduced this for crossed modules in [22, Definition 3.1], but this is not new, for spaces a
+good reference is [13, Section I.F].
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+7
+Definition 1.18. Let L: XMod → XMod be a localization functor. An exact sequence
+T
+Q
+N
+1
+1
+κ
+α
+admits a fiberwise localization if there exists a commutative diagram of horizontal exact
+sequences
+T
+Q
+N
+E
+Q
+LN
+1
+1
+1
+1
+κ
+j
+ℓN
+p
+α
+g
+where g is an L-equivalence.
+The following theorem is a fusion of two results from [22] namely Theorem 3.4 and Corollary
+3.7. From now on, every localization functor that we consider is a regular-epi localization.
+Theorem 1.19. Let L: XMod → XMod be a regular-epi localization functor. An exact se-
+quence of crossed modules
+(4)
+T
+Q
+N
+1
+1
+κ
+α
+admits a fiberwise localization if and only if we have the following inclusion
+(5)
+[κ2(ker(ℓN
+2 )), T1] ⊆ κ1(ker(ℓN
+1 ))
+2. Fiberwise localization and flatness
+In this section, we investigate the fiberwise localization of L-flat exact sequences and their
+pullbacks in the context of regular-epi localization functors of crossed modules L: XMod →
+XMod (even if this notion is not defined only for regular-epi functor as we will see in Propo-
+sition 5.6). This section will be essential to study the link between conditionally flatness
+and admissibility in Section 3. First, let us recall the definitions of L-flat and conditionally
+flatness.
+Definition 2.1. Let L be a localization functor, a short exact sequence
+T
+Q
+N
+1
+1
+κ
+α
+is called L-flat if the sequence
+LT
+LQ
+LN
+L(κ)
+L(α)
+is a short exact sequence.
+Remark 2.2. We recall that limits are computed componentwise in the category of crossed
+modules. In the case of pullbacks in XMod they are built as follows [19]. Let α: T → Q and
+g : Q′ → Q be two morphisms of crossed modules. Then the pullback of α along g is given
+by the following square
+T
+T′
+Q′
+Q
+πT
+g
+πQ′
+α
+The object part T′ of the pullback is built component-wise as in the case of groups
+(T1 ×Q1 Q′
+1, T2 ×Q2 Q′
+2, ∂′),
+
+8
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+where ∂′ and the action are induced by the universal property of the pullbacks in Grp. The
+projections are the natural ones, given also component-wise.
+Following the terminology introduced in [14] for groups and spaces, we define the notion
+of conditional flatness for localization functors in crossed modules.
+Definition 2.3. Let L be a localization functor. We say that this functor is conditionally
+flat if the pullback of any L-flat exact sequence is L-flat.
+In Section 3 we provide a characterization of conditional flatness. To achieve this goal we
+will use a similar strategy to the one applied to groups and topological spaces in [14]. The
+authors exploit heavily the existence of fiberwise localization in the categories of groups and
+spaces. However, in our article [22], we observed that fiberwise localization does not always
+exist for a given localization functor and a given exact sequence in XMod. Fortunately, when
+we work with L-flat exact sequences we can show that it is always possible to construct a
+fiberwise localization.
+Lemma 2.4. Let L be a regular-epi localization. Then any L-flat exact sequence of crossed
+modules admits a fiberwise localization.
+Proof. Let
+T
+Q
+N
+1
+1
+κ
+α
+be an L-flat exact sequence of
+crossed modules. The L-flatness of the sequence implies in particular that Lκ is a monomor-
+phism. Consider the following diagram of exact sequences:
+1
+1
+ker(ℓT)
+(1)
+ker(ℓN)
+N
+T
+LN
+LT
+κ
+Lκ
+ℓN
+ℓT
+We conclude from [2, Lemma 4.2.4.(1)] that (1) is a pullback since Lκ is a monomorphism.
+Then we have that κ(ker(ℓN)) is a normal subcrossed module of T as it can be seen as the
+intersection of the normal subcrossed modules N and ker(ℓT) of T. Therefore, we can apply
+Theorem 1.19
+□
+To understand conditional flatness we must study the pullback of an L-flat exact sequence.
+It will thus be very handy in Section 3 to know that any such pullback admits a fiberwise
+localization.
+Lemma 2.5. Let L be a regular-epi localization. Let
+(6)
+T
+Q
+N
+1
+1
+κ
+α
+be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules.
+Then, we can construct the fiberwise localization of the pullback of (6) along g
+N
+N
+T′
+T
+Q′
+Q
+1
+1
+1
+1
+κ
+πT
+κ′
+g
+πQ′
+α
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+9
+Remark 2.6. In the rest of the article, and in particular in the following proof, we identify
+N with the normal subcrossed module κ(N) of T and with κ′(N), normal subcrossed module
+of T′. We will therefore omit the us of κ and κ′. For example an element of the group N1
+that we want to consider in T′
+1 will be denoted (n1, 1) instead of κ′
+1(n1) = (κ1(n1), 1).
+Proof of Lemma 2.5. We need to verify that ker(ℓN) is a normal crossed module of T′. Since
+N is a subcrossed module of T′, we just need to verify (5) of Theorem 1.19. Let (t1, q1) be an
+element in T ′
+1 and (x2, 1) be an element of ker(ℓN
+2 ), then we have the following equality
+(x2,1)(t1, q1)(t1, q1)−1 = ( x2t1t−1
+1 , q1q−1
+1 ) = ( x2t1t−1
+1 , 1).
+Indeed, by Lemma 2.4 we know that the original sequence (6) admits a fiberwise localization
+which then implies by Theorem 1.19 that [ker(ℓN
+2 ), T1] ⊂ ker(ℓN
+1 ) i.e for any x2 ∈ ker(ℓN
+2 ) and
+t1 ∈ T1 we have x2t1t−1
+1
+∈ ker(ℓN
+1 ). But then, with the notation introduced in Remark 2.6,
+this is equivalent to say that the element ( x2t1t−1
+1 , 1) belongs to ker(ℓN
+1 ).
+□
+This lemma is not trivial since the fiberwise localization of an exact sequence of crossed
+modules does not always exist as we have proved in [22, Theorem 4.5]. If we want the strategy
+for groups and spaces to be also viable in the study of conditional flatness for crossed modules,
+we need a final ingredient, namely a commutation rule for the fiberwise localization and the
+pullback operations.
+Proposition 2.7. Let us consider an L-flat exact sequence where L is a regular-epi localization
+functor.
+Then, the pullback of its fiberwise localization is the fiberwise localization of its
+pullback.
+Proof. Let
+N
+N
+T′
+T
+Q′
+Q
+1
+1
+1
+1
+κ
+πT
+κ′
+g
+πQ′
+α
+be the pullback of an L-flat exact sequence. Then we construct the fiberwise localizations of
+the two sequences by quotienting out the kernel of the localization morphism ℓN as follows.
+N
+1
+T′
+Q′
+1
+N
+1
+T
+Q
+1
+LN
+1
+T′/ker(ℓN)
+Q′
+1
+LN
+1
+T/ker(ℓN)
+Q
+1
+κ′
+πQ′
+κ
+α
+g
+πT
+j
+j′
+p
+p′
+g
+f ′
+f
+ℓN
+ℓN
+We complete the diagram by defining a morphism δ: T′/ker(ℓN) → T/ker(ℓN) via the
+universal property of the cokernel since f ◦ πT ◦ κ′|ker(ℓN) = 1, where κ′|ker(ℓN) : ker(ℓN) → T′ is
+
+10
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+the inclusion of the kernel of ℓN.
+ker(ℓN)
+ker(ℓN)
+T′
+T
+T ′/ker(ℓN)
+T/ker(ℓN)
+1
+1
+1
+1
+κ|ker(ℓN)
+πT
+κ′|ker(ℓN)
+f ′
+f
+δ
+N
+1
+T′
+Q′
+1
+N
+1
+T
+Q
+1
+LN
+1
+T′/ker(ℓN)
+Q′
+1
+LN
+1
+T/ker(ℓN)
+Q
+1
+κ′
+πQ′
+κ
+α
+g
+πT
+j
+j′
+p
+p′
+g
+δ
+f ′
+f
+lN
+lN
+We can check that δ makes the two front faces commute. Indeed, the right and left faces
+commute by using the fact that ℓN and f ′ are epimorphisms respectively.
+The commutativity of the above diagram and Proposition 1.5 implies that
+LN
+T′/ker(ℓN)
+Q′
+1
+1
+j′
+p′
+is the pullback of
+T/ker(ℓN)
+Q
+LN
+1
+1
+j
+p
+along g.
+□
+Remark 2.8. In [14], the construction of the fiberwise localization in the category of groups
+was functorial, therefore from the morphism T′ → T between the pullback sequence and the
+sequence itself we have directly a morphism between the fiberwise localization of the pullback
+sequence and the fiberwise localization of the original sequence. In other words the map δ
+comes for free in contrast to the category of crossed modules where we have to build the map
+δ explicitly.
+3. Conditional flatness and admissibility
+In this section, we develop a simpler characterisation of conditional flatness, thanks to
+the results of the previous section. We introduce the notion of admissibility for the class
+of regular epimorphisms and show that it is equivalent to conditional flatness. With this
+equivalence, we can easily establish conditional flatness for a given localization functor. We
+observe that some properties of localization functors, such as right-exactness, imply directly
+admissibility for the class of regular epimorphism.
+The first step allows us to restrict the definition of conditional flatness (Definition 2.3) to
+fiberwise localizations of L-flat exact sequences (Lemma 3.1). More precisely, we show that
+the pullback of an L-flat exact sequence is L-flat if and only if the pullback of its fiberwise
+localization is so.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+11
+Lemma 3.1. Let L be a regular-epi localization functor. Then L is conditionally flat if and
+only if for any L-flat exact sequence
+T
+Q
+N
+1
+1
+κ
+α
+with N
+an L-local crossed module, the pullback sequence along any morphism Q′ → Q is L-flat.
+Proof. This is clear since f ′ and ℓN are L-equivalences in this diagram:
+LN
+N
+T′
+T′/ker(ℓN)
+Q′
+Q′
+1
+1
+1
+1
+j′
+f ′
+ℓN
+κ′
+πQ′
+p′
+The top row is thus L-flat if and only if so is the bottom row and we conclude by Proposi-
+tion 2.7.
+□
+The previous lemma allows us to follow the approach introduced in [14]. For the sake of
+completeness, we give an explicit proof of the following results even if the arguments are
+similar to the group theoretical ones.
+Proposition 3.2. Let L be a regular-epi localization functor. Then L is conditionally flat if
+and only if the pullback of any exact sequence of L-local objects is L-flat.
+Proof. By the previous lemma it is sufficient to consider exact sequence with an L-local kernel
+LN. Consider thus an L-flat exact sequence
+T
+Q
+LN
+1
+1
+j
+p
+.
+We build the following diagram where g : Q′ → Q is any morphism of crossed modules and
+(1) is a pullback.
+LN
+LN
+LN
+T′
+(1)
+(2)
+T
+LT
+Q′
+Q
+LQ
+1
+1
+1
+1
+1
+1
+j
+L(j)
+ℓT
+ℓQ
+πT
+j′
+g
+πQ′
+L(p)
+p
+We observe that since each row is exact, (2) is a pullback by Proposition 1.5, and then
+(1) + (2) is also a pullback. Hence, the top row is the pullback of the bottom exact sequence
+of L-local objects along the map ℓQ ◦ g, which shows the claim.
+□
+Definition 3.3. A localization functor L is said to be admissible for the class of regular
+epimorphisms if it preserves any pullback of the form
+LT
+T′
+Q
+LQ
+πLT
+ℓQ
+πQ
+α
+
+12
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+where α is a regular epimorphism.
+Theorem 3.4. Let L be a regular-epi localization functor. Then the following statements are
+equivalent
+(1) L is conditionally flat;
+(2) L is admissible for the class of regular epimorphisms.
+Proof. The implication (1) ⇒ (2) is trivial, so let us prove (2) ⇒ (1). Consider any exact
+sequence of L-local objects
+LT
+LQ
+LN
+1
+1
+α
+and any morphism g : A → LQ. By Proposition 3.2 conditional flatness is established if we
+prove that the pullback of the exact sequence along g is L-flat. Let us first observe that this
+morphism g factors through LA via the universal property of the localization:
+A
+LA
+LQ
+ℓA
+g
+˜g
+Hence, we can first construct the pullback of
+LT
+LQ
+LN
+1
+1
+α
+along
+˜g and then pullback the resulting sequence along ℓA:
+LN
+LN
+LN
+T′′
+T′
+LT
+A
+LA
+LQ
+1
+1
+1
+1
+1
+1
+g
+πLT
+˜g
+ℓA
+πA
+α
+πLA
+Since the category of L-local objects is closed under pullbacks, T′ is L-local and we can
+apply condition (2) to conclude that the upper row is L-flat. This observation implies that
+the pullback of
+LT
+LQ
+LN
+1
+1
+α
+along g is an L-flat sequence as
+desired.
+□
+The above theorem gives an easier characterisation of conditionally flatness in the category
+of crossed modules. It will be useful in rest of the article.
+Remark 3.5. Admissibility for the class of regulars epimorphisms in the context of semi-
+abelian categories is studied in [15]. Similar results are proven for functors of localizations that
+admit a functorial fiberwise localization. Note that their result does not imply Theorem 3.4
+since localization functors of crossed modules do not admit functorial fiberwise localizations
+in general. However, the implication “(1) implies (2)”, in Theorem 3.4, holds even for not
+necessarily regular-epi localization functors.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+13
+Proposition 3.6. If L: XMod → XMod is a localization functor that is right exact in XMod,
+then L is admissible for the class of regular epimorphisms.
+Proof. Let us consider the following pullback of an L-flat exact sequence of crossed modules
+along a morphism g : Q′ → Q.
+T′
+Q′
+N
+T
+Q
+N
+1
+1
+1
+1
+(1)
+g
+πT
+κ
+f
+κ′
+πQ′
+By applying L to this diagram, we obtain (since L is right exact) the following diagram
+LT′
+LQ′
+LN
+LT
+LQ
+LN
+1
+1
+1
+L(g)
+L(πT)
+L(κ)
+L(f)
+L(κ′)
+L(πQ′)
+Since L(κ) = L(πT)◦L(κ′) is a (normal) monomorphism, we conclude that L(κ′) is a monomor-
+phism. Normality follows then by right-exactness and we conclude by Theorem 3.4.
+□
+Note that this proof holds in any semi-abelian category.
+Corollary 3.7. The functor of abelianization Ab: XMod → XMod is admissible for the class
+of regular epimorphisms.
+Proof. The functor of abelianization Ab: XMod → XMod is right exact. Since the exactness
+can be shown component-wise, the result follows.
+□
+Sometimes it is handy to rely on our group theoretical knowledge to construct simple
+examples of localization functors and how they behave on crossed modules. The proof of the
+following proposition is based on a counter-example coming from groups via the functor X
+defined in Remark 1.3.
+Proposition 3.8. There are regular-epi localization functors L: XMod → XMod that are not
+admissible for the class of regular epimorphisms.
+Proof. We export via X: Grp → XMod the example in [14, Theorem 5.1] of a localization
+functor in groups that is not admissible for the class of regular epimorphisms.
+Let Lφ be the localization functor induced by the projection φ: C4 → C2, where Cn denotes
+a cyclic group of order n. It gives rise to a localization functor LXφ : XMod → XMod. In
+particular, if we apply X to the extension of Lφ-local groups considered in [14], we obtain an
+exact sequence of LXφ-local crossed modules:
+(1, Z)
+(1, C2)
+(1, Z)
+1
+1
+If we pullback along the morphism of crossed modules Xφ, we obtain the following exact
+sequence
+(1, Z × C2)
+(1, C4)
+(1, Z)
+1
+1
+
+14
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+We conclude from [22, Lemma 1.4] that this exact sequence is not LXφ-flat. Indeed, if it was
+the case we would have a contradiction with the group theoretical observation in [14].
+□
+4. Admissibility and acyclicity
+In the categories of groups and topological spaces, the localization functor L is a nullification
+functor if and only if the kernels of the localization morphisms are L-acyclic (which means
+that Lker(ℓM) is trivial for any M ∈ XMod). This characterisation implies in particular that
+any nullification functor is admissible for the class of regular epimorphisms. It is interesting
+to notice that even if nullification functors of crossed modules do not have acyclic kernels,
+we have a similar result in XMod: the L-acyclicity of the kernels of localization morphisms
+implies the admissibility.
+Proposition 4.1. Let L: XMod → XMod be a regular-epi localization functor such that
+ker(ℓM : M → LM) is L-acyclic for any M ∈ XMod. Then L is admissible for the class of
+regular epimorphisms.
+Proof. Consider the pullback of
+LT
+LQ
+LN
+1
+1
+κ
+f
+along ℓQ: Q →
+LQ:
+LN
+LN
+T′
+LT
+Q
+LQ
+1
+1
+1
+1
+κ
+πLT
+κ′
+ℓQ
+πQ
+f
+We need to prove that πLT is an L-equivalence. Since XMod is a pointed protomodular
+category and ℓQ is a regular epi by assumption, we know that πLT is the cokernel of ker(ℓQ) ∼=
+ker(πLT) → T′. Let Y be a local object, for any g : T′ → Y we have the following diagram:
+ker(ℓQ)
+Lker(ℓQ) = 1
+T′
+LT
+Y
+g′
+πLT
+g
+˜g
+By the universal property of the localization there exists g′: 1 → Y that makes the left square
+commute. Hence, by the universal property of the cokernel there exists a unique ˜g: LT → Y
+such that the triangle commutes and we conclude that πLT is an L-equivalence.
+□
+However, localization functors of crossed modules do not behave like localization functors
+of groups. As explained above, in the category of groups (but also of topological spaces), the
+kernels of the localization morphisms are L-acyclic if and only if L is a nullification functor
+[14]. In the context of crossed modules, we do not have such a characterization of nullification
+functors.
+Remark 4.2. We know by [22, Proposition 4.6] that there are nullification functors, for
+example PXZ defined in Example 1.14, such that the kernels of their localization morphisms
+are not acyclic in general. Still, in the next proposition, we prove that if the kernel of the
+localization morphism is L-acyclic, as in Proposition 4.1, then the localization functor is a
+nullification.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+15
+The cardinal in the next proof is chosen exactly as in Bousfield’s [5, Theorem 4.4] for
+spaces.
+Proposition 4.3. Let f : B → C be a morphism of crossed modules and Lf : XMod → XMod
+be a regular-epi localization functor.
+If the kernels of the localization morphisms are Lf-
+acyclic, then Lf is a nullifcation functor.
+Proof. Our strategy is to construct a crossed module A such that we can compare the functor
+Lf with the nullification functor PA (Definition 1.10) via a natural transformation ψ. We
+choose κ to be the first infinite ordinal greater than the number of chosen generators of B
+and C, i.e., generators of the groups B1, B2, C1 and C2. We construct the crossed module
+A := � Aα, where Aα are all the Lf-acyclic crossed modules with less than 2κ generators, see
+[5, Theorem 4.4].
+The first step of this proof is to show that if a crossed module X is Lf-local then it is A-local.
+Let φ be a morphism in Hom(A, X) and construct by naturality the following commutative
+diagram
+1 = LfA
+A
+X
+LfX
+Lfφ
+∼=
+φ
+By hypothesis, we have an isomorphism between X and LfX and by construction of A, we
+obtain LfA = 1. Therefore, φ factors through the zero object and hence Hom(A, X) = 1,
+which is equivalent to say that X is A-local. Now consider the PA-equivalence pT: T → PAT
+and the Lf-local object LfT. By the above observation, we have that LfT is A-local and by
+the universal property we have the desired morphism ψT
+T
+PAT
+LfT
+pT
+ℓT
+ψT
+We construct next the fiberwise A-nullification of the following exact sequence
+T
+LfT
+ker(ℓT)
+1
+1
+ℓT
+By assumption ker(ℓT) is Lf-acyclic, hence also PA-acyclic by design.
+This implies that
+ker
+�
+pT: ker(ℓT) → PAker(ℓT)
+�
+is equal to ker(ℓT). Hence, the exact sequence satisfies condi-
+tion (5) of Theorem 1.19 and we obtain the following fiberwise nullification
+T
+LfT
+ker(ℓT)
+T/ker(ℓT)
+LfT
+1
+1
+1
+1
+1
+pT
+f
+∼=
+ℓT
+
+16
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+Since f is a PA-equivalence, so is ℓT. Hence, we obtain a morphism ϕT in the following
+commutative diagram:
+T
+LfT
+PAT
+ψT
+ℓT
+pT
+ϕT
+By universal property, we can conclude that the two compositions of ψT and ϕT are isomorphic
+to identities so that LfT ∼= PAT. A similar argument shows the naturality of ψ and ϕ and
+therefore Lf is a nullification functor, namely PA.
+□
+5. Nullification functors and admissibility
+In the category of groups, the fact that kernels of localization morphisms are L-acyclic
+was fundamental to prove that nullification functors are admissible for the class of regular
+epimorphisms. This fact is not true in general for nullification functors in the category of
+crossed modules as shown in [22, Proposition 4.6], it is thus natural to ask whether nullifica-
+tion functors are admissible. We provide an affirmative answer in this final section, but let
+us first prove that our counter-example PXZ is admissible.
+Proposition 5.1. The nullification functor PXZ is admissible for the class of regular epimor-
+phisms.
+Proof. Theorem 5.1 in [15] implies that PXZ is admissible provided that the reflective category
+of PXZ-local objects is a Birkhoff subcategory, i.e., it is closed under regular quotients and
+subobjects. Here PXZ-local objects are crossed modules of the form A → 1 where A is any
+abelian group and the connecting homomorphism is the trivial homomorphism. Therefore it
+is clearly closed under subobjects. Moreover, the quotient of A → 1 by a normal subcrossed
+modules N → 1 is the crossed module A/N → 1 that is PXZ-local.
+□
+The remaining part of the section is devoted to the proof that all nullification functors are
+admissible for the class of regular epmorphisms. Consider a nullification functor PA where
+A = (A1, A2, ∂) is a crossed module. To show the admissibility, it is enough to prove that the
+pullback of an exact sequence of PA-local crossed modules along the coaugmentation map
+is PA-flat, in other words that the map f in the following commutative diagram of crossed
+modules is a PA-equivalence
+W
+Q
+PAN
+PAT
+PAQ
+PAN
+1
+1
+1
+1
+(1)
+pQ
+f
+h
+g
+where (1) is a pullback and g and h are regular epimorphisms. To do so we follow step by
+step the inductive construction of PAQ = colimQβ as presented in [22, Proposition 2.8], see
+also Proposition 1.11. For each successor ordinal β + 1 we obtain Qβ+1 from Qβ by killing all
+morphisms out of A so let us start with the construction of Q1 from Q0 = Q.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+17
+Remark 5.2. Let ϕ : A → Q be a morphism of crossed modules. The crossed module Q1 is
+the quotient of Q by the normal closure KQ in Q of the image of
+ev:
+�
+ϕ∈Hom(A,Q)
+A = M −→ Q
+which is defined by ϕ on the copy of A indexed by ϕ. The idea behind the construction we
+perform next is that we do not need to kill all morphisms from A to the extension W in order
+to construct its nullification PAW, it is sufficient to take care of those factoring through Q.
+Beware that given an extension N → T → Q with N an A-acyclic crossed module, it is not
+true in general that all morphisms from A to T factor through Q.
+By definition of pQ we have the following equality for the composition pQ ◦ ϕ = 1 = h ��� 1
+as below. Therefore, any morphism from A to Q induces one from A to W:
+(7)
+PAT
+W
+Q
+PAQ
+A
+h
+pQ
+f
+g
+1
+ϕ
+∃!ψ
+We call ψ the morphism determined by ϕ and it makes sense now to consider KW, the normal
+closure in W of the image of M → W.
+Lemma 5.3. With the same notation as in Remark 5.2, we have an isomorphism KW ∼= KQ.
+Proof. Limits are computed levelwise for crossed modules, so the pullback W consists of
+compatible pairs (x, q) for x ∈ (PAT)i and q ∈ Qi for i = 1, 2. By construction of ψ we have
+ψ(a) = (1, ϕ(a)).
+Now, we compute the kernels of the cokernels of ev: M → Q and (1, ev): M → W. We
+have the two following descriptions of the kernels.
+KQ =
+�
+ev1(M1)Q2[ev2(M2)Q2, Q1], ev2(M2)Q2, ∂
+�
+KW =
+�
+(1, ev1)(M1)W2[(1, ev2)(M2)W2, W1], (1, ev2)(M2)W2, ∂′�
+The second group of the crossed module KW is the easier one:
+(1, ev2)(M2)W2 = {(t2,q2)(1, ev2(m2)) | (t2, q2) ∈ W2, m2 ∈ M2}
+= {(1, q2ev2(m2)) | q2 ∈ Q2, m2 ∈ M2}
+= 1 × ev2(M2)Q2.
+where the second equality holds since h is surjective. Via similar computations, we see that
+(1, ev1)(M1)W1 = 1 × ev1(M1)Q1, so we are left with proving that
+[(1, ev2)(M2)W2, W1] = 1 × [ev2(M2)Q2, Q1]
+
+18
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+This we do via the following equalities:
+[(1, ev2)(M2)2, W1] = [(1 × ev2(M2)Q2), W1]
+= {(1,x2)(t1, q1)(t1, q1)−1 | x2 ∈ ev2(M2)Q2, (t1, q1) ∈ W1}
+= {(1,x2 q1q−1
+1
+| x2 ∈ ev2(M2)Q2, q1 ∈ Q1}
+= 1 × [(ev2(M2)Q2, Q1]
+So finally we can conclude that KW = 1 × KQ, in particular KW and KQ are isomorphic.
+□
+Proposition 5.4. For any ordinal β, we have a commutative diagram
+PAT
+Wβ
+Qβ
+PAQ
+W
+Q
+(2)
+hβ
+g
+fβ
+pQ
+β
+h
+where (2) is a pullback square, the maps fβ : W → Wβ and pQ
+β : Q → Qβ are PA-equivalences,
+and hβ is a regular epimorphism.
+Proof. We prove it by induction. Since the nullification uses possibly a transfinite construc-
+tion we have to initialize the induction, but the case β = 0 holds by assumption, and then
+check the statement for successor and limit ordinals.
+The successor case Suppose that for an ordinal β the lemma is proved. Then we consider
+the kernels KW
+β and KQ
+β of the cokernels of the evaluation maps ev : �
+Hom(A,Qβ) A −→ Qβ and
+ev : �
+Hom(A,Qβ) A −→ Wβ respectively. They fit in the following diagram of exact rows:
+Wβ+1
+Wβ
+Qβ
+Qβ+1
+KW
+β
+KQ
+β
+(2)
+pQ
+(β→β+1)
+f(β→β+1)
+hβ
+∼=
+iW
+iQ
+∃!hβ+1
+Lemma 5.3 applies here and gives us the isomorphism between KW
+β and KQ
+β . The composition
+pQ
+(β→β+1) ◦ hβ ◦ iW : KW
+β → Qβ+1
+is zero by commutativity, yielding by the universal property of the cokernel the morphism
+hβ+1: Wβ+1 → Qβ+1. The isomorphism between the kernels implies that (2) is a pullback (see
+Proposition 1.5). By induction hypothesis hβ is a regular epimorphism and the composition
+pQ
+(β→β+1) ◦ hβ : Wβ → Qβ+1 is also a regular epimorphism, hence so is hβ+1. We show now
+that pQ
+(β→β+1) and f(β→β+1) are PA-equivalences.
+
+ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES
+19
+For the first one we write the cokernel Qβ+1 as the pushout along the evaluation morphism:
+1
+� A
+Qβ
+Qβ+1
+pQ
+(β→β+1)
+1
+ϕ
+inc
+where the coproduct is taken over Hom(A, Q). The trivial map A → 1 is a PA-equivalence,
+thus so is the pushout pQ
+(β→β+1) : Qβ → Qβ+1 by Lemma 1.8 (1). By composing with the
+PA-equivalence Q → Qβ we see that pQ
+β+1 : Q → Qβ+1 is a PA-equivalence as well. The same
+argument shows that fβ+1 : W → Wβ+1 is also a PA-equivalence. By the universal property
+of the localization, we obtain two maps, one from Wβ+1 to PAT and the other from Qβ+1 to
+PAQ such that (2) commutes:
+PAT
+Wβ+1
+Qβ+1
+PAQ
+W
+Q
+(2)
+(1)
+hβ+1
+g
+fβ+1
+pQ
+β+1
+h
+f
+pQ
+Since (1) and the outer rectangle are pullbacks and hβ+1 is a regular epimorphism, we can
+conclude by Proposition 4.1.4 in [2] that (2) is a pullback.
+The limit case To prove the statement for a general transfinite induction we need to prove
+it for a limit ordinal as well. Let γ be a limit ordinal and
+Qγ = colimα<γQα
+Wγ = colimα<γWα
+We have shown that pQ
+(α−1→α) : Qα−1 → Qα is a PA-equivalence for all α < γ. Hence the
+composition pQ
+α : Q → Qα is also a PA-equivalence and Lemma 1.8 (3), implies that pQ
+γ : Q →
+Qγ is a PA-equivalence. The same reasoning holds for fγ : W → Wγ. The existence of the
+maps f : W → PAT and pQ : Q → PAQ give us two maps Wγ → PAT and Qγ → PAQ as
+shown on the diagram below (8).
+The nullification PAQ is constructed as filtered colimit of the Qα, see Proposition 1.11.
+Filtered colimits commutes with finite limits, in particular with kernels. Therefore
+KQ
+γ := ker(Q → Qγ) ∼= colimα<γker(Q → Qα)
+where ker(Q → Qα) will be denoted KQ
+α. The category XMod is a variety of algebras (also
+called algebra category of fixed type). Hence, by [21, Proposition IX.1.2], we know that the
+forgetful functor U : XMod → Set creates filtered colimits. In other words we have :
+U(colimα<γKQ
+α) = colimα<γUKQ
+α =
+�
+α<γ
+UKQ
+α
+where the colimit in the first term lies in the category of crossed modules and the second
+colimit in the category of sets. This means that we know the structure of colimα<γKQ
+α as a
+
+20
+OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK
+set. Now since KQ
+α ∼= KW
+α for all α < γ and KQ
+γ can be written as a union of KQ
+α (as well as
+KW
+γ ) we conclude that KQ
+γ ∼= KW
+γ . We consider now the diagram:
+(8)
+PAT
+Wγ
+Qγ
+PAQ
+W
+Q
+(1)
+(2)
+hγ
+g
+fγ
+pQ
+γ
+h
+f
+pQ
+Since the kernels of fγ and pQ
+γ are isomorphic we deduce that (2) is a pullback.
+As we
+have shown that every map pQ
+(α→α+1) : Qα → Qα+1 is a regular epimorphism, the morphism
+pQ
+α : Q → Qα is also a regular epimorphism, being a composition of regular epimorphisms in
+a regular category. The colimit functor being a left adjoint functor, it preserves colimits and
+in particular cokernels. In a pointed protomodular category, any regular epimorphism is a
+cokernel, therefore
+pQ
+γ : Q → Qγ
+is a regular epimorphism. The composition pQ
+γ ◦ g is also a regular epimorphism, and we
+conclude that so is hγ. With the same argument as for the successor step, we get that (1) is
+a pullback, which ends the induction proof.
+□
+We are ready now for the main result of this section.
+Theorem 5.5. Let A be any crossed module. The nullification functor PA is admissible for
+the class of regular epimorphisms.
+Proof. Let W be the pullback of a regular epimorphism h: PAT → PAQ between PA-local
+crossed modules along the localization morphism pQ : Q → PAQ. Let λ be the ordinal such
+that Qλ ∼= PAQ (see Proposition 1.11). By Proposition 5.4 we have a diagram:
+PAT
+Wλ
+Qλ
+PAQ
+W
+Q
+(2)
+∼=
+hλ
+g
+h
+fλ
+pQ
+λ
+where the outer rectangle is a pullback, the morphisms fλ and pQ
+λ are PA-equivalences, and
+(2) is a pullback. Since isomorphisms are stable under pullbacks, we have an isomorphism
+Wλ ∼= PAT. We have thus proved that the map f : W → PAT is a PA-equivalence, which
+means that the functor PA is admissible.
+□
+In this article we have focused on regular-epi localization functors because they appear nat-
+urally when studying conditional flatness and admissibility in the category of groups, crossed
+modules, or more general semi-abelian categories. We conclude this section by observing that
+the notion of conditional flatness can also be defined for non regular-epi localization functor.
+The next proposition gives an example of such a localization functor which is conditionally
+
+REFERENCES
+21
+flat. Let us stress that we will not a priori have an equivalence with admissiblity, as was
+the case for regular-epi localization functors by Theorem 3.4. In the proof of the following
+proposition we have thus to verify the more general condition for conditional flatness, as in
+Definition 2.3.
+Proposition 5.6. There exists a non regular-epi localization functor which is nevertheless
+conditionally flat and therefore admissible for the class of regular epimorphisms.
+Proof. We consider the functor I defined in Example 1.16 which sends any crossed module
+(N1, N2, ∂N) to (N2, N2, IdN2). This functor is not regular-epi because if we consider a crossed
+module for which the connecting morphism is not surjective then the localization morphism
+will not be a regular epimorphism.
+We prove now that I is conditional flat. Let
+T
+Q
+N
+1
+1
+κ
+α
+be any exact sequence of crossed modules.
+We see that I((N1, N2, ∂N)) = (N2, N2, IdN2)
+is a normal subcrossed module of (T2, T2, IdT2) = I((T1, T2, ∂T) and that I((Q1, Q2, ∂Q)) =
+(Q2, Q2, IdQ2) is the cokernel of κ: N → T. Therefore any exact sequence of crossed modules
+is I-flat. In particular any pullback along any morphism of crossed modules of an I-flat exact
+sequence is I-flat, hence I is conditionally flat.
+□
+References
+[1]
+A. J. Berrick and E. Dror Farjoun. “Fibrations and nullifications”. In: Israel J. Math.
+135 (2003), pp. 205–220.
+[2]
+F. Borceux and D. Bourn. Mal’cev, protomodular, homological and semi-abelian cate-
+gories. Vol. 566. Springer Science & Business Media, 2004.
+[3]
+D. Bourn. “Normalization Equivalence, Kernel Equivalence, and Affine Categories”. In:
+2006, pp. 43–62.
+[4]
+A. K. Bousfield. “Constructions of factorization systems in categories”. In: J. Pure
+Appl. Algebra 9.2 (1976), pp. 207–220.
+[5]
+A. K. Bousfield. “Homotopical localizations of spaces”. In: Amer. J. Math. 119.6 (1997),
+pp. 1321–1354. issn: 0002-9327. url: http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.6bousfield.pdf.
+[6]
+R. Brown and P. Higgins. “On the connection between the second relative homotopy
+groups of some related spaces”. In: Proceedings of The London Mathematical Society
+(1978), pp. 193–212.
+[7]
+R. Brown and C. Spencer. “G-groupoids, crossed modules and the fundamental groupoid
+of a topological group”. In: Indag. Math. (Proceedings) 79.4 (1976), pp. 296–302.
+[8]
+C. Casacuberta. “Anderson localization from a modern point of view”. In: Contemp.
+Math. 181 (1995), pp. 35–46.
+[9]
+C. Casacuberta. “On structures preserved by idempotent transformations of groups
+and homotopy types”. In: Crystallographic groups and their generalizations (Kortrijk,
+1999). Vol. 262. Contemp. Math. Amer. Math. Soc., Providence, RI, 2000, pp. 39–68.
+[10]
+C. Casacuberta and A. Descheemaeker. “Relative group completions”. In: Journal of
+Algebra 285.2 (2005), pp. 451–469.
+[11]
+C. Cassidy, M. H´ebert, and G. M. Kelly. “Reflective subcategories, localizations and
+factorization systems”. In: J. Austral. Math. Soc. Ser. A 38.3 (1985), pp. 287–329.
+[12]
+T. Everaert, M. Gran, and T. Van der Linden. “Higher Hopf formulae for Homology
+via Galois Theory”. In: Adv. Math. 217 (2008), pp. 2231–2267.
+
+22
+REFERENCES
+[13]
+E. Dror Farjoun. Cellular spaces, null spaces and homotopy localization. Vol. 1622.
+Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996, pp. xiv+199.
+[14]
+E. Dror Farjoun and J. Scherer. “Conditionally flat functors on spaces and groups”. In:
+Collect. Math. 66.1 (2015), pp. 149–160.
+[15]
+M. Gran and J. Scherer. “Conditional flatness and admissibility of a reflector in a
+semi-abelian category”. In: preprint (2022), 15 pages.
+[16]
+Philip S. Hirschhorn. Model categories and their localizations. Vol. 99. Mathemati-
+cal Surveys and Monographs. American Mathematical Society, Providence, RI, 2003,
+pp. xvi+457.
+[17]
+G. Janelidze and G. M. Kelly. “Galois theory and a general notion of central ex-
+tension”. In: J. Pure Appl. Algebra 97.2 (1994), pp. 135–161. issn: 0022-4049. doi:
+10.1016/0022-4049(94)90057-4. url: https://doi.org/10.1016/0022-4049(94)90057-4.
+[18]
+G. Janelidze, L. M´arki, and W. Tholen. “Semi-abelian categories”. In: J. Pure Appl.
+Algebra 168 (2002), pp. 367–386.
+[19]
+M. Ladra and A.R. Grandjean. “Crossed modules and homology”. In: Journal of Pure
+and Applied Algebra 95.1 (1994), pp. 41–55.
+[20]
+M. Ladra, M.P. L´opez L´opez, and E. Rodeja. “Epimorphisms of crossed modules”. In:
+Southeast Asian Bulletin of Mathematics 28 (Jan. 2004).
+[21]
+S. Mac Lane. Categories for the Working Mathematicians. 2nd ed. Springer, 1997.
+[22]
+O. Monjon, J. Scherer, and F. Sterck. Non-existence of fiberwise localization for crossed
+modules. arxiv:2207.09702, to appear in Israel J. Math. 2022.
+[23]
+K. Norrie. “Actions and automorphisms of crossed modules”. In: Bulletin de la Soci´et´e
+Math´ematique de France 118.2 (1990), pp. 129–146.
+[24]
+K. Norrie. “Crossed modules and analogues of group theorems”. PhD thesis. King’s
+College, University of London, 1987.
+[25]
+J. H. C. Whitehead. “Combinatorial homotopy II”. In: Bull. Amer. Math. Soc 55 (1949),
+pp. 453–496.
+Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland
+Email address: [email protected]
+Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland
+Email address: [email protected]
+Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain,
+Belgium
+Email address:
+[email protected]
+

JdAyT4oBgHgl3EQffviy/content/tmp_files/2301.00347v1.pdf.txt

@@ -0,0 +1,692 @@
+JWST high redshift galaxy observations have a strong tension with Planck CMB
+measurements
+Deng Wang∗ and Yizhou Liu
+National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China
+JWST high redshift galaxy observations predict a higher star formation efficiency that the stan-
+dard cosmology, which poses a new tension to ΛCDM. We find that the situation is worse than
+expected.
+The true situation is that the Planck CMB measurement has a strong tension with
+JWST high redshift galaxy observations. Specifically, we make a trial to alleviate this tension by
+considering alternative cosmological models including dark matter-baryon interaction, f(R) gravity
+and dynamical dark energy. Within current cosmological constraints from Planck-2018 CMB data,
+we find that these models all fail to explain such a large tension. A possible scenario to escape
+from cosmological constraints is the extended Press-Schechter formalism, where we consider the
+local environmental effect on the early formation of massive galaxies. Interestingly, we find that an
+appropriate value of nonlinear environmental overdensity of a high redshift halo can well explain
+this tension.
+I.
+INTRODUCTION
+Since the cosmic acceleration is discovered by Type Ia supernovae (SNe Ia) [1, 2] and confirmed by two independent
+probes cosmic microwave background (CMB) [3–5] and baryon acoustic oscillations (BAO) [6, 7], the standard 6-
+parameter cosmological model, Λ-cold dark matter (ΛCDM) has achieved great success in characterizing the physical
+phenomena across multiple scales at the background and perturbation levels. However, the validity of ΛCDM is
+challenged by various kinds of new observations for a long time, and consequently new puzzles emerge such as the
+so-called Hubble constant (H0) tension (see [8, 9] for recent reviews). It is noteworthy that, so far, we can not study
+effectively the correctness of ΛCDM around redshift z ∼ 10, since currently mainstream probes BAO and SNe Ia
+can not give direct observations at high redshifts. The lack of stable high redshift observations will prevent us from
+testing ΛCDM more completely during the early stage of the evolution of our universe.
+Very excitingly, the recent released high redshift galaxy observations [10–13] in the range z ∈ [7, 11] by JWST,
+which contains a population of surprisingly massive galaxy candidates with stellar masses of order of 109M⊙, can
+help explore whether ΛCDM is valid at high redshifts.
+In the literature, Refs.[10, 11, 14, 15] have reported the
+cumulative stellar mass density (CSMD) estimated from early JWST data is higher than that predicted by ΛCDM
+within z ∈ [7, 11]. Ref.[16] points out that dynamical dark energy (DDE) can explain this anomalous signal and the
+corresponding constraint on DDE is displayed. Subsequently, if the nature of dark matter (DM) is fuzzy, this high
+SMD can be recovered [17]. Furthermore, Ref.[18] discusses under which circumstances primordial non-Gaussianity
+can act as a solution.
+Since these high redshift galaxy observations from JWST have important implications on cosmology and astro-
+physics, we attempt to probe whether early JWST data indicates any possible signal of new physics. Specifically, we
+study three classes of beyond ΛCDM cosmological models, i.e., DM-baryon interaction (DMBI), modified gravity (MG)
+and DDE. In addition, we consider the case of the extended halo mass function (HMF). We find that Within current
+cosmological constraints from Planck-2018 CMB obervations, these three models all fail to explain this large tension.
+A possibly successful scenario to escape from cosmological constraints is the extended Press-Schechter formalism.
+This study is outlined in the following manner. In the next section, we introduce the basic formula of CSMD. In
+Section III, we review briefly the alternative cosmological models and extended Press-Schechter HMF. In Section IV,
+numerical results are displayed. The discussions and conclusions are presented in the final section.
+II.
+BASIC FORMULA
+As shown in Ref.[10], the CSMD from early JWST data has a large excess relative to that predicted by ΛCDM.
+To explain this excess, we shall briefly introduce the basic formula of the cumulative SMD. The HMF for a given
+∗Electronic address: [email protected]
+arXiv:2301.00347v1  [astro-ph.CO]  1 Jan 2023
+
+2
+cosmological model reads as
+dn
+dM = F(ν) ρm
+M 2
+����
+d ln σ
+d ln M
+���� ,
+(1)
+where the function F(ν) for the Press-Schechter HMF [19] is expressed as
+F(ν) =
+�
+2
+π νe− ν2
+2 ,
+(2)
+and ρm denotes the average background matter density, M the halo mass, σ the variance of smoothed linear matter
+density field and reads as
+σ2(R) =
+1
+2π2
+� ∞
+0
+k2P(k)W 2(kR)dk,
+(3)
+where k is the comoving wavenumber, P(k) the matter power spectrum, W(kR) = 3(sin kR − kR cos kR)/(kR)3 the
+Fourier transformation of a spherical top-hat filter with radius R = [3M/(4π¯ρ0)]1/3, ν = δc/[D(z) σ] [20] (δc = 1.686
+is the critical collapsed density) and D(z) = g(z)/[g(0)(1 + z)] the linear growth factor for a specific cosmological
+model, where g(z) for ΛCDM reads as
+g(z) = 5
+2Ωm(z)
+�
+Ωm(z)
+4
+7 − ΩΛ(z) +
+�
+1 + Ωm(z)
+2
+� �
+1 + ΩΛ(z)
+70
+��−1
+,
+(4)
+where Ωm(z) and ΩΛ(z) are energy densities of matter and dark energy (DE) at a given redshift, respectively.
+An effective quantity to study the validity of the ΛCDM model is the CSMD ρ⋆, which can be characterized by a
+fraction of baryon mass contained within a given DM halo above a certain mass scale M⋆ and reads as
+ρ⋆(> M⋆, z) = ϵfb
+� z2
+z1
+� ∞
+M⋆
+ϵfb
+dn
+dM MdM dV
+dz
+dz
+V (z1, z2),
+(5)
+where ϵ is the star formation efficiency, fb the baryon fraction and V (z1, z2) the comoving volume in the redshift range
+z ∈ [z1, z2].
+III.
+ALTERNATIVE MODELS
+A.
+Dark matter-baryon interaction
+Up to now, the standard cosmological paradigm indicates that DM is cold, collisionless and only participates in
+gravitational interactions [9]. In light of the lack of experimental detections of DM and emergent cosmological tensions
+in recent years, the scenario beyond the standard DM assumption becomes more and more attractive. An interesting
+category is interactions between DM and the Standard Model particles such as baryons, photons and neutrinos. In
+this study, we consider the case of DMBI.
+The interaction between DM and baryons produces a momentum exchange proportional to momentum transfer
+cross section, which can be shown as
+σT =
+�
+(1 − cos θ)dΩ d¯σ
+dΩ,
+(6)
+In the weakly coupled theory, σT can just depend on even powers of DM-baryon relative velocity v and, in general, it is
+a power law function of v. Here we adopt σT = σDM−bvnb and denote the DMBI cross section as σDM−b. Specifically,
+we study the mini-charged DM (DM particle with a fractional electric charge) corresponding to the case of nb = −4,
+which has been used to explain the anomalous 21 cm signal from EDGES [21].
+For this model, we introduce two basic assumptions: (i) DM and baryons obey the Maxwell velocity distribution; (ii)
+both species are non-relativistic. As a consequence, the Euler equation of DM can obtain an extra term ΓDM−b(θb −
+θDM), where ΓDM−b is the conformal DM-baryon momentum exchange rate, and θDM and θb represent the velocities
+of DM and baryons, respectively. At leading order, ΓDM−b is expressed in terms of DM bulk velocity and reads as
+[22]
+ΓDM−b = aρbfHeσDM−bc−4
+mDM + mb
+� TDM
+mDM
++ Tb
+mb
++ V 2
+RMS
+3
+�−1.5
+,
+(7)
+
+3
+where a is the scale factor, ρb the average baryon energy density, fHe ≃ 0.76, c−4 = 0.27 the integration constant
+(see [22, 23] for details), and Ti and mi denote the temperature and average mass of species i, respectively. The bulk
+velocity dispersion can be shown as [24]
+V 2
+RMS =
+�
+�
+�
+10−8,
+z > 103
+(1 + z)2
+10
+,
+z ≤ 103 .
+(8)
+The interaction between DM and baryons can produce the energy and momentum exchange. It is clear that DMBI
+reduces to ΛCDM when σDM−b = 0. There is a possibility that DMBI can increase the baryon fraction and conse-
+quently give a large star formation efficiency. This indicates that DMBI can act as a potential solution to the recent
+puzzle from JWST data.
+B.
+Modified gravity
+Since general relativity (GR) can not explain current cosmic expansion in the absence of cosmological constant, the
+modifications in the gravity sector on cosmic scales has inspired a broad interest in order to describe this anomalous
+phenomenon. Here we shall consider the simplest extension to GR, f(R) gravity, where the modification is a function
+of Ricci scalar R. f(R) gravity was firstly introduced by Buchdahl [25] in 1970 and more detailed information can be
+found in recent reviews [26, 27]. Its action is written as
+S =
+�
+d4x√−g
+�f(R)
+2
++ Lm
+�
+,
+(9)
+where Lm and g denote the matter Lagrangian and the trace of a given metric, respectively.
+For the late-time universe, a viable f(R) gravity scenario should explain the cosmic expansion, pass the local
+gravity test and satisfy the stability conditions. To investigate whether MG can explain the high redshift galaxy
+data from JWST, in this study, we consider the so-called Hu-Sawicki f(R) model (hereafter HS model) [28], which is
+characterized by
+f(R) = R −
+2ΛR¯n
+R¯n + µ2¯n ,
+(10)
+where ¯n and µ are two free parameters characterizing this model. By taking R ≫ µ2, the approximate f(R) function
+can be expressed as
+f(R) = R − 2Λ − fR0
+¯n
+R¯n+1
+0
+R¯n ,
+(11)
+where R0 is the present-day value of Ricci scalar and fR0 = −2Λµ2/R2
+0. Note that HS f(R) gravity reduces to ΛCDM
+when fR0 = 0.
+An intriguing question is whether recent JWST anomaly is a signal of beyond GR. We will carefully analyze this
+possibility in this study.
+C.
+Dynamical dark energy
+Although Ref.[16] has claimed that DDE can explain the large CSMD from JWST, we think their method is
+inappropriate and consequently their result maybe incorrect. We need to reanalyze the case of DDE.
+As is well known, the equation of state (EoS) of DE w = −1 in the standard cosmological model. However, starting
+from observations, the doubt about the correctness of ΛCDM stimulates the community to explore whether DE is
+dynamical over time or not. In general, one depicts the DDE model by a simple Taylor expansion of DE EoS, i.e.,
+ω(a) = ω0 + (1 − a)ωa [29, 30], where ωa characterizes the time evolution of DE EoS. The dimensionless Hubble
+parameter is expressed as
+EDDE(z) =
+�
+Ωm(1 + z)3 + (1 − Ωm)(1 + z)3(1+ω0+ωa)e
+−3ωaz
+1+z
+� 1
+2
+.
+(12)
+Note that this model is a two-parameter extension to ΛCDM and it reduces to ΛCDM when ω0 = −1 and ωa = 0.
+
+4
+D.
+Extended halo mass function
+When applied into a complicated gravity system, the function of Press-Schechter HMF is limited, since it does
+not consider the nonlinear environmental effects. To overcome this shortcoming, the extended Press-Schechter (EPS)
+HMF is proposed in Ref.[31] and reads as
+dn(M1, z|M2, δ2)
+dM1
+= M2
+M1
+fm(S1, δ1|S2, δ2)
+����
+dS1
+dM1
+���� ,
+(13)
+where the mass variance S1 = σ2(M1) and S2 = σ2(M2) (see Eq.(3)), and one can obtain the average number of
+progenitors at time t1 in the mass range (M1, M1 + dM1) which by time t2 (t2 > t1) have merged to form a large halo
+of mass M2. The multiplicity function fm is expressed as
+fm(S1, δ1|S2, δ2) =
+1
+√
+2π
+δ1 − δ2
+(S1 − S2)3/2 exp
+�
+− (δ1 − δ2)2
+2(S1 − S2)
+�
+dS1.
+(14)
+δ1 and δ2 are, respectively, the linear overdensities in spherical regions of masses M1 and M2. To study the environ-
+mental impacts on the high redshift HMF, we choose M2 as a present-day halo corresponding to current overdensity
+δ2. To compute δ2, one should transform the nonlinear overdensity δnl at redshift z in Eulerian space into the linear
+overdensity in Lagrangian space. The corresponding analytic fitting formula based on spherical collapse model is
+[32, 33]
+δ2(δnl, z) =
+δ1
+1.68647
+�
+1.68647 −
+1.35
+(1 + δnl)2/3 −
+1.12431
+(1 + δnl)1/2 +
+0.78785
+(1 + δnl)0.58661
+�
+.
+(15)
+Since there is a possibility that the excessively high CSMD from JWST is caused by nonlinear environmental effect,
+we attempt to explain it using the EPS formalism.
+IV.
+METHODS AND RESULTS
+At first, we employ the best fits from current cosmological constraints as our baseline values for four models. Since
+we hope that the following calculations can be permitted by present-day observations, our discussions and results
+will mainly focus on the allowed parameter space. Then, for different models, we use different Boltzmann codes to
+calculate their background evolution, growth factors and matter power spectrum at different redshifts. Specifically,
+we take CLASS [22, 23, 34] for DMBI and use modified CAMB [35, 36] for f(R) gravity, DDE and EPS scenarios. Note
+that ΛCDM is adopted in the EPS scenario. Subsequently, we compute the HMF at different redshifts for the above
+four models. Finally, we work out the maximal CSMD for each model according to the permitted parameter space,
+and check whether these scenarios are consistent with the latest JWST data. Notice that Eq.(4) is only used in the
+EPS model and the growth factors of the other three models are obtained from the corresponding software package.
+Our numerical analysis results are presented in Figs.1-3. At first, we display the CSMD of ΛCDM in the redshift
+range z ∈ [7, 9] and see its performance. In general, the SFE ϵ is about 10% according to current observational
+constraints [11]. Nonetheless, one can see that in the top left panel of Fig.1, 10% is nowhere near enough to reach
+the lower bounds of JWST data points in ΛCDM. One needs the star formation rate in galaxies to be at least 50%
+in order to explain the inconsistency. In the meanwhile, one can easily find that ϵ = 0.8 can successfully explain two
+data points but 100% SFE can not. Except for ΛCDM, we all calculate the maximal CSMD in the other models, i.e.,
+assuming ϵ = 1.
+In the second place, we make a trial to explore whether alternative cosmological models can alleviate even solve
+the tension between JWST and Planck CMB observations. In the DMBI case, we attempt to acquire a higher baryon
+fraction by the coupling between DM and baryons, and consequently explain this discrepancy occurred in ΛCDM.
+However, we find that varying coupling strength σDM−b hardly affects the CSMD, and only the variation of interaction
+DM fraction Ωidm affects significantly the CSMD. When assuming the DM particle mass mDM = 100 GeV, the cross
+section σDM−b = 10−42 cm2 and choosing the fraction Ωidm = 0.01, 0.03 and 0.05, this tension can be efficiently
+relieved but it seems that this model is difficult to explain both data points. However, if considering the current
+cosmological constraint that gives a very small Ωidm [24], DMBI still behaves like ΛCDM and can not resolve this
+discrepancy. We have also studied the impacts of mDM and find different DM particle masses also can not explain
+JWST data.
+In f(R) gravity, we find small fR0 such as 0.1 and 1 can not expalin the anomaly but a very large value fR0 = 10
+can do. This implies that one needs a large deviation from GR to be responsible for JWST data. Unfortunately, the
+
+5
+109
+1010
+1011
+M
+102
+103
+104
+105
+106
+107
+108
+( > M )[M
+Mpc
+3]
+CDM, = 1
+CDM, = 0.8
+CDM, = 0.5
+CDM, = 0.1
+109
+1010
+1011
+M
+103
+104
+105
+106
+107
+108
+( > M )[M
+Mpc
+3]
+CDM
+idm=0.01, mDM=100 GeV
+idm=0.03, mDM=100 GeV
+idm=0.05, mDM=100 GeV
+109
+1010
+1011
+M
+103
+104
+105
+106
+107
+108
+( > M )[M
+Mpc
+3]
+CDM
+fR0 = 0.1
+fR0 = 1
+fR0 = 10
+109
+1010
+1011
+M
+102
+103
+104
+105
+106
+107
+108
+( > M )[M
+Mpc
+3]
+nl = 1
+nl = 0.7
+nl = 0.6
+nl = 0.5
+nl = 0.4
+nl = 0.1
+CDM
+FIG. 1: The CSMDs for the ΛCDM, DMBI, f(R) gravity and EPS models are shown from top to bottom and left to right,
+respectively. Note that for ΛCDM, we compute the CSMDs in the redshift range z ∈ [7, 9] by choosing different values of the
+SFE ϵ. For the other models, we calculate the CSMDs in the redshift range z ∈ [9, 11] when ϵ = 1.
+latest cosmological constraint gives log10 fR0 < −6.32 at the 2 σ confidence level [37], which is much smaller than 10.
+Therefore, similar to DMBI, f(R) gravity also fails to alleviate this tension. Interestingly, this gives us a hint that, if
+two galaxies observed by JWST are located in the low density region of the universe where MG effect is very large,
+the data can be appropriately explained.
+Furthermore, we are interested in whether the nature an simple extension to ΛCDM, DDE, can explain the incon-
+sistency. As mentioned above, Ref.[16] claimed that JWST data can clearly constrain DDE. However, within current
+constraining precision, we query this conclusion. To ensure the validity of our conclusion, we constrain ΛCDM and
+DDE models using the Planck-2018 CMB temperature and polarization data (see Fig.2), and then obtain the best
+fitting values of parameters of these two models. One can easily find the constrained values of model parameters
+of ΛCDM in Ref.[5]. For DDE, we obtain current baryon and CDM densities Ωbh2 = 0.0225 and Ωch2 = 0.1184,
+the ratio between angular diameter distance and sound horizon at the redshift of last scattering θMC = 1.04109,
+the optical depth due to the reionization τ = 0.06, the amplitude and spectral index of primordial power spectrum
+As = 2.114 × 10−9 and ns = 0.9698, and two DE EoS parameters ω0 = −0.38 and ωa = −4.8. Same as DMBI
+and f(R) gravity models, we use the same method to work out the CSMD of DDE, and find that the variation of
+the CSMD is largely dominated by the values of six basic parameters Ωbh2, Ωch2, θMC, τ, As and ns. Although
+ω0 and ωa is loosely constrained by CMB data (constrained ω0-ωa parameter space is large), different values of ω0
+and ωa hardly affect the CSMD. For instance, in the left panel of Fig.3, ω0 = −0.38 and ωa = −4.8 plus the ΛCDM
+and DDE best fits gives completely different CSMDs. Choosing the ΛCDM best fit, (ω0, ωa) = (−0.38, −4.8) and
+
+6
+0.0220
+0.0224
+0.0228
+bh2
+0.7
+0.8
+0.9
+1.0
+1.1
+8
+0.2
+0.3
+0.4
+0.5
+m
+50
+60
+70
+80
+90
+H0
+8
+6
+4
+2
+0
+2
+4
+wa
+2
+1
+0
+w
+3.00
+3.05
+3.10
+ln(1010As)
+0.96
+0.97
+0.98
+ns
+0.02
+0.04
+0.06
+0.08
+0.10
+1.0400
+1.0405
+1.0410
+1.0415
+1.0420
+100
+MC
+0.114
+0.116
+0.118
+0.120
+0.122
+0.124
+ch2
+0.114
+0.117
+0.120
+0.123
+ch2
+1.040
+1.041
+1.042
+100
+MC
+0.02
+0.04
+0.06
+0.08
+0.10
+0.96
+0.97
+0.98
+ns
+3.00
+3.05
+3.10
+ln(1010As)
+2
+1
+0
+w
+8
+4
+0
+4
+wa
+50
+60
+70
+80
+90
+H0
+0.2
+0.3
+0.4
+0.5
+m
+0.7
+0.8
+0.9
+1.0
+1.1
+8
+DDE
+CDM
+FIG. 2: The marginalized posterior probability distributions of the ΛCDM and DDE models from the Planck-2018 CMB
+constraints are shown.
+(ω0, ωa) = (−1, −1) gives very similar results in the logarithmic space. In the medium and right panels of Fig.3, we
+verifies that taking same best fits of ΛCDM and DDE, respectively, choosing different DE EoS parameter pair just
+produces very limited differences. After scanning the DDE parameter space, we find clearly that DDE also can not
+explain this tension, but its best fit can help increase the value of CSMD and become closer to JWST data points (see
+the left panel of Fig.3). The reason that the result in Ref.[16] is different from ours is that they do not implement an
+appropriate cosmological constraint based on the Planck CMB data.
+The result from f(R) gravity prompts us to study the environmental effect of JWST galaxies on the CSMD. The
+most straightforward method is replacing the Press-Schechter HMF with the EPS formalism in the framework of
+ΛCDM, where the sole parameter δnl characterizes the nonlinear environmental effect of a high redshift halo. In the
+bottom right panel, we calculate the maximal CSMDs in the redshift range z ∈ [9, 11] for the EPS model. We find
+that neither overlarge (δnl = 1) nor too small (δnl = 0.1) explain JWST observations and that the larger δnl is, the
+larger the CSMD is. Since the total sky area covered by the JWST initial observation is large enough (∼ 40 armin2)
+
+7
+109
+1010
+1011
+M
+103
+104
+105
+106
+107
+108
+( > M )[M
+Mpc
+3]
+CDM
+w0=-0.38, wa=-4.8 + DDE best fit
+w0=-0.38, wa=-4.8 + CDM best fit
+w0=-1, wa=-1 + CDM best fit
+3.60
+3.62
+3.64
+3.66
+3.68
+3.70
+M
+1e10
+12000
+12200
+12400
+12600
+12800
+13000
+( > M )[M
+Mpc
+3]
+DDE w0=-0.38, wa=-4.8
+DDE w0=-1, wa=-5
+DDE w0=-1, wa=-1
+DDE w0=-1, wa=0
+3.60
+3.62
+3.64
+3.66
+3.68
+3.70
+M
+1e10
+42000
+42500
+43000
+43500
+44000
+44500
+45000
+( > M )[M
+Mpc
+3]
+DDE w0=-0.38, wa=-4.8
+DDE w0=-1, wa=-5
+DDE w0=-1, wa=-1
+DDE w0=-1, wa=0
+FIG. 3: The CSMDs of the DDE model computed at in the redshift range z ∈ [9, 11] are shown when assuming ϵ = 1. Left:
+Different combinations of parameter values and best fits from constraints, respectively. Medium: Only the ΛCDM best fit;
+Right: Only the DDE best fit.
+[10], we can not rule out this possibly local environmental effect. However, unfortunately, there is no δnl passing two
+data points simultaneously.
+V.
+DISCUSSIONS AND CONCLUSIONS
+Recently, the early data release of JWST reveals the possible existence of high redshift galaxies. What is interesting
+is these galaxies in the redshift range z ∈ [7, 11] exhibit the overlarge star formation rate, which is incompatible with
+the prediction of the standard cosmology. This may indicate that JWST data contain the signal of new physics.
+In this study, we try to resolve this tension with alternative cosmological models including DMBI, f(R) gravity
+and DDE. We find that in light of the precision of current cosmological constraint from Planck-2018 CMB data, these
+models all fail to explain this large tension. Specifically, for DMBI, the coupling strength σDM−b between DM and
+baryons hardly affects the CSMD. For f(R) gravity, the effect of varying fR0 on the CSMD is too small to relive the
+tension. For DDE, although the constrained DE EoS parameter space is large, different parameter pair (ω0, ωa) just
+produces very limited differences in the CSMD. Interestingly, a large interacting DM fraction and a large deviation
+from Einstein’s gravity can both generate a large CSMD.
+A possible scenario to escape from current cosmological constraints is the EPS formalism, where we consider the
+local environmental effect on the CSMD. We find that an appropriate value of nonlinear environmental overdensity
+of a high redshift halo can well explain the CSMD discrepancy. However, we do not find an EPS model that can
+simultaneously explain two data points.
+In the near future, JWST will bring more useful data to human beings, so that we can extract more physical
+information to uncover the mysterious veil of nature.
+Acknowledgments
+DW warmly thanks Liang Gao, Jie Wang and Qi Guo for helpful discussions. We thank Hang Yang for letting us
+notice the JWST related works. This study is supported by the National Nature Science Foundation of China under
+Grants No.11988101 and No.11851301.
+[1] A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae for an accelerating universe and a
+cosmological constant,” Astron. J. 116, 1009-1038 (1998).
+[2] S. Perlmutter et al. [Supernova Cosmology Project], “Measurements of Ω and Λ from 42 high redshift supernovae,”
+Astrophys. J. 517, 565-586 (1999).
+[3] D. N. Spergel et al. [WMAP Collaboration], “First year Wilkinson Microwave Anisotropy Probe (WMAP) observations:
+Determination of cosmological parameters,” Astrophys. J. Suppl. 148, 175-194 (2003).
+[4] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. XVI. Cosmological parameters,” Astron. Astrophys. 571,
+A16 (2014).
+
+8
+[5] N. Aghanim et al. [Planck Collaboration], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641,
+A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)].
+[6] C. Blake and K. Glazebrook, “Probing dark energy using baryonic oscillations in the galaxy power spectrum as a cosmo-
+logical ruler,” Astrophys. J. 594, 665-673 (2003).
+[7] H. J. Seo and D. J. Eisenstein, “Probing dark energy with baryonic acoustic oscillations from future large galaxy redshift
+surveys,” Astrophys. J. 598, 720 (2003).
+[8] E. Di Valentino et al., “Snowmass2021 - Letter of interest cosmology intertwined I: Perspectives for the next decade,”
+Astropart. Phys. 131, 102606 (2021).
+[9] E. Abdalla et al., “Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with
+the cosmological tensions and anomalies,” JHEAp 34, 49-211 (2022).
+[10] I. Labbe et al., “A very early onset of massive galaxy formation,” [arXiv:2207.12446 [astro-ph.GA]].
+[11] Y. Harikane et al., “A Comprehensive Study on Galaxies at z 9-16 Found in the Early JWST Data: UV Luminosity
+Functions and Cosmic Star-Formation History at the Pre-Reionization Epoch,” [arXiv:2208.01612 [astro-ph.GA]].
+[12] R. Naidu et al., “Two Remarkably Luminous Galaxy Candidates at z ≃ 10 − 12 Revealed by JWST,” [arXiv:2207.09434
+[astro-ph.GA]].
+[13] M. Castellano et al., “Early results from GLASS-JWST. III: Galaxy candidates at z ∼ 9 − 15,” [arXiv:2207.09436 [astro-
+ph.GA]].
+[14] M. Boylan-Kolchin, “Stress Testing ΛCDM with High-redshift Galaxy Candidates,” [arXiv:2208.01611 [astro-ph.CO]].
+[15] C. Lovell et al., “Extreme Value Statistics of the Halo and Stellar Mass Distributions at High Redshift: are JWST Results
+in Tension with ΛCDM?,” [arXiv:2208.10479 [astro-ph.GA]].
+[16] N. Menci et al., “High-redshift Galaxies from Early JWST Observations: Constraints on Dark Energy Models,” Astrophys.
+J. Lett. 938, no.1, L5 (2022).
+[17] Y. Gong, B. Yue, Y. Cao and X. Chen, “Fuzzy Dark Matter as a Solution to the Stellar Mass Density of High-z Massive
+Galaxies,” [arXiv:2209.13757 [astro-ph.CO]].
+[18] M. Biagetti, G. Franciolini and A. Riotto, “The JWST High Redshift Observations and Primordial Non-Gaussianity,”
+[arXiv:2210.04812 [astro-ph.CO]].
+[19] W. H. Press and P. Schechter, “Formation of galaxies and clusters of galaxies by selfsimilar gravitational condensation,”
+Astrophys. J. 187, 425-438 (1974).
+[20] S. M. Carroll, W. H. Press and E. L. Turner, “The Cosmological constant,” Ann. Rev. Astron. Astrophys. 30, 499-542
+(1992).
+[21] R. Barkana, “Possible interaction between baryons and dark-matter particles revealed by the first stars,” Nature 555,
+no.7694, 71-74 (2018).
+[22] C. Dvorkin, K. Blum and M. Kamionkowski, “Constraining Dark Matter-Baryon Scattering with Linear Cosmology,” Phys.
+Rev. D 89, no.2, 023519 (2014).
+[23] W. L. Xu, C. Dvorkin and A. Chael, “Probing sub-GeV Dark Matter-Baryon Scattering with Cosmological Observables,”
+Phys. Rev. D 97, no.10, 103530 (2018).
+[24] N. Becker et al., “Cosmological constraints on multi-interacting dark matter,” JCAP 02, 019 (2021).
+[25] H. A. Buchdahl, “Non-linear Lagrangians and cosmological theory,” Mon. Not. Roy. Astron. Soc. 150, 1 (1970).
+[26] A. De Felice and S. Tsujikawa, “f(R) theories,” Living Rev. Rel. 13, 3 (2010).
+[27] T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. 82, 451-497 (2010).
+[28] W. Hu and I. Sawicki, “Models of f(R) Cosmic Acceleration that Evade Solar-System Tests,” Phys. Rev. D 76, 064004
+(2007).
+[29] M. Chevallier and D. Polarski, “Accelerating universes with scaling dark matter,” Int. J. Mod. Phys. D 10, 213-224 (2001).
+[30] E. V. Linder, “Exploring the expansion history of the universe,” Phys. Rev. Lett. 90, 091301 (2003).
+[31] J. R. Bond, S. Cole, G. Efstathiou and N. Kaiser, “Excursion set mass functions for hierarchical Gaussian fluctuations,”
+Astrophys. J. 379, 440 (1991).
+[32] H. J. Mo and S. D. M. White, “An Analytic model for the spatial clustering of dark matter halos,” Mon. Not. Roy. Astron.
+Soc. 282, 347 (1996).
+[33] R. K. Sheth and G. Tormen, “An Excursion Set Model of Hierarchical Clustering : Ellipsoidal Collapse and the Moving
+Barrier,” Mon. Not. Roy. Astron. Soc. 329, 61 (2002).
+[34] T. R. Slatyer and C. L. Wu, “Early-Universe constraints on dark matter-baryon scattering and their implications for a
+global 21 cm signal,” Phys. Rev. D 98, no.2, 023013 (2018).
+[35] A. Lewis, A. Challinor and A. Lasenby, “Efficient computation of CMB anisotropies in closed FRW models,” Astrophys.
+J. 538, 473-476 (2000).
+[36] A. Lewis, “Efficient sampling of fast and slow cosmological parameters,” Phys. Rev. D 87, no.10, 103529 (2013).
+[37] D. Wang, “Pantheon+ constraints on dark energy and modified gravity: An evidence of dynamical dark energy,” Phys.
+Rev. D 106, no.6, 6 (2022).
+