It is shown that an LTI system is a relaxation system if and only if its
Hankel operator is cyclic monotone.  Cyclic monotonicity of the Hankel
operator implies the existence of a storage function whose gradient is the
Hankel operator.  This storage is a function of past inputs alone, is
independent of the state space realization, and admits a generalization to
nonlinear circuit elements.

§ INTRODUCTION

Relaxation systems are a class of LTI systems which first arose in the study of relaxation phenomena in viscoelastic
materials, and, in the finite dimensional case, correspond to RC and RL circuits
Meixner1964.  Relaxation systems are highly structured.  They correspond
to systems with completely monotonic impulse responses, with transfer functions which are sums of first order lags Bernstein1928,
Widder1941, Meixner1964 and it was shown by Willems1972a that they admit
state space realizations which are both externally symmetric, corresponding to the
circuit property of reciprocity, and internally symmetric, encoding the fact that all
the energy storage elements are of the same type.
There has been a recent revival of interest in relaxation systems
Pates2019, Pates2022, Grussler2022, Grussler2022a, Bar-Shalom2023, Yafaev2015,
Yafaev2015a, Margaliot2019. For example, it was observed by Pates et al.
Pates2019, Pates2022 that they
admit very simple $H_\infty$-optimal controllers, with highly structured circuit

Dissipativity theory Willems1972 connects the circuit theory of passivity to the dynamical
systems theory of stability via the storage function, which represents the
energy stored in a system.
For a relaxation system, there exists a
storage function which is completely determined by the Hankel operator, that is, the
future output in response to a past input Willems1972a. Relaxation
systems therefore represent a class of systems for which
the storage can be defined externally, as a function of past input only.

Existing characterizations of relaxation systems rely on linearity
and time invariance.
We are motivated by a characterization that is not
limited to LTI systems. This paper presents some preliminary steps in this direction,
through connections to monotone operator theory.  The property of
monotonicity was originally introduced in efforts to generalize the property of
passivity to networks of nonlinear resistors Duffin1946, Minty1960,
Minty1961, Minty1962.  Monotone operator theory now forms a pillar of convex
optimization theory Rockafellar1976, Bauschke2011, Combettes2018, Ryu2022a,
owing to the fact that the gradient of a convex function is a monotone operator.

An early question in the theory of monotone operators was when the converse is true,
when is a monotone operator the gradient of a convex function?  This question was
answered by Rockafellar1966, Rockafellar1970b, who showed that a stronger
property than monotonicity is required: cyclic monotonicity.

In this paper, we reconnect the property of cyclic monotonicity with its
circuit theoretic origins, showing that cyclic monotonicity corresponds precisely to
relaxation, that is, to circuits with a single type of energy storage element.  Our
main result shows that an equivalent characterization of relaxation is that a
system's Hankel operator is cyclic monotone. For single input, single output LTI
operators, this equivalence was shown independently in the recent work of Yafaev
Yafaev2015, Yafaev2015a. Our proof is MIMO, and uses a state space
representation.  Cyclic monotonicity of the Hankel operator
implies that it is the gradient of some convex functional, and we show that this convex
functional is precisely the intrinsic storage of a relaxation system observed by
Willems.  Because cyclic monotonicity is not restricted to linear systems, our
characterization opens the way to a nonlinear concept of relaxation.

Cyclic monotonicity has previously been studied in the context of Lur'e
systems Adly2017a, Scherer2022, multi-agent systems
Sharf2019 and recently in the context of incrementally
port-Hamiltonian systems Camlibel2023, where it was shown that
a port-Hamiltonian system with a maximal cyclic monotone Dirac structure may be defined
in terms of a convex function of the state and input.  In contrast, we consider
cyclic monotonicity of an external map, the Hankel operator, that maps past inputs to future outputs.

The remainder of this paper is structured as follows.  In Section <ref>,
we introduce the necessary preliminary material from the theory of passivity and
monotone operators.
In Section <ref>, we give the first of our
main results, that relaxation is equivalent to cyclic monotonicity of the Hankel
operator.  In Section <ref>, we introduce a new notion of an
intrinsic storage functional and show that the convex functional whose
gradient is the Hankel operator is the intrinsic storage of Willems. Conclusions
and directions for future work are given in Section <ref>.

§ PRELIMINARIES

§.§ State space systems and Hankel operators

We study linear, time-invariant state space systems of the form
ẋ(t)    =    A x(t) + B u(t)

y(t)    =    C x(t) + D u(t),

where $x(t) \in \R^n$, $u(t) \in \R^m$, $y(t) \in \R^p$ $A \in \R^{n\times n}$,  $B \in \R^{n
\times m}$, $C \in \R^{p \times n}$ and $D \in \R^{p \times m}$.
A system is said to be stable if $A$ is Hurwitz, and minimal if
$(A, B)$ is controllable and $(A, C)$ is observable.  The transfer function of
system (<ref>) is given by $H(s) := C(sI - A)^{-1}B + D$, and the impulse response is given by
h(t) := D\delta(t) + Ce^{A t}B,
$ where $\delta(t)$ denotes the Dirac delta.  We also define $g(t) := Ce^{A t}B$ to be
the impulse response of the system with no feedthrough term.

A complete inner product space is called a Hilbert space.
The space $L_{2}(\R, \R^n)$ is the set of signals $u:
\R \to \R^n$ such that
∫_-∞^∞u(t)u(t) t < ∞.

This space forms a Hilbert space of equivalence classes of
functions when equipped with the inner product
uy := ∫_-∞^∞u(t)y(t) t,

which induces the norm $\norm{u} := \sqrt{\ip{u}{u}}$.  We define $L_2(\R_{\geq 0},
\R^n)$ and
$L_2(\R_{\leq 0}, \R^n)$ similarly, but with  time axes of $[0, \infty)$ and  $(-\infty, 0]$,
respectively.  We will use the shorthand notation $L_2^n$ for $L_2(\R_{\geq 0},
\R^n)$.

A stable system admits a Hankel operator, which maps an input
on $L_{2}(\R_{\leq 0}, \R)$ to the corresponding output on $L_{2}(\R_{\geq 0}, \R)$,
assuming zero input from time $0$.  Given an impulse response $h$ and input $\bar{u} \in L_{2}(\R_{\leq 0}, \R)$, the
output of the Hankel operator $\Gamma_h$ at time $t$ is given by
y(t)    =    ∫_-∞^0 h(t - τ)u̅(τ) τ.

Letting $u(t) := \bar{u}(-t)$,
the Hankel operator has the expression
(Γ_h u)(t)    :=    ∫_0^∞h(t + τ)u(τ) τ,

and defines an operator on $L_{2}(\R_{\geq 0}, \R)$.  If the system is stable, the
Hankel operator is continuous [Prop. 4.1]Partington1988.

For the remainder of this paper, we will consider systems which are square,
that is, the input dimension $m$ is equal to the output dimension $p$.

§.§ Passivity, reciprocity and relaxation

Passivity is a formalization of the notion that a system can be realized without any
internal power source.  Central to the theory of passivity is the storage function,
which represents the energy stored within a system. We recall the following
definition of passivity.

A system of the form (<ref>) is said to be passive if, for
any input/output trajectory $(u, y)$ of the system and $t_0 \in \R$, there
exists a $K \in \R$ such that, if $(\hat u, \hat y)$ is also an input/output
trajectory of the system and $(\hat u(t), \hat y(t)) = (u(t), v(t))$ for all
$t < t_0$, then
-∫_t_0^t_1 û(t)ŷ(t) t ≤K

for all $t_1 \geq t_0$.

It is shown in [Thm. 13]Hughes2017c that, for a (not necessarily minimal)
system of the form (<ref>), Definition <ref> is
equivalent to the existence of a matrix $Q = Q\tran \succeq 0$ satisfying the linear matrix
AQ + Q A     Q B - C

B Q - C     - D - D ≼0.

This is precisely the condition given by [Thm. 3]Willems1972a in the
context of minimal LTI state space systems.

A signature matrix is a diagonal matrix whose diagonal entries are either $1$ or

A system of the form (<ref>) is said to be (externally)
reciprocal with respect to the signature matrix $\Sigma_e$ if  $\Sigma_e
H(s) = \Sigma_e H(s)\tran$, where $H(s)$ is the transfer matrix of

Reciprocal systems admit internally reciprocal state space realizations.

A system of the form (<ref>) is reciprocal if and only if it admits a state space realization
$(A, B, C, D)$ such that
Σ_i     0
0     Σ_e
-A     -B
C     D
-A    C
-B    D
Σ_i     0
0     Σ_e ,

where $\Sigma_i$ is a signature matrix.

We now define relaxation systems, the main subject of this paper.

A system of the form (<ref>) is said to be a relaxation system if
$D = D\tran \succeq 0$ and $g(t) = Ce^{A t}B$ is a completely
monotonic function for $t \in [0, \infty)$:
g(t)    =    g(t) for all  t ≥0,

(-1)^k ^kt^k g(t)    ≽     0  for all  k = 1, 2, …
and  t ≥0.

Relaxation systems first arose in the context of viscoelastic materials
Meixner1964, and, in the context of electrical circuits, correspond to the
impedances of RC circuits and the admittances of RL circuits.
Several equivalent characterizations of relaxation systems are
known in the literature Meixner1964, Willems1972a, Willems1976,
Marcus1975, Bernstein1928, Aissen1951, which we collect in the following theorem.

Consider a system of the form (<ref>). Then the following are

* the system is a relaxation system.

* $H(s)$ admits the form
H(s)    =    G_0 + G_1/s + ∑_i=2^n G_i/s + λ_i,

where $G_i = G_i\tran \succeq 0$ for all $i$ and $0 \leq \lambda_0 < \lambda_1 < \ldots
< \lambda_N$, for some $N \in \mathbb{Z}_{\geq 0}$.

* $H(s)$ admits a minimal state space realization $(A_1, B_1, C_1, D_1)$ such that
A_1    =    A_1≼0

B_1    =    C_1

D_1    =    D_1≽0.

* $D  \succeq 0$,
CB     CAB     …    CA^n-1B

CAB     CA^2 B     …    CA^n B

⋮    ⋮    ⋱    ⋮

CA^n-1B     CA^nB         CA^2n-2B
≽    0

CAB     CA^2 B     …    CA^n B

CA^2B     CA^3 B     …    CA^n+1 B

⋮    ⋮    ⋱    ⋮

CA^nB     CA^n+1B         CA^2n-1B
≼    0,

and all three of these matrices are symmetric.

§.§ Cyclic monotonicity

In this section, we introduce the notions of monotonicity and cyclic monotonicity,
for operators on a Hilbert space $\hil$.

Given an operator $A: \hil \to \hil$, the graph of $A$ is the set
$\gra{A} \subseteq \hil \times \hil$ defined by
A    :=    { (u, y)   |   u ∈, y = A(u) }.

An operator $A: \hil \to \hil$ is said to be
monotone if, for all $u_1, u_2 \in \hil, y_1 = A(u_1), y_2 = A(u_2)$,
u_1 - u_2y_1 - y_2 ≥0.

If $\gra{A}$ is not properly contained within the graph of any other
monotone operator, $A$ is said to be maximal monotone.

An operator $A: \hil \to \hil$ is said to be
$n$-cyclic monotone if, for all sets of input/output pairs $\{(u_i,
y_i) \; | \; u_i \in \hil, y_i = A(u_i), i = 0, \ldots, n\}$,
y_0u_0 - u_1 + y_1u_1 - u_2 + …+ y_nu_n - u_0 ≥0.

If $A$ is $n$-cyclic monotone  for all $n \geq 1$, $A$ is said to be
cyclic monotone.  If $\gra{A}$ is not contained within the
graph of any other monotone operator, $A$ is said to be maximal
cyclic monotone.

Maximality is guaranteed for continuous operators [Cor.
20.25]Bauschke2011, so the Hankel operators associated with  the  stable linear operators considered in
this paper are automatically maximal.

An operator $A: \hil \to \hil$ is said to be self-adjoint if, for all $u, y
\in \hil$,
A (u)y    =    uA (y).

Asplund1970 gives the following characterization of the cyclic
monotonicity of a linear operator.  Given a linear operator $A: \hil \to \hil$, we define
the complexification of $A$, denoted $A_c$, by
A_c(u + jw) := A(u) + jA(w).

This operates on the complexification of $\hil$, denoted $\hil_c$. We endow this
space with the inner product
u + jwy + jv_c := uy + wv + j(wy - uv).

The numerical range of an operator $A_c$ on $\hil_c$ is defined as
W(A_c) := {A_c (z)z_c/z  |  z ∈A_c, z ≠0}.

A linear operator $A$ on $\hil$ is $n$-cyclic monotone if and only if, for
all $z \in W(A_c)$, $\arg{z} \leq \pi/n$.

For the limiting case of cyclic monotonicity, we have the following corollary.

A linear operator $A$ on $\hil$ is cyclic monotone if and only if it is
self-adjoint and, for all $u \in \dom{A}$, $\ip{A(u)}{u} \geq 0$.

$n$-cyclic monotonicity for all $n$ implies that $\arg{z} = 0$ for all $z \in
W(A_c)$.  Equivalently, $\arg{\ip{A_c (z)}{z}} = 0$ for all $z = u + jw \in
\dom{A_c}, \norm{z} \neq 0$.  Expanding the inner product:
(uA(u) + wA(w) +

j(wA(u) - uA(w)))    =    0

so  uA(u) + wA(w)    ≥    0

and  A(w)u = wA(u).

A function $f: \hil \to \R\cup\{\infty\}$ is said to be proper if its value is
never $-\infty$ and is finite somewhere, closed if
its epigraph is closed and convex if, for all $x, y \in \hil$
and $\theta \in (0, 1)$,
f(θx + (1-θ)y)    ≤    θf(x) + (1-θ)f(y).

Our interest in cyclic monotonicity stems from the following theorem of Rockafellar.

A continuous operator $A: \hil \to \hil$ is maximal cyclic monotone if and only if it is the
gradient of a closed, convex and proper function from $\hil$
to $(-\infty, \infty]$.  Moreover, this function is uniquely determined by
$A$ up to an additive constant.

§ RELAXATION AND CYCLIC MONOTONICITY

In this section, we establish the relationship between relaxation systems and cyclic
monotone operators, and add a fifth equivalence to
Theorem <ref>: relaxation is equivalent to cyclic
monotonicity of the Hankel operator.  The following theorem generalizes
[Cor. 1.2]Yafaev2015 to multiple input, multiple output operators,
assuming a finite-dimensional state space realization.

Consider the system (<ref>) and assume that $A$ is Hurwitz.  The
system is a relaxation system if and only if its Hankel operator $\Gamma_h$ is cyclic monotone and $D = D\tran \succeq

We begin by showing that relaxation implies cyclic monotonicity of the Hankel
operator (the condition on $D$ being immediate from the definition of
By Corollary <ref>, cyclic monotonicity of $\Gamma_h$ is
equivalent to the following two conditions, for all $u, w \in L_2^m$:
Γ_h wu    =    wΓ_h u

uΓ_h u     ≥    0.

We begin by showing (<ref>).  Note that relaxation implies
reciprocity with respect to $\Sigma_e = I$, and this in turn implies symmetry
of the impulse responses $h(t)$ and $g(t)$.

We also note that, for any $u, w \in L_2^m$,
∫_0^∞u(t)(∫_0^∞h(t + τ) w(τ)

=    ∫_0^∞u(t)(∫_0^∞g(t + τ) w(τ)

∫_0^∞u(t)(∫_0^∞h(t + τ) w(τ)

=    ∫_0^∞u(t)∫_0^∞C e^At e^Aτ B w(τ) + D w(τ) δ(t + τ)

=     ∫_0^∞u(t)(∫_0^∞C e^At e^Aτ B w(τ) τ) t

+ ∫_0^∞u(t)D w̅(t) t,

w̅(t)    :=     w(t)     t = 0

0     otherwise.

We then have
∫_0^∞u(t)D w̅(t) t    =    0,

so (<ref>) implies (<ref>).
Using symmetry of the inner product, (<ref>) is equivalent to
∫_0^∞u(t)(∫_0^∞g(t + τ) w(τ)

= ∫_0^∞w(t)(∫_0^∞g(t + τ) u(τ)

To show that $g(t) = g(t)\tran$ implies (<ref>), take the left
hand side of (<ref>), transpose and apply Fubini's theorem:
∫_0^∞u(t)(∫_0^∞g(t + τ) w(τ)

= ∫_0^∞(∫_0^∞w(τ)g(t + τ)τ)  u(t)

= ∫_0^∞(∫_0^∞w(τ)g(t + τ)u(t) t)

= ∫_0^∞w(t)(∫_0^∞g(t + τ)u(t) t)

We next show that relaxation implies (<ref>). Let $(A_1, B_1, C_1, D_1)$ be a state space
realization of the form of Theorem <ref>, 3), with
impulse response $h(t)$.  Then, using (<ref>),
uΓ_h u    =    ∫_0^∞u(t) ∫_0^∞h(t + τ)
u(τ) τ t

=    ∫_0^∞u(t) ∫_0^∞C_1e^A_1t e^A_1
τ B_1 u(τ)

=    (∫_0^∞e^A_1t B_1 u(t) t )∫_0^∞e^A_1t B_1
u(t) t

≥    0.

This establishes that relaxation implies cyclic monotonicity of the Hankel operator.

We now show the converse, that cyclic monotonicity of the Hankel operator and
$D = D\tran \succeq 0$ together imply relaxation.
We begin by showing that (<ref>) implies symmetry of
$g(t)$ for all $t \geq 0$.  Indeed, let
$v(\tau) = \delta(\tau)e_j$
and $u(t) = \delta(t - t_0) e_i$, where $t_0 \in [0, \infty)$, $e_i$ denotes
the $i^{\text{th}}$ canonical basis vector of $\R^n$ and $\delta$ denotes the Dirac
delta.  Substituting these signals into (<ref>) gives
e_ig(t_0) e_j    =    e_jg(t_0) e_i,

that is, $g(t_0)$ is symmetric for all $t_0 \in [0, \infty)$, which is
equivalent to symmetry of $h(t)$ under the assumption $D = D\tran$.  This in
turn is equivalent to reciprocity with respect to $\Sigma_e = I$.

Finally, we show that reciprocity, (<ref>) and $D = D\tran \succeq 0$ imply relaxation.
Let  be a stable system with $\hat D = \hat
D \tran \succeq 0$ and Hankel operator
$\Gamma_h$ which satisfies (<ref>) and (<ref>).
Let $(\hat A, \hat B, \hat C, D)$ be a minimal system with transfer function
equal to $D\delta(t) + Ce^{At}B$.
By reciprocity, it follows from [Lem 3]Willems1972a that there exists a
unique, invertible, symmetric matrix $T$ such that
ÂT    =    T Â

TB̂    =    Ĉ.

We claim that $T \geq 0$.  Suppose, on the contrary, that $T$ has a negative
eigenvalue.  Let $x_0$ be a corresponding eigenvector.  Let $\bar{u}:
(-\infty, 0] \to \R^n$ be an input that drives the system from $x=0$ at
$t=-\infty$ to $x(0) = x_0$.  Such an input exists, as $(\hat A, \hat B)$ is
controllable.  Let $u(t) = \bar{u}(-t)$.  By positivity of $\Gamma_h$, we
0    ≤    uΓ_h u

=    ∫_0^∞u(t)∫_0^∞Ĉe^Â(t +

=    ∫_0^∞u(t)Ĉe^Ât ∫_-∞^0 e^-Âτ B̂
u̅(τ) τt

=    ∫_0^∞u(t)Ĉ e^Ât x_0 t

=    ∫_0^∞u(t)B̂T e^Ât x_0 t

=    ∫_-∞^0 u̅(t)B̂e^-Ât t T x_0

=    x_0T x_0 < 0,

which is a contradiction.  Hence $T \geq 0$.  It follows from
Lemma <ref> in the Appendix that the system is passive.  It then follows from
[Thm. 7]Willems1972a that there exists a minimal realization $(A_1, B_1,
C_1, D_1)$ of the system which satisfies
Σ_i A_1    =    A_1 Σ_i

C_1   =    -Σ_i B_1

D_1    =    D_1≽0,

where $\Sigma_i$ is a signature matrix.
It follows from Equation (<ref>) and positivity of $\Gamma_h$
∫_0^∞u(t)C_1 e^At t ∫_0^∞e^A_1τ B_1 u(τ)
τ ≥0

for all $u$.  Hence
-∫_0^∞u(t)B_1e^A_1t t Σ_i ∫_0^∞e^A_1τ B_1 u(τ) τ ≥0

for all $u$.
Suppose that $\Sigma_i$ has  entry $(j, j)$ equal to $1$.  By controllability
of $(A_1, B_1)$, we can choose an input such that
∫_0^∞e^A_1τ B_1 u(τ) τ = e_j.

But then $-e_j\tran \Sigma_i e_j \prec 0$, which contradicts
(<ref>).  Hence $\Sigma_i = -I$, so the system is of the relaxation

§ INTRINSIC STORAGES FOR RELAXATION SYSTEMS

Theorem <ref> establishes the equivalence of relaxation and
cyclic monotonicity of the Hankel operator.  It then follows from Rockafellar's theorem
that the Hankel operator is the gradient of a closed, convex and proper functional
mapping $L_2^m \to \R$. It
turns out that this convex functional is precisely the input/output storage
observed by [$\S10$]Willems1972a.

Before formalizing this result, we show that passivity is guaranteed by the existence
of a nonnegative functional of the past input to the system.  We call this object an
intrinsic storage functional.
We then give a simple, illustrative example.

Consider a system of the form (<ref>).  Given a signal $u \in L_2(\R,
\R^m)$ and time $t\in\R$, denote by $u_t$ the truncation of $u$ to the time axis $(-\infty,
t]$.  If there exists a functional $V$ mapping a truncated signal $u_t$ into
$\R_{\geq 0}$ and satisfying
Vt(u_t) ≤u(t)y(t),

for all $t \in \R$ and input/output trajectories $(u, y)$ of the system, then
the system is passive.

Let $t_0, t_1 \in \R$, $t_1 \geq t_0$.  Integrating (<ref>) from
$t_0$ to $t_1$ gives
V(u_t_0) - V(u_t_1) ≥-∫_t_0^t_1 u(t)y(t) t.

Passivity then follows from nonnegativity of $V(u_{t_1})$, with $K$ in
Definition <ref> equal to $V(u_{t_0})$.

Consider the linear RC circuit shown in Figure <ref>.  Denoting the
voltage on the capacitor by $v_c$, we have the following state space model
for the impedance of the circuit:
t v_c    =    -1/R_1 C v_c +  1/C     1/C   i_1
i_2 ,

v_2     =     1
1 v_2 +  0     0
0     R_2
i_2 .

A two-port RC circuit.

We consider the following experiment: time-varying current sources,
$\bar{i}_{1t}(\cdot)$ and $\bar{i}_{2t}(\cdot)$, are
attached to the ports from time $-\infty$, when there is no charge on the capacitor, to
time $t \in \R$.  The current sources are then replaced by voltmeters, which read
voltages $\bar{v}_1(\cdot)$ and $\bar{v}_2(\cdot)$.  We define $i_{nt}(\tau) = \bar{i}_{nt}(t - \tau)$ and
$v_n(\zeta) = \bar{v}_n(\zeta + t)$ for $n = 1,2$. Define $v =
\begin{pmatrix} v_1 & v_2 \end{pmatrix}\tran$ and $i_t =
\begin{pmatrix} i_{1t} & i_{2t} \end{pmatrix}\tran$.  Solving the state space
model gives the Hankel operator
v(ζ)    =     ∫_0^∞1
1 e^-1/R_1C(ζ+ τ)  1/C     1/C i_t(τ) τ,

plus an additional term $R_2 i_{2t}(0)$ when $\zeta = 0$.
Computing the inner product $(1/2)\ip{i_t}{v}$ over $L_2$ gives
1/2∫_0^∞i_t(ζ)∫_0^∞ 1
e^-1/R_1C(ζ+ τ)  1/C     1/C i_t(τ)τζ

1/2C( ∫_0^∞(i_1t(ζ) + i_2t(ζ))e^-1/R_1Cζζ.

.∫_0^∞(i_1t(τ) + i_2t(τ))

= 1/2C q_c(0)^2,

where $q_c = \frac{1}{C} v_c$ is the charge on the capacitor and the last
line follows by solving the state space equations with zero initial
This expression is the energy stored in the capacitor at time $\tau = 0$.
Taking the derivative with respect to time gives
t1/2i_tv    =    1/C q_c(0)t q_c(0).

Let $\eta(t, \tau) := t - \tau$.  Then
t i_nt    =    t i̅_nt(η(t, τ)) = i̅_ntη ηt =  i̅_nt^'(t - τ)

and    τ i_nt    =    τ i̅_nt(η(t, τ)) = i̅_ntη ητ = - i̅_nt^'(t - τ),

so $\td{}{t} i_{nt} =  -\td{}{\tau}
i_{nt}$. We then have:
t q_c(0)    =    ∫_0^∞e^-1/R_1Cτ t(i_1t(τ) + i_2t(τ))

=    -∫_0^∞e^-1/R_1C τ τ (i_1t(τ) + i_2t(τ))

Integrating by parts then gives
tq_c(0)    =    -[e^-1/R_1Cτ(i_1t(τ) + i_2t(τ))]_0^∞
1/R_1C∫_0^∞e^-1/R_1Cτ (i_1t + i_2t)(τ)τ

=    i_1t(0) + i_2t(0) - 1/R_1C q_c(0),  so

t1/2i_tv    =    v_c(0)(i_1t(0) + i_2t(0)) -

≤    v_c(0) (i_1t(0) + i_2t(0)) + R_2 i_2t(0)^2

=    v̅(t)i̅_t(t).

The variables $\bar v$ and $\bar i_t$ correspond to a particular experiment, however, the
right hand side of this
dissipation inequality only involves the value of $\bar i_t$ and $\bar v$ at
time $t$, the instant in the experiment when both the current source and the
voltmeter are connected.  These can thus be considered samples of an
arbitrary current/voltage trajectory.
The functional $(1/2)\ip{i_t}{v}$ is thus an intrinsic storage functional for the system,
and is expressed purely in terms of the input $i$ and output $v$.
Furthermore, the derivative of this functional with respect to $i_t$ is
the Hankel operator of the system.  The quantity $(1/R_1C^2)q_c(0)^2$ is the
instantaneous power dissipated by the resistor $R_1$.

In order to generalize the construction of the intrinsic storage in Example <ref> to
arbitrary relaxation systems, we require a notion of gradient on $L_2^m$.  This is
given by the functional derivative, $\partial V/\partial u$, which we define via the
first variation:
Vuϕ    :=    [ϵ(V(u + ϵϕ))]_ϵ= 0.

Let $h$ be the impulse response of a relaxation system, and
$\Gamma_h$ be the corresponding Hankel operator.  Then
$\Gamma_h$ is the functional derivative of
V(u) := 1/2uΓ_h u.

Computing the functional derivative gives
Vuϕ    =    1/2[ϵ u + ϵϕΓ_h(u + ϵϕ)]_ϵ= 0

=    1/2ϕΓ_h u + 1/2uΓ_h ϕ

=    Γ_h uϕ,

where the final inequality follows from self-adjointness of $\Gamma_h$.  It
then follows that $\partial V/ \partial u = \Gamma_h$.

The following theorem establishes that the function of Lemma <ref> is in
fact an intrinsic storage functional.

Let $h$ be the impulse response of a relaxation system, and
$\Gamma_h$ be the corresponding Hankel operator.  Then
the system is passive with intrinsic storage functional
V(u) := 1/2uΓ_h u.

The proof of Theorem <ref> makes use of the following lemma,
which establishes a recursive property of relaxation systems with respect to the
derivative.  This is a generalization of the fact that the power dissipated by the
resistor $R_1$ in
Example <ref> is positive.

Let $g(t) = Ce^{At}B$ be the impulse response of a relaxation
system, without the direct component $D\delta(t)$.  Then any
system with impulse response $-\td{}{t} g$ is also a relaxation system.

By Definition <ref>, $g$ is completely monotonic, so
(-1)^k ^kt^k g(t) ≥0

for all $k = 1, 2, \ldots$ This implies complete monotonicity of $-\td{}{t}

Nonnegativity of $V$ follows from positivity of $\Gamma_h$
(Theorem <ref>).  It remains to show that $V$ satisfies
the dissipation inequality (<ref>).
Let the input trajectory be $\bar{u} \in L_{2}(\R, \R^m)$
and define the past input corresponding to time $t \in \R$ by
u_t(τ)    :=    u̅ (t - τ),   τ∈[0, ∞).

Let $\eta(t, \tau) := t - \tau$.  Then
t u_t(τ)    =    t u̅_t(η(t, τ)) = u̅_tη ηt =  u̅_t^'(t - τ)

and    τ u_t    =    τ u̅_t(η(t, τ)) = u̅_tη ητ =  - u̅_t^'(t - τ),

t u_t    =     -τ

We then have
Vt(u_t)    =    Vu(u_t)u_tt

=    ∫_0^∞∫_0^∞u_t(τ)h(ζ+ τ) τ u_tt(ζ) ζ

=    -∫_0^∞y(ζ)u_tζ (ζ) ζ,

where the final line uses Lemma <ref> and
Equation (<ref>).  Integration by parts then gives
Vt(u_t) = -[y(ζ)u_t(ζ)]_0^∞+ ∫_0^∞ζy(ζ)u_t(ζ) ζ

= y(0)u_t(0) +
∫_0^∞ ∫_0^∞u_t(τ)hζ(τ+
ζ)τ u_t(ζ) ζ,

where (<ref>) uses (<ref>) in the proof of Thm.
Denote $\mathrm{d}g/\mathrm{d}\zeta$ by $g^\prime$.  Then the rightmost term in
can be written as
-Γ_(-g^') u_tu_t ≤0,

where the inequality follows from the fact that that $\Gamma_{(-g^\prime)}$ is
the Hankel operator of a relaxation system (Lemma <ref>),
hence cyclic monotone
(Theorem <ref>). Substituting in
(<ref>) gives
Vt(u_t)    ≤    u_t(0)y(0) = u̅_t(t)y̅(t).

A consequence of Rocakfellar's theorem is that the storage $V(u_t)$ is uniquely
determined by the Hankel operator $\Gamma_h$, up to an additive constant.  It was
observed in Willems1972a that this same storage is also uniquely determined
by the requirements of passivity and internal reciprocity.

§ CONCLUSIONS

We have shown that a system being of the relaxation type is equivalent to cyclic
monotonicity of the Hankel operator.  Rockafellar's theorem allows us to construct a
convex storage functional, whose gradient is the Hankel operator, which is completely
determined by input/output measurements.

Cyclic monotonicity is equally well-defined for the Hankel operators of nonlinear
systems, and this allows us to construct intrinsic storages for nonlinear systems.
This will be a topic of future research.

Consider a stable system  of the form (<ref>).
Suppose that $D = D\tran  \succeq 0$ and there exists a matrix $T = T\tran \succeq 0$ such that
AT    =    TA

T B    =    C.

Then the system is passive.

It suffices to show that $T$ satisfies (<ref>), which reduces to $TA
\preceq 0$ since $TA$ is symmetric.
We can factorize $T$ as follows:
T = V Λ_1     0
0     0 V,

where $\Lambda_1 > 0$ and $V$ is orthogonal, so
VTAV   =     Λ_1     0
0     0 VAV.

Since $TA$ is symmetric, $VTAV\tran = (VTAV\tran)\tran$.  Define $\bar A
:= VAV\tran$.  Note that $\bar{A}$ is Hurwitz, as $A$ is Hurwitz.
Partition $\bar{A}$ into
A̅    =     A̅_11     A̅_12

A̅_21     A̅_22.

Λ_1     0
0     0 A̅    =
Λ_1 A̅_11     Λ_2 A̅_12

0     0

Since this matrix is symmetric, $\Lambda_1 \bar{A}_{12} = 0$, which implies
$\bar{A}_{12} = 0$.  Hence $\bar{A}$ is lower block triangular, and so
$\bar{A}_{11}$ is Hurwitz.  We claim that $\Lambda_1 \bar{A}_{11} < 0$.

Let $\operatorname{In}(A)$ denote the inertia of the matrix $A$.
Since $\Lambda_1 \bar{A}_{11}$ is symmetric, it follows from Sylvester's law
of inertia that
In(Λ_1A̅_11)    =
In(Λ_1^-1/2 Λ_1 A̅_11

=    In(Λ_1^1/2 A̅_11

This equals the inertia of $\bar{A}_{11}$, so we have that all the
eigenvalues of $\bar{A}_{11}$ are real and negative, and $\Lambda_1
\bar{A}_{11} \prec 0$.  Hence
Λ_1     0
0     0  ≼0,

so $TA \preceq 0$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleArchive for Rational Mechanics and Analysis
shortjournalArch. Rational Mech. Anal.
titleOn the Theory of Linear Passive Systems
tlc37/Zotero/storage/B23AZA9F/Meixner - 1964 - On the theory of linear passive systems.pdf

journaltitleActa Mathematica
titleSur Les Fonctions Absolument Monotones

Princeton University Press

seriesPrinceton Mathematical Series
titleThe Laplace Transform

journaltitleArchive for Rational Mechanics and Analysis
shortjournalArch. Rational Mech. Anal.
shorttitleDissipative Dynamical Systems Part II
titleDissipative Dynamical Systems Part II: Linear Systems with Quadratic Supply Rates
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Willems1972/Willems1972.pdf

booktitle2019 IEEE 58th Conference on Decision and Control (CDC)
eventtitle2019 IEEE 58th Conference on Decision and Control (CDC)
titleOn the Optimal Control of Relaxation Systems
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Pates2019/Pates2019.pdf;/Users/tlc37/Zotero/storage/GTPBIPLW/9029933.html
Capacitors,Energy storage,Linear systems,Optimal control,Physics,Transfer functions

journaltitleIEEE Control Systems Magazine
titlePassive and Reciprocal Networks: From Simple Models to Simple Optimal Controllers
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Pates2022/Pates2022.pdf;/Users/tlc37/Zotero/storage/Z5NJUAGH/9781270.html
Behavioral sciences,Capacitors,Inductors,Mathematical models,Optimal control,Passive networks,PD control,Power systems,Renewable energy resources,Resistors,Riccati equations,Scalability,State-space methods

abstractThis article considers balanced truncation of discrete-time Hankel $k$-positive systems, characterized by Hankel matrices whose minors up to order $k$ are nonnegative. Our main result shows that if the truncated system has order $k$ or less, then it is Hankel totally positive ($ınfty$-positive), meaning that it is a sum of first-order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive ($k=1$), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.
eventtitleIEEE Transactions on Automatic Control
journaltitleIEEE Transactions on Automatic Control
titleBalanced Truncation of $k$-Positive Systems
tlc37/Dropbox (Cambridge University)/Zotero attachments/Grussler et al2022/Grussler et al2022.pdf;/Users/tlc37/Zotero/storage/TTRM3AAF/stamp.html
$k$ -positivity,Analytical models,Biological system modeling,Controllability,Discrete-time systems,Eigenvalues and eigenfunctions,External positivity,internal positivity,MIMO communication,model order reduction,nonnegative matrix factorization,Observability,positive systems,total positivity

abstractThis paper studies the variation diminishing property of k-positive linear time-invariant (LTI) systems, which diminish the number of sign changes (variation) from input to output, if the input variation is at most k−1. We characterize this property for the discrete-time Toeplitz and Hankel operators of finite-dimensional causal systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k first order positive systems. This is shown by expressing the k-positivity of a LTI system as the external positivity (that is, 1-positivity) of k compound LTI systems. Our characterization generalizes well known properties of externally positive systems (k=1) and totally positive systems (k=∞; also known as relaxation systems in case of the Hankel operator). All results readily extend to continuous-time systems by considering sampled impulse responses.
titleVariation Diminishing Linear Time-Invariant Systems
tlc37/Dropbox (Cambridge University)/Zotero attachments/Grussler et al2022/Grussler et al22.pdf

abstractThe multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, graph theory, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a growing interest in applications of these compounds, and their generalizations, in systems and control theory. The goal of this tutorial paper is to provide a gentle and self-contained introduction to these topics with an emphasis on the geometric interpretation of the compounds, and to describe some of their recent applications including several non-trivial generalizations of positive systems, cooperative systems, contracting systems, and more.
journaltitleMathematics of Control, Signals, and Systems
shortjournalMath. Control Signals Syst.
shorttitleCompound Matrices in Systems and Control Theory
titleCompound Matrices in Systems and Control Theory: A Tutorial
tlc37/Zotero/storage/AG35LKAP/Bar-Shalom et al. - 2023 - Compound matrices in systems and control theory a.pdf

abstractFor self-adjoint Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra. We also show that very strong perturbations, for example, a perturbation by the Carleman operator, do not change the total number of negative eigenvalues of finite rank Hankel operators. As a by-product of our considerations, we obtain an explicit description of the group of unitary automorphisms of all bounded Hankel operators.
journaltitleJournal of Functional Analysis
shortjournalJournal of Functional Analysis
titleOn Finite Rank Hankel Operators
tlc37/Dropbox (Cambridge University)/Zotero attachments/Yafaev2015/Yafaev2015.pdf;/Users/tlc37/Zotero/storage/6S762PP5/S0022123614005229.html
Necessary and sufficient conditions for the sign-definiteness,The Carleman operator and its perturbations,The sign-function,Total multiplicity of the positive and negative spectra

journaltitleAnalysis & PDE
shortjournalAnal. PDE
titleCriteria for Hankel Operators to Be Sign-Definite
tlc37/Zotero/storage/WPECVQFY/Yafaev - 2015 - Criteria for Hankel operators to be sign-definite.pdf

shorttitleRevisiting Totally Positive Differential Systems
titleRevisiting Totally Positive Differential Systems: A Tutorial and New Results
tlc37/Dropbox (Cambridge University)/Zotero attachments/Margaliot et al2019/Margaliot et al2019.pdf

journaltitleArchive for Rational Mechanics and Analysis
titleDissipative Dynamical Systems Part I: General Theory

journaltitleBulletin of the American Mathematical Society
titleNonlinear Networks. I
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Duffin1946/Duffin1946.pdf

journaltitleProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
shortjournalProc. R. Soc. Lond. A
titleMonotone Networks
tlc37/Dropbox (Cambridge University)/Zotero attachments/Minty1960/Minty1960.pdf

eventtitleIRE Transactions on Circuit Theory
journaltitleIRE Transactions on Circuit Theory
titleSolving Steady-State Nonlinear Networks of 'Monotone' Elements
tlc37/Dropbox (Cambridge University)/Zotero attachments/Minty1961/Minty1961.pdf;/Users/tlc37/Zotero/storage/CCNKWLZ2/1086765.html
Band pass filters,Circuits,Diodes,Frequency,H infinity control,Impedance,Phase change materials,Rectifiers,Steady-state,Voltage

journaltitleDuke Mathematical Journal
shortjournalDuke Math. J.
titleMonotone (Nonlinear) Operators in Hilbert Space
tlc37/Dropbox (Cambridge University)/Zotero attachments/Minty/Minty1962.pdf;/Users/tlc37/Zotero/storage/FGCNK92R/1077470254.html

journaltitleSIAM Journal on Control and Optimization
shortjournalSIAM J. Control Optim.
titleMonotone Operators and the Proximal Point Algorithm
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Rockafellar1976/Rockafellar1976.pdf

New York, NY

Springer New York

seriesCMS Books in Mathematics
titleConvex Analysis and Monotone Operator Theory in Hilbert Spaces
tlc37/Zotero/storage/ZP2B26GB/Bauschke and Combettes - 2011 - Convex Analysis and Monotone Operator Theory in Hi.pdf

journaltitleMathematical Programming
shortjournalMath. Program.
titleMonotone Operator Theory in Convex Optimization
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Combettes2018/Combettes2018.pdf

Cambridge University Press

shorttitleLarge-Scale Convex Optimization
titleLarge-Scale Convex Optimization: Algorithms & Analysis via Monotone Operators
tlc37/Dropbox (Cambridge University)/Zotero attachments/Ryu et al2022/Ryu et al2022.pdf;/Users/tlc37/Zotero/storage/5JT9BGQD/2A7F8E7428BFA4EDB8AFACA11AB97E4C.html

journaltitlePacific Journal of Mathematics
shortjournalPacific J. Math.
titleCharacterization of the Subdifferentials of Convex Functions
tlc37/Zotero/storage/95H2QVCS/Rockafellar - 1966 - Characterization of the subdifferentials of convex.pdf

journaltitlePacific Journal of Mathematics
shortjournalPacific J. Math.
titleOn the Maximal Monotonicity of Subdifferential Mappings
tlc37/Zotero/storage/298SQG28/Rockafellar - 1970 - On the maximal monotonicity of subdifferential map.pdf

journaltitleJournal of Mathematical Analysis and Applications
shortjournalJournal of Mathematical Analysis and Applications
titleMaximal Monotonicity and Cyclic Monotonicity Arising in Nonsmooth Lur'e Dynamical Systems
tlc37/Zotero/storage/RPXV4AV4/Adly et al. - 2017 - Maximal monotonicity and cyclic monotonicity arisi.pdf

abstractWe develop a novel convex parametrization of integral quadratic constraints with a terminal cost for subdifferentials of convex functions, involving general O'Shea-Zames-Falb multipliers. We show the benefit of our results for the reduction of conservatism of existing techniques, and sketch applications to the analysis of optimization algorithms or the stability analysis of neural network controllers. The development is prepared by providing a novel link between the convex integrability of a multivariable mapping and dissipativity theory.
series25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022
titleDissipativity, Convexity and Tight O'Shea-Zames-Falb Multipliers for Safety Guarantees
tlc37/Dropbox (Cambridge University)/Zotero attachments/Scherer2022/Scherer2022.pdf;/Users/tlc37/Zotero/storage/57DPYVJE/S2405896322026775.html

abstractThis paper studies analysis and synthesis problems for diffusively coupled multi-agent systems. We focus on networks comprised of multi-input multi-output nonlinear systems that possess a property we term maximal equilibrium-independent cyclically monotone passivity (MEICMP), which is an extension of recent passivity results. We demonstrate that networks comprised of MEICMP systems are related to a pair of dual network optimization problems. In particular, we show that the steady-state behavior of the multi-agent system correspond to the minimizers of appropriately defined network optimization problems. This optimization perspective leads to a synthesis procedure for designing the network controllers to achieve a desired output. We provide detailed examples of networked systems satisfying these properties and demonstrate the results for a network of damped planar oscillators.
eventtitleIEEE Transactions on Automatic Control
journaltitleIEEE Transactions on Automatic Control
titleAnalysis and Synthesis of MIMO Multi-Agent Systems Using Network Optimization
tlc37/Dropbox (Cambridge University)/Zotero attachments/Sharf et al2019/Sharf et al2019.pdf;/Users/tlc37/Zotero/storage/VP8QGXZ5/8676217.html
automatic control,closed loop systems,control system analysis,control system synthesis,Control systems,convergence,Convergence,convex functions,couplings,Couplings,decentralized control,feedback,graph theory,MIMO,MIMO communication,Multi-agent systems,multidimensional systems,nonlinear control systems,nonlinear dynamical systems,optimization,Optimization,Steady-state

journaltitleSIAM Journal on Control and Optimization
shortjournalSIAM J. Control Optim.
titlePort-Hamiltonian Systems Theory and Monotonicity

Cambridge ; New York

Cambridge University Press

isbn978-0-521-36611-3 978-0-521-36791-2
seriesLondon Mathematical Society Student Texts
titleAn Introduction to Hankel Operators
Hankel operators

titleA Theory of Passive Linear Systems with No Assumptions
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Hughes/Hughes3.pdf
Electrical Engineering and Systems Science - Systems and Control

journaltitleJournal of the Franklin Institute
shortjournalJournal of the Franklin Institute
titleRealization of Systems with Internal Passivity and Symmetry Constraints
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Willems1976/Willems1976.pdf

journaltitleIEEE Transactions on Circuits and Systems
shortjournalIEEE Trans. Circuits Syst.
titleNonstationary Network Synthesis via State-Space Techniques
tlc37/Dropbox (Cambridge University)/control/Zotero attachments/Marcus1975/Marcus1975.pdf;/Users/tlc37/Zotero/storage/IDTI6DRC/Marcus and Willems - 1975 - Nonstationary network synthesis via state-space te.pdf;/Users/tlc37/Zotero/storage/MLMC56E4/1084120.html
Circuits and systems,Equations,Laboratories,Linear systems,NASA,Network synthesis,Space technology,Time varying systems

journaltitleProceedings of the National Academy of Sciences of the United States of America
shortjournalProc Natl Acad Sci U S A
titleOn the Generating Functions of Totally Positive Sequences
tlc37/Dropbox (Cambridge University)/Zotero attachments/Aissen et al1951/Aissen et al1951.pdf

American Mathematical Society

booktitleNonlinear Functional Analysis, Proceedings of Symposia in Pure Mathematics
titleA Monotone Convergence Theorem for Sequences of Nonlinear Mappings


[table]capposition=top capbtabboxtable[][]

11institutetext: College of Physics and Information Engineering, Fuzhou
University, Fuzhou, China 11email<EMAIL_ADDRESS>
# Multi-Depth Boundary-Aware Left Atrial Scar Segmentation Network

Mengjun Wu The two authors have equal contributions to the paper. Wangbin Ding
⋆ Mingjin Yang ${{}^{(\textrm{{\char 0\relax}})}}$ Liqin Huang

###### Abstract

Automatic segmentation of left atrial (LA) scars from late gadolinium enhanced
CMR images is a crucial step for atrial fibrillation (AF) recurrence analysis.
However, delineating LA scars is tedious and error-prone due to the variation
of scar shapes. In this work, we propose a boundary-aware LA scar segmentation
network, which is composed of two branches to segment LA and LA scars,
respectively. We explore the inherent spatial relationship between LA and LA
scars. By introducing a Sobel fusion module between the two segmentation
branches, the spatial information of LA boundaries can be propagated from the
LA branch to the scar branch. Thus, LA scar segmentation can be performed
condition on the LA boundaries regions. In our experiments, 40 labeled images
were used to train the proposed network, and the remaining 20 labeled images
were used for evaluation. The network achieved an average Dice score of 0.608
for LA scar segmentation.

###### Keywords:

Left Atrial Scar Multi-Depth Segmentation Boundary-Aware.

##  1 Introduction

Atrial fibrillation (AF) is the most common arrhythmia, occurring at any age,
from children to the elderly [3]. Clinically, catheter ablation (CA) [8] is a
widely used invasive procedure for AF treatment, but with a 45% recurrence
rate [1]. Recent studies demonstrated the relationship between the recurrence
of AF and left atrial (LA) scars after CA [15, 5]. Late gadolinium enhanced
(LGE) cardiac MR has emerged as one of the promising techniques for imaging LA
scars [14]. Delineating scarring regions from LGE images could analyze the
formation of LA scars, and benefit the monitoring and management of AF
patients.

Conventional scar segmentation methods are mainly based on thresholding,
region-growing and graph-cut algorithms [7]. Deep-learning (DL) based methods
have recently been widely studied for LA scar segmentation tasks. Most DL-
based methods explore employing LA or LA walls to improve the scar
segmentation. For instance, Chen et al.[2] presented a multi-task segmentation
methods, where LA and scars were jointly predicted with an attention model; Li
et al.[9, 11] formulated the spatial relationship between LA walls and scars
as loss function, which could force the network to focus on objective regions
during inference.

Generally, the size of LA scars is varied largely. In the training dataset of
LAscarQS 2022 [11, 12, 10], each LGE image contains average 41.17 scars, and
the size of scars are ranged from 0.98 mm3 to 7545.89 mm3. Table 1 presents
the statistical information of the scars in LAscarQS 2022. One can observe,
76.1% of scars sizes are within 50 mm3, and they occupy 16.17% of total scars
volume; whereas only 2.8% of scars sizes are larger than 500 mm3, but they
cover 48% of total scars volume in the whole dataset. For the tiny objects, a
shallower network could outperform the deep U-Net; For the large objects, a
deeper network could outperform the shallower network [16]. The optimal depth
of a segmentation network can vary due to the variety of sizes, which poses an
additional challenge in performing scar segmentation.

Range (mm3) | 0-50 | 50-100 | 100-150 | 150-200 | 200-250 | 250-300 | 300-350 | 350-400 | 400-450 | 450-500 |  $>$500
---|---|---|---|---|---|---|---|---|---|---|---
Number of Scar | 1881 | 262 | 121 | 39 | 24 | 17 | 20 | 14 | 14 | 7 | 71
Percentage (%) | 76.15 | 10.61 | 4.89 | 1.58 | 0.97 | 0.69 | 0.809 | 0.566 | 0.566 | 0.28 | 2.87
Total Number | 2470
Scar Volume | 33014 | 18624 | 14981 | 6778 | 5237 | 4632 | 6510 | 5273 | 5898 | 3321 | 99918
Percentage (%) | 16.17 | 9.12 | 7.34 | 3.32 | 2.56 | 2.27 | 3.19 | 2.58 | 2.89 | 1.626 | 48.93
Total Scar Volume | 204191

Table 1: statistical information of scarring regions in the training dataset
of LAScarQS2022

As shown in Figure 1, we propose a multi-depth boundary-aware network, namely
MDBAnet, to segment different sizes of LA scars. The main contribution of this
work includes: (1) We present a multi-depth segmentation network to segment
multiple sizes of scars. (2) We propose a plug-and-play Sobel [13] fusion
module, which aims to extract LA boundary information to improve scar
segmentation.

Figure 1: The architecture of multi-depth boundary-aware network (MDBAnet). It
consists of two segmentation branches, i.e., the scar branch and the left
atrial (LA) branch. In both branches, we introduce multiple U-Nets with
different depths to perform scar and LA segmentation. Furthermore, we propose
a Sobel fusion module to extract and propagate LA boundaries information from
the LA branch to the scar branch. For conciseness, we only reserve the data
flow between the deepest LA decoder path and scar decoder, and all skip
connections are omitted.

##  2 Method

###  2.1 Network Architecture

MDBAnet comprises two branches, which segment LA and LA scars, respectively.
Scars are distributed on the LA wall, and the size of scars varies largely, as
seen from our statistical information (Table 1). In order to achieve scar
segmentation of different sizes, the scar branch is stacked with multiple
U-Nets that share the same encoder but with different decoder depths. We
expect that the shallow networks will focus on segmenting small-size scars,
while the deep networks will focus on segmenting large-size scars. Finally, we
fuse segmentation results of each U-Net as follows:

$\hat{Y}_{Scar}=\frac{1}{N}\sum_{n=1}^{N}\hat{Y}_{n}$ (1)

where $N$ is the number of U-Nets with different depths, and $\hat{Y}_{n}$ is
the output of the corresponding U-Net.

Furthermore, we aim to improve the performance of scar segmentation by jointly
performing LA segmentation. Thus, as shown in Figure 1, we introduce the LA
branch for LA segmentation. The network architecture of the LA branch is
symmetric to the scar branch, which is stacked with multiple U-Nets. It
outputs LA regions for LGE images.

###  2.2 Sobel Fusion Module

We explore the inherent spatial relationship between LA and LA scars.
Generally, LA scars are distributed around LA boundaries. We introduce a Sobel
[6] operator to extract the boundary information of feature maps. Then a Sobel
fusion module (SFM) is proposed to take full advantage of the spatial
relationship between boundary information and LA scars. The input of SFM
includes the feature maps of LA decoder, the previous layer of scar decoder
and scar encoder. The output of SFM can be calculated as follows:

${F}^{out}=\left(F^{Dec}_{Scar}\otimes S\left(F^{Dec}_{LA}\right)\right)\oplus
F^{Enc}_{Scar},$ (2)

where $\otimes$ and $\oplus$ represent element-wise multiplication and
concatenation, respectively, $S$ represents 3D Sobel operation, $F^{Enc}$ and
$F^{Dec}$ are the feature map from the encoder path and decoder path,
respectively. Here, Sobel operation is implemented by a fixed kernel
convolution layer, which consists of three 3D Sobel kernels. Following Xu et
al.[13], each 3D Sobel Kernel can be described as a 3×3×3 matrix, as shown in
Figure 2. They can be used to extract the boundary information from the axial,
sagittal and coronal views of image.

Figure 2: Our 3D Sobel kernel

Based on the SFM, the feature map from the decoder path of the LA branch could
be passed to the 3D Sobel Kernel to get the boundary information of LA. Then
we re-calibrate the feature map of the scar branch with the boundary
information, which provides spatial attention and forces the network to focus
on the LA boundaries region.

###  2.3 Loss Function

We employ Dice loss and cross-entropy loss to jointly optimize the
segmentation results of the network. The total loss of our network is:

$\begin{split}\mathcal{L}=&-DCS(\hat{Y}_{Scar},Y_{Scar})+CE(\hat{Y}_{Scar},Y_{Scar})\\\
&-DCS(\hat{Y}_{LA},Y_{LA})+CE(\hat{Y}_{LA},Y_{LA}),\end{split}$ (3)

where $\hat{Y}_{\\{Scar,LA\\}}$ and $Y_{\\{Scar,LA\\}}$ are the predicted and
gold standard labels, respectively; $DCS(a,b)$ calculate the Dice score (DS)
between a and b; and $CE(a,b)$ calculate the cross-entropy loss.

##  3 Experiment

###  3.1 Dataset

We trained and evaluated our method on the Left Atrial and Scar Quantification
& Segmentation Challenge 2022 (LAScarQS 2022) dataset, which aimed to segment
LA and LA scars from LGE CMR images. The challenge dataset provides a total of
60 labeled and 10 unlabeled LGE CMR images, and gold standard labels include:
LA and LA scars. In our experiment, we split the labeled images into a
training set of 40 cases, and the remaining 20 cases for evaluation. Finally,
the performance of the network was evaluated on 10 unlabeled images.

###  3.2 Implementations

Our network was implemented in PyTorch, using two NVIDIA GeForce RTX 3080
GPUs. We used SGD optimizer to adjust the network parameters (batch size=2,
weight decay=0.00003, momentum=0.99). The initial learning rate was set 0.01
and decayed exponentially. During training, enhancement techniques, i.e.,
random rotation, random scaling, random elastic deformation, gamma-corrected
enhancement and mirroring, were applied on the fly.

###  3.3 Result

We compared our method to three different segmentation methods:

  * •

nnU-Net [4]: One of the state-of-the-art segmentation networks. We trained it
with 3D LGE images as well as corresponding scar labels.

  * •

MDnet: A multi-depth segmentation network based on U-Net, which is the scar
branch of MDBAnet.

  * •

MDBAnetmul: A variation of MDBAnet. We implement a multiplication fusion
module to propagate information from the LA branch to the scar branch.

  * •

MDBAnet: The proposed network. It consists two branches with multi-depth
network to segment LA and scars. We implement a Sobel fusion module to
propagate information from the LA branch to the scar branch.

To evaluate methods, DS and Hausdorff distance (HD) were calculated between
the prediction results and the gold standard label.

Methods | LA | Scar
---|---|---
DS | HD (mm) | DS | HD (mm)
nnU-Net | - | - | 0.488(0.090) | 39.62(12.81)
MDnet | - | - | 0.504(0.085) | 40.66(12.88)
MDBAnetmul | 0.926(0.020) | 17.83(11.33) | 0.504(0.087) | 33.21(10.43)
MDBAnet | 0.923(0.027) | 19.18(11.10) | 0.512(0.083) | 31.67(10.86)

Table 2: The performance of different methods. DS: Dice score; HD: Hausdorff
distance. Note that MDBAnet and MDBAnetmul jointly produce scar and LA
segmentation, we thus list the LA segmentation results of this two methods.

Table 2 shows the segmentation performance of different methods. Methods using
a multi-depth strategy (i.e., MDBAnet, MDBAnetmul and MDnet ) had obtained
better segmentation performance compared to single-depth network, i.e., nnU-
Net. Particularly, MDnet improved Dice and HD by 0.01 and 1 for scar
segmentation, respectively. It proves the effectiveness of using multi-depth
strategy. Meanwhile, scar segmentation could be further improved by jointly
performing LA segmentation. One can see that both MDBAnetmul and MDBAnet
improve the scar segmentation results by utilizing information from the LA
branch. For example, MDBAnet achieved an improvement DS by 0.8% (p$=$0.125),
and significantly reduced HD from 40.66 to 31.67 (p$<$0.05) against MDnet.
Besides, MDBAnetmul propagated the entire feature maps of the LA branch, while
the MDBAnet extracted the boundary information via SFM. MDBAnet could obtain
better DS and HD for scar segmentation. This implied the benefit of SFM.

In Figure 3, we showed four typical cases for visualization. nnU-Net may
failed to perform segmentation for tiny scars (yellow Boxes), which is consist
to the quantanity result of Table 2. Moreover, MDBAnet achieved better results
for some difficult cases, such as ambiguity scars (yellow Arrows). This was
probably due to the usage of SFM, which could force the scar branch to focus
on boundary regions.

Figure 3: Visualization of different scar segmentation methods. Yellow Arrows
mark the advantage of MDBAnet, while yellow Boxes denote the tiny scars.

##  4 Conclusion

In this work, we have proposed a multi-depth boundary-aware LA scar
segmentation network. It consists of two segmentation branches based on multi-
depth strategy. Meanwhile, we implemented a SFM to propagate information from
LA branch to scar branch. The experimental results showed that multi-depth
network has a positive effect on scar segmentation, and SFM was capable of
further improving scar segmentation performance. The network achieved a DS of
0.608 on validation data of LAScarQS 2022.

## References

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# Minimal free resolution of generalized repunit algebras

Isabel Colaço  Departamento de Matemática e Ciências Físicas, Instituto
Politécnico de Beja, 7800-295 Beja, Portugal<EMAIL_ADDRESS>and
Ignacio Ojeda  Departamento de Matemáticas, Universidad de Extremadura, 06071
Badajoz, Spain<EMAIL_ADDRESS>
###### Abstract.

Let $\Bbbk$ be an arbitrary field and let $b>1,n>1$ and $a$ be three positive
integers. In this paper we explicitly describe a minimal $S-$graded free
resolution of the semigroup algebra $\Bbbk[S]$ when $S$ is a generalized
repunit numerical semigroup, that is, when $S$ is the submonoid of
$\mathbb{N}$ generated by $\\{a_{1},a_{2},\ldots,a_{n}\\}$ where
$a_{1}=\sum_{j=0}^{n-1}b^{j}$ and $a_{i}-a_{i-1}=a\,b^{i-2},\ i=2,\ldots,n$,
with $\gcd(a,a_{1})=1$.

###### Key words and phrases:

Numerical Semigroup, monmial curve, semigroup algebra, minimal free
resolution, Betti numbers

###### 2020 Mathematics Subject Classification:

Primary: 16W50, 13D02 secondary: 20M14

The authors are partially supported by Proyecto de Excelencia de la Junta de
Andalucía (ProyExcel_00868) and by Proyecto de investigación del Plan Propio -
UCA 2022-2023 (PR2022-011). The second author is partially supported by grant
PID2022-138906NB-C21 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A
way of making Europe”, and by research group FQM024 funded by Junta de
Extremadura (Spain)/FEDER funds.

##  1\. Introduction

Let $\Bbbk[\mathbf{x}]=\Bbbk[x_{1},\ldots,x_{n}]$ be the polynomial ring in
$n$ indeterminates over an arbitrary field $\Bbbk$, let $S$ be the numerical
semigroup with minimal system of generators
$A=\\{{a}_{1},\ldots,{a}_{n}\\}\subset\mathbb{N}$ (see [12] for details on
numerical semigroups) and let $\Bbbk[S]:=\bigoplus_{a\in S}\Bbbk\chi^{a}$ be
the semigroup $\Bbbk-$algebra of $S$.

Considering the ring $\Bbbk[\mathbf{x}]$ graded by $S$ via
$\deg(x_{i})=a_{i},\ i=1,\ldots,n,$ we have that the kernel of the
$\Bbbk-$algebra homomorphism

$\varphi_{A}:\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[S],x_{i}\mapsto\chi^{a_{i}}$

determines a presentation of $\Bbbk[S]$ as $S-$graded
$\Bbbk[\mathbf{x}]-$module. Indeed, the so-called _toric ideal_
$I_{A}:=\ker(\varphi_{A})$ is known (see, e.g. [14, Lemma 4.1]) to be
generated by

$\left\\{\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\ \mid\
\sum_{i=1}^{n}u_{i}a_{i}=\sum_{i=1}^{n}v_{i}a_{i},\
\mathbf{u}=(u_{1},\ldots,u_{n}),\mathbf{v}=(v_{1},\ldots,v_{n})\in\mathbb{N}^{n}\right\\},$

where $\mathbf{x}^{\mathbf{u}}:=x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$. In
particular, it is homogeneous for the grading determined by $S$.

So, if
$\left\\{f_{i}:=\mathbf{x}^{\mathbf{u}_{i}}-\mathbf{x}^{\mathbf{v}_{i}}\ \mid\
i=1,\ldots,\beta_{1}\right\\}$ is a minimal generating system of $I_{A}$ and
$\varphi_{0}$ denotes the corresponding canonical projection, then

$\Bbbk[\mathbf{x}]^{\beta_{1}}\stackrel{{\scriptstyle\varphi_{1}:=(f_{1},\ldots,f_{\beta_{1}})}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\stackrel{{\scriptstyle\varphi_{0}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

is exact and $S-$graded by suitable degree shiftings of the leftmost free
module. Now, one can compute a minimal system of generators of the kernel of
$\varphi_{1}$, say
$\\{\mathbf{f}_{12},\ldots,\mathbf{f}_{\beta_{2}2}\\}\subset\Bbbk[\mathbf{x}]^{\beta_{1}}$,
so that the sequence

$\Bbbk[\mathbf{x}]^{\beta_{2}}\stackrel{{\scriptstyle\varphi_{2}:=(\mathbf{f}_{12}|\ldots|\mathbf{f}_{\beta_{2}2})}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{\beta_{1}}\stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\stackrel{{\scriptstyle\varphi_{0}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

is exact and, after the appropriate degree shifts, $S-$graded. So, by
repeating this process as many times as necessary until reaching
$\ker\varphi_{p}=0$, which is guaranteed by the Hilbert syzygy theorem (see,
e.g., [5, Theorem 1.13]), we obtain a _minimal $S-$graded free resolution of
$\Bbbk[S]$_. The minimal free resolution is unique up to isomorphism (see [5,
Section 20.1]). The $\beta_{i},\ i=1,\ldots,p$, are called _Betti numbers of
$\Bbbk[S]$_ (see Remark 5 for more details).

Computing a minimal free resolution of $\Bbbk[S]$ is possible using Groebner
bases techniques. Other related tasks are to characterize the minimal free
resolution $S-$graded in terms of the combinatorics within $S$ (see, for
example, [3, 11]) or, for special cases of $S$, to describe explicitly a
minimal $S-$graded free resolution of $S$ in terms of $S$ basically (see e.g.
[8]). This article is about the latter.

Let $b$ and $n$ be two integers greater than one and let $S$ be the submonoid
of $\mathbb{N}$ generated by $\\{a_{1},a_{2},\ldots\\}\subset\mathbb{N}$,
where

$a_{1}=\sum_{j=0}^{n-1}b^{j}\ \text{}\ a_{i}-a_{i-1}=a\,b^{i-2},\ i\geq 2,$

for $a\in\mathbb{Z}_{+}$ relatively prime with $a_{1}$. In [2], it is proved
that $S$ is a numerical semigroup whose minimal generating system is
$A:=\\{a_{1},\ldots,a_{n}\\}$. These numerical semigroups are called
_generalized repunit numerical semigroups_ (see [1, 2]) as they generalize the
repunit numerical semigroups introduced in [13].

The aim of this paper is to explicitly describe a minimal $S-$graded free of
resolution of $\Bbbk[S]$ when $S$ is a generalized repunit numerical
semigroup. In what follows, we consider $S$ to be a generalized repunit
numerical semigroup and refer $\Bbbk[S]$ as a generalized repunit
$\Bbbk-$algebra.

We finally notice that if $b=1$, then $S$ is generated by an arithmetic
sequence. In this case, the $S-$graded free of resolution of $\Bbbk[S]$ is
fully described by P. Gimenez et al. in [8]. The minimal free resolution of
numerical semigroups generated by arithmetic sequences has its own interest
as, for instance, the Betti numbers of $\Bbbk[S]$ and the coordinate ring of
its tangent cone ring coincide. We emphasize that, by [1, Corollary 2] and [6,
Theorem 3.12], generalized repunit $\Bbbk-$algebras also have this property.

##  2\. The minimal free resolution

Let $b>1,n>1$ and $a>1$ be three fixed integer numbers such that $a$ and
$a_{1}=\sum_{j=0}^{n-1}b^{j}$ are relatively prime. With the same notation as
in the introduction, let $S$ be the generalized repunit numerical semigroup
generated by $A=\\{a_{1},\ldots,a_{n}\\}$.

In [1] it is proved that $I_{A}$ is minimally generated by $2\times 2-$minors
of the matrix

(1)
$X:=(x_{ij})=\left(\begin{array}[]{cccc}x_{1}^{b}&\cdots&x_{n-1}^{b}&x_{n}^{b}\\\
x_{2}&\cdots&x_{n}&x_{1}^{a+1}\end{array}\right).$

Therefore, since $I_{A}$ is a determinantal ideal, the generalized repunit
$\Bbbk-$algebra $\Bbbk[S]$ can be resolved by the Eagon-Northcott complex
introduced in [4] and described below.

Let $y_{1},y_{2}$ be two indeterminates and let $M_{j}$ be the
$\Bbbk[\mathbf{x}]-$submodule of $\Bbbk[\mathbf{x}][y_{1},y_{2}]$ generated by
the monomials in $y_{1}$ and $y_{2}$ of degree $j$. Define

$\Bbbk[\mathbf{x}]^{X}_{j}:=\bigwedge^{j+1}\Bbbk[\mathbf{x}]^{n}\otimes_{\Bbbk[\mathbf{x}]}M_{j-1},\
j=1,\ldots,n-1,$

where $\bigwedge^{j+1}\Bbbk[\mathbf{x}]^{n}$ is the degree $j+1$ component of
the exterior algebra of the free $\Bbbk[x]-$module $\Bbbk[x]^{n}$. Thus, if
$\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{n}\\}$ is the usual basis of
$\Bbbk[\mathbf{x}]^{n}$; that is, the basis of $\Bbbk[\mathbf{x}]^{n}$ such
that $\mathbf{e}_{i}$ has a one in place $i$ and zeros elsewhere, for each
$i\in\\{1,\ldots,n\\}$, then the $\Bbbk[\mathbf{x}]-$module
$\bigwedge^{j+1}\Bbbk[\mathbf{x}]^{n}$ is generated by
$\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}$, for each $1\leq
i_{1}<\cdots<i_{j+1}\leq n$, for each $j\in\\{1,\ldots,n-1\\}$.

Now, since the codimension of $I_{A}$ is $n-1$, because $I_{A}$ defines an
irreducible monomial curve in the $n-$dimensional affine space over $\Bbbk$,
by [4, Theorem 2], we conclude that

$0\to\Bbbk[\mathbf{x}]^{X}_{n-1}\stackrel{{\scriptstyle
d_{n-1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{X}_{n-2}\stackrel{{\scriptstyle
d_{n-2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{X}_{1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

is a minimal free resolution of $\Bbbk[S]$, with

(2) $d_{1}(\mathbf{e}_{i}\wedge\mathbf{e}_{j}\otimes
1)=\left|\begin{array}[]{cc}x_{1i}&x_{1j}\\\ x_{2i}&x_{2j}\end{array}\right|,\
\text{for every}\ 1\leq i<j\leq n,$

and

(3) $d_{j}(\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}})=\sum_{k\stackrel{{\scriptstyle*}}{{=}}1}^{2}\sum_{l=1}^{j+1}(-1)^{l+1}x_{ki_{l}}\mathbf{e}_{i_{1}}\wedge\cdots\wedge\widehat{\mathbf{e}_{i_{l}}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}}y_{k}^{-1},$

for every $1\leq i_{1}<\cdots<i_{j+1}\leq n,\ u_{1},u_{2}\in\mathbb{N}$ such
that $u_{1}+u_{2}=j-1$ and $j\in\\{2,\ldots,n-1\\}$, where the asterisk means
that we only sum over those $k$ for which $u_{k}>0$ and
$\widehat{\mathbf{e}_{i_{l}}}$ means omitting $\mathbf{e}_{i_{l}}$.

######  Lemma 1.

For each $j\in\\{1,\ldots,n-1\\}$, the $\Bbbk[\mathbf{x}]-$module
$\Bbbk[\mathbf{x}]^{X}_{j}$ is isomorphic to
$\Bbbk[\mathbf{x}]^{j\binom{n}{j+1}}$.

###### Proof.

Since $\bigwedge^{j+1}\Bbbk[\mathbf{x}]^{n}$ and $M_{j-1}$ are isomorphic as
$\Bbbk[\mathbf{x}]-$modules to $\Bbbk[\mathbf{x}]^{\binom{n}{j+1}}$ and
$\Bbbk[\mathbf{x}]^{j}$, respectively, for each $j\in\\{1,\ldots,n-1\\}$, we
have that

$\Bbbk[\mathbf{x}]^{X}_{j}=\bigwedge^{j+1}\Bbbk[\mathbf{x}]^{n}\otimes_{\Bbbk[\mathbf{x}]}M_{j-1}\cong\Bbbk[\mathbf{x}]^{\binom{n}{j+1}}\otimes_{\Bbbk[\mathbf{x}]}\Bbbk[\mathbf{x}]^{j}\cong\Bbbk[\mathbf{x}]^{j\binom{n}{j+1}},$

for each $j=1,\ldots,n-1$. ∎

So, applying the previous lemma, we have the following.

######  Proposition 2.

Let $\phi_{0}$ be the identity map of $\Bbbk[\mathbf{x}]$ and, for each
$j\in\\{1,\ldots,n-1\\}$, fix a $\Bbbk[\mathbf{x}]-$module isomorphism
$\phi_{j}:\Bbbk[\mathbf{x}]^{X}_{j}\to\Bbbk[\mathbf{x}]^{j\binom{n}{j+1}}$. If
$\beta_{j}=j\binom{n}{j+1}$ and $\delta_{j}=\phi_{j-1}\circ
d_{j}\circ\phi_{j}^{-1},\ j=1,\ldots,n-1$, then

(4)
$0\to\Bbbk[\mathbf{x}]^{\beta_{n-1}}\stackrel{{\scriptstyle\delta_{n-1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{\beta_{n-2}}\stackrel{{\scriptstyle\delta_{n-2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\delta_{2}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{\beta_{1}}\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

is a minimal free resolution of $\Bbbk[S]$.

For a clearer understanding of Proposition 2, we provide, as an illustrative
example, the well-known minimal free resolution of $\Bbbk[S]$ for $n=3$ (see,
for example, [10, Theorem 2.3] or, in broader generality, the Hilbert-Burch
theorem [5, Theorem 20.15]).

######  Example 3.

Let $S$ be the numerical semigroup generated by
$a_{1}=1+b+b^{2},a_{2}=1+b+b^{2}+a$ and $a_{3}=1+b+b^{2}+a(1+b)$. In this
case, a minimal free resolution of $\Bbbk[S]$ is equal to

$0\to\Bbbk[\mathbf{x}]^{2}\stackrel{{\scriptstyle\delta_{2}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]^{3}\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[S]\to
0$

where $\delta_{2}$ and $\delta_{1}$ are the $\Bbbk[\mathbf{x}]-$module
homomorphisms whose matrices with respect to the corresponding usual bases are

$A_{2}=\begin{pmatrix}\vskip 6.0pt plus 2.0pt minus 2.0ptx_{1}^{b}&x_{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0ptx_{2}^{b}&x_{3}\\\
x_{3}^{b}&x_{1}^{a+1}\end{pmatrix}\quad\text{and}\quad
A_{1}=\left(\begin{array}[]{ccc}x_{2}^{b}x_{1}^{a+1}-x_{3}^{b+1}&-x_{1}^{a+b+1}+x_{2}x_{3}^{b}&x_{1}^{b}x_{3}-x_{2}^{b+1}\end{array}\right),$

respectively. Clearly, in this case, $\beta_{1}=3$ and $\beta_{2}=2$.

##  3\. The minimal $S-$graded free resolution

The minimal free resolution (4), given in the previous section by using Eagon-
Northcott, is not $S-$graded in general. The reason for this is that the maps
$\delta_{i},\ i=1,\ldots,n-1$, described in Proposition 2, are not necessarily
$S-$homogeneous of degree $0$.

To achieve a minimal free resolution $S-$graded of $\Bbbk[S]$, we must
appropriately shift the free $\Bbbk[\mathbf{x}]-$modules that appear in
$\Bbbk[\mathbf{x}]^{\beta_{j}},\ j=1,\ldots,n$, in such a way the maps
$\delta_{j},\ j=1,\ldots,n$, defined in Proposition 2 become $S$-homogeneous
of degree $0$. More precisely, we need to find positive integers $s_{j,k},\
k=1,\ldots,\beta_{j},\ j=1,\dots,n-1,$ such that the maps in the minimal free
resolution

$\displaystyle 0\to$
$\displaystyle\bigoplus_{k=1}^{\beta_{n-1}}\Bbbk[\mathbf{x}](-s_{n-1,k})\stackrel{{\scriptstyle\delta_{n-1}}}{{\longrightarrow}}\bigoplus_{k=1}^{\beta_{n-2}}\Bbbk[\mathbf{x}](-s_{n-2,k})\stackrel{{\scriptstyle\delta_{n-2}}}{{\longrightarrow}}\cdots$
$\displaystyle\cdots\stackrel{{\scriptstyle\delta_{2}}}{{\longrightarrow}}\bigoplus_{k=1}^{\beta_{1}}\Bbbk[\mathbf{x}](-s_{1,k})\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

are $S-$homogeneous of degree $0$. Recall that $\Bbbk[\mathbf{x}](-s)$ means
that the basis elements of $\Bbbk[\mathbf{x}](-s)$ as $\Bbbk[x]-$module, say
$1$, have degree $s$. Thus, for example, $x_{1}\in\Bbbk[\mathbf{x}](-s)$ has
$S-$degree $a_{1}+s$.

######  Example 4.

By considering the degree-shift isomorphisms

$\Bbbk[\mathbf{x}]^{2}\cong\Bbbk[\mathbf{x}](-b\,a_{1}-(b+1)a_{3})\bigoplus\Bbbk[\mathbf{x}](-a_{2}-(b+1)a_{3})$

and

$\Bbbk[\mathbf{x}]^{3}\cong\Bbbk[\mathbf{x}](-(b+1)a_{3})\bigoplus\Bbbk[\mathbf{x}](-a_{2}-b\,a_{3})\bigoplus\Bbbk[\mathbf{x}](-(b+1)a_{2})$

in Example 3, we obtain a minimal $S-$graded free resolution of $\Bbbk[S]$
because these shifts make $\delta_{2}$ and $\delta_{1}$ $S-$homogeneous of
degree $0$.

######  Remark 5.

Given $j\in\\{1,\ldots,n-1\\}$, we have that
$\operatorname{Tor}^{\Bbbk[\mathbf{x}]}_{j}(\Bbbk,\Bbbk[S])_{s}\neq 0$ if and
only if $s=s_{j,k}$ for some $1\leq k\leq\beta_{j}$ (see, e.g. [9, Lemma
1.32]); in fact, the number of $s_{j,k}$’s that are equal to a given $s\in S$
is $\dim(\operatorname{Tor}^{\Bbbk[\mathbf{x}]}_{j}(\Bbbk,\Bbbk[S])_{s})$.
Summarizing, the integers $s_{j,k}$ are uniquely determined by $\Bbbk[S]$.

Our goal is to compute the integers $s_{i,k}$. To start, we introduce
additional notation. From now on we will write $a_{n+1}=(a+1)a_{1}$ and
$c=b^{n}-1-a$.

######  Lemma 6.

With the notation above, $b\,a_{i}=c+a_{i+1},\ i=1,\ldots,n$.

###### Proof.

Clearly,
$b\,a_{1}=b\left(\sum_{j=0}^{n-1}b^{j}\right)=b^{n}-1+a_{1}=b^{n}-1-a+a_{2}=c+a_{2}.$
Now, if $i\in\\{2,\ldots,n\\}$, then

$\displaystyle b\,a_{i}$
$\displaystyle=b\,a_{1}+b\,a\left(\sum_{j=0}^{i-2}b^{j}\right)=c+a_{2}+a\left(\sum_{j=1}^{i-1}b^{j}\right)$
$\displaystyle=c+a_{1}+a+a\left(\sum_{j=1}^{i-1}b^{j}\right)=c+a_{1}+a\left(\sum_{j=0}^{i-1}b^{j}\right)=c+a_{i+1},$

and we are done. ∎

######  Proposition 7.

The maps in the exact sequence

$\bigoplus_{i=1}^{n-1}\left(\bigoplus_{j=0}^{i-1}\Bbbk[\mathbf{x}](-a_{n-i+1}-b\,a_{n-j})\right)\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\Bbbk[\mathbf{x}]\longrightarrow\Bbbk[\mathbf{x}]/I_{A}\cong\Bbbk[S]\to
0$

are $S-$homogeneous of degree $0$. In particular, the set $\left\\{s_{1,k}\mid
k=1,\ldots,\beta_{1}=\binom{n}{2}\right\\}$ is equal to
$\left\\{c+a_{i_{1}}+a_{i_{2}}\mid 1<i_{1}<i_{2}\leq n+1\right\\}.$

###### Proof.

In [1] it is proved that $I_{A}$ is minimally generated by $2\times 2-$minors
of the matrix $X$ defined in (1). Then first part follows straightforward from
the definition of $d_{1}$ (see (2)). Now, the second part is an immediate
consequence of Lemma 6. ∎

Recall that $x\in\mathbb{N}\setminus S$ is said to be a _pseudo-Frobenius
element of $S$_ if $x+s\in S$ for every $s\in S\setminus\\{0\\}$. This set is
known to be finite and is denoted by $\operatorname{PS}(S)$ (see [12, Section
2.4] for more details).

######  Lemma 8.

The set $\\{s_{n-1,k}\mid k=1,\ldots,\beta_{n-1}=n-1\\}$ is equal to

$\left\\{k\,c+\sum_{i=2}^{n+1}a_{i}\mid
k\in\\{1,\ldots,\beta_{n-1}=n-1\\}\right\\}.$

###### Proof.

By [7, Corollary 17], we have that

$\operatorname{PF}(S)=\left\\{s-\sum_{i=1}^{n}a_{i}\mid
s\in\\{s_{n-1,1},\ldots,s_{n-1,n-1}\\}\right\\}.$

Now since, by [2, Corollary 30], $\operatorname{PF}(S)=\\{k\,c+a\,a_{1}\ \mid\
k=1,\ldots,n-1\\}$, our claim follows. ∎

######  Proposition 9.

The maps in the exact sequence

$0\to\bigoplus_{k=1}^{n-1}\Bbbk[\mathbf{x}](-s_{n-1,k})\stackrel{{\scriptstyle\delta_{n-1}}}{{\longrightarrow}}\bigoplus_{k=2}^{n-1}\left(\bigoplus_{j=1}^{n}\Bbbk[\mathbf{x}](b\,a_{j}-s_{n-1,k})\right)$

are $S-$homogeneous of degree $0$. Moreover, the set $\left\\{s_{n-2,k}\mid
k=1,\ldots,\beta_{n-2}=(n-2)n\right\\}$ is equal to

$\left\\{k\,c+\sum_{\begin{subarray}{c}i=2\\\ i\neq
j+1\end{subarray}}^{n+1}a_{i}\mid k\in\\{1,\ldots,n-2\\}\ \text{and}\
j\in\\{1,\ldots,n\\}\right\\}.$

###### Proof.

For the first part, it suffices to observe the matrix of the map $d_{n-1}$
defined in (3) with respect to the usual bases is

$\left(\begin{array}[]{ccccccc}x_{1}^{b}&x_{2}&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ x_{n-1}^{b}&x_{n}&0&\ldots&0&0\\\
x_{n}^{b}&x_{1}^{a+1}&0&\ldots&0&0\\\ 0&x_{1}^{b}&x_{2}&\ldots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&x_{n-1}^{b}&x_{n}&\ldots&0&0\\\
0&x_{n}^{b}&x_{1}^{a+1}&\ldots&0&0\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\ldots&x_{1}^{b}&x_{2}\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\ldots&x_{n-1}^{b}&x_{n}\\\ 0&0&0&\ldots&x_{n}^{b}&x_{1}^{a+1}\\\
\end{array}\right).$

For the second part, we first observe that, by Lemma 8, we can suppose
$s_{n-1,k}=k\,c+\sum_{i=2}^{n+1}a_{i},\ k=2,\ldots,n-1$. Therefore, by Lemma
6, we have that

$\displaystyle s_{n-1,k}-b\,a_{j}$
$\displaystyle=k\,c+\sum_{i=2}^{n+1}a_{i}-b\,a_{j}=k\,c+\sum_{i=2}^{n+1}a_{i}-c-a_{j+1}$
$\displaystyle=(k-1)c+\sum_{i=2}^{n+1}a_{i}-a_{j+1}=(k-1)c+\sum_{\begin{subarray}{c}i=2\\\
i\neq j+1\end{subarray}}^{n+1}a_{i},$

for every $j\in\\{1,\ldots,n\\}$ and $k\in\\{2,\ldots,n-1\\}$. ∎

Now, we can finally state and prove the main theorem about the minimal
$S-$graded free resolution of the generalized repunit $\Bbbk-$algebra
$\Bbbk[S]$.

######  Theorem 10.

The set $B_{j}:=\\{s_{j,k}\mid k=1,\ldots,\beta_{j}=j\binom{n}{j+1}\\}$ is
equal to

$B^{\prime}_{j}:=\left\\{k\,c+a_{i_{1}}+\cdots+a_{i_{j+1}}\ \mid\
k\in\\{1,\ldots,j\\}\ \text{and}\ 1<i_{1}<\cdots<i_{j+1}\leq n+1\right\\},$

for every $j\in\\{1,\ldots,n-1\\}$.

###### Proof.

First, we note that, by Propositions 7 and 9, we already know that the result
is true for $j=1$ and $j=n-1$, respectively.

Let $j\in\\{2,\ldots,n-2\\}$. If we fix a bijection $\sigma_{j}:B_{j}\to
B^{\prime}_{j}$, then we have the isomorphism of free
$\Bbbk[\mathbf{x}]-$modules

$\bigoplus_{k=1}^{\beta_{j}}\Bbbk[\mathbf{x}](-s_{j,k})\longrightarrow
F_{j}:=\bigoplus_{1<i_{1}<\cdots<i_{j+1}\leq
n+1}\left(\bigoplus_{k=1}^{j}\Bbbk[\mathbf{x}](-k\,c-a_{i_{1}}-\cdots-
a_{i_{j+1}})\right)$

such that the element in the usual basis of left-hand module which has a $1$
at place $\Bbbk[\mathbf{x}](-s_{j,k})$ and zeros elsewhere is sent to the
element in the usual basis of $F_{j}$ which has a $1$ at place
$\Bbbk[\mathbf{x}](-\sigma(s_{j,k}))$ and zeros elsewhere . So, if we prove
that, for each $j\in\\{2,\ldots,n-2\\}$, there exist
$\Bbbk[\mathbf{x}]-$module isomorphisms $\phi_{j}:\Bbbk[\mathbf{x}]_{j}^{X}\to
F_{j}$ and $\phi_{j-1}:\Bbbk[\mathbf{x}]_{j-1}^{X}\to F_{j-1}$ such that

$\delta_{j}:=\phi_{j-1}\circ d_{j}\circ\phi_{j}^{-1}:F_{j}\to F_{j-1}$

is $S-$homogeneous of degree $0$, where
$d_{j}:\Bbbk[\mathbf{x}]_{j}^{X}\to\Bbbk[\mathbf{x}]_{j-1}^{X}$ is the
$\Bbbk[\mathbf{x}]-$module homomorphism defined in Section 2, we are done
since the $s_{j,k}$ are uniquely defined (see Remark 5).

Let $j\in\\{2,\ldots,n-2\\}$ and let us define the $\Bbbk[\mathbf{x}]-$module
isomorphism $\phi_{j}:\Bbbk[\mathbf{x}]_{j}^{X}\to F_{j}$ such that
$\phi_{j}(\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}})$ is equal to the element that has a $1$ in the
coordinate corresponding to the direct summand
$\Bbbk[\mathbf{x}](-(u_{1}+1)\,c-a_{i_{1}+1}-\cdots-a_{i_{j+1}+1})$ and zeros
elsewhere (recall that $u_{1},u_{2}\in\mathbb{N}$ verifies $u_{1}+u_{2}=j-1$).
Since, given $j\in\\{2,\ldots,n-2\\}$ and $l\in\\{1,\ldots,j\\}$, the
$S-$degree of the $k\,i_{j}-$th entry of the matrix $X$ defined in (1) is

$\left\\{\begin{array}[]{lcl}b\,a_{i_{l}}&\text{if}&k=1;\\\
a_{i_{l}+1}&\text{if}&k=2,\end{array}\right.$

we have that the $S-$degree of
$\phi_{j-1}(x_{ki_{l}}\mathbf{e}_{i_{1}}\wedge\cdots\wedge\widehat{\mathbf{e}_{i_{l}}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}}y_{k}^{-1})$ is

$\left\\{\begin{array}[]{lcl}b\,a_{i_{l}}+u_{1}\,c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}-a_{i_{l}+1}&\text{if}&k=1;\\\
a_{i_{l}+1}+(u_{1}+1)\,c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}-a_{i_{l}+1}&\text{if}&k=2,\end{array}\right.$

In the first case, by Lemma 6, we have that

$b\,a_{i_{l}}+u_{1}\,c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}-a_{i_{l}+1}=$
$c+a_{i_{l}+1}+u_{1}c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}-a_{i_{l}+1}=$
$(u_{1}+1)c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}.$

and, in the second case, we have that

$a_{i_{l}+1}+(u_{1}+1)c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}-a_{i_{l}+1}=$
$(u_{1}+1)c+a_{i_{1}+1}+\cdots+a_{i_{j+1}+1}.$

So, the $S-$degree
$\phi_{j-1}(x_{ki_{l}}\mathbf{e}_{i_{1}}\wedge\cdots\wedge\widehat{\mathbf{e}_{i_{l}}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}}y_{k}^{-1})$ is equal to the $S-$degree of
$\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}}$, for every $j\in\\{2,\ldots,n-2\\}$ and
$l\in\\{1,\ldots,j\\}.$ Therefore, since

$d_{j}(\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}})=\sum_{k\stackrel{{\scriptstyle*}}{{=}}1}^{2}\sum_{l=1}^{j+1}(-1)^{l+1}x_{ki_{l}}\mathbf{e}_{i_{1}}\wedge\cdots\wedge\widehat{\mathbf{e}_{i_{l}}}\wedge\cdots\wedge\mathbf{e}_{i_{j+1}}\otimes
y_{1}^{u_{1}}y_{2}^{u_{2}}y_{k}^{-1},$

for every $j\in\\{2,\ldots,n-1\\}$, we conclude that, for our choice of the
isomorphisms $\phi_{j},\ j=2,\ldots,n-2$, the maps $\delta_{j}=\phi_{j-1}\circ
d_{j}\circ\phi_{j}$ are $S-$homogeneous of degree zero, for every
$j=2,\ldots,n-2$. ∎

COI Statement. The authors declared that they have no conflict of interest.

Acknowledgments. The authors would like to thank Manuel B. Branco for his
valuable suggestions that improved the presentation of this article.

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