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The analysis of wave patterns in a structure which possesses periodicity in the spatial and temporal dimensions is presented. The topic of imperfect chiral interfaces is also considered. Although causality is fundamental for physical processes, natural wave phenomena can be observed when a wave is split at a temporal interface. A wave split at a spatial interface is a more common occurrence; however, when the coefficients of the governing equations are time-dependent, the temporal interface becomes important. Here, the associated frontal waves are studied, and regimes are analysed where the growth of the solution in time is found. Imperfect interfaces, across which the displacements are discontinuous, are also considered in the vector case of chiral elastic systems. Analytical study and asymptotic approximations are supplied with illustrative numerical examples. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase composites with general geometries have the same set of possible effective tensors as those of hierarchical laminates. Other questions involve the conductivity and elasticity of composites. Finally, some future directions for wave and other equations are mentioned. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways., Competing Interests: We declare we have no competing interests.

Field patterns occur in space-time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space-time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of [Formula: see text]-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integer p . If p takes its maximum value, then we have a complete analytic material. Otherwise, it is incomplete analytic material of rank p . For two-dimensional materials, further progress can be made in the identification of analytic materials by using the well-known fact that a 90 ° rotation applied to a divergence-free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 2 . We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1 = ω 2 and [Formula: see text] where Λ =(λ 1 λ 2 …λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a ⊙ b is defined to be the vector ( a 1 b 1 , a 2 b 2 ,…, a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

The quasistatic field around a circular hole in a two-dimensional hyperbolic medium is studied. As the loss parameter goes to zero, it is found that the electric field diverges along four lines each tangent to the hole. In this limit, the power dissipated by the field in the vicinity of these lines, per unit length of the line, goes to zero but extends further and further out so that the net power dissipated remains finite. Additionally the interaction between polarizable dipoles in a hyperbolic medium is studied. It is shown that a dipole with small polarizability can dramatically influence the dipole moment of a distant polarizable dipole, if it is appropriately placed. We call this the searchlight effect, as the enhancement depends on the orientation of the line joining the polarizable dipoles and can be varied by changing the frequency. For some particular polarizabilities the enhancement can actually increase the further the polarizable dipoles are apart.

It is shown how a recently proposed method of cloaking is effective over a broad range of frequencies. The method is based on three or more active devices. The devices, while not radiating significantly, create a "quiet zone" between the devices where the wave amplitude is small. Objects placed within this region are virtually invisible. The cloaking is demonstrated by simulations with a broadband incident pulse., ((c) 2009 Optical Society of America)

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.

Background: The late Dr. Vincent McGovern (1915 to 1983) was an international authority on melanoma pathology and one of the first to suggest that assessment of tumor mitotic rate (TMR) might provide useful prognostic information. Data for a large cohort of patients, now with extended follow-up, whose tumors had been assessed by Dr. McGovern were analyzed to reassess the independent prognostic value of TMR in primary localized, cutaneous melanoma., Methods: Information was extracted from the Sydney Melanoma Unit database for 1317 patients treated between 1957 and 1982 for whom there was complete clinical information and whose primary lesion pathology, which included tumor thickness, ulcerative state, and TMR, had been assessed by Dr. McGovern. All these assessments were made according to the recommendations of the Eighth International Pigment Cell Conference, held in Sydney in 1972 under the auspices of the International Union Against Cancer. Factors predicting melanoma-specific survival were analyzed with the Cox proportional hazards regression model., Results: Stage, according to the recently revised American Joint Committee on Cancer Staging System (which is based on tumor thickness and ulceration) was the most predictive factor for survival (P<.0001). This was followed by primary lesion site (P<.0001), patient age (P=.0005), and TMR (P=.008)., Conclusions: TMR was confirmed to be an important independent predictor of survival of patients with primary cutaneous melanoma. However, its predictive value was less than it was when assessed according to the 1982 revisions of the 1972 TMR recommendations.

Purpose: To analyze prognostic factors, effects of treatment, and survival for patients with cerebral metastases from melanoma., Patients and Methods: All melanoma patients with cerebral metastases treated at the Sydney Melanoma Unit between 1952 and 2000 were identified. From 1985 to 2000, patients were diagnosed and treated using consistent modern techniques and this cohort was analyzed in detail. Multivariate analysis of prognostic factors for survival was performed., Results: A total of 1137 patients with cerebral metastases were identified; 686 were treated between 1985 and 2000. For these 686 patients, the median time from primary diagnosis to cerebral metastasis was 3.1 years (range, 0 to 41 years). A total of 646 patients (94%) have died as a result of melanoma. The median survival from the time of diagnosis of cerebral metastasis was 4.1 months (range, 0 to 17.2 years). Treatment was as follows: surgery and postoperative radiotherapy, 158 patients; surgery alone, 47 patients; radiotherapy alone, 236 patients; and supportive care alone, 210 patients. Median survival according to treatment received for these four groups was 8.9, 8.7, 3.4, and 2.1 months, respectively; the differences between surgery and nonsurgery groups were statistically significant. On multivariate analysis, significant factors associated with improved survival were surgical treatment (P <.0001), no concurrent extracerebral metastases (P <.0001), younger age (P =.0007), and longer disease-free interval (P =.036). Prognostic factors analysis confirmed the important influence of patient selection on treatment received., Conclusion: This large series documents the characteristics of patients who developed cerebral metastases from melanoma. Median survival was dependent on treatment, which in turn was dependent on patient selection.

The undoubted success of the SMU as a specialist multidisciplinary melanoma treatment center has clearly been the result of many factors. Perhaps chief among these was the vision and commitment that led Dr. Milton to establish it in the first place, and the sharing of that vision and commitment by those who were associated with him and by those who joined the SMU later. Another vitally important element, however, has been the continuing sense of unity and purpose fostered by the weekly SMU clinical meetings, which are truly multidisciplinary, in which all staff are encouraged to participate, and at which the desirability of adherence to agreed, evidence-based treatment guidelines is emphasized. A further influential factor has been the SMU's strong commitment to clinical and basic research as a concomitant of high quality clinical care, with stimulation, encouragement, and advice provided at its monthly multidisciplinary research meetings, where all current and proposed clinical and laboratory studies are discussed. As a result of these activities, despite an ever-increasing number of people working within it, the SMU has been able to present to referring doctors, to patients, and to the community a unified commitment to the best possible patient care and to high quality clinical and laboratory research. These groups have responded by recognizing the SMU as the major referral center for melanoma in Australia, as evidenced by the steadily increasing number of patients referred to it for treatment each year. Melanoma is a more pressing health problem in Australia than elsewhere, because it is the third most common cancer in women (after breast cancer and colorectal cancer), and the fourth most common cancer in men (after prostate cancer, colorectal cancer, and lung cancer). Nevertheless the experiences of the SMU as a large multidisciplinary melanoma treatment center are likely to have relevance and application in other countries, where the incidence of melanoma is lower but continues to rise, and may within a few years approach rates currently recorded in Australia.

Clinical diagnosis of melanoma can be difficult. A review of the accuracy of clinical diagnosis of melanoma, over a 12 month period, was undertaken at the Skin and Cancer Foundation Australia. The overall accuracy rate was 65.6% with seborrhoeic keratosis, melanocytic naevi and basal cell carcinoma the most common clinical misdiagnoses given to melanoma. Specialist doctors with more than 10 years experience had a higher rate of correct diagnosis than trainee doctors with 0-5 years experience.

The incidence and prognosis of cutaneous melanoma in children under 13 years of age has always been difficult to assess. The main reasons for this difficulty include the rarity of melanoma in children, referral biases from investigating institutions, changing conceptions in the histological diagnosis of true melanoma, and the lack of large enough study groups with sufficient follow-up to estimate 10-year survival rates. The present study documents 32 cases of childhood cutaneous melanoma drawn from the records of two large referral centres in New South Wales, Australia: the Sydney Melanoma Unit and the Newcastle Melanoma Unit in order to demonstrate some of the difficulties in the assessment of incidence and prognosis in children.

Malignant melanoma recurring deep in the body can present with a considerable range of symptoms and although routine CT Scan and chest x-ray are probably not indicated as they are expensive, time consuming and give very meagre returns, the minute a patient complains of any of these symptoms full scale investigation is imperative. Although it may not be possible to do very much about the symptoms, sometimes suitable surgical removal will give the patient if not cure at least prolonged palliation.

The personal experience with 5 rare types of malignant melanoma is reviewed to point out some of the practical problems in the diagnosis and management of these tumors. The rare forms discussed are conjunctival, nasal, oral, vulvar, and penile melanomas. All pigmented lesions in the oral cavity, but not the penis or vulva, should be prophylactically excised as lesions in the mouth have a higher malignant potential. Local excision of all 5 forms of primary melanomas, no matter how locally advanced they may be, is the sole treatment. Nevertheless, anatomic constraints often preclude surgery with generous margins and consequently local recurrence, particularly for conjunctival, nasal, and oral primary lesions, is usually the major first failure in treatment. Lymph node dissection is only performed if the regional nodes are palpable at the time of first presentation. Elective lymph node dissections are not performed since the patients are often elderly, the lymphatic drainage is usually ambiguous or multiple, and the disease tends to spread hematogenously rather than lymphatically. Surgery still remains the cornerstone of treatment for these rare forms of melanoma but prognosis is very poor since surgery is often a palliative rather than a curative measure. Improved survival may depend on identifying more effective chemotherapeutic and immunologic agents.