For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product form... more For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form det H ((I H − A)(I H − B)) = det H (I H − A)det H (I H − B). When trace class operators are replaced by Hilbert-Schmidt operators A, B ∈ B 2 (H) and the Fredholm determinant det H (I H − A), A ∈ B 1 (H), by the 2nd regularized Fredholm determinant det H,2 (I H − A) = det H ((I H − A) exp(A)), A ∈ B 2 (H), the product formula must be replaced by det H,2 ((I H − A)(I H − B)) = det H,2 (I H − A)det H,2 (I H − B) × exp(− tr H (AB)). The product formula for the case of higher regularized Fredholm determinants det H,k (I H −A), A ∈ B k (H), k ∈ N, k 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.
ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interve... more ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interventions in disrupting and dismantling criminal networks. We tested three law enforcement interventions that targeted social capital in criminal networks (betweenness, degree and cut-set) and two interventions that targeted human capital (actors who possess money and those who possess precursor chemicals). These five interventions are compared with each other and with random (opportunistic) removal of actors in two settings: (i) with network adaptation incorporated into the simulations and (ii) without network adaptation. Results illustrate that the removal of actors based on betweenness centrality was the most efficient strategy, leading to network disruption in the least number of steps and was relatively consistent across replications. Targeting actors who possessed money was the second most effective overall and was also relatively consistent in its disruptive effect.
Restoring missing ecological interactions by reintroducing locally extinct species or ecological ... more Restoring missing ecological interactions by reintroducing locally extinct species or ecological surrogates for extinct species has been mooted as an approach to restore ecosystems. Australia's apex predator, the dingo, is subject to culling in order to prevent attacks on livestock. Dingo culling has been linked to ecological cascades evidenced by irruptions of herbivores and introduced mesopredators and declines of small and medium sized mammals. Maintenance of dingo populations is untenable for land-managers in many parts of Australia owing to their depredations on livestock. However, it may be possible to fill the apex predator niche with the Tasmanian devil which has less impact on livestock. Devils once occurred throughout Australia, but became extinct from the mainland about 3000 years ago, but are now threatened by a disease epidemic in Tasmania. To explore the feasibility of reintroducing devils to mainland Australia we used species distribution models (SDMs) to determine if suitable climatic conditions for devils exist and fuzzy cognitive mapping (FCM) to predict the effects of devil reintroduction. Based on devils' current distribution, our SDM indicates that suitable areas for devils exist in southeastern Australia. Our FCM examined ecosystem responses to predator-management scenarios by manipulating the abundances of devils, dingoes and foxes. Our FCMs showed devils would have cascading effects similar to, but weaker than those of dingoes. Devil introduction was linked to lower abundances of introduced mesopredators and herbivores. Abundances of small and medium sized mammals and understorey vegetation complexity increased with devil introduction. However, threatened species vulnerable to fox predation benefited little from devil introduction. Our study suggests that reintroducing ecological surrogates for apex predators may yield benefits for biodiversity conservation.
The known There are no systematically reported national data on the structure and characteristics... more The known There are no systematically reported national data on the structure and characteristics of general medical practice in Australia. The new Network analysis of 21 years of Medicare claims indicates that general practice communities have generally increased in size, continuity of care and patient loyalty have remained stable, and greater sharing of patients by GPs is associated with greater patient loyalty. The implications Our new approach to analysing routinely collected data allows continuous monitoring of the characteristics of Australian general practices and how these characteristics affect patient care.
Extending previous results in the literature, random colored substitution networks are introduced... more Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced. Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.
IEEE Transactions on Information Theory, Sep 1, 2010
It is proved that the set of higher weight enumerators of a linear code over a finite field is eq... more It is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the Tutte polynomial associated to the code. An explicit expression for the Tutte polynomial is given in terms of the subcode weights. Generalizations of these results are proved and are applied to codeword m-tuples. These general results are used to prove a very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity. In addition, a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented.
In this paper we provide a 4-GDD of type 2 2 5 5 , thereby solving the existence question for the... more In this paper we provide a 4-GDD of type 2 2 5 5 , thereby solving the existence question for the last remaining feasible type for a 4-GDD with no more than 30 points. We then show that 4-GDDs of type 2 t 5 s exist for all but a finite specified set of feasible pairs (t, s).
IEEE Transactions on Information Theory, May 1, 2016
The critical exponent of a matroid is one of the important parameters in matroid theory and is re... more The critical exponent of a matroid is one of the important parameters in matroid theory and is related to the Rota and Crapo's Critical Problem. This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid. An upper bound on the covering dimension is conjectured and nearly proven, improving a classical bound for the critical exponent. Finally, a construction is given of linear codes that attain equality in the covering dimension bound.
We present new values and bounds on the (normalised) closeness centralityCC of connected graphs a... more We present new values and bounds on the (normalised) closeness centralityCC of connected graphs and on its productlCC with the mean distancel of these graphs. Our main result presents the fundamental bounds 1 ≤lCC < 2. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions forCC andlCC for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values nCC Pn and of nCC Ln both equal π, and the asymptotic limits oflCC for these families of graphs are both equal to π/3. We conjecture that the set of valueslCC for all connected graphs is dense in the interval [1, 2).
Supplemental material giving details on the extraction of clinics of General Practitioners from n... more Supplemental material giving details on the extraction of clinics of General Practitioners from networks formed by national claims data.
Extending previous results in the literature, random colored substitution networks are introduced... more Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced. Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.
ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interve... more ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interventions in disrupting and dismantling criminal networks. We tested three law enforcement interventions that targeted social capital in criminal networks (betweenness, degree and cut-set) and two interventions that targeted human capital (actors who possess money and those who possess precursor chemicals). These five interventions are compared with each other and with random (opportunistic) removal of actors in two settings: (i) with network adaptation incorporated into the simulations and (ii) without network adaptation. Results illustrate that the removal of actors based on betweenness centrality was the most efficient strategy, leading to network disruption in the least number of steps and was relatively consistent across replications. Targeting actors who possessed money was the second most effective overall and was also relatively consistent in its disruptive effect.
In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes... more In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for m-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic m-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic m-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic mtuple weight enumerators of self-dual codes over finite fields.
For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product form... more For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form det H ((I H − A)(I H − B)) = det H (I H − A)det H (I H − B). When trace class operators are replaced by Hilbert-Schmidt operators A, B ∈ B 2 (H) and the Fredholm determinant det H (I H − A), A ∈ B 1 (H), by the 2nd regularized Fredholm determinant det H,2 (I H − A) = det H ((I H − A) exp(A)), A ∈ B 2 (H), the product formula must be replaced by det H,2 ((I H − A)(I H − B)) = det H,2 (I H − A)det H,2 (I H − B) × exp(− tr H (AB)). The product formula for the case of higher regularized Fredholm determinants det H,k (I H −A), A ∈ B k (H), k ∈ N, k 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.
ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interve... more ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interventions in disrupting and dismantling criminal networks. We tested three law enforcement interventions that targeted social capital in criminal networks (betweenness, degree and cut-set) and two interventions that targeted human capital (actors who possess money and those who possess precursor chemicals). These five interventions are compared with each other and with random (opportunistic) removal of actors in two settings: (i) with network adaptation incorporated into the simulations and (ii) without network adaptation. Results illustrate that the removal of actors based on betweenness centrality was the most efficient strategy, leading to network disruption in the least number of steps and was relatively consistent across replications. Targeting actors who possessed money was the second most effective overall and was also relatively consistent in its disruptive effect.
Restoring missing ecological interactions by reintroducing locally extinct species or ecological ... more Restoring missing ecological interactions by reintroducing locally extinct species or ecological surrogates for extinct species has been mooted as an approach to restore ecosystems. Australia's apex predator, the dingo, is subject to culling in order to prevent attacks on livestock. Dingo culling has been linked to ecological cascades evidenced by irruptions of herbivores and introduced mesopredators and declines of small and medium sized mammals. Maintenance of dingo populations is untenable for land-managers in many parts of Australia owing to their depredations on livestock. However, it may be possible to fill the apex predator niche with the Tasmanian devil which has less impact on livestock. Devils once occurred throughout Australia, but became extinct from the mainland about 3000 years ago, but are now threatened by a disease epidemic in Tasmania. To explore the feasibility of reintroducing devils to mainland Australia we used species distribution models (SDMs) to determine if suitable climatic conditions for devils exist and fuzzy cognitive mapping (FCM) to predict the effects of devil reintroduction. Based on devils' current distribution, our SDM indicates that suitable areas for devils exist in southeastern Australia. Our FCM examined ecosystem responses to predator-management scenarios by manipulating the abundances of devils, dingoes and foxes. Our FCMs showed devils would have cascading effects similar to, but weaker than those of dingoes. Devil introduction was linked to lower abundances of introduced mesopredators and herbivores. Abundances of small and medium sized mammals and understorey vegetation complexity increased with devil introduction. However, threatened species vulnerable to fox predation benefited little from devil introduction. Our study suggests that reintroducing ecological surrogates for apex predators may yield benefits for biodiversity conservation.
The known There are no systematically reported national data on the structure and characteristics... more The known There are no systematically reported national data on the structure and characteristics of general medical practice in Australia. The new Network analysis of 21 years of Medicare claims indicates that general practice communities have generally increased in size, continuity of care and patient loyalty have remained stable, and greater sharing of patients by GPs is associated with greater patient loyalty. The implications Our new approach to analysing routinely collected data allows continuous monitoring of the characteristics of Australian general practices and how these characteristics affect patient care.
Extending previous results in the literature, random colored substitution networks are introduced... more Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced. Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.
IEEE Transactions on Information Theory, Sep 1, 2010
It is proved that the set of higher weight enumerators of a linear code over a finite field is eq... more It is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the Tutte polynomial associated to the code. An explicit expression for the Tutte polynomial is given in terms of the subcode weights. Generalizations of these results are proved and are applied to codeword m-tuples. These general results are used to prove a very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity. In addition, a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented.
In this paper we provide a 4-GDD of type 2 2 5 5 , thereby solving the existence question for the... more In this paper we provide a 4-GDD of type 2 2 5 5 , thereby solving the existence question for the last remaining feasible type for a 4-GDD with no more than 30 points. We then show that 4-GDDs of type 2 t 5 s exist for all but a finite specified set of feasible pairs (t, s).
IEEE Transactions on Information Theory, May 1, 2016
The critical exponent of a matroid is one of the important parameters in matroid theory and is re... more The critical exponent of a matroid is one of the important parameters in matroid theory and is related to the Rota and Crapo's Critical Problem. This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid. An upper bound on the covering dimension is conjectured and nearly proven, improving a classical bound for the critical exponent. Finally, a construction is given of linear codes that attain equality in the covering dimension bound.
We present new values and bounds on the (normalised) closeness centralityCC of connected graphs a... more We present new values and bounds on the (normalised) closeness centralityCC of connected graphs and on its productlCC with the mean distancel of these graphs. Our main result presents the fundamental bounds 1 ≤lCC < 2. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions forCC andlCC for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values nCC Pn and of nCC Ln both equal π, and the asymptotic limits oflCC for these families of graphs are both equal to π/3. We conjecture that the set of valueslCC for all connected graphs is dense in the interval [1, 2).
Supplemental material giving details on the extraction of clinics of General Practitioners from n... more Supplemental material giving details on the extraction of clinics of General Practitioners from networks formed by national claims data.
Extending previous results in the literature, random colored substitution networks are introduced... more Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced. Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.
ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interve... more ABSTRACT The current paper aimed to investigate the effectiveness of five law enforcement interventions in disrupting and dismantling criminal networks. We tested three law enforcement interventions that targeted social capital in criminal networks (betweenness, degree and cut-set) and two interventions that targeted human capital (actors who possess money and those who possess precursor chemicals). These five interventions are compared with each other and with random (opportunistic) removal of actors in two settings: (i) with network adaptation incorporated into the simulations and (ii) without network adaptation. Results illustrate that the removal of actors based on betweenness centrality was the most efficient strategy, leading to network disruption in the least number of steps and was relatively consistent across replications. Targeting actors who possessed money was the second most effective overall and was also relatively consistent in its disruptive effect.
In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes... more In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for m-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic m-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic m-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic mtuple weight enumerators of self-dual codes over finite fields.
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