Datasets:

Monketoo
/

math-docs-dataset

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+# Optimal Control Using Nonholonomic Integrators
+Marin Kobilarov and Gaurav Sukhatme
+Robotic Embedded Systems Laboratory
+University of Southern California
+Los Angeles, CA, USA
+Email: mkobilar | gaurav@robotics.usc.edu
+**Abstract**— This paper addresses the optimal control of nonholonomic systems through provably correct discretization of the system dynamics. The essence of the approach lies in the discretization of the Lagrange-d'Alembert principle which results in a set of forced discrete Euler-Lagrange equations and discrete nonholonomic constraints that serve as equality constraints for the optimization of a given cost functional. The method is used to investigate optimal trajectories of wheeled robots.
+## I. INTRODUCTION
+We study motion planning for mechanical systems with *nonholonomic* (e.g. rolling) constraints. That is, we consider the problem of computing the controls necessary to move the system from an initial state to a goal state while optimizing some criterion such as the total energy consumed or the distance travelled while satisfying task constraints such as obstacles in the environment.
+For this purpose, we develop optimization algorithms based on structure-preserving numerical integrators derived from *discrete mechanics*. In particular, we employ a discrete *Lagrange-d'Alembert* principle which provides a variational framework to study mechanical systems with constraints and external forces. The resulting discrete systems inherit the structure of the continuous systems and respect the work-energy balance which is important for numerical accuracy and stability.
+The method is illustrated using two simple car-like models. Experimental comparison with a standard approach (i.e. direct collocation) demonstrates the computational advantages of the proposed framework.
+### A. Related work
+Trajectory design and motion control of robotic systems have been studied from many different perspectives. A particularly interesting point of view is the geometric approach, that uses symmetry and reduction techniques (see [1], [2]). The authors of [3] use reduction by symmetry to simplify the optimal control problem and provide a framework for computing motions for general nonholonomic systems. A related approach, applied to an underwater eel-like robot, involves finding approximate solutions using truncated basis of cyclic input functions [4]. Other applications of symmetry in robotics include motion planning of dynamic variable inertia mechanical systems [5], improving the precision and efficiency of randomized kinodynamic planning [6], efficient motion planning using maneuver primitives [7], [8].
+The systems we consider fall in the class of underactuated nonlinear systems with drift. The motion planning problem for these systems cannot be solved in closed form and one has to resort to numerical optimization techniques. One such approach relies on the differential flatness of systems [9], [10] or on performing system inversion [11]. Several motion planners have been implemented in mobile robotics to optimize trajectories among obstacles. We mention the path deformation method in [12] that could be extended to systems with drift; the method based on standard parametrized optimal control for autonomous navigation [13]; and various methods applied to wheeled robots [14], [15], [16], [17]; as well as a nonlinear optimization method for local planning in a randomized search framework [18] (see [19] for additional references).
+The recent work on Discrete Mechanics and Optimal Control (DMOC) ([20], [21], [22]) employs a discretization strategy different from the standard optimization methods (i.e. collocation, shooting, or multiple shooting, see [23] for a taxonomy of various methods). It is based on *variational integrators* [24] which, unlike standard integration methods, can preserve energy, momenta, and symplectic structure, or in the presence of forces, compute the exact change in these quantities. The approach can be extended to the nonholonomic case to yield *nonholonomic integrators* (see [25], [26]). We use such integrators in a DMOC framework as a basis for this work.
+In the context of mobile robotics, our proposed method may seem related to existing variational approaches such as [3] and [27], where the continuous equations of motion are first obtained, enforced as constraints, and subsequently discretized. However, the main difference is that in the DMOC framework variational principles determining the system dynamics are discretized *first* so as to ensure provably-good numerical behavior; the resulting discrete equations are then used as constraints along with a discrete cost function to form the control problem.
+### B. The Optimization problem
+We consider a finite-dimensional mechanical system with configuration space $Q$ and a distribution $\mathcal{D}$ that describes the nonholonomic constraints of interest. The distribution $\mathcal{D}$ is a collection of linear subspaces denoted $\mathcal{D}_q \subset T_q Q$ of the tangent space $T_q Q$ for each $q \in Q$. A curve $q(t) \in Q$ is said

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+Fig. 4. Final state achievement error
+Fig. 5. Trajectory execution error
+REFERENCES
+[1] S. Kelly and R. Murray, "Geometric phases and robotic locomotion," *Journal of Robotic Systems*, vol. 12, no. 6, pp. 417-431, 1995.
+[2] J. Ostrowski, "Computing reduced equations for robotic systems with constraints and symmetries," *IEEE Transactions on Robotics and Automation*, pp. 111-123, 1999.
+[3] J. P. Ostrowski, J. P. Desai, and V. Kumar, "Optimal gait selection for nonholonomic locomotion systems," *The International Journal of Robotics Research*, vol. 19, no. 3, pp. 225-237, 2000.
+[4] J. Cortes, S. Martinez, J. P. Ostrowski, and K. A. McIsaac, "Optimal gaits for dynamic robotic locomotion," *The International Journal of Robotics Research*, vol. 20, no. 9, pp. 707-728, 2001.
+[5] E. A. Shammas, H. Choset, and A. A. Rizzi, "Motion planning for dynamic variable inertia mechanical systems with non-holonomic constraints," in *International Workshop on the Algorithmic Foundations of Robotics*, 2006.
+[6] P. Cheng, E. Frazzoli, and S. LaValle, "Exploiting group symmetries to improve precision in kinodynamic and nonholonomic planning," in *International Conference on Intelligent Robots and Systems*, vol. 1, 2003, pp. 631-636.
+[7] E. Frazzoli, M. A. Dahleh, and E. Feron, "Maneuver-based motion planning for nonlinear systems with symmetries," *IEEE Transactions on Robotics*, vol. 21, no. 6, pp. 1077-1091, dec 2005.
+[8] C. Dever, B. Mettler, E. Feron, and J. Popovic, "Nonlinear trajectory generation for autonomous vehicles via parameterized maneuver classes," *Journal of Guidance, Control, and Dynamics*, vol. 29, no. 2, pp. 289-302, 2006.
+[9] M. B. Milam, K. Mushambi, and R. M. Murray, "A new computational approach to real-time trajectory generation for constrained mechanical systems," in *IEEE Conference on Decision and Control*, vol. 1, 2000, pp. 845-851.
+[10] N. Faiz, S. Agrawal, and R. Murray, "Differentially flat systems with inequality constraints: An approach to real-time feasible trajectory
+generation," *Journal of Guidance, Control, and Dynamics*, vol. 24, no. 2, pp. 219-227, 2001.
+[11] N. Petit, M. B. Milam, and R. M. Murray, "Inversion based trajectory optimization," in *IFAC Symposium on Nonlinear Control Systems Design*, 2001.
+[12] F. Lamiraux, D. Bonnafous, and O. Lefebvre, "Reactive path deformation for nonholonomic mobile robots," *IEEE Transactions on Robotics*, vol. 20, no. 6, pp. 967-977, 2004.
+[13] A. Kelly and B. Nagy, "Reactive nonholonomic trajectory generation via parametric optimal control," *The International Journal of Robotics Research*, vol. 22, no. 7-8, pp. 583-601, 2003.
+[14] A. W. Divelbiss and J. T. Wen, "A path space approach to non-holonomic motion planning in the presence of obstacles," *IEEE Transaction on Robotics and Automation*, vol. 13, no. 1, pp. 443-451, 1997.
+[15] K. Kondak, G. Hommel, S. Horstmann, and S. Kristensen, "Computation of optimal and collision-free movements for mobile robots," in *International Conference on Intelligent Robots and Systems*, 2000, pp. 1906-1911.
+[16] J.-S. Liu, L.-S. Wang, and L.-S. Tsai, "A nonlinear programming approach to nonholonomic motion planning with obstacle avoidance," in *International Conference on Robotics and Automation*, vol. 1, 1994, pp. 70-75.
+[17] D. Kogan and R. Murray, "Optimization-based navigation for the DARPA grand challenge," 2006.
+[18] T. Karatas and F. Bullo, "Randomized searches and nonlinear programming in trajectory planning," in *IEEE Conference on Decision and Control*, vol. 5, 2001, pp. 5032-5037.
+[19] S. M. LaValle, *Planning Algorithms*. Cambridge University Press, Cambridge, U.K., 2006.
+[20] O. Junge, J. Marsden, and S. Ober-Blöbaum, "Discrete mechanics and optimal control," in *Proceedings of the 16th IFAC World Congress*, 2005.
+[21] E. Kanso and J. Marsden, "Optimal motion of an articulated body in a perfect fluid," in *IEEE Conference on Decision and Control*, 2005, pp. 2511-2516.
+[22] T. Lee, N. McClamroch, and M. Leok, "Optimal control of a rigid body using geometrically exact computations on SE(3)," in *Proc. IEEE Conf. on Decision and Control*, 2006.
+[23] J. Betts, "Survey of numerical methods for trajectory optimization," *Journal of Guidance, Control, and Dynamics*, vol. 21, no. 2, pp. 193-207, 1998.
+[24] J. Marsden and M. West, "Discrete mechanics and variational integrators," *Acta Numerica*, vol. 10, pp. 357-514, 2001.
+[25] J. C. Monforte, *Geometric Control and Numerical Aspects of Non-holonomic Systems*. Springer, 2002.
+[26] R. McLachlan and M. Perlmutter, "Integrators for nonholonomic mechanical systems," *Journal of NonLinear Science*, vol. 16, pp. 283-328, aug 2006.
+[27] J. P. Desai and V. Kumar, "Motion planning for cooperating mobile manipulators," *Journal of Robotic Systems*, vol. 16, no. 10, pp. 557-579, 1999.
+[28] A. Bloch, *Nonholonomic Mechanics and Control*. Springer, 2003.
+[29] J. E. Marsden and T. S. Ratiu, *Introduction to Mechanics and Symmetry*. Springer, 1999.
+[30] N. Getz and J. Marsden, "Control for an autonomous bicycle," in *IEEE International Conference on Robotics and Automation*, 1995, pp. 1397-1402.

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+(2) let $A_{k+d}$ be the admissible subsurface of $D_{d,r}^{\infty}$ obtained from $A_k$ by adding a d-leg pants and $\phi(A_{k+d}) = A_{k+d}'$. Then the associated oriented ribbon braid (resp. the ribbon braid) of $\phi$ can be obtained from $\tau_{\phi}$ by splitting the corresponding band into d bands. Conversely, if we split one band of the ribbon braids into d bands, the isotopy class of the corresponding asymptotic rigid (resp. quasi-rigid) homeomorphism does not change.
+Note that for any $d \ge 2$, $r \ge 1$, we have an natural embedding $\iota_{d,r}: D_{d,r}^{\infty} \to D_{d,r+1}^{\infty}$ by mapping
+the rooted boundaries of $D_{d,r}^{\infty}$ to the first $r$ rooted boundaries according to the order. This
+induces an embedding of groups $i_{\mathcal{H},d,r}: \mathcal{H}V_{d,r} \to \mathcal{H}V_{d,r+1}$, $i_{\mathcal{B},d,r}: \mathcal{B}V_{d,r} \to \mathcal{B}V_{d,r+1}$. On the other
+hand, we also have natural embeddings $i_{R,d,r}: RV_{d,r} \to RV_{d,r+1}$ and $i_{R+,d,r}: RV_{d,r}^{+} \to RV_{d,r+1}^{+}$
+induced by inclusion of roots. We have the following.
+**Theorem 3.24.** There exist isomorphisms $f_{d,r} : \mathcal{H}V_{d,r} \to RV_{d,r}$ such that $f_{d,r+1} \circ i_{\mathcal{H},d,r} = i_{\mathcal{H},d,r+1} \circ f_{d,r}$. Restricting to the subgroups $\mathcal{B}V_{d,r}$, one gets isomorphisms $f_{d,r} : \mathcal{B}V_{d,r} \to RV_{d,r}^{+}$ with the same property.
+**Proof.** Since the two cases are parallel, we will only prove the theorem for $\mathcal{B}V_{d,r}$. We will define two maps $f_{d,r} : \mathcal{B}V_{d,r} \to RV_{d,r}^{+}$ and $g_{d,r} : RV_{d,r}^{+} \to \mathcal{B}V_{d,r}$ such that $f_{d,r} \circ g_{d,r} = \text{id}$ and $g_{d,r} \circ f_{d,r} = \text{id}$.
+Given an element $x \in \mathcal{B}V_{d,r}$, we can define $f_{d,r}$ as follows. Let $\varphi_x$ be an asymptotically rigid homeomorphism of $D_{d,r}^\infty$ representing $x$ with support $A_k$, where $k$ is the number of admissible loops. By Proposition 3.17, $\pi(x)$ provides an element $[F_-, \sigma, F_+]$ in the Higman–Thompson group $V_{d,r}$, where $F_-$ and $F_+$ are $(d,r)$-forests with $k$ leaves. But what we want is a ribbon braid connecting the $k$ leaves. For this we simply apply Lemma 3.23 (1) to the map $\varphi_x$ with support $A_k$, denote the corresponding element in $RB_k^+$ by $\tau_{\varphi_x}$. We define $f_{d,r}(x) = [F_-, \tau_{\varphi_x}, F_+]$.
+Now given $y \in RV_{d,r}^{+}$, one can define an element in $\mathcal{B}V_{d,r}$ as follows. Suppose $(F_-, \mathbf{r}_y, F_+)$ is a representative for $y$, where $F_-$ and $F_+$ are $(d, r)$-forests and $\mathbf{r}$ is a ribbon braid between the leaves of $F_-$ and $F_+$. Add a root to $F_-$ (resp. $F_+$) with an edge connecting to the root of each tree in $F_-$ (resp. $F_+$) and then throw away the open half edge connecting to the leaves. Denote the resulting tree by $T_-$ (resp. $T_+$). Now $q^{-1}(T_-)$, $q^{-1}(T_+)$ gives us two admissible subsurfaces $A_k$, $A'_k$ in $D_{d,r}^\infty$ where $k$ is the number of leaves for $F_-$. And by Lemma 3.23 (1), the ribbon element $\mathbf{r}_y$ in $RB_k^+$ give us an asymptotic rigid homeomorphism $\psi_y$ with support $A_k$ and maps $A_k$ to $A'_k$.
+Now one can check that $f_{d,r} \circ g_{d,r} = \text{id}$ and $g_{d,r} \circ f_{d,r} = \text{id}$. Therefore, the two groups are isomorphic. The fact that the diagram commutes is immediate from the definition. $\square$
+# 4. HOMOLOGICAL STABILITY OF RIBBON HIGMAN–THOMPSON GROUPS
+In this section, we show the homological stability for oriented ribbon Higman–Thompson
+groups and explain at the end how the same proof applies to the ribbon Higman–Thompson
+groups.
+4.1. **Homogeneous categories and homological stability.** In this subsection, we review the basics of homogeneous categories and refer the reader to [RWW17] for more details. Note that we adopt their convention of identifying objects of a category with their identity morphisms.
+**Definition 4.1** ([RWW17, Definition 1.3]). A monoidal category $(\mathcal{C}, \oplus, 0)$ is called *homogeneous* if 0 is initial in $\mathcal{C}$ and if the following two properties hold.
+**H1** Hom$(A, B)$ is a transitive Aut$(B)$-set under postcomposition.
+**H2** The map Aut$(A) \to \mathrm{Aut}(A \oplus B)$ taking $f$ to $f \oplus \mathrm{id}_B$ is injective with image
+$$
+\mathrm{Fix}(B) := \{\phi \in \mathrm{Aut}(A \oplus B) \mid \phi \circ (\iota_A \oplus \mathrm{id}_B) = \iota_A \oplus \mathrm{id}_B\}
+ $$
+where $\iota_A: 0 \rightarrow A$ is the unique map.

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+# RINGS WHOSE MODULES ARE FINITELY GENERATED OVER THEIR ENDOMORPHISM RINGS
+BY
+NGUYEN VIET DUNG (Zanesville, OH) and JOSÉ LUIS GARCÍA (Murcia)
+**Abstract.** A module *M* is called finendo (cofinendo) if *M* is finitely generated (respectively, finitely cogenerated) over its endomorphism ring. It is proved that if *R* is any hereditary ring, then the following conditions are equivalent: (a) Every right *R*-module is finendo; (b) Every left *R*-module is cofinendo; (c) *R* is left pure semisimple and every finitely generated indecomposable left *R*-module is cofinendo; (d) *R* is left pure semisimple and every finitely generated indecomposable left *R*-module is finendo; (e) *R* is of finite representation type. Moreover, if *R* is an arbitrary ring, then (a)⇒(b)⇔(c), and any ring *R* satisfying (c) has a right Morita duality.
+**1. Introduction.** Modules which are finitely generated over their endomorphism rings, also called finendo modules, were introduced by Faith [16], who used them to characterize several classes of rings (e.g. right self-injective rings, right pseudo-Frobenius rings, prime right Goldie rings). The finendo condition occurs naturally in several contexts, in general module theory and representation theory. Special classes of finendo modules, studied recently in the literature, include endofinite modules [11], product-complete modules [27], endocoherent modules [2], and tilting modules [3, 4].
+One of the main questions we will consider in this paper is to characterize rings *R* with the property that all right *R*-modules are finendo. More specifically, we will try to answer the question raised in [13] whether such a ring *R* has *finite representation type*, i.e. *R* is an artinian ring with finitely many isomorphism classes of finitely generated indecomposable left and right modules. Our study of this question is related to and motivated by works of several authors that investigate the relationship between representation-theoretic properties of rings and endoproperties of their modules.
+A right *R*-module *M* is called *endofinite* if *M* has finite length as a left module over its endomorphism ring. An important starting point, obtained independently by Crawley-Boevey [10], Huisgen-Zimmermann and Zimmermann [26] and Prest [29], is that every right *R*-module is endofinite if and
+2000 Mathematics Subject Classification: 16G10, 16D70, 16D90.
+Key words and phrases: finendo module, cofinendo module, pure semisimple ring, ring of finite representation type, local duality.

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+modules $\operatorname{Hom}_H(IX, \operatorname{ann}_U(K))$ and $\operatorname{Hom}_H(\bar{X}, \operatorname{ann}_U(K))$ is finitely generated over the endomorphism ring of $\operatorname{ann}_U(K)$.
+By hypothesis, $\operatorname{Hom}_E(X, U)$ is finitely generated over $\operatorname{End}(U_E)$. Thus there exist elements $f_1, \dots, f_s \in \operatorname{Hom}_E(X, U)$ so that each $f : X_E \to U_E$ has the form $f = \sum_{i=1}^s \alpha_i \circ f_i$ for certain $\alpha_i \in \operatorname{End}(U_E)$. By noting that $K = \operatorname{Hom}_R(X, \operatorname{ann}_X(I))$, it follows easily that any $f : X_E \to U_E$ restricts to a homomorphism $\hat{f} : (IX)_E \to (\operatorname{ann}_U(K))_E$. On the other hand, every homomorphism $g : (IX)_E \to (\operatorname{ann}_U(K))_E$ can be extended to $X_E \to U_E$, because $U_E$ is injective. It follows immediately that $\hat{f}_1, \dots, \hat{f}_s$ is a system of generators of $\operatorname{Hom}_E(IX, \operatorname{ann}_U(K)) = \operatorname{Hom}_H(IX, \operatorname{ann}_U(K))$ as a module over $\operatorname{End}(\operatorname{ann}_U(K)_H)$.
+Finally, we consider $\operatorname{Hom}_H(\overline{X}, \operatorname{ann}_U(K))$. Take a set $a_1, \dots, a_r$ of generators of $I$ as a left $R/J$-module. Then each $\hat{f}_i \otimes a_j$ gives $\mu(\hat{f}_i \otimes a_j) = \phi_{ij} \in \operatorname{Hom}_H(\overline{X}, \operatorname{ann}_U(K))$. Each element $g \in \operatorname{Hom}_H(\overline{X}, \operatorname{ann}_U(K))$ can be written as $g = \sum_{j=1}^r \mu(h_j \otimes a_j)$. In turn, $h_j = \sum_{i=1}^s \alpha_{ij} \hat{f}_i$, for certain endomorphisms $\alpha_{ij}$ of $\operatorname{ann}_U(K)$, in view of the preceding paragraph. Therefore
+$$g = \sum_{i,j} \alpha_{ij} \mu(\hat{f}_i \otimes a_j) = \sum_{i,j} \alpha_{ij} \phi_{ij},$$
+which shows that the module $\operatorname{Hom}_H(\overline{X}, \operatorname{ann}_U(K))$ is finitely generated over $\operatorname{End}(\operatorname{ann}_U(K))$, as was to be shown. ■
+We are now ready to deduce the central result of this section.
+**THEOREM 4.2.** If $R$ is a left pure semisimple ring such that every finitely generated indecomposable left $R$-module is cofinendo, then $R$ is right Morita.
+*Proof*. Suppose, to the contrary, that there is some left pure semisimple ring $A$ which is not right Morita and such that every finitely generated indecomposable left $A$-module is cofinendo.
+Note that every quotient ring $Q$ of $A$ has the same property, i.e., every finitely generated indecomposable left $Q$-module is cofinendo. This is because the restriction of the scalars functor $Q$-Mod $\to A$-Mod views each left $Q$-module as a left $A$-module, with the same endomorphisms. As in [21, Section 5], we may use the fact that $A$ is left artinian to obtain a quotient $R$ of $A$ which is left pure semisimple and not right Morita, and such that all the proper quotients of $R$ are right Morita. Then every finitely generated indecomposable left $R$-module is cofinendo.
+We now follow the proof of [21, Proposition 5.8] to show that this ring $R$ has a unique minimal two-sided ideal $I$. Indeed, if there are non-zero two-sided ideals $I_1, I_2$ of $R$ with $I_1 \cap I_2 = 0$, and $E$ is the injective hull of a simple right $R$-module, then the natural monomorphism of right $R$-modules

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+$$R \to R/I_1 \oplus R/I_2 \text{ induces an epimorphism}$$
+$$p : \operatorname{Hom}_R(R/I_1, E) \oplus \operatorname{Hom}_R(R/I_2, E) \to E.$$
+But $E_j = \operatorname{Im}(p|_{R/I_j})$ is a submodule of $E$ which is indecomposable injective as a right $R/I_j$-module. Since $R/I_j$ is right Morita, $E_j$ is finitely generated also as a right $R$-module. Hence $E$ is finitely generated, and $R$ is right Morita, which is a contradiction.
+We next claim that if $R$ any left pure semisimple ring such that every finitely generated indecomposable left $R$-module is cofinendo, and $I \subseteq J$ is a two-sided ideal of $R$ such that $IJ = JI = 0$ and $R/I$ is right Morita, then $R$ is also right Morita. The application of this claim to the left pure semisimple and non-right Morita ring $R$ above will yield the desired contradiction and complete our proof.
+The proof of the claim could also be obtained by adapting the proofs of [21, Proposition 6.2, Theorem 6.3], but we give a direct proof for the convenience of the reader. First we note that, under the assumptions in the claim, $T_I(R)$ is left pure semisimple. By [22, Corollary 2], it is enough to show that each left $T_I(R)$-module is a direct sum of indecomposable modules. In turn, it will suffice to show the property for Grassmannian modules, according to Lemma 4.1. But this follows easily from the facts that the functor $\operatorname{Gr}$ preserves indecomposable modules [5] and direct sums, and every left $R$-module is a direct sum of indecomposable modules.
+By Lemma 4.1, every finitely generated indecomposable left $T_I(R)$-module is cofinendo. As $T_I(R)$ is a hereditary ring, we infer by Theorem 3.5 that $T_I(R)$ is of finite representation type. Note that $R$ is right artinian, by Corollary 2.9. Now let $M$ be any indecomposable injective right $R$-module. Since $\operatorname{ann}_M(I)$ is an injective indecomposable right $R/I$-module [16, Proposition 8], it is finitely generated, as $R/I$ is right Morita. Moreover, $\operatorname{Gr}(M)$ is indecomposable by [21, Theorem 6.1]. Therefore $\operatorname{Gr}(M)$ is finitely generated, because $T_I(R)$ is of finite representation type, hence $M/\operatorname{ann}_M(I)$ is finitely generated as a right $R$-module, and hence so is $M$. This shows that $R$ is right Morita. ■
+**PROPOSITION 4.3.** Let $R$ be a left pure semisimple ring such that every finitely generated indecomposable left $R$-module is cofinendo. Then the quotient ring $R/(J(R))^2$ is of finite representation type.
+*Proof.* Set $R' = R/(J(R))^2$. Thus $R'$ is a left pure semisimple ring such that every finitely generated indecomposable left $R'$-module is cofinendo. If $J$ is the Jacobson radical of $R'$, then $J^2 = 0$, and we may consider the matrix ring $M_J(R') = \begin{pmatrix} R'/J & 0 \\ J & R'/J \end{pmatrix}$.

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+only if $R$ is of finite representation type. In view of this result, it is natural to ask if finite representation type of a ring $R$ can be characterized by more restricted endo-chain conditions on right (or left) $R$-modules. In this direction, it has been proved recently that a ring $R$ has finite representation type if and only if every pure-projective right $R$-module is endoartinian [14, Proposition 4.10], if and only if every pure-injective right $R$-module is endonoetherian [15, Theorem 5.8]. On the other hand, the ascending (descending) chain condition on finite matrix subgroups of all right $R$-modules is equivalent to $R$ having left (respectively, right) pure global dimension zero, i.e. $R$ is a left (respectively, right) pure semisimple ring (see [26, 29]).
+As a dual notion to finendo modules, a left (or right) $R$-module $M$ is said to be *cofinendo* if $M$ is finitely cogenerated as a module over its endomorphism ring. Thus the finendo and cofinendo conditions are generalizations of the endonoetherian and endoartinian conditions, respectively. It was shown in [13, Theorem 3.10] that a ring $R$ has finite representation type if and only if every right $R$-module is both finendo and cofinendo. However, it was left open whether the finendo condition or cofinendo condition alone would be sufficient for the result to hold (see [13, Questions 1 and 2, pp. 122–123]).
+We will see in this paper that, for any ring $R$, if all right $R$-modules are finendo, then all left $R$-modules are cofinendo, and the latter condition holds precisely when $R$ is left pure semisimple and every finitely generated indecomposable left $R$-module is cofinendo (Theorem 2.11, Corollary 2.12).
+Recall that a ring $R$ is *left pure semisimple* if every left $R$-module is a direct sum of finitely generated modules (see, e.g., [18, 33, 34]). Left and right pure semisimple rings are precisely the rings of finite representation type (see [6, 19, 31]). However, it has been a long-standing open problem, known as the *pure semisimplicity conjecture*, whether left pure semisimple rings are also right pure semisimple (see [24, 38] for historical surveys on the conjecture). Therefore, the question of characterizing rings $R$ with all right $R$-modules finendo can be regarded as a restricted form of the pure semisimplicity conjecture.
+A central part of this paper is devoted to the study of left pure semisimple rings $R$ such that every finitely generated indecomposable left $R$-module is cofinendo or finendo, respectively. We show that if $R$ is a left pure semisimple hereditary ring, then either the cofinendo or finendo condition on all finitely generated indecomposable left $R$-modules implies finite representation type. As a consequence, we obtain positive answers to [13, Questions 1 and 2, pp. 122–123] in the hereditary case (Theorem 3.5). For general rings $R$, the situation seems more complicated, and we are able to show that if $R$ is left pure semisimple with all finitely generated indecomposable left $R$-modules cofinendo, then $R$ has a right Morita duality, and the quotient ring $R/(J(R))^2$ is of finite representation type (Corollaries 4.2 and 4.3).

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+By Lemma 4.1, every finitely generated indecomposable left $M_J(R')$-module is cofinendo. Moreover, $M_J(R')$ is left pure semisimple, and it is of infinite representation type if $R'$ is of infinite representation type, by [5, Corollary 4.3]. But it is hereditary, and this contradicts Theorem 3.5. Therefore $R'$ is of finite representation type. $\blacksquare$
+We summarize our findings for the class of rings of our main interest, in
+the general case.
+COROLLARY 4.4. Let $R$ be a ring such that every right $R$-module is finendo. Then $R$ is a left pure semisimple ring with a right Morita duality, and the quotient ring $R/(J(R))^2$ is of finite representation type.
+*Proof*. The result follows by combining Corollary 2.12, Theorem 2.11, Theorem 4.2, and Proposition 4.3. $\blacksquare$
+We conclude the paper with the following remark (observed indepen-
+dently by Professor Daniel Simson, who suggested including it here).
+REMARK 4.5. Following Simson [39], a right artinian ring $R$ is said to have an *infinite right Morita sequence* if there is an infinite sequence of right artinian rings $\{R_n\}_{n=0}^{\infty}$ such that $R_0 = R$, and there is a Morita duality $\mathcal{D}_n$: mod-$R_n \rightarrow R_{n+1}$-mod for each $n \ge 0$. Such a sequence, if it exists, is uniquely determined by the ring $R$. Examples of artinian rings having an infinite right Morita sequence include artinian rings with self-duality, artinian PI-rings, and rings of finite representation type. Simson [39, Theorem 2.5] has shown that left pure semisimple rings $R$ having an infinite right Morita sequence are of finite representation type. Now, suppose that $R$ is a ring with all right $R$-modules finendo. By Corollary 4.4, $R$ is right Morita, hence there is a ring $R_1$ and a Morita duality $\mathcal{D}$: mod-$R \rightarrow R_1$-mod, and clearly $R_1$ is left pure semisimple. If the finendo property of all right $R$-modules is transferred through the Morita duality to the right $R_1$-modules, that would show the existence of an infinite right Morita sequence for such a ring $R$. This in turn would imply that rings $R$ with all right $R$-modules finendo must have finite representation type, in view of Simson's result above. However, we do not know if the finendo property of all right $R$-modules is preserved through the Morita duality in this situation.
+**Acknowledgments.** The authors would like to thank Professor Daniel Simson and the referee for several helpful comments and suggestions.
+REFERENCES
+[1] F. W. Anderson and K. R. Fuller, *Rings and Categories of Modules*, 2nd ed., Springer, 1992.
+[2] L. Angeleri Hügel, *Endocoherent modules*, Pacific J. Math. 212 (2003), 1–11.

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+[3] L. Angeleri Hügel and F. U. Coelho, *Infinitely generated tilting modules of finite projective dimension*, Forum Math. 13 (2001), 239–250.
+[4] L. Angeleri Hügel and H. Valenta, *A duality result for almost split sequences*, Colloq. Math. 80 (1999), 267–292.
+[5] M. Auslander, *Representation dimension of Artin algebras*, Queen Mary College Notes, 1971.
+[6] —, *Representation theory of Artin algebras II*, Comm. Algebra 1 (1974), 269–310.
+[7] —, *Functors and morphisms determined by objects*, in: Representation Theory of Algebras (Philadelphia PA, 1976), Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978, 1–244.
+[8] M. Auslander, I. Reiten and S. Smalø, *Representation Theory of Artin Algebras*, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995.
+[9] J. A. Beachy, *On quasi-artinian rings*, J. London Math. Soc. (2) 3 (1971), 449–452.
+[10] W. W. Crawley-Boevey, *Tame algebras and generic modules*, Proc. London Math. Soc. 63 (1991), 241–265.
+[11] —, *Modules of finite length over their endomorphism rings*, in: Representations of Algebras and Related Topics, S. Brenner and H. Tachikawa (eds.), London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press, 1992, 127–184.
+[12] N. V. Dung, *Strong preinjective partitions and almost split morphisms*, J. Pure Appl. Algebra 158 (2001), 131–150.
+[13] —, *Contravariant finiteness and pure semisimple rings*, in: Algebra and Its Applications D. V. Huynh et al. (eds.), Contemp. Math. 419, Amer. Math. Soc., 2006, 111–124.
+[14] N. V. Dung and J. L. García, *Endofinite modules and pure semisimple rings*, J. Algebra 289 (2005), 574–593.
+[15] —, —, *Endoproperties of modules and local duality*, J. Algebra 316 (2007), 368–391.
+[16] C. Faith, *Modules finite over endomorphism ring*, in: Lecture Notes in Math. 246, Springer, 1972, 145–189.
+[17] —, *Injective Modules and Injective Quotient Rings*, Lecture Notes in Pure Appl. Math. 72, Dekker, New York, 1982.
+[18] K. R. Fuller, *On rings whose left modules are direct sums of finitely generated modules*, Proc. Amer. Math. Soc. 54 (1976), 39–44.
+[19] K. R. Fuller and I. Reiten, *Note on rings of finite representation type and decompositions of modules*, Proc. Amer. Math. Soc. 50 (1975), 92–94.
+[20] P. Gabriel, *Indecomposable representations II*, Symposia Mat. Inst. Naz. Alta Mat. 11 (1973), 81–104.
+[21] I. Herzog, *A test for finite representation type*, J. Pure Appl. Algebra 95 (1994), 151–182.
+[22] B. Huisgen-Zimmermann, *Rings whose right modules are direct sums of indecomposable modules*, Proc. Amer. Math. Soc. 77 (1979), 191–197.
+[23] —, *Strong preinjective partitions and representation type of artinian rings*, ibid. 109 (1990), 309–322.
+[24] —, *Purity, algebraic compactness, direct sum decompositions, and representation type*, in: Infinite Length Modules, H. Krause et al. (eds.), Trends Math., Birkhäuser, 2000, 331–367.
+[25] B. Huisgen-Zimmermann and M. Saorín, *Direct sums of representations as modules over their endomorphism rings*, J. Algebra 250 (2002), 67–89.
+[26] B. Huisgen-Zimmermann and W. Zimmermann, *On the sparsity of representations of rings of pure global dimension zero*, Trans. Amer. Math. Soc. 320 (1990), 695–711.

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+Let $M = \sum_{j \in J} X_j$ be a sum of finitely generated endosubmodules of $M$. Since finitely generated endosubmodules are matrix subgroups, the join in the lattice of matrix subgroups of the $X_j$ is again $M$, and thus $M$ is the join of a finite subfamily $\{X_j \mid j \in F\}$. But the usual sum of finitely many finitely generated endosubmodules of $M$ is a finitely generated endosubmodule, hence a matrix subgroup of $M$, so the join of the finite subfamily $\{X_j \mid j \in F\}$ is $\sum_{j \in F} X_j$. Therefore $M = \sum_{j \in F} X_j$ is finitely generated as a module over its endomorphism ring.
+Now, if $N = \bigoplus_{i \in I} N_i$ is a direct sum of finitely presented left $R$-modules, and $M = \bigoplus_{i \in I} D(N_i)$, then [15, Theorem 4.1] implies that the lattice of endosubmodules of $N$ is anti-isomorphic to the lattice of matrix subgroups of $M$, and thus the second assertion of the proposition follows. $\blacksquare$
+**REMARK 2.7.** We note that the finendo and cofinendo properties are not preserved under direct summands, in general. Indeed, if $M$ is any left module over an arbitrary ring $R$ and $M$ is not finendo, then the left $R$-module $R \oplus M$ is a generator in $R$-Mod, hence it is finendo by Corollary 2.2. This shows that direct summands of finendo modules need not be finendo, in general. As for the cofinendo property, let $R$ be a left artinian hereditary ring that is not right artinian (for example, $R = (\begin{smallmatrix} F & F \\ 0 & K \end{smallmatrix})$, where $K$ is a subfield of a field $F$, and $F$ is infinite-dimensional over $K$). Then there is a finitely generated projective left $R$-module $M$ that is not finendo (see Lemma 3.3), and the left $R$-module $L = R \oplus M$ is finendo, as above. By Proposition 2.4, the local dual $D(M)$ of $M$ is not cofinendo. On the other hand, it follows from [32, Theorem 1.6] that $D(L) = D(R \oplus M) \cong D(M) \oplus D(R)$, because the endomorphism ring of $R \oplus M$ is semiprime. Note that, since $L$ is finitely generated over its semiprime endomorphism ring, $L$ is semisimple modulo its endoradical (see, e.g., [43, 42.3]). Applying again Proposition 2.4, we find that $D(R \oplus M)$ is cofinendo. Hence the cofinendo right $R$-module $D(M) \oplus D(R)$ contains a direct summand $D(M)$ that is not cofinendo.
+In this paper, we are interested in characterizing rings over which all or certain classes of right (or left) modules are finendo or cofinendo. The following results will be used.
+**LEMMA 2.8.** Let $R$ be a ring. Then $R$ is right artinian if and only if every quotient ring $R/I$ for a two-sided ideal $I$ of $R$ has a finitely generated essential right socle.
+*Proof.* See Beachy [9]. $\blacksquare$
+**COROLLARY 2.9.** Let $R$ be a left artinian ring such that every finitely generated indecomposable left $R$-module is cofinendo. Then $R$ is right artinian.

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+*Proof.* By Lemma 2.3, every finitely generated left $R$-module is cofinendo. In particular, for every two-sided ideal $I$ of $R$, the ring $R/I$ is finitely generated as a right module over itself. Hence Lemma 2.8 shows that $R$ is right artinian. $\blacksquare$
+**LEMMA 2.10.** *Let $R$ be a right artinian ring. Then $R$ is left pure semisimple if and only if every pure-projective right $R$-module is finendo.*
+*Proof.* See [13, Lemma 3.8]. $\blacksquare$
+We now give a characterization of rings over which every left module is cofinendo.
+**THEOREM 2.11.** *The following conditions are equivalent for a ring $R$:*
+(a) Every left $R$-module is cofinendo.
+(b) $R$ is left pure semisimple, and every finitely generated indecomposable left $R$-module is cofinendo.
+(c) $R$ is left pure semisimple, and every indecomposable pure-injective right $R$-module is finendo.
+(d) $R$ is left pure semisimple, and every pure-injective right $R$-module is finendo.
+(e) $R$ is left artinian, and every direct sum of a family of indecomposable pure-injective right $R$-modules is finendo.
+*Proof.* (a)⇒(b). Suppose that (a) holds. Then, in particular, the quotient ring $R/I$ has a finitely generated essential right socle for every two-sided ideal $I$ of $R$. Hence $R$ is right artinian by Lemma 2.8. Let $N = \bigoplus_{i \in I} N_i$ be any direct sum of finitely presented right $R$-modules $N_i$. Consider the direct sum $M = \bigoplus_{i \in I} M_i$, where $M_i = D(N_i)$ is the local dual of $N_i$. Then by Proposition 2.4, the fact that the left $R$-module $M$ is cofinendo implies that $N$ is finendo. Therefore, $R$ is right artinian and every pure-projective right $R$-module is finendo (keep in mind that over the right artinian ring $R$, pure-projective right $R$-modules are precisely the direct sums of finitely presented right $R$-modules). Applying Lemma 2.10, we conclude that $R$ is left pure semisimple.
+(b)⇒(a). Assume that $R$ is left pure semisimple and every finitely generated indecomposable left $R$-module is cofinendo. Let $M$ be any left $R$-module; we need to show that $M$ is finitely cogenerated over its endomorphism ring. By [25, Lemma A], because $M$ is $\Sigma$-pure-injective, $M$ is semiartinian as a module over its endomorphism ring. In particular, $M$ has an essential endosocle. Therefore it is sufficient to show that the endosocle $B$ of $M$ is finitely generated. Let $M = \bigoplus_{i \in I} M_i$ be an indecomposable decomposition of $M$, each $M_i$ having a local endomorphism ring. By [25, Lemma B(2)], there is a direct sum decomposition $B = \bigoplus_{i \in I} B_i$, where $B_i = B \cap M_i$,

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+and $B_i$ is the End$(M_i)$-submodule of $M_i$ consisting of all elements of $M_i$ annihilated by all non-isomorphisms in $\bigcup_{j \in I} \text{Hom}_R(M_i, M_j)$. We will first show that there are only finitely many non-isomorphic modules $M_i$ such that $B_i$ is non-zero.
+Since $R$ is left pure semisimple, by [23, Theorem A] (cf. [12, Corollary 3.7]), the family $\{M_i \mid i \in I\}$ has a finite cogenerating set $\mathcal{A} = \{M_{i_1}, \dots, M_{i_n}\}$, i.e. each $M_i$ can be embedded into a finite direct sum of modules in $\mathcal{A}$. Suppose that some $M_i$ is not isomorphic to any of the modules in $\mathcal{A}$. Then the reject of the family $\mathcal{A}$ in $M_i$ is zero, i.e. the intersection of all kernels of homomorphisms from $M_i$ to the modules in $\mathcal{A}$ is zero. Note that homomorphisms from $M_i$ to the modules in $\mathcal{A}$ are non-isomorphisms. It follows from the description of $B_i$ above that $B_i$ is zero, proving the claim.
+By grouping the isomorphic indecomposable modules $M_i$ together, we obtain a disjoint partition $I = \bigcup_{\alpha \in \Omega} I_\alpha$ such that if $i, j \in I_\alpha$ then $M_i \cong M_j$, and if $i \in I_\alpha$, $j \in I_\beta$ with $\alpha \neq \beta$, then $M_i \not\cong M_j$. Set $N_\alpha = \bigoplus_{i \in I_\alpha} M_i$. Then $M = \bigoplus_{\alpha \in \Omega} N_\alpha$. Moreover, as observed above, $B \cap N_\alpha \neq 0$ for only finitely many distinct $\alpha \in \Omega$. Note that, by hypothesis, each $M_i$ has a finitely generated endoscle, and [25, Lemma B(1)] implies that for any index set $K$, $\text{Esoc}(M_i^{(K)}) = (\text{Esoc}(M_i))^{(K)}$ is also finitely generated. Thus each $N_\alpha$ has a finitely generated endoscle. For each $i \in I$, because the endoscle of $M_i$ is the intersection of all kernels of non-isomorphisms from $M_i$ to $M_i$, it is clear that $B_i$ is contained in the endoscle of $M_i$. Hence
+$$B \cap N_\alpha = \bigoplus_{i \in I_\alpha} (B \cap M_i) \subseteq \bigoplus_{i \in I_\alpha} \text{Esoc}(M_i) = \text{Esoc}\left(\bigoplus_{i \in I_\alpha} M_i\right) = \text{Esoc}(N_\alpha)$$
+and so $B \cap N_\alpha$ is a finitely generated endosubmodule of the endoscle of $N_\alpha$. Keeping in mind that each endomorphism of $N_\alpha$ can also be identified in a natural way with an endomorphism of $M$, it follows easily that $B = \bigoplus_{\alpha \in \Omega} (B \cap N_\alpha)$, with only finitely many non-zero terms, is indeed finitely generated over the endomorphism ring $S$ of $M$.
+(b) ⇒ (c). Suppose that (b) holds. Since $R$ is left pure semisimple, we know that every indecomposable pure-injective right $R$-module is the local dual of a finitely presented indecomposable left $R$-module (see [15, Proposition 5.6]). Thus it follows from Proposition 2.6 that every indecomposable pure-injective right $R$-module is finendo.
+(c) ⇒ (d). Assume that (c) holds. If $N$ is any finitely generated indecomposable left $R$-module, then its local dual $D(N)$ is an indecomposable pure-injective right $R$-module. Since $D(N)$ is finendo by hypothesis, Proposition 2.5 implies that $N$ is cofinendo. This shows that (b) holds.
+Now let $M$ be any pure-injective right $R$-module. By [15, Proposition 5.6] $M$ is the pure-injective envelope of a direct sum $\bigoplus_{i \in I} D(N_i)$, each $N_i$ being a finitely presented indecomposable left $R$-module. By the implication

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+(b) ⇒ (a) proved above, the left R-module $\bigoplus_{i \in I} N_i$ is cofinendo. Let A be
+the left functor ring of R, and T : Mod(R) → Mod(A) be the canonical em-
+bedding functor (see, e.g., [15, p. 375] for definitions and basic properties).
+Note that T(M) is the injective hull of $T(\bigoplus_{i \in I} D(N_i)) \cong \bigoplus_{i \in I} T(D(N_i))$.
+By [15, Lemma 2.1], the torsion theory of Mod(A) cogenerated by $T(M)$
+is the same as the torsion theory cogenerated by $\bigoplus_{i \in I} T(D(N_i))$. By ap-
+plying [15, Proposition 3.2] to M and $\bigoplus_{i \in I} D(N_i)$, we see that the lattice
+of matrix subgroups of M is isomorphic to the lattice of matrix subgroups
+of $\bigoplus_{i \in I} D(N_i)$, which in turn is anti-isomorphic to the lattice of endosub-
+modules of $N = \bigoplus_{i \in I} N_i$ (see [15, Theorem 4.1]). Since N is cofinendo, we
+conclude that M is finitely generated as a module over its endomorphism
+ring, by Proposition 2.6.
+(d) ⇒ (e). Suppose that (d) holds. Clearly (d) ⇒ (c), and we have already established (c) ⇒ (b) in the proof of (c) ⇒ (d) above. Hence (b) holds. Then we know by the implication (b) ⇒ (a) that every left R-module is cofinendo. On the other hand, because R is left pure semisimple, we know by [15, Proposition 5.6] that every direct sum of indecomposable pure-injective right R-modules is a direct sum of local duals of finitely presented indecomposable left R-modules. Therefore it follows from Proposition 2.6 that every direct sum of indecomposable pure-injective right R-modules is finendo, proving (e).
+(e) ⇒ (a). Suppose that (e) holds. Let $N$ be any pure-projective left $R$-module. Because $R$ is left artinian, there is a direct sum decomposition $N = \bigoplus_{i \in I} N_i$ with each $N_i$ finitely presented with local endomorphism ring. Take the local dual $M_i = D(N_i)$ of each $N_i$, and set $M = \bigoplus_{i \in I} M_i$. It is well-known that each $M_i$ is an indecomposable pure-injective right $R$-module. Hence $M$ is finendo by (e), and Proposition 2.5 implies that $N$ is cofinendo. Thus $R$ is a left artinian ring with the property that each pure-projective left $R$-module is cofinendo. It follows from [13, Lemma 3.9] that $R$ is left pure semisimple. Thus every left $R$-module is pure-projective and (a) follows. $\blacksquare$
+We deduce the following consequence that gives a relationship between
+the two open questions mentioned in the introduction.
+COROLLARY 2.12. Let $R$ be a ring, and consider the following two conditions.
+(a) Every right R-module is finendo.
+(b) Every left R-module is cofinendo.
+Then we have the implication (a) ⇒ (b).
+*Proof.* Suppose that (a) holds. In particular, every quasi-injective right *R*-module is finendo, hence we know by Faith [16, Theorem 17A] that *R* is

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+We now turn to the case when the homomorphism $\bar{\mu}: U \to \operatorname{Hom}(M, V)$ is not injective. But, since the module $X$ is indecomposable, we have $V = 0$ and $U \cong G$, and the module is then isomorphic to $(G, 0, 0)$. And we have already seen in part (a) that this is a finendo module. Hence we have shown that every finitely generated indecomposable left $R_M$-module is finendo. ■
+We are now in a position to prove the main result of this section, giving positive answers to Questions 1 and 2 in [13, pp. 122–123] for hereditary rings. The implications (c)⇒(e) and (d)⇒(e) also sharpen [14, Theorem 4.1] in the hereditary case, where finitely generated indecomposable left $R$-modules were assumed to be endofinite (see also [35, Corollary 3.2]).
+**THEOREM 3.5.** Let $R$ be any hereditary ring. Then the following conditions are equivalent.
+(a) Every right $R$-module is finendo.
+(b) Every left $R$-module is cofinendo.
+(c) $R$ is left pure semisimple, and every finitely generated indecomposable left $R$-module is cofinendo.
+(d) $R$ is left pure semisimple, and every finitely generated indecomposable left $R$-module is finendo.
+(e) $R$ is of finite representation type.
+*Proof.* (a)⇒(b) and (b)⇔(c) follow from Corollary 2.12 and Theorem 2.11, respectively (even without the hereditary hypothesis on the ring $R$).
+(c)⇒(e). Suppose that (c) holds, i.e. $R$ is left pure semisimple and every finitely generated indecomposable left $R$-module is cofinendo. Then, by Lemma 2.3, every finitely generated left $R$-module is cofinendo.
+Now assume on the contrary that $R$ is not of finite representation type. In view of Lemma 3.1, we can take $R$ to be a basic and indecomposable ring. Moreover, we know by Lemma 3.2 that there exist a pair of indecomposable projective direct summands $P_i \not\approx P_j$ of $RR$ and a ring isomorphism
+$$ R_B = \begin{pmatrix} F & 0 \\ B & G \end{pmatrix} \cong \operatorname{End}(P_i \oplus P_j) $$
+where $B = \operatorname{Hom}_R(P_i, P_j)$, $F = \operatorname{End}_R(P_i)$ and $G = \operatorname{End}_R(P_j)$ are division rings, such that the ring $R_B$ is not of finite representation type. Note that, by Simson [36, Theorem 17.46], there is a fully faithful embedding functor $T$: $R_B$-mod $\to$ $R$-mod. It follows easily that $R_B$ is a left pure semisimple ring. Note also that every finitely generated indecomposable left $R_B$-module can be seen as a finitely generated left $R$-module having the same endomorphism ring. Consequently, every finitely generated indecomposable left $R_B$-module is cofinendo. By Corollary 2.9 we see that $R_B$ is right artinian.

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+Grassmannian or isomorphic to (X, 0, 0). In this second case, it is obvious
+that X has to be a simple left R/J-module.
+Suppose next that the finitely generated indecomposable left $T_I(R)$-module $M$ is Grassmannian. By the density of the functor $J$ in [5, Theorem 3.1, b)], there exists a left $R$-module $X$ with injective hull $Q$ so that $M$ is isomorphic to $(X/ann_X(I), \text{soc}(Q), f)$, where the mapping $f: I \otimes_{R/J} (X/ann_X(I)) \to \text{soc}(Q)$ is canonical. Since $\text{soc}(Q) = \text{soc}(X)$, we infer that $M$ is isomorphic to $\text{Gr}(X)$.
+Now, suppose that $X$ is not indecomposable. Then $X = X_1 \oplus X_2$ implies that $\text{soc}(X) = \text{soc}(X_1) \oplus \text{soc}(X_2)$, hence the given $T_I(R)$-module would be a non-trivial direct sum, since $R$ is left artinian. This proves (a).
+(b) We start by noting that every indecomposable left $T_I(R)$-module of the form $(X, 0, 0)$ is isomorphic to $X$ as a module over its endomorphism ring. Since $X$ is an $R/J$-module and $R/J$ is semisimple, we deduce that it is endofinite, and, in particular, cofinendo.
+By (a), it remains to see that any indecomposable Grassmannian left $T_I(R)$-module $M$ is cofinendo. In view of [5, Theorem 3.1], the restriction of the functor $\text{Gr}$ to those left $R$-modules $X$ such that $\text{ann}_X(I)$ is injective as a left $R/I$-module is a full functor. Since it is also dense, we can choose an indecomposable finitely generated left $R$-module $X$ such that $\text{Gr}(X) \cong M$ and there is a surjective ring homomorphism
+$$\phi : E = \text{End}_{(R,X)} \to H = \text{End}_{(T_I(R)\text{Gr}(X))}.$$
+Let $K = \text{Ker}(\phi)$. Both $E$ and $H$ are local rings. If $U_E$ is the injective hull of the unique simple right $E$-module, then it is not hard to see that $\text{ann}_U(K)_H$ is the injective hull of the unique simple right $H$-module. Therefore $D(X) = \text{Hom}_E(X,U)$ and $D(\text{Gr}(X)) = \text{Hom}_H(\text{Gr}(X), \text{ann}_U(K))$.
+Our plan for the proof is the following. Since $RX$ is cofinendo, it follows from Proposition 2.6 that $D(X) = \text{Hom}_E(X,U)$ is finendo. We will use this to show that $D(\text{Gr}(X))$ is a finendo right $T_I(R)$-module, and by Proposition 2.5 we will conclude that $\text{Gr}(X)$ is cofinendo.
+We know from [23, Remark 2, p. 312] that the endomorphism ring of $D(X)$ is isomorphic to $\text{End}(U_E)$ in the natural way, and, similarly, the endomorphism ring of $D(\text{Gr}(X))$ is isomorphic to $\text{End}(\text{ann}_U(K)_H)$. Now, $\text{Gr}(X)$ is, as a right $H$-module, isomorphic to the direct sum of the modules $\overline{X} = X/\text{ann}_X(I)$ and $\text{soc}(X) = IX$. Specifically, the structure of $D(\text{Gr}(X))$ as a right $T_I(R)$-module is easily seen to be given by the triple $(\text{Hom}_H(IX, \text{ann}_U(K)), \text{Hom}_H(\overline{X}, \text{ann}_U(K)), \mu)$, where $\mu$ is the surjective canonical map
+$$\mu : \text{Hom}_H(IX, \text{ann}_U(K)) \otimes_{R/J} I \to \text{Hom}_H(\overline{X}, \text{ann}_U(K)).$$
+Thus, to see that $D(\text{Gr}(X))$ is finendo, it will suffice to show that each of the

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+# OPTIMIZATION AND SIMULATION FOR AIRPORT EMERGENCY INVENTORY WITH REPLACEMENT
+Meng, Q. C.*; Guo, Y.*; Zhao, P. X.*#; Lu, T. X.*; Wan, X. L.*; Rong, X. X.** & Pan, W.***
+* School of Management, Shandong University, Jinan 250100, China
+** School of Mathematics, Shandong University, Jinan 250100, China
+*** School of Economics and Management, Wuhan University, Wuhan 430072, China
+E-Mail: pxzhao@sdu.edu.cn (# Corresponding author)
+## Abstract
+The paper assumes that the accident occurrence time is stochastic and emergency supplies is perishable, then two replacement stochastic models based on remaining lifetime and remaining quantity are first proposed. In order to identify the effectiveness of replacement strategy, two replacement-based stochastic models are compared with the general stochastic model that is non-replacement, measured by inventory level and total costs. A discrete-event simulation model is developed to demonstrate effects of occurrence time uncertainty, replacement ratios and distributed functions in occurrence time and demand. Sensitive analysis shows that the optimal decision is more sensitive to remaining quantity ratio as compared to remaining lifetime ratio. The paper shows that when decision-makers ignore occurrence time uncertainty and limited warehousing time, they may significantly miss better decisions. Further, simulation results demonstrate that different distributed functions in both occurrence time and demand lead to different inventory strategies.
+(Received, processed and accepted by the Chinese Representative Office.)
+**Key Words:** Occurrence Time Uncertainty, Emergency Supplies, Replacement strategy, Inventory Optimization
+## 1. INTRODUCTION
+Although civil aviation accident is event of small probability, the losses and fatalities are serious. For example, there were 92 accidents and 474 fatalities last year, which leads to big losses for companies and society. According to safety report from International Civil Aviation Organization (ICAO), about 70% civil aviation accidents occurred around the airports, so the ICAO formulated a series of regulations for emergency supplies in the airport. The airports always reserve much more emergency supplies and replenish emergency supplies immediately after emergency response. Therefore, investments in airport emergency supplies are significant and a cause of financial concern and subsequent economic losses. Thus, minimizing the losses and the costs, while maintaining the constraint of service level, is the main purpose of the airport.
+In this paper, we first propose replacement strategy and develop an integrated model of occurrence time uncertainty and limited warehousing time, capturing the effects of occurrence time uncertainty and replacement strategy on service level and total costs. Our designs of alternative inventory system are derived from two sources: (1) observations of emerging practices in returns processing and (2) previous researches on emergency supplies inventory management. We analyse a benchmark system where an airport emergency planner decides the inventory level with replacement strategy to minimize the total costs. We find that the inventory level in this benchmark system is not always higher or less than that in non-replacement setting. To understand how occurrence time uncertainty affects the inventory level, we setup the model with deterministic occurrence time and unlimited warehousing time, concluding that consideration of occurrence time uncertainty and limited warehousing time induces or reduces emergency inventory.

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+when stochastic variables conform to different distributions such as exponential distributions and normal distributions. In the absence of realistic data, we estimate parameters by taking advantage of relative literature [9, 14, 20]. Several mid-size problems are designed.
+Table II: Specific value of parameters.
+<table><thead><tr><th>a</th><th>b</th><th>c</th><th>d</th><th>T</th><th>s</th><th>e</th><th>θ</th><th>k</th></tr></thead><tbody><tr><td>1</td><td>3</td><td>10</td><td>110</td><td>2</td><td>20</td><td>12</td><td>0.5</td><td>10</td></tr></tbody></table>
+From results in Table III and Table IV, we find that considering occurrence time uncertainty and limited warehousing time reduces service level by 4.3 %, while reduces total costs by 51 %. We also show that the value interval will affect the optimal solutions, as well as distributed functions. Furthermore, the replacement strategy based on remaining lifetime seems to be more reliable facing occurrence time uncertainty and demand uncertainty, resulting from total costs variance between replacement strategy based on remaining lifetime (0.32 % – 0.37 % in example 1; 0.36 % – 2.67 % in example 2) and replacement strategy based on remaining quantity (-181.54 % – 0.96 % in example 1; 12.34 % in example 2). The simulation results and explanations of distribution functions are presented in Figs. 1 to 6.
+Table III: Optimal solutions to all models for $c = 100$ example.
+<table><thead><tr><th>$q_1/q_2$</th><th>0</th><th>0.1</th><th>0.2</th><th>0.3</th><th>0.4</th><th>0.5</th><th>0.6</th><th>0.7</th><th>0.8</th><th>0.9</th><th>1</th></tr></thead><tbody><tr><th>$I_1^*$</th><td>103.3</td><td>103.4</td><td>103.6</td><td>103.7</td><td>103.8</td><td>104</td><td>104.2</td><td>104.3</td><td>104.5</td><td>104.8</td><td>105</td></tr><tr><th>$ETC_1^*$</th><td>646.1</td><td>646</td><td>646.1</td><td>646</td><td>645.9</td><td>646</td><td>646.1</td><td>645.9</td><td>646</td><td>646.2</td><td>646.3</td></tr><tr><th>$I_2^*$</th><td>110</td><td>107.6</td><td>105</td><td>102.7</td><td>100.8</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr><tr><th>$ETC_2^*$</th><td>-525</td><td>-274</td><td>-56.9</td><td>127.9</td><td>284.5</td><td>400</td><td>490</td><td>560</td><td>610</td><td>640</td><td>650</td></tr><tr><th>$I_3^*$</th><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td><td>103.1</td></tr><tr><th>$ETC_3^*$</th><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td><td>643.8</td></tr><tr><th>$I_2$</th><td>123</td><td>119.6</td><td>116.6</td><td>114.1</td><td>112</td><td>110.3</td><td>109</td><td>108</td><td>107.2</td><td>106.8</td><td>106.7</td></tr><tr><th>$ETC_2$</th><td>-1116</td><td>-597</td><td>-159</td><td>209.6</td><td>516.8</td><td>768.4</td><td>969.3</td><td>1122</td><td>1228</td><td>1292</td><td>1313</td></tr><tr><th>$I_3$</th><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td><td>107.7</td></tr><tr><th>$ETC_3$</th><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td><td>1316</td></tr></tbody></table>
+Figure 1: Stochastic occurrence time and stochastic demand conform to uniform distribution.
+Figure 2: Stochastic occurrence time and stochastic demand conform to uniform distribution and exponential distribution respectively with $\lambda = 1/105$.

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+Table IV: Optimal solutions to all models for $c = 10$ example.
+<table><thead><tr><th>q<sub>1</sub>/q<sub>2</sub></th><th>0.1</th><th>0.2</th><th>0.3</th><th>0.4</th><th>0.5</th><th>0.6</th><th>0.7</th><th>0.8</th><th>0.9</th></tr></thead><tbody><tr><th>I<sub>1</sub>*</th><td>34.1</td><td>34.7</td><td>35.4</td><td>36</td><td>36.7</td><td>37.4</td><td>38.2</td><td>39</td><td>39.9</td></tr><tr><th>ETC<sub>1</sub>*</th><td>511.8</td><td>510.6</td><td>509.4</td><td>508.1</td><td>506.7</td><td>505.2</td><td>503.7</td><td>502</td><td>500.3</td></tr><tr><th>I<sub>2</sub>*</th><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><th>ETC<sub>2</sub>*</th><td>560</td><td>560</td><td>560</td><td>560</td><td>560</td><td>560</td><td>560</td><td>560</td><td>560</td></tr><tr><th>I<sub>3</sub>*</th><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td><td>40.8</td></tr><tr><th>ETC<sub>3</sub>*</th><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td><td>498.5</td></tr></tbody></table>
+Figure 3: Exponential distribution of stochastic occurrence time with $\lambda = 1$ and uniform distribution of stochastic demand.
+Figure 4: Exponential distribution of stochastic occurrence time with $\lambda = \frac{1}{2}$ and uniform distribution of stochastic demand.
+Figure 5: Uniform distribution of stochastic occurrence time and normal distribution of stochastic demand with $\mu = 105$, $\sigma = 1/105^2$.
+Figure 6: Normal distribution of stochastic occurrence time with $\mu = \frac{1}{2}$, $\sigma = \frac{1}{4}$ and uniform distribution of stochastic demand.

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+The results in Figs. 1 to 6 show that different stochastic distribution leads to different inventory strategies and the service levels are more likely to be enhanced with replacement strategy when stochastic demand conforms to normal distribution. $\lambda$ represents the frequency within per unit time and the large value of $\lambda$ means the high probability of civil aviation accident, leading to airport reserves more emergency supplies to deal with emergency and risk. Furthermore, the effect of distributed function in stochastic occurrence time is not so great comparatively.
+# 6. CONCLUSIONS
+The paper studies an inventory control system with stochastic accident occurrence time and short lifetime of perishable emergency supplies, then first proposes two stochastic models with replacement strategy based on remaining lifetime and remaining quantity of emergency supplies respectively at the end of emergency preparation phase.
+The effectiveness of replacement strategy is not only correlated to unit replacement cost, but also to the value of replacement ratio. The accident occurrence time uncertainty and limited warehousing lifetime lower the optimal inventory level, and simulation results show that considering occurrence time uncertainty and limited warehousing time can reduce total costs by 51 % though reduce service level by 4.3 %. Simulation results demonstrate that the distribution interval affects optimal solutions, and that different distributed functions in both occurrence time and demand lead to different decisions. Further, solving a replacement model based on remaining lifetime can obtain more reliable solutions.
+Our work helps enrich researches on risk management in theory and provide advice and suggestions for airport in practice. However, there is much future work to perfect our research content. Obviously, the network designs and supply chain management between airports and suppliers should be studied deeply. Then, a game model should be proposed between airports, government, society and companies for coordination mechanism. Thirdly, a multi-stage stochastic programming or dynamic planning should be researched on emergency supplies inventory management.
+# ACKNOWLEDGEMENTS
+This study is supported by the National Nature Science Foundation of China (NSFC) (Grant no. 71572096, 71373188, U1333115).
+# REFERENCES
+[1] Mete, H. O.; Zabinsky, Z. B. (2010). Stochastic optimization of medical supply location and distribution in disaster management, *International Journal of Production Economics*, Vol. 126, No. 1, 76-84, doi:10.1016/j.ijpe.2009.10.004
+[2] Whybark, D. C. (2007). Issues in managing disaster relief inventories, *International Journal of Production Economics*, Vol. 108, No. 1-2, 228-235, doi:10.1016/j.ijpe.2006.12.012
+[3] Tang, M.; Gong, D.; Liu, S.; Zhang, H. (2016). Applying multi-phase particle swarm optimization to solve bulk cargo port scheduling problem, *Advances in Production Engineering & Management*, Vol. 11, No. 4, 299-310, doi:10.14743/apem2016.4.228
+[4] Yu, G. D. (2016). Modelling for emergency manufacturing resources schedule to unexpected events, *International Journal of Simulation Modelling*, Vol. 15, No. 2, 313-326, doi:10.2507/IJSIMM15(2)10.348
+[5] Taskin, S.; Lodree Jr., E. J. (2010). Inventory decisions for emergency supplies based on hurricane count predictions, *International Journal of Production Economics*, Vol. 126, No. 1, 66-75, doi:10.1016/j.ijpe.2009.10.008

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+[25] Qi, Y.; Tang, M.; Zhang, M. (2014). Mass customization in flat organization: The mediating role of supply chain planning and corporation coordination, *Journal of Applied Research and Technology*, Vol. 12, No. 2, 171-181, doi:10.1016/S1665-6423(14)72333-8
+[26] Luscombe, R.; Kozan, E. (2016). Dynamic resource allocation to improve emergency department efficiency in real time, *European Journal of Operational Research*, Vol. 255, No. 2, 593-603, doi:10.1016/j.ejor.2016.05.039
+[27] Guide Jr., V. D. R.; Souza, G. C.; Van Wassenhove, L. N.; Blackburn, J. D. (2006). Time value of commercial product returns, *Management Science*, Vol. 52, No. 8, 1200-1214, doi:10.1287/mnsc.1060.0522
+[28] Shyur, H.-J. (2008). A quantitative model for aviation safety risk assessment, *Computers & Industrial Engineering*, Vol. 54, No. 1, 34-44, doi:10.1016/j.cie.2007.06.032
+[29] Skorupski, J. (2016). The simulation-fuzzy method of assessing the risk of air traffic accidents using the fuzzy risk matrix, *Safety Science*, Vol. 88, 76-87, doi:10.1016/j.ssci.2016.04.025
+[30] Sun, R. S.; Meng, L. H. (2013). Time distribution of aviation accident, *Journal of Transport Information and Safety*, Vol. 31, No. 2, 83-87

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+Researches on emergency supplies management obtain many achievements in preposition, optimal inventory, planning and distribution [1-4], and help reduce losses of life and property in practice [5-8]. Emergency means uncertainty, so stochastic models, with stochastic lead time, demand, supply, procurement cost, transport cost etc., were proposed to fulfil purposes of minimizing cost and maximizing satisfaction degree of people in affected areas [9-10]. Paper [11] set up a three-level model containing supply, storage and demand for the emergency planning at the preparation and response stage, and the objective of this model is minimizing the social cost. Research [12] proposed a coordination system among emergency warehouses. In this system, a stochastic model was built with constraints of service equality, traffic congestion and response time. In [13], authors established a stochastic model to decide optimal storage time. Pan et al. [14] proposed a stochastic inventory model with stochastic occurrence time, stochastic demand and perishable emergency supplies.
+Civil aviation emergency decision is a prospective issue, but return and replacement of emergency supplies is insignificant, except for commercial products [15]. In [16], authors came up with a novel reverse logistics system for post-disaster debris capable of systematically minimizing total reverse logistical costs, risk-induced cost and psychological cost. The effectiveness of the proposed system was demonstrated by applying it to a case study of Wenchuan earthquake. Paper [17] proposed an effective location models for sorting recyclables in public management, and provided an optimal location planning design for recycling urban solid wastes given the uncertainty future outcome of economic factors, consumer behaviour and environmental awareness. Paper [18] described an exploratory study of reverse exchange systems used for medical devices in the UK National Health Service.
+Emergency management for civil aviation mainly includes designing information systems, simulation of emergency evacuation, and formulating emergence planning. Paper [19] analysed Singapore Airlines Flight 006 aircraft accident in 2000, and suggested that plan, execute, and support systems are important for medical emergency supplies in airport. Pan and Guo [20] studied the compensation mechanism based on bargaining game strategy on the background of MH370. In [21], authors proposed an accident rescue model for railroad system, and considered the uncertainty of demand, idle probability of ambulance, and risk level in a section of railway. Similarly to railway rescue system, the airport rescue systems are couple with uncertainty. ICAO stipulates that the airport should reserve emergency supplies to enhance response capability. The emergency supplies include alcohol, blood, plasma and some other perishable medical materials [22]. Inventory management of perishable products also has attracted much attention [23-25].
+Emergency decisions for disasters need to be capable of adapting to random and highly dynamic change [26], and decision-makers have a growing interest in improved strategies from reverse logistics [27]. Thus, a significant difference between our model and previous research on emergency supplies management is that we introduce the replacement strategy facing occurrence time uncertainty into the inventory control system. The optimal inventory levels and total costs of models are compared to show the effects of the replacement strategy and the occurrence time uncertainty. The deficient research about return of emergency supplies and the important practice of emergency inventory make our work valuable.
+This paper is organized as follows. In section 2, the researched problem is defined and described. Replacement model based on remaining time and remaining quantity are presented with unknown stochastic distribution functions in section 3. In section 4, we compare inventory level and total costs of different models and present analyses about effectiveness of replacement strategy and occurrence time uncertainty. In section 5, we develop a discrete-event simulation model to demonstrate and analyse decisions, applying the results to some simple numerical cases. Finally, conclusions are reached in section 6.

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+## 2. PROBLEM DESCRIPTION
+In practice, emergency supplies for civil aviation accidents generally come from airport's inventory and some specified rescue units. The emergency supplies ordering system is different from quantitative order method or regular order method. The ending time of a single emergency phase is min(*t*, *T*), among which *t* represents stochastic civil aviation occurrence time and *T* represents shelf life of emergency supplies. In other words, the preparation phase will enter into a new circulate phase when emergency response is end or emergency supplies are expired. We assume the stochastic demand and the stochastic occurrence time are mutually independent.
+For the purpose of our analysis, the following notations are used:
+* *e* – the unit expired cost of emergency supplies,
+* *s* – the unit shortage cost of emergency supplies,
+* *k* – the unit exchange or replacement cost of emergency supplies,
+* θ – the ratio of expected expired quantity to remaining quantity, $0 ≤ θ ≤ 1$,
+* *x* – the stochastic demand of emergency supplies at stage of emergency response,
+* *I* – the optimal inventory level, a decision variable,
+* *t* – stochastic civil aviation occurrence time,
+* *T* – shelf life of emergency supplies,
+* *a*, *b* – the lower and upper bound of stochastic occurrence time *t*,
+* *c*, *d* – the lower and upper bound of stochastic demand *x*,
+* *q*₁ – the replacement lifetime ratio, defined by time at replacement dividing lifetime,
+* *q*₂ – the replacement quantity ratio, defined by amount of usage dividing total inventory,
+* *f(x)* – probability density function of stochastic demand,
+* *g(t)* – probability density function of stochastic occurrence time,
+* *F(x)* – cumulative density function of stochastic demand,
+* *G(t)* – cumulative density function of stochastic occurrence time,
+* (*A*)⁺ – max (0, *A*), that is, (*A*)⁺ = max (0, *A*).
+The two replacement scenarios are proposed based on checking system of emergency supplies in airport practically. At the end of emergency phase, airport will check inventory. If the remaining lifetime is short or remaining quantity is large, in order to avoid the huge expired losses, airport will ask suppliers to return and exchange the surplus materials.
+**Scenario 1:** airport and suppliers reach an agreement on replacement lifetime ratio $q_1$, (1) if civil aviation accident occurs within $q_1T$, that is, when the remaining life time is longer than $(1 - q_1)T$, new emergency supplies will be replenished to the inventory level *I* without replacement strategy; (2) if civil aviation accident occurs beyond $q_1T$, and the remaining life time is shorter than $(1 - q_1)T$, all surplus materials will be replaced by suppliers and amount *I* of new emergency supplies will be transported to airport; (3) if no civil aviation accident occurs within shelf life *T*, all emergency supplies will be expired and airport affords the expired losses. Scenario 1 is the replacement strategy based on remaining lifetime.
+**Scenario 2:** airport and suppliers reach an agreement on replacement quantity ratio $q_2$, (1) if civil aviation accident demand is less than the amount $q_2I$, in other words, when the remaining quantity is larger than $(1-q_2)I$, all surplus materials will be replaced by suppliers and amount *I* of new emergency supplies will be transported to airport; (2) if civil aviation accident demand exceeds the amount $q_2I$, and the remaining quantity is smaller than $(1-q_2)I$, new emergency supplies will be replenished to the inventory level *I* without replacement strategy; (3) if no civil aviation accident occurs within shelf life *T*, all emergency supplies will be expired and airport affords the expired losses. Scenario 2 is the replacement strategy based on remaining quantity.

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+# 4. ANALYSIS FOR REPLACEMENT STRATEGY
+To demonstrate the effectiveness of replacement-based strategy, the general storage model with non-replacement strategy is proposed at the same time.
+## 4.1 General model with non-replacement
+$$ ETC_3 = \int_a^T \int_I^d s(x-I)f(x)g(t)dxdt + \int_a^T \int_c^I \theta e(I-x)f(x)g(t)dxdt + \int_T^b \int_c^d eIf(x)g(t)dxdt \quad (5) $$
+The second-order derivative is greater than zero, and the unique optimal solution of Eq. (7) can be calculated through letting first-order derivative being equal to zero.
+$$ I_3^* = F^{-1} \left( \frac{(s+e)G(T)-e}{(s+\theta e)G(T)} \right)^{+} $$
+**Proposition 4:** $\frac{\partial I_3^*}{\partial T} \ge 0$, $\frac{\partial I_3^*}{\partial s} \ge 0$, $\frac{\partial I_3^*}{\partial e} \le 0$, $\frac{\partial I_3^*}{\partial \theta} \le 0$.
+Proposition 4 shows that the optimal inventory level of Eq. (5) increases with the increase of the shelf life and unit shortage cost, and decreases with the increase of unit expired cost and possible expired rate.
+## 4.2 Comparison and analysis
+Considering replacement strategy complicates the inventory management problem and raises new issues when determining the optimal inventory strategies. To provide closed-form expressions, we assume both stochastic demand and stochastic occurrence time conform to a specific distribution. In order to testify the different effects of different distribution on the optimal inventory strategies, the different distributed functions in stochastic variable are extended in and numerical simulation section.
+Three popular stochastic distribution types are uniform distribution, normal distribution, and exponential distribution. Normal distribution requires a large amount of history data to define its parameters [9]. Exponential distribution seems to demonstrate the fundamental characteristics of random occurrence time [28], and it also requires mutually independent data to define parameters. However, for civil aviation accidents, the historical data are often sparse, and accidents are not mutually independent because of ex post measures [29]. In addition, the real data in paper [30] reveals that occurrence time conforms to uniform distribution in practice. Therefore, assumptions of uniform distributed function in stochastic occurrence time and uniformed distributed function in stochastic demand are reasonable and feasible. Then the solutions of Eqs. (1), (4), and (5) are obtained.
+$$ I_1^* = \left( \frac{(s+e)G(T)-e}{(s+k)G(T)+(\theta e-k)G(q_1 T)} \right)^{+} (d-c) + c $$
+$$ I_2^* = \max\left( \frac{(s+k)(T-a)c - e(b-T)(d-c)}{(T-a)(2kq_2 - kq_2^2 + s + \theta e - 2\theta e q_2 + \theta e q_2^2)}, c \right) $$
+$$ I_3^* = \left( \frac{(s+e)G(T)-e}{(s+\theta e)G(T)} \right)^{+} (d-c) + c $$
+## 4.3 Replacement model based on remaining lifetime versus general model
+**Lemma 1:** When $\theta e > k$, then $\frac{\partial(ETC_3 - ETC_1)}{\partial q_1} < 0$ and $ETC_3 \ge ETC_1$. When $\theta e \le k$, then $\frac{\partial(ETC_3 - ETC_1)}{\partial q_1} \ge 0$ and $ETC_3 \le ETC_1$.

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+**Proof:** $\frac{\partial(ETC_3 - ETC_1)}{\partial q_1} = \frac{1}{(b-a)(d-c)} \cdot (\theta e - k)(I - c)(\frac{I+c}{2} - I)$, and $I \ge c$.
+With a larger $q_1$, the probability of civil aviation accident occurring within $q_1T$ is larger and the remaining lifetime is shorter when asked to replace the surplus materials, leading to the larger probability of being expiration and not replacing surplus materials. Thus, if unit possible expired loss is higher than unit replacement cost, the expected losses will increase with the increase of $q_1$. Otherwise, the expected losses $ETC_1$ will decrease with the increase of $q_1$ if unit replacement cost is higher.
+**Lemma 2:** When $\theta e > k$, then $I_3^* < I_1^*$ and $\partial|I_1^*-I_3^*|/\partial q_1 \le 0$. When $\theta e \le k$, then $I_3^* \ge I_1^*$ and $\partial|I_1^*-I_3^*|/\partial q_1 \le 0$.
+**Proof:** Due to $1/I_3^* - 1/I_1^* = \frac{(\theta e - k)(G(T) - G(q_1 T))}{((s+e)G(T) - e)(d-c)}$ and $G(T) - G(q_1 T) \ge 0$.
+Thus, when $\theta e > k$, then $1/I_3^* > 1/I_1^*$; and when $\theta e \le k$, then $1/I_3^* \le 1/I_1^*$.
+In addition, when $\theta e > k$, then:
+$$ \frac{\partial |I_1^* - I_3^*|}{\partial q_1} = \partial \left( \frac{(s+e)G(T) - e}{(s+k)G(T) + (\theta e - k)G(q_1 T)} - \frac{(s+e)G(T) - e}{(s+\theta e)G(T)} \right) / \partial q_1 \le 0. $$
+and when $\theta e \le k$, then:
+$$ \frac{\partial |I_1^* - I_3^*|}{\partial q_1} = \partial \left( \frac{(s+e)G(T) - e}{(s+\theta e)G(T)} - \frac{(s+e)G(T) - e}{(s+k)G(T) + (\theta e - k)G(q_1 T)} \right) / \partial q_1 \le 0. $$
+Airport tends to store more emergency supplies to reduce shortage losses if unit replacement cost is lower. Contrarily, the airport tends to store less emergency supplies to reduce the high replacement cost if unit replacement cost is higher than unit possible expired loss. In addition, $q_1$ cannot change the relationship between $I_1^*$ and $I_3^*$.
+**Theorem 1:** The replacement strategy based on remaining lifetime can improve emergency supplies inventory strategies when $\theta e > k$.
+Theorem 1 can be demonstrated by Lemma 1 and Lemma 2. When unit replacement cost is lower than unit possible expired loss, the replacement strategy can reduce the expected losses and enhance the inventory level no matter what the value of $q_1$ is. Therefore, the airport should bargain with suppliers to lowing replacement cost, and can benefit from the replacement strategy based on remaining lifetime.
+## 4.4 Replacement model based on remaining quantity versus general model
+**Lemma 3:** When $\theta e > k$, $\frac{\partial(ETC_3 - ETC_2)}{\partial q_2} > 0$. When $\theta e \le k$, $\frac{\partial(ETC_3 - ETC_2)}{\partial q_2} \le 0$. There exists a turning point $\tilde{q}_2 = c/I$, making $(ETC_3 - ETC_2)_{q_2=c/I} = 0$ ($\tilde{q}_2 = 2-c/I \ge 1$ is rejected).
+**Proof:** $\frac{\partial(ETC_3 - ETC_2)}{\partial q_2} = \frac{1}{(b-a)(d-c)} \cdot (\theta e - k)I^2(1-q_2)(T-a)$, and $1-q_2 \ge 0, T-a \ge 0$.
+For a larger $q_2$, the probability of replacing surplus materials is larger and the probability of expiration is smaller. Thus, if unit possible expired loss is higher than unit replacement cost, the expected losses will increase with the increase of $q_2$. Otherwise, the expected losses $ETC_2$ will decrease with the increase of $q_2$. So the value of $q_2$ can change the decision whether to select the replacement strategy based on remaining quantity or not.

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+# Simulating Quantum Computations with Tutte Polynomials
+Ryan L. Mann¹,²,*
+¹School of Mathematics, University of Bristol, Bristol, BS8 1UG, United Kingdom
+²Centre for Quantum Computation and Communication Technology,
+Centre for Quantum Software and Information,
+Faculty of Engineering & Information Technology,
+University of Technology Sydney, NSW 2007, Australia
+We establish a classical heuristic algorithm for exactly computing quantum probability amplitudes. Our algorithm is based on mapping output probability amplitudes of quantum circuits to evaluations of the Tutte polynomial of graphic matroids. The algorithm evaluates the Tutte polynomial recursively using the deletion-contraction property while attempting to exploit structural properties of the matroid. We consider several variations of our algorithm and present experimental results comparing their performance on two classes of random quantum circuits. Further, we obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants and an alternative efficient classical algorithm for computing the output probability amplitudes of Clifford circuits.
+## I. INTRODUCTION
+There is a natural relationship between quantum computation and evaluations of Tutte polynomials [1, 2]. In particular, quantum probability amplitudes are proportional to evaluations of the Tutte polynomial of graphic matroids. In this paper we use this relationship to establish a classical heuristic algorithm for exactly computing quantum probability amplitudes. While this problem is known to be #P-hard in general [3], our algorithm focuses on exploiting structural properties of an instance to achieve an improved runtime over traditional methods. Previously it was known that this problem can be solved in time exponential in the treewidth of the underlying graph [4].
+The basis of our algorithm is a mapping between output probability amplitudes of quantum circuits and evaluations of the Tutte polynomial of graphic matroids [2, 5, 6]. Our algorithm proceeds to evaluate the Tutte polynomial recursively using the deletion-contraction property. At each step in the recursion, our algorithm computes certain structural properties of the matroid in order to attempt to prune the computational tree. This approach to computing Tutte polynomials was first studied by Haggard, Pearce, and Royle [7]. Our algorithm can be seen as an adaption of their work to special points of the Tutte plane where we can exploit additional structural properties.
+The performance of algorithms for computing Tutte polynomials based on the deletion-contraction property depends on the heuristic used to decide the ordering of the recursion [7–9]. We consider several heuristics introduced by Pearce, Haggard, and Royle [8] and a new heuristic, which is specific to our algorithm. We present some experimental results comparing the performance of these heuristics on two classes of random quantum circuits corresponding to dense and sparse instances.
+The correspondence between output probability amplitudes of quantum circuits and evaluations of Tutte polynomials also allows us to obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants by a theorem of Pendavingh [10]. This gives rise to an alternative efficient classical algorithm for computing output probability amplitudes of Clifford circuits.
+This paper is structured as follows. We introduce matroid theory in Section II and the Tutte polynomial in Section III. In Sections IV, V, and VI, we establish a mapping between output probability amplitudes of quantum circuits and evaluations of the Tutte polynomial of graphic matroids. This is achieved by introducing the Potts model partition function in Section IV, Instantaneous Quantum Polynomial-time circuits in Section V, and a class of universal quantum circuits in Section VI. In Section VII, we use this mapping to obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants. We also obtain an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. We describe our algorithm in Section VIII and present some experimental results in Section IX. Finally, we conclude in Section X.
+## II. MATROID THEORY
+We shall now briefly introduce the theory of matroids. The interested reader is referred to the classic textbooks of Welsh [11] and Oxley [12] for a detailed treatment. Matroids were introduced by Whitney [13] as a structure that generalises the notion of linear dependence. There are many equivalent ways to define a matroid. We shall define a matroid by the independence axioms.
+**Definition 1 (Matroid).** A matroid is a pair $M = (S, \mathcal{I})$ consisting of a finite set $S$, known as the *ground set*, and a collection $\mathcal{I}$ of subsets of $S$, known as the *independent sets*, such that the following axioms are satisfied.
+1. The empty set is a member of $\mathcal{I}$.

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+**Definition 15** (Brown's invariant). If $|\text{supp}(x)| \equiv 0 \pmod 4$ for all $x \in V \cup V^\perp$, then Brown's invariant $\sigma(V)$ is defined to be the smallest integer such that
+$$\sum_{x \in V} i^{|\text{supp}(x)|} = \sqrt{2}^{d(V)+\dim(V)} e^{\frac{i\pi}{4}\sigma(V)}.$$
+The following theorem of Pendavingh [10] provides an explicit form for the Tutte polynomial of a binary matroid at $(-i, i)$ in terms of the bicycle dimension and Brown's invariant.
+**Theorem 11** (Pendavingh [10]). Let $V$ be a linear subspace of $\mathbb{F}_2^2$ and let $M(V)$ be the corresponding binary matroid with ground set $S$. If $|\text{supp}(x)| \equiv 0 \pmod 4$ for all $x \in V \cup V^\perp$, then,
+$$T(M(V); -i, i) = \sqrt{2}^{d(V)} e^{\frac{i\pi}{4}(2|S|-3r(M)-\sigma(V))}.$$
+*Otherwise, $T(M(V); -i, i) = 0$. Further, $T(M(V); -i, i)$ can be evaluated in polynomial time.*
+As an immediate consequence of Theorem 11, we obtain an explicit form for Clifford circuit amplitudes in terms the bicycle dimension and Brown's invariant of the corresponding matroid. Furthermore, we obtain an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. For similar results of this flavour see Refs. [22, 23].
+## VIII. ALGORITHM OVERVIEW
+We shall now use the correspondence between quantum computation and evaluations of the Tutte polynomial to establish a heuristic algorithm for computing quantum probability amplitudes. To compute a probability amplitude, it is sufficient to compute the Tutte polynomial of a graphic matroid at $x = -icot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$ for an integer $k \ge 2$ [2, 5, 6]. Our algorithm will use the deletion-contraction property to recursively compute the Tutte polynomial. At each step in the recursion, the algorithm will compute certain structural properties of the graph in order to attempt to prune the computational tree. Our algorithm can be seen an adaption of the work of Haggard, Pearce, and Royle [7] to special points of the Tutte plane. We proceed by describing the key aspects of our algorithm.
+### A. Multigraph Deletion-Contraction Formula
+To improve the performance of our algorithm, we shall use the following deletion-contraction formula for multigraphs.
+**Proposition 12.** Let $G = (V, E)$ be a multigraph and let $e$ be a multiedge of $G$ with multiplicity $|e|$. If $e$ is a loop, then
+$$T(G; x, y) = y^{|e|} T(G \setminus \{e\}; x, y).$$
+If $e$ is a coloop, then
+$$T(G; x, y) = \left( x + \sum_{i=1}^{|e|-1} y^i \right) T(G/\{e\}; x, y).$$
+Finally, if $e$ is neither a loop nor a coloop, then
+$$T(G; x, y) = T(G\setminus\{e\}; x, y) + \left( \sum_{i=0}^{|e|-1} y^i \right) T(G/\{e\}; x, y).$$
+**Proposition 12** can easily be proven from the deletion-contraction formula by induction; we omit the proof.
+If $U$ is the underlying graph of $G$, then the number of recursive calls may be bounded by $O(2^{|E(U)|})$. Alternatively, we may bound the number of recursive calls in terms of the number of vertices plus the number of edges $s = |V(U)| + |E(U)|$ in the underlying graph. The number of recursive calls $R_s$ is then bounded by $R_s \le R_{s-1} + R_{s-2}$, which is precisely the Fibonacci recurrence. Hence the number of recursive calls is bounded by $O(\phi^{|V(U)|+|E(U)|})$, where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio [24]. A careful analysis shows that the number of recursive steps is bounded by $O(\tau(U) \cdot |E(U)|)$, where $\tau(U)$ denotes the number of spanning trees in $U$ [14].
+At each step in the recursion, we use the multigraph deletion-contraction formula to remove all multiedges that correspond to either a loop or a coloop in the underlying graph. This process contributes a multiplicative factor to the proceeding evaluation. Notice that when $G$ is a graph whose underlying graph is a looped forest, then every edge in the underlying graph is either a loop or a coloop. Hence, we obtain the following formula for the Tutte polynomial of $G$.
+**Corollary 13.** Let $G = (V, E)$ be a multigraph whose underlying graph $U$ is a looped forest. Further, for each edge $e$ in $U$, let $|e|$ denote its multiplicity in $G$. Then,
+$$T(G; x, y) = \prod_{e \in E(U)} y^{|e|} \prod_{coloop} \left( x + \sum_{i=1}^{|e|-1} y^i \right).$$
+*Proof.* The proof follows immediately from Proposition 12. ■
+### B. Graph Simplification
+There are a number of techniques that we can use to simplify the graph at each step in the recursion. Firstly, we may remove any isolated vertices, since they do not contribute to the evaluation.
+Secondly, when $x = -icot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$ for an integer $k \in \mathbb{Z}^+$, we may replace each multiedge with a multiedge of equal multiplicity modulo $8k$. To account for this, we multiply the proceeding evaluation by a efficiently computable factor. Specifically, we invoke the following proposition.

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+**Proposition 14.** Fix $k \in \mathbb{Z}^+$. Let $G = (V, E)$ be a multigraph and let $G' = (V', E')$ be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ modulo 8k. Then,
+$$T(G; x, y) = \left( i e^{\frac{i\pi}{4k}} \sin\left(\frac{\pi}{4k}\right) \right)^{\kappa(E)-\kappa(E')} T(G'; x, y),$$
+where $x = -i \cot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$.
+We prove Proposition 14 in Appendix E.
+### C. Vertigan Graphs
+The Tutte polynomial of a multigraph whose edge multiplicities are all integer multiples of an integer $k \in \mathbb{Z}^+$ may be evaluated at the point $x = -i \cot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$ in polynomial time. This can be seen by the following proposition.
+**Proposition 15.** Fix $k \in \mathbb{Z}^+$. Let $G = (V, E)$ be a multigraph whose edge multiplicities are all integer multiples of $k$. Further let $G' = (V', E')$ be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ divided by $k$. Then,
+$$T(G; x, y) = \left( \sqrt{2} e^{\frac{i\pi(1-k)}{4k}} \sin\left(\frac{\pi}{4k}\right) \right)^{-r(G)} T(G'; -i, i),$$
+where $x = -i \cot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$.
+We prove Proposition 15 in Appendix F and note that this is a special consequence of the *k*-thickening approach of Jaeger, Vertigan, and Welsh [16]. The Tutte polynomial may then be efficiently computed by Vertigan's algorithm [17]; we call such a multigraph a *Vertigan graph*. We may therefore prune the computational tree whenever the graph is a Vertigan graph with respect to $k$. Note that this corresponds to quantum circuits comprising gates from the Clifford group.
+### D. Connected Components
+The Tutte polynomial factorises over components.
+**Proposition 16.** Let $G = (V, E)$ be a graph with connected components $C = \{C_i\}_{i=1}^k$, then,
+$$T(G; x, y) = \prod_{i=1}^{k} T(C_i; x, y).$$
+**Proposition 17** can easily be proven from the deletion-contraction formula. For a proof, we refer the reader to Ref. [25, Section 3]. Similarly to the connected component case, we may use this property to prune the computational tree and improve performance. Note that the biconnected components of a graph may be listed in time linear in the number of edges via depth-first search [26].
+### E. Biconnected Components
+An identical result holds for biconnected components.
+**Proposition 17 (Tutte [25]).** Let $G = (V, E)$ be a graph with biconnected components $B = \{B_i\}_{i=1}^k$, then,
+$$T(G; x, y) = \prod_{i=1}^{k} T(B_i; x, y).$$
+**Proposition 18** (Haggard, Pearce, and Royle [7]). Let $G = (V, E)$ be a multigraph whose underlying graph $U$ is an $n$-cycle with edges indexed by the positive integers. Further, for each edge $e$ in $U$, let $|e|$ denote its multiplicity in $G$. Then,
+$$T(G; x, y) = \sum_{k=1}^{n-2} \left( \prod_{j=k+1}^{n} y_x(|e_j|) \prod_{j=1}^{k-1} y_1(|e_j|) + y_x(|e_n| + |e_{n-1}|) \prod_{j=1}^{n-2} y_1(|e_j|) \right),$$
+where $y_x(j) := x + \sum_{i=1}^{j-1} y_i$.
+**Proposition 19** can easily be proven from the deletion-contraction formula. For a proof, we refer the reader to Ref. [7, Theorem 4]. We may use this proposition to prune the computational tree whenever the underlying graph is a cycle.
+### F. Multi-Cycles
+The Tutte polynomial of a multigraph whose underlying graph is a cycle may be computed in polynomial time by invoking the following proposition.
+### G. Planar Graphs
+The Tutte polynomial of a planar graph along the hyperbola $(x-1)(y-1) = 2$ may be evaluated in polynomial time via the Fisher-Kasteleyn-Temperley (FKT) algorithm [27–29]. We may therefore use this algorithm to prune the computational tree whenever the underlying graph is planar. Note that we may test whether a graph is planar in time linear in the number of vertices [30].

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+## H. Edge-Selection Heuristics
+The performance of our algorithm depends on the heuristic used to select edges. We shall consider six edge-selection heuristics: *vertex order*, *minimum degree*, *maximum degree*, *minimum degree sum*, *maximum degree sum*, and *non-Vertigan*. These edge-selection heuristics were first studied by Pearce, Haggard, and Royle [8], with the exception of non-Vertigan, which is specific to our algorithm.
+*Vertex order*: The vertices of the graph are assigned an ordering. A multiedge is selected from those incident to the lowest vertex in the ordering and whose other endpoint is also the lowest vertex of any incident in the ordering. For contractions, the vertex inherits the lowest of the positions in the ordering.
+*Minimum degree*: A multiedge is selected from those incident to a vertex with minimal degree in the underlying graph.
+*Maximum degree*: A multiedge is selected from those incident to a vertex with maximal degree in the underlying graph.
+*Minimum degree sum*: A multiedge is selected from those whose sum of degrees of its endpoints is minimal in the underlying graph.
+*Maximum degree sum*: A multiedge is selected from those whose sum of degrees of its endpoints is maximal in the underlying graph.
+*Non-Vertigan*: A multiedge is selected from those whose multiplicity is not an integer multiple of $k$; we call such a multiedge *non-Vertigan*. Using this edge-selection heuristic, the number of recursive calls may be bounded by $O(2^{\nu(G)})$, where $\nu(G)$ denotes the number of non-Vertigan multiedges in $G$. This is due to the fact that both the deletion and contraction operation reduce the number of non-Vertigan multiedges by at least one. We note that this is similar to the *Sum-over-Cliffords* approach studied in Refs. [31–33].
+## I. Other Methods
+There are many other methods that may improve the performance of our algorithm, which we do not study. We shall proceed by discussing some of these.
+*Isomorphism testing*: During the computation the graphs encountered and the evaluation of their Tutte polynomial is stored. At each recursive step, we test whether the graph is isomorphic to one already encountered, and if so, we use the evaluation of the isomorphic graph instead. Haggard, Pearce, and Royle [7] showed that isomorphism testing can lead to an improvement in the performance of computing Tutte polynomials. Note that this may not be as effective when the input is a multigraph.
+*Almost planar*: At each step in the recursion, we may test whether the graph is close to being planar, and if so, select edges in such a way that the deletion and
+contraction operations give rise to a planar graph. For example, if the graph is *apex*, that is, it can be made planar by the removal of a single vertex, then we may select a multiedge incident to such a vertex. Similarly, if the underlying graph is *edge apex* or *contraction apex* [34], then we may select a multiedge such that the deletion or the contraction operation gives rise to a planar graph.
+$k$-connected components: Similarly to the connected and biconnected component case, we may compute the Tutte polynomial in terms of its $k$-connected components [35, 36].
+## IX. EXPERIMENTAL RESULTS
+In this section we present some experimental results comparing the performance of the edge-selection heuristics described in Section VIII H on two classes of random quantum circuits. Our experiments were performed using SageMath 9.0 [37]. The source code and experimental data are available at Ref. [38].
+The first class we consider corresponds to random instances of IQP circuits induced by dense graphs. Specifically, an instance is an IQP circuit induced by a complete graph with edge weights chosen uniformly at random from the set $\{\frac{m\pi}{8} | m \in \mathbb{Z}/8\}$. This class of IQP circuits is precisely that studied in Ref. [39], where it is conjectured that approximating the corresponding amplitudes up to a multiplicative error is #P-hard on average.
+The second class we consider corresponds to random instances of IQP circuits induced by sparse graphs. Specifically, an instance is an IQP circuit induced by a random graph where each of the possible edges is included independently with probability 1/2 and with edge weights chosen uniformly at random from the set $\{\frac{m\pi}{8} | m \in \mathbb{Z}/8\}$.
+We run our algorithm using each of the edge-selection heuristics to compute the principal probability amplitude of 64 random instances of both the dense and sparse class on 12 vertices. The performance of each edge-selection heuristic is measured by counting the number of leaves in the computational tree. Our experimental data is presented in Appendix G. We find that the non-Vertigan edge-selection heuristic performs particularly well for the dense class and the maximum degree sum edge-selection heuristic performs particularly well for the sparse class.
+## X. CONCLUSION & OUTLOOK
+We established a classical heuristic algorithm for exactly computing quantum probability amplitudes. Our algorithm is based on mapping output probability amplitudes of quantum circuits to evaluations of the Tutte polynomial of graphic matroids. The algorithm evaluates the Tutte polynomial recursively using the deletion-contraction property while attempting to exploit structural properties of the matroid. We considered several

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+edge-selection heuristics and presented experimental results comparing their performance on two classes of random quantum circuits. Further, we obtained an explicit form for Clifford circuit amplitudes in terms of matroid invariants and an alternative efficient classical algorithm for computing the output probability amplitudes of Clifford circuits.
+## ACKNOWLEDGEMENTS
+We thank Michael Bremner, Adrian Chapman, Iain Moffatt, Ashley Montanaro, Rudi Pendavingh, and Dan Shepherd for helpful discussions. This research was supported by the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme (QuantAlgo project), EPSRC grants EP/L021005/1 and EP/R043957/1, and the ARC Centre of Excellence for Quantum Computation and Communication Technology (CQC2T), project number CE170100012. Data are available at the University of Bristol data repository, data.bris, at https://doi.org/10.5523/bris.kbhgclva863q21tjkqpyr5uvq.
+## Appendix A: Proof of Proposition 3
+**Proposition 3 (restatement).**
+$$Z_{\text{Potts}}(G; q, \Omega, \Upsilon) = q^{-\kappa(E)} Z_{\text{Potts}}(G'; q, \Omega \cup \Upsilon, 0).$$
+*Proof.* By definition,
+$$
+\begin{aligned}
+q^{\kappa(E)} Z_{\text{Potts}}(G; q, \Omega, \Upsilon) &= q^{\kappa(E)} \sum_{\sigma \in \mathbb{Z}_q^V} \exp \left( \sum_{\{u,v\} \in E} \omega_{\{u,v\}} \delta(\sigma_u, \sigma_v) + \sum_{v \in V} v_v \delta(\sigma_v) \right) \\
+&= q^{\kappa(E)} \prod_{i=1}^{\kappa(E)} \sum_{\sigma \in \mathbb{Z}_q^V(C_i)} \exp \left( \sum_{\{u,v\} \in E(C_i)} \omega_{\{u,v\}} \delta(\sigma_u, \sigma_v) + \sum_{v \in V(C_i)} v_v \delta(\sigma_v) \right).
+\end{aligned}
+$$
+Now, by combining terms that are invariant under a $\mathbb{Z}_q$ symmetry, we have
+$$
+\begin{aligned}
+q^{\kappa(E)} Z_{\text{Potts}}(G; q, \Omega, \Upsilon) &= \prod_{i=1}^{\kappa(E)} \sum_{\sigma \in \mathbb{Z}_q^V(C_i)} \exp \left( \sum_{\{u,v\} \in E(C_i)} \omega_{\{u,v\}} \delta(\sigma_u, \sigma_v) \right) \sum_{\sigma' \in \mathbb{Z}_q} \exp \left( \sum_{v \in V(C_i)} v_v \delta(\sigma_v, \sigma') \right) \\
+&= \sum_{\sigma \in \mathbb{Z}_q^{V'}} \exp \left( \sum_{\{u,v\} \in E} \omega_{\{u,v\}} \delta(\sigma_u, \sigma_v) + \sum_{\{u,v\} \in E' \setminus E} v_v \delta(\sigma_u, \sigma_v) \right) \\
+&= Z_{\text{Potts}}(G'; q, \Omega \cup \Upsilon, 0).
+\end{aligned}
+$$
+This completes the proof. ■
+## Appendix B: Proof of Proposition 7
+**Proposition 7 (restatement).** Let $G = (V, E)$ be a graph with the weights $\Omega = \{\omega_e \in [-\pi, \pi]\}_{e \in E}$ assigned to its edges and the weights $\Upsilon = \{v_v \in [-\pi, \pi]\}_{v \in V}$ assigned to its vertices, then,
+$$\psi_{\chi_G}(0^{|V|}) = \frac{1}{2^{|V|}} Z_{\text{Ising}}(G; i\Omega, i\Upsilon).$$

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+*Proof.* By definition,
+$$
+\begin{align*}
+\psi_{\chi_G}(0^{|V|}) &= \langle 0^{|V|} | \exp \left( i \sum_{\{u,v\} \in E} \omega_{\{u,v\}} X_u X_v + i \sum_{v \in V} v_v X_v \right) | 0^{|V|} \rangle \\
+&= \langle +^{|V|} | \exp \left( i \sum_{\{u,v\} \in E} \omega_{\{u,v\}} Z_u Z_v + i \sum_{v \in V} v_v Z_v \right) | +^{|V|} \rangle \\
+&= \frac{1}{2^{|V|}} \sum_{x,y \in \{0,1\}^V} \langle y | \exp \left( i \sum_{\{u,v\} \in E} \omega_{\{u,v\}} Z_u Z_v + i \sum_{v \in V} v_v Z_v \right) | x \rangle \\
+&= \frac{1}{2^{|V|}} \sum_{x \in \{0,1\}^V} \exp \left( i \sum_{\{u,v\} \in E} \omega_{\{u,v\}} (-1)^{x_u \oplus x_v} + i \sum_{v \in V} v_v (-1)^{x_v} \right) \\
+&= \frac{1}{2^{|V|}} \sum_{z \in \{-1,+1\}^V} \exp \left( i \sum_{\{u,v\} \in E} \omega_{\{u,v\}} z_u z_v + i \sum_{v \in V} v_v z_v \right) \\
+&= \frac{1}{2^{|V|}} Z_{\text{Ising}}(G; i\Omega, i\Upsilon).
+\end{align*}
+$$
+This completes the proof.
+## Appendix C: Proof of Proposition 8
+**Proposition 8 (restatement).**
+$$
+\psi_{X_{G(\theta)}}(0^{|V|}) = \psi_{X_{G'(\theta)}}(0^{|V'|}).
+$$
+*Proof.*
+$$
+\begin{align*}
+\psi_{X_G(\theta)}(0^{|V|}) &= \frac{1}{2^{|V|}} Z_{\text{Ising}}(G; i\theta, i\theta) && (\text{by Proposition 7}) \\
+&= \frac{1}{2^{|V|}} e^{-i\theta(|E|+|V|)} Z_{\text{Potts}}(G; 2, 2i\theta, 2i\theta) && (\text{by Proposition 5}) \\
+&= \frac{1}{2^{|V|+\kappa(E)}} e^{-i\theta(|E|+|V|)} Z_{\text{Potts}}(G'; 2, 2i\theta, 0) && (\text{by Proposition 3}) \\
+&= \frac{1}{2^{|V|+\kappa(E)}} Z_{\text{Ising}}(G'; i\theta, 0) && (\text{by Proposition 5}) \\
+&= \psi_{X_{G'}(\theta)}(0^{|V'|}) && (\text{by Proposition 7}).
+\end{align*}
+$$
+This completes the proof.
+## Appendix D: Proof of Proposition 9
+**Proposition 9 (restatement).**
+$$
+\psi_{X_{G'}}(\theta)(0^{|V'|}) = e^{i\theta(r(G') - |E'|)} (i\sin(\theta))^{r(G')} T(G'; x, y),
+$$
+where $x = -i \cot(\theta)$ and $y = e^{2i\theta}$.

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+*Proof.*
+$$
+\begin{align*}
+\psi_{\chi_{G'}(\theta)} (0^{|V'|}) &= \frac{1}{2^{|V'|}} Z_{\text{Ising}}(G'; i\theta, 0) && (\text{by Proposition 7}) \\
+&= \frac{1}{2^{|V'|}} e^{-i\theta|E'|} Z_{\text{Potts}}(G'; 2, 2i\theta, 0) && (\text{by Proposition 5}) \\
+&= \frac{1}{2^{r(G')}} e^{-i\theta|E'|} (e^{2i\theta} - 1)^{r(G')} \mathrm{T}(G'; -i \cot(\theta), e^{2i\theta}) && (\text{by Proposition 4}) \\
+&= e^{i\theta(r(G') - |E'|)} (i \sin(\theta))^{r(G')} \mathrm{T}(G'; -i \cot(\theta), e^{2i\theta}).
+\end{align*}
+$$
+This completes the proof.
+## Appendix E: Proof of Proposition 14
+**Proposition 14** (restatement). Fix $k \in \mathbb{Z}^+$. Let $G = (V, E)$ be a multigraph and let $G' = (V', E')$ be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ modulo $8k$. Then,
+$$ T(G; x, y) = \left( i e^{\frac{i\pi}{4k}} \sin\left(\frac{\pi}{4k}\right) \right)^{\kappa(E)-\kappa(E')} T(G'; x, y), $$
+where $x = -i \cot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$.
+*Proof.*
+$$
+\begin{align*}
+T(G; x, y) &= e^{\frac{i\pi}{4k}(|E|-r(G))} \left(i \sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)} \psi_{\chi_G\left(\frac{\pi}{4k}\right)}(0^{|V|}) && \text{(by Proposition 9)} \\
+&= e^{\frac{i\pi}{4k}(|E|-r(G))} \left(i \sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)} \psi_{\chi_{G'}\left(\frac{\pi}{4k}\right)}(0^{|V'|}) \\
+&= e^{\frac{i\pi}{4k}(|E|-|E'|)} \left(i e^{\frac{i\pi}{4k}} \sin\left(\frac{\pi}{4k}\right)\right)^{r(G')-r(G)} T(G'; x, y) && \text{(by Proposition 9)} \\
+&= \left(i e^{\frac{i\pi}{4k}} \sin\left(\frac{\pi}{4k}\right)\right)^{\kappa(E)-\kappa(E')} T(G'; x, y).
+\end{align*}
+$$
+This completes the proof.
+## Appendix F: Proof of Proposition 15
+**Proposition 15** (restatement). Fix $k \in \mathbb{Z}^+$. Let $G = (V, E)$ be a multigraph whose edge multiplicities are all integer multiples of $k$. Further let $G' = (V', E')$ be the graph formed from $G$ by taking the multiplicity of each multiedge in $G$ divided by $k$. Then,
+$$ T(G; x, y) = \left( \sqrt{2} e^{\frac{i\pi(1-k)}{4k}} \sin\left(\frac{\pi}{4k}\right) \right)^{-r(G)} T(G'; -i, i), $$
+where $x = -i \cot(\frac{\pi}{4k})$ and $y = e^{\frac{i\pi}{2k}}$.
+*Proof.*
+$$
+\begin{align*}
+T(G; x, y) &= e^{\frac{i\pi}{4k}(|E|-r(G))} \left(i \sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)} \psi_{\chi_G\left(\frac{\pi}{4k}\right)}(0^{|V|}) && \text{(by Proposition 9)} \\
+&= e^{\frac{i\pi}{4k}(|E|-r(G))} \left(i \sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)} \psi_{\chi_{G'}\left(\frac{\pi}{4k}\right)}(0^{|V'|}) \\
+&= \left(\sqrt{2} e^{\frac{i\pi(1-k)}{4k}} \sin\left(\frac{\pi}{4k}\right)\right)^{-r(G)} T(G'; -i, i) && \text{(by Proposition 9)}
+\end{align*}
+$$
+This completes the proof.

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+## Appendix G: Tables of Experimental Data
+We present our experimental data in Table I and Table II. The rows of the tables represent edge-selection heuristics and the columns represent quantities relating to the number of leaves in the computational trees.
+<table><thead><tr><th></th><th>Sum</th><th>Mean</th><th>Mean Deviation</th><th>#Empty</th><th>#Vertigan</th><th>#Multicycle</th><th>#Planar</th></tr></thead><tbody><tr><td>Non-Vertigan</td><td>8888920</td><td>138889</td><td>36225</td><td>912</td><td>2124367</td><td>50097</td><td>6713544</td></tr><tr><td>Vertex Order</td><td>37173344</td><td>580834</td><td>150982</td><td>186</td><td>167486</td><td>553618</td><td>36452054</td></tr><tr><td>Minimum Degree</td><td>57014650</td><td>890854</td><td>162094</td><td>0</td><td>353780</td><td>1238429</td><td>55422441</td></tr><tr><td>Maximum Degree</td><td>28604576</td><td>446947</td><td>119407</td><td>950</td><td>86125</td><td>412215</td><td>28105286</td></tr><tr><td>Minimum Degree Sum</td><td>50716183</td><td>792440</td><td>170360</td><td>0</td><td>243290</td><td>1010284</td><td>49462609</td></tr><tr><td>Maximum Degree Sum</td><td>10993306</td><td>171770</td><td>39409</td><td>6971</td><td>8998</td><td>78030</td><td>10899307</td></tr></tbody></table>
+TABLE I. Performance of the edge-selection heuristics on 64 random instances of the dense class on 12 vertices.
+<table><thead><tr><th></th><th>Sum</th><th>Mean</th><th>Mean Deviation</th><th>#Empty</th><th>#Vertigan</th><th>#Multicycle</th><th>#Planar</th></tr></thead><tbody><tr><td>Non-Vertigan</td><td>63958</td><td>999</td><td>973</td><td>30</td><td>7994</td><td>467</td><td>55467</td></tr><tr><td>Vertex Order</td><td>93642</td><td>1463</td><td>1518</td><td>88</td><td>74</td><td>813</td><td>92667</td></tr><tr><td>Minimum Degree</td><td>412557</td><td>6446</td><td>6316</td><td>0</td><td>2158</td><td>7285</td><td>403114</td></tr><tr><td>Maximum Degree</td><td>91218</td><td>1425</td><td>1476</td><td>16</td><td>87</td><td>933</td><td>90182</td></tr><tr><td>Minimum Degree Sum</td><td>291763</td><td>4559</td><td>4630</td><td>0</td><td>1138</td><td>4169</td><td>286456</td></tr><tr><td>Maximum Degree Sum</td><td>50375</td><td>787</td><td>808</td><td>14</td><td>25</td><td>415</td><td>49921</td></tr></tbody></table>
+TABLE II. Performance of the edge-selection heuristics on 64 random instances of the sparse class on 12 vertices.
+[1] D. Aharonov, I. Arad, E. Eban, and Z. Landau, arXiv eprints (2007), arXiv:quant-ph/0702008.
+[2] D. Shepherd, arXiv eprints (2010), arXiv:1005.1744.
+[3] S. Fenner, F. Green, S. Homer, and R. Pruim, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, 3953 (1999), arXiv:quant-ph/9812056.
+[4] I. L. Markov and Y. Shi, SIAM Journal on Computing 38, 963 (2008), arXiv:quant-ph/0511069.
+[5] D. Shepherd and M. J. Bremner, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 465, 1413 (2009), arXiv:0809.0847.
+[6] M. J. Bremner, R. Jozsa, and D. J. Shepherd, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (The Royal Society, 2010) p. rspa20100301, arXiv:1005.1407.
+[7] G. Haggard, D. J. Pearce, and G. Royle, ACM Transactions on Mathematical Software (TOMS) 37, 24 (2010).
+[8] D. J. Pearce, G. Haggard, and G. Royle, in Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory-Volume 94 (Australian Computer Society, Inc., 2009) pp. 153–162.
+[9] M. Monagan, arXiv eprints (2012), arXiv:1209.5160.
+[10] R. Pendavingh, Journal of Algebraic Combinatorics 39, 141 (2014), arXiv:1203.0910.
+[11] D. J. Welsh, Matroid theory (Academic Press, 1976).
+[12] J. G. Oxley, Matroid theory, Vol. 3 (Oxford University Press, USA, 2006).
+[13] H. Whitney, American Journal of Mathematics 57, 509 (1935).
+[14] K. Sekine, H. Imai, and S. Tani, in International Symposium on Algorithms and Computation (Springer, 1995) pp. 224–233.
+[15] A. Björklund, T. Husfeldt, P. Kaski, and M. Koivisto, in 49th Annual IEEE Symposium on Foundations of Computer Science (IEEE, 2008) pp. 677–686, arXiv:0711.2585.
+[16] F. Jaeger, D. L. Vertigan, and D. J. Welsh, in Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 108 (Cambridge Univ Press, 1990) pp. 35–53.
+[17] D. Vertigan, Journal of Combinatorial Theory, Series B 74, 378 (1998).
+[18] M. Snook, The Electronic Journal of Combinatorics 19, 41 (2012).
+[19] D. J. Welsh, London Mathematical Society Lecture Note Series 186, 372 (1993).
+[20] S. Iblisdir, M. Cirio, O. Boada, and G. Brennen, Annals of Physics 340, 205 (2014), arXiv:1208.3918.
+[21] K. Fujii and T. Morimae, New Journal of Physics 19, 033003 (2017), arXiv:1311.2128.
+[22] C. Guan and K. W. Regan, arXiv eprints (2019), arXiv:1904.00101.
+[23] D. Gosset, D. Grier, A. Kerzner, and L. Schaeffer, arXiv eprints (2020), arXiv:2009.03218.
+[24] H. S. Wilf, Algorithms and Complexity (AK Peters/CRC Press, 2002).
+[25] W. T. Tutte, Canadian Journal of Mathematics 6, 80 (1954).

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+[26] J. Hopcroft and R. Tarjan, *Communications of the ACM* **16**, 372 (1973).
+[27] H. N. Temperley and M. E. Fisher, *Philosophical Magazine* **6**, 1061 (1961).
+[28] P. W. Kasteleyn, *Journal of Mathematical Physics* **4**, 287 (1963).
+[29] P. Kasteleyn, *Graph Theory and Theoretical Physics*, 43 (1967).
+[30] J. Hopcroft and R. Tarjan, *Journal of the ACM (JACM)* **21**, 549 (1974).
+[31] S. Bravyi, G. Smith, and J. A. Smolin, *Physical Review X* **6**, 021043 (2016), arXiv:1506.01396.
+[32] S. Bravyi and D. Gosset, *Physical Review Letters* **116**, 250501 (2016), arXiv:1601.07601.
+[33] S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, *Quantum* **3**, 181 (2019), arXiv:1808.00128.
+[34] M. Lipton, E. Mackall, T. W. Mattman, M. Pierce, S. Robinson, J. Thomas, and I. Weinschelbaum, *Involve*, a Journal of Mathematics **11**, 413 (2017), arXiv:1608.01973.
+[35] A. Andrzejak, *Journal of Combinatorial Theory, Series B* **70**, 346 (1997).
+[36] J. Bonin and A. De Mier, *Advances in Applied Mathematics* **32**, 31 (2004).
+[37] The Sage Developers, *SageMath, the Sage Mathematics Software System (Version 9.0)* (2020).
+[38] R. L. Mann, *Data from “Simulating Quantum Computations with Tutte Polynomials”* (2021).
+[39] M. J. Bremner, A. Montanaro, and D. J. Shepherd, *Physical Review Letters* **117**, 080501 (2016), arXiv:1504.07999.

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+2. Every subset of a member of $\mathcal{I}$ is a member of $\mathcal{I}$.
+3. If $A$ and $B$ are members of $\mathcal{I}$ and $|A| > |B|$, then there exists an $x \in A \setminus B$ such that $B \cup \{x\}$ is a member of $\mathcal{I}$.
+The rank of a subset $A$ of $\mathcal{S}$ is given by the rank function $r: 2^{\mathcal{S}} \to \mathbb{N}$ of the matroid defined by $r(A) := \max(|X| \mid X \subseteq A, X \in \mathcal{I})$. The rank of a matroid $M$, denoted $r(M)$, is the rank of the set $S$.
+The archetypal class of matroids are *vector matroids*. A vector matroid $M = (S, \mathcal{I})$ is a matroid whose ground set $S$ is a subset of a vector space over a field $\mathbb{F}$ and whose independent sets $\mathcal{I}$ are the linearly independent subsets of $S$. The rank of a subset of a vector matroid is the dimension of the subspace spanned by the corresponding vectors. We say that a matroid is $\mathbb{F}$-representable if it is isomorphic to a vector matroid over the field $\mathbb{F}$. A matroid is a *binary matroid* if it is $\mathbb{F}_2$-representable and is a *ternary matroid* if it is $\mathbb{F}_3$-representable. A matroid that is representable over every field is called a *regular matroid*.
+Every finite graph $G = (V, E)$ induces a matroid $M(G) = (S, \mathcal{I})$ as follows. Let the ground set $S$ be the set of edges $E$ and let the independent sets $\mathcal{I}$ be the subsets of $E$ that are a forest, i.e., they do not contain a simple cycle. It is easy to check that $M(G)$ satisfies the independence axioms. The rank of a subset $A$ of a cycle matroid is $|V| - \kappa(A)$, where $\kappa(A)$ denotes the number of connected components of the subgraph with edge set $A$. The rank of the cycle matroid $M(G)$, denoted $r(M(G))$ or simply $r(G)$, is the rank of the set $E$. The matroid $M(G)$ is called the *cycle matroid* of $G$. We say that a matroid is *graphic* if it is isomorphic to the cycle matroid of a graph.
+Graphic matroids are regular. To see this consider assigning to the graph $G$ an arbitrary orientation $D(G)$, that is, for each edge $e = \{u, v\}$ in $G$, we choose one of $u$ and $v$ to be the *positive end* and the other one to be the *negative end*. Then construct the *oriented incidence matrix* of $G$ with respect to the orientation $D(G)$.
+**Definition 2** (Oriented incidence matrix). Let $G = (V, E)$ be a graph and let $D(G)$ be an orientation of $G$. Then the oriented incidence matrix of $G$ with respect to $D(G)$ is the $|V| \times |E|$ matrix $A_{D(G)} = (a_{ve})_{|V| \times |E|}$ whose entries are
+$$a_{ve} = \begin{cases} +1, & \text{if } v \text{ is the positive end of } e; \\ -1, & \text{if } v \text{ is the negative end of } e; \\ 0, & \text{otherwise.} \end{cases}$$
+The rows of the oriented incidence matrix $A_{D(G)}$ correspond to the vertices of $G$ and the columns correspond to the edges of $G$. Each column contains exactly one +1 and exactly one -1 representing the positive and negative ends of the corresponding edge. If the column space of $A_{D(G)}$ is the ground set of a vector matroid, then it is easy to see that a subset is independent if and only if
+it is a forest in $G$. Hence, the oriented incidence matrix provides a representation of a graphic matroid over every field.
+A minor of a matroid $M$ is a matroid that is obtained from $M$ by a sequence of deletion and contraction operations.
+**Definition 3** (Deletion). Let $M = (S, \mathcal{I})$ be a matroid and let $e$ be an element of the ground set. Then the deletion of $M$ with respect to $e$ is the matroid $M\setminus\{e\} = (S', \mathcal{I}')$ whose ground set is $S' = S\setminus\{e\}$ and whose independent sets are $\mathcal{I}' = \{I \subseteq S\setminus\{e\} \mid I \in \mathcal{I}\}$.
+The deletion of an element from the cycle matroid of a graph corresponds to removing an edge from the graph.
+**Definition 4** (Contraction). Let $M = (S, \mathcal{I})$ be a matroid and let $e$ be an element of the ground set. Then the contraction of $M$ with respect to $e$ is the matroid $M/\{e\} = (S', \mathcal{I}')$ whose ground set is $S' = S\setminus\{e\}$ and whose independent sets are $\mathcal{I}' = \{I \subseteq S\setminus\{e\} \mid I \cup \{e\} \in \mathcal{I}\}$.
+The contraction of an element from the cycle matroid of a graph corresponds to removing an edge from the graph and merging its two endpoints.
+An element $e$ of a matroid is said to be a *loop* if $\{e\}$ is not an independent set and said to be a *coloop* if $e$ is contained in every maximally independent set. If an element $e$ of a matroid is either a loop or a coloop then the deletion and contraction of $e$ are equivalent.
+### III. THE TUTTE POLYNOMIAL
+We shall now briefly introduce the Tutte polynomial, which is a well-known invariant in matroid and graph theory.
+**Definition 5** (Tutte polynomial of a matroid). Let $M = (S, \mathcal{I})$ be a matroid with rank function $r: 2^{\mathcal{S}} \to \mathbb{N}$. Then the Tutte polynomial of $M$ is the bivariate polynomial defined by
+$$T(M; x, y) := \sum_{A \subseteq S} (x-1)^{r(M)-r(A)} (y-1)^{|A|-r(A)}.$$
+The Tutte polynomial may also be defined recursively by the *deletion-contraction property*.
+**Definition 6** (Deletion-contraction property). Let $M = (S, \mathcal{I})$ be a matroid. If $M$ is the empty matroid, i.e., $S = \emptyset$, then
+$$T(M; x, y) = 1.$$
+Otherwise, let $e$ be an element of the ground set. If $e$ is a loop, then
+$$T(M; x, y) = yT(M\setminus\{e\}; x, y).$$

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+If $e$ is a coloop, then
+$$T(M; x, y) = xT(M/\{e\}; x, y).$$
+Finally, if $e$ is neither a loop nor a coloop, then
+$$T(M; x, y) = T(M \setminus \{e\}; x, y) + T(M/\{e\}; x, y).$$
+The deletion-contraction property immediately gives an algorithm for recursively computing the Tutte polynomial. This algorithm is in general inefficient, but the performance may be improved by using isomorphism testing to reduce the number of recursive calls [14]. The performance of this algorithm depends on the heuristic used to choose elements of the ground set [7–9]. Björklund et al. [15] showed that the Tutte polynomial can be computed in time exponential in the number of vertices.
+The Tutte polynomial of a graph may be recovered by considering the Tutte polynomial of the cycle matroid of a graph and using the fact that the rank of a subset $A$ of a cycle matroid is $|V| - \kappa(A)$, where $\kappa(A)$ denotes the number of connected components of the subgraph with edge set $A$.
+**Definition 7** (Tutte Polynomial of a graph). Let $G = (V, E)$ be a graph and let $\kappa(A)$ denote the number of connected components of the subgraph with edge set $A$. Then the Tutte polynomial of $G$ is a polynomial in $x$ and $y$, defined by
+$$T(G; x, y) := \sum_{A \subseteq E} (x-1)^{\kappa(A)-\kappa(E)} (y-1)^{\kappa(A)+|A|-|V|}.$$
+The Tutte polynomial is trivial to evaluate along the hyperbola $(x-1)(y-1) = 1$ for any matroid. In the case of graphic matroids, Jaeger, Vertigan, and Welsh [16] showed that the Tutte polynomial is #P-hard to evaluate, except along this hyperbola and when $(x,y)$ equals one of nine special points.
+**Theorem 1** (Jaeger, Vertigan, and Welsh [16]). *The problem of evaluating the Tutte polynomial of a graphic matroid at an algebraic point in the (x,y)-plane is #P-hard except when $(x-1)(y-1) = 1$ or when $(x,y)$ equals one of $(1,1), (-1,-1), (0,-1), (-1,0), (i,-i), (-i,i), (j,j^2), (j,j^2), \text{ or } (j^2,j), \text{ where } j = \exp(2\pi i/3)$. In each of these exceptional cases the evaluation can be done in polynomial time.*
+Vertigan [17] extended this result to vector matroids.
+**Theorem 2** (Vertigan [17]). *The problem of evaluating the Tutte polynomial of a vector matroid over a field $\mathbb{F}$ at an algebraic point in the (x,y)-plane is #P-hard except when $(x-1)(y-1) = 1$, $(x,y)$ equals (1,1), or when*
+1. $|\mathbb{F}| = 2$ and $(x, y)$ equals one of $(-1, -1)$, $(0, -1)$, $(-1, 0)$, $(i, -i)$, or $(-i, i)$;
+2. $|\mathbb{F}| = 3$ and $(x, y)$ equals one of $(j, j^2)$ or $(j^2, j)$, where $j = \exp(2\pi i/3)$; or
+3. $|\mathbb{F}| = 4$ and $(x, y)$ equals $(-1, -1)$.
+In each of these exceptional cases, except when $(x,y)$ equals (1,1), the evaluation can be done in polynomial time.
+Snook [18] showed that when $(x,y)$ equals (1,1) and $\mathbb{F}$ is either a finite field of fixed characteristic or a fixed infinite field, then evaluating the Tutte polynomial is #P-hard. It is an open problem to understand the complexity of evaluating the Tutte polynomial at (1,1) over any fixed field.
+## IV. THE POTTS MODEL PARTITION FUNCTION
+The Potts model is a statistical physical model described by an integer $q \in \mathbb{Z}^+$ and a graph $G = (V, E)$, with the vertices representing spins and the edges representing interactions between them. A set of edge weights $\{\omega_e\}_{e \in E}$ characterise the interactions and a set of vertex weights $\{v_v\}_{v \in V}$ characterise the external fields at each spin. A configuration of the model is an assignment $\sigma$ of each spin to one of $q$ possible states. The Potts model partition function is defined as follows.
+**Definition 8** (Potts model partition function). Let $q \in \mathbb{Z}^+$ be an integer and let $G = (V, E)$ be a graph with the weights $\Omega = \{\omega_e\}_{e \in E}$ assigned to its edges and the weights $\Upsilon = \{v_v\}_{v \in V}$ assigned to its vertices. Then the $q$-state Potts model partition function is defined by
+$$Z_{\text{Potts}}(G; q, \Omega, \Upsilon) := \sum_{\sigma \in \mathbb{Z}_q^V} w_G(\sigma),$$
+where
+$$w_G(\sigma) = \exp \left( \sum_{\{u,v\} \in E} \omega_{\{u,v\}} \delta(\sigma_u, \sigma_v) + \sum_{v \in V} v_v \delta(\sigma_v) \right).$$
+The Potts model partition function with an external field is equivalent to the zero-field case on an augmented graph $G' = (V', E')$. To construct $G'$ from $G$, for each of the connected components $\{C_i\}_{i=1}^{\kappa(E)}$ of $G$ add a new vertex $u_i$ and for every vertex $v \in V(C_i)$ add an edge $e_v = \{u_i, v\}$ with the weight $v_v$ assigned to it. Then we have the following proposition.
+**Proposition 3.**
+$$Z_{\text{Potts}}(G; q, \Omega, \Upsilon) = q^{-\kappa(E)} Z_{\text{Potts}}(G'; q, \Omega \cup \Upsilon, 0).$$
+A similar proposition appears in Welsh's monograph [19]; we prove Proposition 3 in Appendix A.
+It will be convenient to consider the Potts model with weights that are all positive integer multiples of a complex number $\theta$. We shall implement this model on the augmented graph $G'$ with all weights equal to $\theta$ by replacing

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+each edge with the appropriate number of parallel edges. Let us denote the partition function of this model by $Z_{\text{Potts}}(G'; q, \theta)$. Then, we have the following proposition relating the partition function of this model to the Tutte polynomial of the augmented graph $G'$.
+**Proposition 4.**
+$$ Z_{\text{Potts}}(G'; q, \theta) = q^{\kappa(E')}(e^{\theta} - 1)^{r(G')} \mathbf{T}(G'; x, y), $$
+where $x = \frac{e^{\theta+q-1}}{e^{\theta}-1}$ and $y = e^{\theta}$.
+In particular, the *q*-state Potts model partition function is related to the Tutte polynomial along the hyperbola $(x-1)(y-1)=q$. For a proof of Proposition 4, we refer the reader to Welsh's monograph [19, Section 4.4].
+The 2-state Potts model partition function specialises to the *Ising model* partition function.
+**Definition 9 (Ising model partition function).** Let $G = (V, E)$ be a graph with the weights $\Omega = \{\omega_e\}_{e \in E}$ assigned to its edges and the weights $\Upsilon = \{v_v\}_{v \in V}$ assigned to its vertices. Then the Ising model partition function is defined by
+$$ Z_{\text{Ising}}(G; \Omega, \Upsilon) := \sum_{\sigma \in \{-1, +1\}^V} w_G(\sigma), $$
+where
+$$ w_G(\sigma) = \exp \left( \sum_{\{u,v\} \in E} \omega_{\{u,v\}} \sigma_u \sigma_v + \sum_{v \in V} v_v \sigma_v \right). $$
+**Proposition 5.**
+$$ Z_{\text{Potts}}(G; 2, \Omega, \Upsilon) = w_G Z_{\text{Ising}} \left( G; \frac{\Omega}{2}, \frac{\Upsilon}{2} \right), $$
+where $w_G = \exp(\frac{1}{2} \sum_{e \in E} \omega_e + \frac{1}{2} \sum_{v \in V} v_v)$.
+*Proof.* The proof follows from some simple algebra. ■
+# V. INSTANTANEOUS QUANTUM POLYNOMIAL TIME
+We shall now briefly introduce the class of commuting quantum circuits, known as *Instantaneous Quantum Polynomial-time* (IQP) circuits [5]. These circuits exhibit many interesting mathematical properties. In particular, the output probability amplitudes of IQP circuits are proportional to evaluations of the Tutte polynomial of binary matroids [2]. IQP circuits comprise only gates that are diagonal in the Pauli-X basis and are described by an X-program.
+**Definition 10 (X-program).** An X-program is a pair $(P, \theta)$, where $P = (p_{ij})_{m \times n}$ is a binary matrix and $\theta \in [-\pi, \pi]$ is a real angle. The matrix $P$ is used to construct a Hamiltonian of $m$ commuting terms acting
+on $n$ qubits, where each term in the Hamiltonian is a product of Pauli-X operators,
+$$ H_{(P,\theta)} := -\theta \sum_{i=1}^{m} \bigotimes_{j=1}^{n} X_j^{p_{ij}}. $$
+Thus, the columns of $P$ correspond to qubits and the rows of $P$ correspond to interactions in the Hamiltonian.
+An X-program induces a probability distribution $P_{(P,\theta)}$ known as an *IQP distribution*.
+**Definition 11 ($\mathcal{P}_{(P,\theta)}$).** For an X-program $(P, \theta)$ with $P = (p_{ij})_{m \times n}$, we define $\mathcal{P}_{(P,\theta)}$ to be the probability distribution over binary strings $x \in \{0,1\}^n$, given by
+$$ \Pr[x] := |\psi_{(P,\theta)}(x)|^2, $$
+where
+$$ \psi_{(P,\theta)}(x) = \langle x | \exp(-iH_{(P,\theta)}) | 0^n \rangle. $$
+The principal probability amplitude $\psi_{(P,\theta)}(0^n)$ of an IQP distribution is directly related to an evaluation of the Tutte polynomial of the binary matroid whose ground set is the row space of $P$.
+**Proposition 6.** Let $(P, \theta)$ be an X-program with $P = (p_{ij})_{m \times n}$. Let $M = (S, \mathcal{I})$ be the binary matroid whose ground set $S$ is the row space of $P$, then,
+$$ \psi_{(P,\theta)}(0) = e^{i\theta(r(M)-m)} (i\sin(\theta))^{r(M)} \mathbf{T}(M; x, y), $$
+where $x = -i \cot(\theta)$ and $y = e^{2i\theta}$.
+A similar result may be obtained for the other probability amplitudes. This can easily be seen when $\theta = \pi/(2k)$ for $k \in \mathbb{Z}^+$, by firstly letting $P|^{k}x$ be the matrix obtained from $P$ by appending $k$ rows identical to $x$, and then observing that $\psi_{(P,\theta)}(x) = -i\psi_{(P|^{k}x,\theta)}(0^n)$. For a proof of Proposition 6 and a treatment of the general $\theta$ case, we refer the reader to Ref. [2, Section 3].
+We shall consider X-programs that are induced by a weighted graph.
+**Definition 12 (Graph-induced X-program).** For a graph $G = (V, E)$ with the weights $\{\omega_e \in [-\pi, \pi]\}_{e \in E}$ assigned to its edges and the weights $\{v_v \in [-\pi, \pi]\}_{v \in V}$ assigned to its vertices, we define the X-program induced by $G$ to be an X-program $\mathcal{X}_G$ such that
+$$ H_{\mathcal{X}_G} = - \sum_{\{u,v\} \in E} \omega_{\{u,v\}} X_u X_v - \sum_{v \in V} v_v X_v. $$
+Any X-program can be efficiently represented by a graph-induced X-program [5]. The principal probability amplitude $\psi_{\mathcal{X}_G}(0^n)$ of the IQP distribution generated by a graph-induced X-program is directly related to the Ising model partition function of the graph with imaginary weights.

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+# Relationships Between Six Incircles
+STANLEY RABINOWITZ
+545 Elm St Unit 1, Milford, New Hampshire 03055, USA
+e-mail: stan.rabinowitz@comcast.net
+web: http://www.StanleyRabinowitz.com/
+**Abstract.** If $P$ is a point inside $\triangle ABC$, then the cevians through $P$ divide $\triangle ABC$ into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.
+**Keywords.** Euclidean geometry, triangle geometry, incircles, inradii, cevians.
+**Mathematics Subject Classification (2010).** 51M04, 51-04.
+## 1. INTRODUCTION
+Japanese Mathematicians of the Edo period were fond of finding relationships between the radii of circles associated with triangles. For example, the 1781 book "Seiyo Sampo" [2, no. 84] gives the relationship $r = \sqrt{r_1r_2} + \sqrt{r_2r_3} + \sqrt{r_3r_1}$ between the radii of the circles in Figure 1 below. This result was later inscribed on an 1814 tablet in the Chiba prefecture [3, p. 30].
+FIGURE 1.
+¹This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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+*Proof.* We have the following six equations for the perimeters of the six triangles.
+$$2s_1 = PB + PD + BD, \quad 2s_2 = PD + PC + DC,$$
+$$2s_3 = PC + PE + CE, \quad 2s_4 = PE + PA + EA,$$
+$$2s_5 = PA + PF + AF, \quad 2s_6 = PF + PB + FB.$$
+Thus $2s_1+2s_3+2s_5-(2s_2+2s_4+2s_6) = (BD-DC)+(CE-EA)+(AF-FB) = 0$ and the lemma follows. $\square$
+We can now state the relationship between the $r_i$ when P is the centroid. This theorem is attributed to Reidt in [1, p. 618].
+**Theorem 4.1.** If P is the centroid of $\triangle ABC$, then
+$$\frac{1}{r_1} + \frac{1}{r_3} + \frac{1}{r_5} = \frac{1}{r_2} + \frac{1}{r_4} + \frac{1}{r_6}.$$
+*Proof.* Recall that if r, s, and K are the inradius, semiperimeter, and area of a triangle, respectively, then $r = K/s$. From Lemma 4.1, the six triangles have the same area. Call this area K. From Lemma 4.2, $s_1+s_3+s_5 = s_2+s_4+s_6$. Divide both sides of this equation by K and use the fact that $1/r_i = s_i/K_i$ to get the desired identity. $\square$
+We note a similar result from [1, p. 618].
+**Theorem 4.2.** If P is the centroid of $\triangle ABC$, then $R_1R_3R_5 = R_2R_4R_6$.
+## 5. THE CIRCUMCENTER
+Now we will consider the case when P is the circumcenter.
+**Lemma 5.1.** Let P be the circumcenter of $\triangle ABC$ and let R be its circumradius.
+Let $\angle PAB = \angle PBA = \alpha$ and $\angle PBC = \beta$ (Figure 5). Then $R/r_1 = \cot\alpha + \cot\frac{\beta}{2}$.
+FIGURE 5.

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+FIGURE 9.
+Applying the Law of Sines to $\triangle ABD$ gives $BD/\sin a = AB/\sin(a + 2c)$. This allows us to compute $BD$. In a similar manner, we get the following.
+$$
+\begin{aligned}
+BD &= \frac{\sin a \sin 2c}{\sin(a + 2c)}, & CE &= \frac{\sin b \sin 2a}{\sin(b + 2a)}, & AF &= \frac{\sin c \sin 2b}{\sin(c + 2b)}, \\
+CD &= \frac{\sin a \sin 2b}{\sin(a + 2b)}, & AE &= \frac{\sin b \sin 2c}{\sin(b + 2c)}, & BF &= \frac{\sin c \sin 2a}{\sin(c + 2a)}.
+\end{aligned}
+ $$
+Note that $\angle BPD = a + b$. Applying the Law of Sines to $\triangle BPD$ allows us to compute the values of $PB$ and $PD$. In the same way, we can compute $PC$, $PE$, $PA$, and $PF$. We get the following.
+$$
+\begin{aligned}
+PA &= \frac{\sin c \sin 2b}{\sin(c+a)}, & PD &= \frac{\sin a \sin b \sin 2c}{\sin(a+b)\sin(a+2c)}, \\
+PB &= \frac{\sin a \sin 2c}{\sin(a+b)}, & PE &= \frac{\sin b \sin c \sin 2a}{\sin(b+c)\sin(b+2a)}, \\
+PC &= \frac{\sin b \sin 2a}{\sin(b+c)}, & PF &= \frac{\sin c \sin a \sin 2b}{\sin(c+a)\sin(c+2b)}.
+\end{aligned}
+ $$
+We now have expressions for the length of every line segment in the figure in terms of $a$, $b$, and $c$. Thus, the perimeters of all the triangles are known and we have expressed each of the $s_i$ in terms of $a$, $b$, and $c$. The areas of the triangles can also be found. For example, $K_1 = \frac{1}{2}PB \cdot BD \sin b$. Knowing all the $s_i$ and $K_i$ lets us find the values of all the $r_i$, since $r_i = K_i/s_i$.
+We can plug these values for the $r_i$ into the expression
+$$ \frac{1}{r_1} + \frac{1}{r_4} + \frac{1}{r_5} - \left( \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_6} \right). $$
+Letting $a = \pi/2 - b - c$ and $b = \pi/6$ then gives us an expression with $c$ as the only variable. Simplifying this expression (using a symbolic algebra system), we find that the result is 0, thus proving our theorem. $\square$

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+**Theorem 6.2.** If P is the incenter of $\triangle ABC$ and $\angle ABC = 120^{\circ}$ (Figure 10), then
+$$r_1r_2r_3 + r_3r_4r_5 + r_5r_6 = r_1r_3r_4 + r_2r_3r_4 + r_4r_5r_6.$$
+This can also be written as $r_1 + r_2 + \frac{r_5r_6}{r_3} = r_5 + r_6 + \frac{r_1r_2}{r_4}$.
+FIGURE 10. $r_1r_2r_3 + r_3r_4r_5 + r_5r_6 = r_1r_3r_4 + r_2r_3r_4 + r_4r_5r_6$
+*Proof*. This theorem can be proven using the same procedure that was used to prove Theorem 6.1. The details are omitted. $\square$
+**Open Question 1.** Is there a simple relationship between the $r_i$ that holds for all triangles when P is the incenter?
+Note that there are two independent variables, *a* and *b*, and six equations representing the values of the $r_i$. Thus, variables *a* and *b* can be eliminated resulting in an equation relating the $r_i$. Since there are so many more equations than variables, multiple relationships can be found. For example, in the 30°-60°-90° right triangle, we have a number of simple relationships between the $r_i$, as shown by the following theorem.
+**Theorem 6.3.** If P is the incenter of $\triangle ABC$ and $\angle ABC = 30^{\circ}$ and $\angle ACB = 60^{\circ}$, then the $r_i$ are related to each other by each of the following equations.
+$$
+\begin{aligned}
+& \frac{2}{r_1^4} + \frac{8}{r_3^4} + \frac{6}{r_4^4} + \frac{5}{r_5^4} = \frac{14}{r_2^4} + \frac{20}{r_6^4}, \\
+& \frac{3}{r_1^2} + \frac{3}{r_2^2} + \frac{3}{r_4^2} = \frac{3}{r_3^2} + \frac{2}{r_5^2} + \frac{3}{r_6^2}, \\
+& \frac{2}{r_1} + \frac{2}{r_2} + \frac{1}{r_5} + \frac{1}{r_6} = \frac{3}{r_3} + \frac{2}{r_4}, \\
+& 3r_1 + 5r_3 + 55r_4 + 22r_6 = 4r_2 + 75r_5, \\
+& 2r_1r_3 + 3r_2r_4 + 9r_5r_1 + 9r_6r_2 = 27r_3r_5 + r_4r_6.
+\end{aligned}
+$$

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+*Proof.* For this triangle, the values of the $r_i$ are as follows.
+$$
+\begin{aligned}
+r_1 &= \frac{1}{4}(\sqrt{2}-1)(\sqrt{3}-1), \\
+r_2 &= \frac{1}{4}(-10-7\sqrt{2}+6\sqrt{3}+4\sqrt{6}), \\
+r_3 &= \frac{1}{4}(9-7\sqrt{2}-5\sqrt{3}+4\sqrt{6}), \\
+r_4 &= \frac{1}{8}(-4-\sqrt{2}+2\sqrt{3}+\sqrt{6}), \\
+r_5 &= \frac{1}{24}(-3\sqrt{2}+2\sqrt{3}+\sqrt{6}), \\
+r_6 &= \frac{1}{4}(1+\sqrt{3}-\sqrt{6}).
+\end{aligned}
+ $$
+These values can be substituted into the stated equations to verify the results (using computer simplification, as necessary). $\square$
+Here is an approach that might be used to find the general relationship between the $r_i$ when P is the incenter. To avoid fractions, we will replace a, b, and c from Figure 9 by 2a, 2b, and 2c to get Figure 11. We can express PB as the sum of two lengths using circles 6 and 1 in two different ways. Equating these expressions gives the following equation.
+$$ (1) \qquad \frac{r_6}{r_1} = \frac{\cot b + \cot(a+b)}{\cot b + \cot(b+c)}. $$
+Since $4a + 4b + 4c = 180^\circ$, $c = 45^\circ - a - b$. Substitute this value of $c$ into equation (1). Then use the addition formula for cotangent,
+$$ \cot(x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y}, $$
+to write all trigonometric expressions in terms of $\cot a$ and $\cot b$.
+In a similar manner we can form two other equations for $r_2/r_3$ and $r_4/r_5$. This gives us the following three equations for $u = r_6/r_1$, $v = r_2/r_3$, and $w = r_4/r_5$ in terms of the two unknowns $C_a = \cot a$ and $C_b = \cot b$.
+$$
+\begin{aligned}
+u &= \frac{(C_a - 1)(C_b^2 + 2C_a C_b - 1)}{(C_a + C_b)(1 + C_a - C_b + C_a C_b)}, \\
+v &= \frac{(C_b - 1)(C_a C_b^2 - 2C_a - C_b)}{(C_a - 1)(C_a C_b^2 - 2C_b - C_a)}, \\
+w &= \frac{(C_a + C_b)(1 - C_a + C_b + C_a C_b)}{(C_b - 1)(C_a^2 + 2C_a C_b - 1)}.
+\end{aligned}
+ $$
+Clearing fractions gives us three polynomial equations in the variables $C_a$ and $C_b$. In theory, we should be able to eliminate $C_a$ and $C_b$ from these three equations, leaving us with a single equation relating $u$, $v$, and $w$. This equation would be

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+FIGURE 12. $5r_1 + 6r_2 + r_4 = r_3 + 3r_5 + 15r_6$
+*Proof.* We follow the same general procedure that was used in the proof of Theorem 6.1. Let $\angle ABE = a$, $\angle EBC = b$, $\angle BCF = c$, and $\angle FCA = d$, as shown in Figure 13. Note that $\angle BAC = \pi - a - b - c - d$, $\angle AFC = a + b + c$, $\angle AEB = b + c + d$, and $\angle EPC = b + c$.
+Without loss of generality, we may assume that $R = 1/2$, where $R$ is the radius of the circumcircle of $\triangle ABC$. By the Extended Law of Sines we get the following.
+$$AB = \sin(c+d), \quad AC = \sin(a+b), \quad BC = \sin(a+b+c+d).$$
+FIGURE 13.

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+We then use the Law of Sines to get the following.
+$$AF = \frac{\sin(a+b)\sin d}{\sin(a+b+c)}, \quad BF = \frac{\sin(a+b+c+d)\sin c}{\sin(a+b+c)}$$
+$$AE = \frac{\sin(c+d)\sin a}{\sin(b+c+d)}, \quad CE = \frac{\sin(a+b+c+d)\sin b}{\sin(b+c+d)}.$$
+Applying the Law of Sines again gives the following.
+$$PB = \frac{\sin(a+b+c+d)\sin c}{\sin(b+c)}, \quad PE = \frac{CE \sin d}{\sin(b+c)}$$
+$$PC = \frac{\sin(a+b+c+d)\sin b}{\sin(b+c)}, \quad PF = \frac{BF \sin a}{\sin(b+c)}.$$
+By Ceva's Theorem, $\frac{BD}{CD} = \frac{BF \cdot AE}{AF \cdot CE}$. This gives the following.
+$$BD = \frac{BF \cdot AE \cdot BC}{BF \cdot AE + AF \cdot CE}$$
+$$CD = \frac{AF \cdot CE \cdot BC}{BF \cdot AE + AF \cdot CE}$$
+Length PA is calculated using the Law of Cosines in triangle APB. We get the following.
+$$PA = \sqrt{AB^2 + PB^2 - 2 \cdot AB \cdot PB \cdot \cos a}.$$
+To find the length of PD without introducing another square root, we can apply Menelaus' Theorem to transversal BPE in △ADC. We get the following.
+$$PD = \frac{PA \cdot BD \cdot CE}{BC \cdot AE}.$$
+We now have expressions for the length of every line segment in the figure in terms of *a*, *b*, *c*, and *d*. We can therefore calculate all the $s_i$, $K_i$, and $r_i$.
+We can plug these values for the $r_i$ into the expression
+$$5r_1 + 6r_2 + r_4 - (r_3 + 3r_5 + 15r_6).$$
+Letting $a = 10^\circ$, $b = 30^\circ$, $c = 80^\circ$, and $d = 20^\circ$ gives us an expression with no variables. Simplifying this expression (using a symbolic algebra system), we find that the result is 0, thus proving our theorem. $\square$
+Sometimes the relationship between the $r_i$ is more regular as in the following two theorems. The proofs are similar to the proof of Theorem 7.1. When the formulas for the lengths of the radii previously found are applied to the equation to be proved, the result is a trigonometric equation that can be proven to be an identity using symbolic algebra computation. The details are omitted.

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+**Theorem 7.2.** If P is a point inside $\triangle ABC$, and $\angle PBA$, $\angle PBC$, $\angle PCB$, and $\angle PCA$ are as shown in Figure 14, then
+$$ \frac{1}{r_1} + \frac{1}{r_3} + \frac{1}{r_4} = \frac{1}{r_2} + \frac{1}{r_5} + \frac{1}{r_6}. $$
+FIGURE 14. $\frac{1}{r_1} + \frac{1}{r_3} + \frac{1}{r_4} = \frac{1}{r_2} + \frac{1}{r_5} + \frac{1}{r_6}$
+**Theorem 7.3.** If P is a point inside $\triangle ABC$, and $\angle PBA$, $\angle PBC$, $\angle PCB$, and $\angle PCA$ are as shown in any of the triangles depicted in Figure 15, then
+$$ \frac{1}{r_1} + \frac{1}{r_4} + \frac{1}{r_6} = \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_5}. $$
+Note that in each of these examples, $t$ is an arbitrary parameter.
+The reader may be wondering how I came up with these results. Here is the procedure I used.
+Let $f(r)$ denote some function of $r$ such as $r$, $r^2$, or $1/r$. I varied all combinations of $a, b, c$, and $d$ (see Figure 13) through all multiples of $1^\circ$ and calculated $f(r_1)$ through $f(r_6)$. Using the Mathematica® function FindIntegerNullVector, I looked for any linear relationships between these six numbers. If a linear relationship existed which involved all six numbers, I logged the quadruple $\langle a, b, c, d \rangle$ along with the coefficients of the relationship into a database. After all quadruples were examined, I looked at all pairs of entries in the database that had the same set of coefficients. If $Q_1 = \langle a_1, b_1, c_1, d_1 \rangle$ and $Q_2 = \langle a_2, b_2, c_2, d_2 \rangle$ were two such quadruples, I then formed the quadruple $Q_3$ such that $\langle Q_1, Q_2, Q_3 \rangle$ formed an arithmetic progression in each component. Then I examined $Q_3$ to see if it also satisfied the same linear combination. If it did, then this suggested a one-parameter family of solutions (which had to be confirmed).
+Note that many solutions were found for various functions $f$, but no one-parameter families of solutions were found except when $f(r) = 1/r$. It is not clear why this should be the case.

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+FIGURE 15. $\frac{1}{r_1} + \frac{1}{r_4} + \frac{1}{r_6} = \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_5}$
+We note several related results that hold whenever $P$ is an arbitrary point inside $\triangle ABC$.
+**Theorem 7.4.** If $P$ is any point inside $\triangle ABC$ (Figure 16), then $K_1K_3K_5 = K_2K_4K_6$.
+FIGURE 16. $K_1K_3K_5 = K_2K_4K_6$

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+In the spirit of Wasan, we investigate the relationships between the radii of six
+circles associated with a triangle and the three cevians through a point inside that
+triangle.
+## 2. NOTATION
+Let $P$ be any point inside a triangle $ABC$. The cevians through $P$ divide $\triangle ABC$ into six smaller triangles. Circles are inscribed in these six triangles. The six triangles and their incircles are numbered from 1 to 6 counterclockwise as shown in Figure 2, where the first triangle is formed by the lines $AP$, $BP$, and $BC$. The i-th triangle has inradius $r_i$, semiperimeter $s_i$, circumradius $R_i$, and area $K_i$.
+FIGURE 2. numbering
+If X and Y are points, then we use the notation $XY$ to denote either the line segment joining X and Y or the length of that line segment, depending on the context.
+## 3. THE ORTHOCENTER
+We start with a known result [4] giving the relationship between the $r_i$ when $P$ is the orthocenter.
+**Theorem 3.1.** If $P$ is the orthocenter of $\triangle ABC$ (Figure 3), then $r_1r_3r_5 = r_2r_4r_6$.
+FIGURE 3. $r_1r_3r_5 = r_2r_4r_6$

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+Figure 4. The renormalized estimation $s_{||\varphi}$ having two local maxima at location and time of two simultaneous releases (black crosses) at $t_0$.
+The computations with artificial model data emphasize a surprising result about the minimum number of concentration measurements required for identifying $m > 1$ simultaneous point releases. In order to reduce the computational volume, time dimension has been suppressed by considering point sources $s(x,y)$ (figure 4 is built without this simplification, i.e. with $s=s(x,y,t)$). This brings some minor modification in the fundamental geometry corresponding to the Euclidean geometry of the ground only. At first glance, the minimum number of measurements required for identifying the various releases should be 3m because each release is characterized by two parameters for horizontal location and intensity. However, we have verified in synthetic experiments that a lesser number is sufficient. We are able to retrieve two simultaneous releases from at least five synthetic measurements and three simultaneous releases from at least six synthetic measurements. The pair or triplet of locations and intensities are retrieved exactly in all of many repeated trials except when one of the releases is placed in a weakly seen region. This surprising result may be partly understood. When n measurements are performed, it seems that the adjoint vectors $a(x_l, y_l), a(x_2, y_2), \dots, a(x_m, y_n)$ computed at n locations should be, in general, linearly independent.
+Notice that m releases of intensity $q_1, q_2, \dots, q_m$ at locations $(\xi_l, \zeta_l), (\xi_2, \zeta_2), \dots, (\xi_m, \zeta_m)$ generate a measurement vector ideally free from noises :
+$$ \boldsymbol{\mu}_0 = q_1 a(\xi_1, \zeta_1) + q_2 a(\xi_2, \zeta_2) + \dots + q_m a(\xi_m, \zeta_m) \quad (10) $$
+The identification of the releases amounts to identify the above decomposition of the measurement vector. We argue that this decomposition is unique, in general, provided that $n \ge 2m$. Indeed, if this decomposition is not unique, we can write two alternative decompositions:
+$$ \boldsymbol{\mu}_0 = q_1 a(\xi_1, \zeta_1) + q_2 a(\xi_2, \zeta_2) + \dots + q_m a(\xi_m, \zeta_m) = q'_1 a(\xi'_1, \zeta'_1) + q'_2 a(\xi'_2, \zeta'_2) + \dots + q'_m a(\xi'_m, \zeta'_m) \quad (11) $$
+Then, by defining $(x_b, y_i) = (\xi_b^-, \zeta_i^-)$ and $(x_{m+i}, y_{m+i}) = (\xi_b^+, \zeta_i^+)$ for $i = 1, 2, \dots, m$, we obtain a family $a(x_l, y_l), a(x_2, y_2), \dots, a(x_{2m}, y_{2m})$ of less than $n$ adjoint vectors corresponding to different locations. According to equation (11) they are linearly dependent. However, this is in contradiction with the aforementioned generic property. The decomposition is unique and this implies that it may be identified (for instance by looping over all $m$-tuples of locations). If the number of measurements is sufficiently large, i.e. $n \ge 2m$, the $m$ releases and their locations can be identified. The computations show that the six parameters corresponding to the location and intensity of three releases may be identified from five measurements. For identifying the six parameters associated with two point releases, at least five measurements are required in the computations. This is less than the number of parameters but does not coincide exactly with the condition $n \ge 2m$. Further investigations are required to fully understand the numerical facts. This will help in designing monitoring networks with the least number of samplers.
+## CONCLUSIONS
+The model PANEPR for fluid advection diffusion was used with winds calculated around the buildings for the prescribed boundary condition corresponding to a Pasquill class C stability and a free wind of 3 ms⁻¹ from SE. The adjoint functions were obtained from a backward integration of the retrograde transport equation on a grid of 5.0×5.0 m² and a time step of 5s. Due to the constant wind conditions, the adjoint functions of the successive measurements at each given detector are identical except for a time shift. The computations made with artificial measurements produced free from noises illustrate the usefulness of the proposed non-Bayesian framework for monitoring atmospheric contaminations. Contaminations due to industrial accidents, most of the time, may be regarded as point releases. The proposed technique allows to identify the origin of such releases. The advantages of the proposed technique, compared to earlier ones, are many. First, the computations are fully and simply realizable without summoning any arbitrary simplifying assumptions. Second, the visibility functions $\varphi$ clearly indicates the regions well or badly seen by the monitoring network and the contribution of each detector to the global performance. Third, the technique turns the set of measurements into visual information showing the structure of the emissions. This visualization is limited by the resolution capability of the monitoring network. In particular, several simultaneous contaminations may be discriminated. In an industrial context, this is important to discriminate several minor releases from a major accident. In addition, it is possible to clarify responsibilities.
+This study raises issues of practical and theoretical importance. The effect of the noise is not described in the present abstract but a more extensive study (Sharan, M. et al., 2009) shows acceptable performance of the noisy inversion. In principle, the proposed technique is based on the simultaneous processing of the whole set of measurements available at all times. However, this is not practically realizable because the continuous flow of measurements is rapidly enormous. Accordingly, it

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+will be necessary to investigate the fading correlations between measurements separated by large time intervals. Finally, a surprising result is observed numerically: the number of measurements required for identifying several simultaneous point releases is strictly less than the number of parameters required to describe them. A tentative explanation has been proposed. This result, relying on the global properties of the informational system is contrary to all classical descriptions. Its complete understanding is left for further investigations but will be probably helpful in optimizing the design and exploitation of monitoring networks.
+## REFERENCES
+- Issartel, J.-P. and Baverel, J., 2003: Adjoint backtracking for the verification of the CTBT, *Atmospheric Chemistry and Physics*, **3**, 475-486.
+- Issartel, J.-P., 2005: Emergence of a tracer source from air concentration measurements: a new strategy for linear assimilation, *Atmospheric Chemistry and Physics*, **5**, 249-273.
+- Issartel, J.-P., Sharan, M., Modani, M., 2007: An inversion technique to retrieve the source of a tracer with an application to synthetic satellite measurements, *Proceedings of the Royal Society*, **463**, 2863-2886.
+- Sharan, M., Issartel, J.-P., Singh, S. K., Kumar, P., 2009: An inversion technique for the retrieval of single point emissions from atmospheric concentration measurements, *Proceedings of the Royal Society*, **465**, 2069-2088.

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+# Principle of Mathematical Induction
+Let $P(n)$ be a statement that depends on $n \in \mathbb{N}$. If the following two conditions hold
+1. $P(1)$ is true. (This is called the base case)
+2. $P(k)$ is true implies $P(k+1)$ is true. (This is called the induction hypothesis and induction step)
+then $P(n)$ is true for all $n \in \mathbb{N}$.
+## Example 1
+Prove that $1+2+3+\cdots+n = \frac{n(n+1)}{2}$ for $n \in \mathbb{N}$.
+### Solution
+Since this is our first example we will be more tedious and show more steps than necessary.
+Let $P(n)$ be the statement $1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$.
+Base Case: $n=1$
+$P(1)$ is true because $1 = \frac{1(1+1)}{2}$.
+Induction Hypothesis: Assume that $P(k)$ is true for some $k \in \mathbb{N}$. That is $1+2+3+\cdots+k = \frac{k(k+1)}{2}$.
+Induction Step: Now, we will show that $P(k+1)$ is true.
+Starting with the Induction Hypothesis, we add $(k+1)$ to both sides and algebraically manipulate the right side to get the desired result.
+$$
+\begin{aligned}
+1 + 2 + 3 + \cdots + k + (k + 1) &= \frac{k(k+1)}{2} + (k+1) \\
+&= \frac{k^2 + k + 2k + 2}{2} \\
+&= \frac{(k+1)((k+1) + 1)}{2}
+\end{aligned}
+ $$
+Hence, $P(k+1)$ is true. Therefore, by mathematical induction, $1+2+3+\cdots+k = \frac{k(k+1)}{2}$ for all $n \in \mathbb{N}$.
+## Remark 1
+This is a common type of problem where you are asked to prove the closed form formula. The only hard part is to get the right hand side to equal to the desired result. Instead of a summation, it could be a product or an recurrence relation. Sometimes, the problems requires you to first find a closed form formula then prove that it is correct.
+## Example 2
+Prove that $2^{n-1} \le n!$ for all $n \in \mathbb{N}$.
+### Solution
+For this example, we will show less steps.
+We will prove this inequality by induction on $n$.
+Base Case: $n=1$ is true because $1 \le 1$.
+Induction Step: Assume the result is true for $n=k$. We will prove $n=k+1$ is also true.
+Starting with the induction hypothesis and multiply both sides of the inequality by 2, we have
+$$
+\begin{aligned}
+2^{(k+1)-1} &\le 2 \cdot k! \\
+&\le (k+1)k! \\
+&= (k+1)!
+\end{aligned}
+ $$
+Therefore by induction, $2^{n-1} \le n!$ for all $n \in \mathbb{N}$.
+## Remark 2
+This is another common type of problem where you are asked to prove an inequality. The only hard part to this example is to recognize the second inequality, $2 \le k+1$.