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Cutaneous adverse events (AEs) following cancer immunotherapy, targeted therapy, and chemotherapy have been well-documented in the literature. A number of case reports have identified phototherapy, a form of light therapy that mimics sunlight exposure, as a noninvasive treatment modality for these cutaneous toxicities. By inducing local suppression of the immune system, phototherapy is a skin-directed treatment with minimal effect on tumor response. Phototherapy may therefore be a viable treatment option for cutaneous AEs from cancer therapies.We reviewed the literature for patients treated with phototherapy for cutaneous AEs following cancer immunotherapy, targeted therapy, or chemotherapy. We also included three previously unpublished cases from our own institution.We identified 24 patients (80% male, mean age 67 years, range 49-75 years). Patients received the following phototherapy types: NB-UVB (n = 17), PUVA (n = 6), or PDT (n = 1). A topical steroid was used in conjunction with phototherapy in seven patients. At phototherapy onset, cancer treatment was either continued, temporarily discontinued, or discontinued (n = 9, 6, 7, respectively; in two cases, the cancer treatment course was unknown). Improvement of cutaneous AEs was observed in 96% of patients.Phototherapy resulted in full or partial improvement in all but one patient. A topical steroid was used in nearly a third of patients, suggesting some oncodermatologists co-administer topicals to further boost response. Continuation of cancer therapy in the majority of patients highlights an additional advantage of phototherapy. We believe phototherapy may be an effective adjunctive treatment to topical steroids when treating these cutaneous toxicities.

The parameters S and M were introduced in (1), (3) and (4) in connection with the stochastic rank and the term rank of a matrix. The results of this paper involve these same parameters. Two of these results are the following. Let A be a matrix of non-negative integers with sum S and maximum row or column sum M and let a = [S/M] (the greatest integer in S/M). It is shown in theorem 1 that A is expressible as a sum of subpermutat ion matrices of rank a and a sub-permutation matrix of rank q, 0 ~< q < a. Further, let K be a bipartite graph with finite vertex sets X and Y. Let S be the total number of edges and let M be the maximum number of edges which have the same vertex. It is shown as a consequence of theorem 3 that K has a transversal G of order a = [S/M] with vertex sets U and V such that every vertex of X which has M edges is an element of U and every vertex of Y which has M edges is an element of V. Theorem 1 may be proved as a corollary of theorem 3 but the connection of theorem 1 with the doubly stochastic extension of a matrix is interesting. It is on this connection that the proof of theorem 1 in section 3 is based.

Von Neumann's fundamental theorem of the theory of games has been extended by various authors and recently in two different directions by Kneser (8) and Nikaidô (9). We present here a form of the theorem, which is more general than that of both these authors. We develop some consequences of this theorem, which make it easier to decide whether certain classes of games have a value and we give several illustrative examples

For the purpose of analysing bipartite graphs (hereinafter called simply graphs) the concept of an exterior covering is introduced. In terms of this concept it is possible in a natural way to decompose any graph into two parts, an inadmissible part and a core. It is also possible to decompose the core into irreducible parts and thus obtain a canonical reduction of the graph. The concept of irreducibility is very easily and naturally expressed in terms of exterior coverings. The role of the inadmissible edges of a graph is to obstruct certain natural coverings of the graph.

Corresponding to every graph, bipartite graph, or directed bipartite graph there exists a directed graph which is connected if and only if the original graph is connected.In this paper, it is shown that for every directed graph there exists a certain bipartite graph such that the directed graph is connected if and only if the bipartite graph is irreducible. Other connections between reducibility and connectivity are established.

If S 1, S 2, S 3, … , S n are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that ai is in S i for (i = 1, 2, 3, … , n) and such that ai ≠ aj for i ≠ j , is that for k = 1, 2, 3, … , n, the union of any k of the sets S1, S 2, … , S n, contains at least k elements. The theorem has a number of consequences amongst which we list the following.

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v wherewith p1 p2, … , Pr being distinct primes and

The theory of non-negative matrices was initiated by Perron (7) and Frobenius (4). Wielandt in (11) gives an elegant exposition of the subject.It is well known to workers in the field that if a matrixAhas all its entries non-negative real numbers, then the pattern of zeros and non-zeros ofAcompletely determines the pattern of zeros and non-zeros in every power ofA. Ptak in (8) and Ptak and Sedlacek in (9) describe this behaviour in terms of some combinatorial constructs.

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

choice of E. The usual procedure is to divide the range of integration into two parts, integrate outwards for a solution satisfying one boundary condition, integrate inwards for a solution satisfying the other boundary condition, match the solutions at an intermediate point and adjust E so that the derivatives also agree [1], [2]. The inward integration may be avoided with the procedure described earlier. A convenient way of dividing the range is according to the sign of f(r). For some r, f(r) < 0 so that condition (ii) is not satisfied: the procedure described here is not always numerically stable when f(r) < 0 [3]; in fact, for some values of i, I d, I < 1. Of a series of standard methods, the Numerov method

Marcus [2] has proved the following theorem.Suppose A is a non-negative normal matrix satisfying p(A) = 0 in which p(λ) is a monic polynomial no two of whose non-zero roots have the same modulus. Then there exists a permutation matrix P such that PAP* is a direct sum, PAP* = A1 ⊕ A2 ⊕ … ⊕ Am, in which each Ai is either O or primitive.This note gives a generalisation of this result, dropping the non-negative assumption and weakening the normality assumption.