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Olkin & Vaeth (1981) consider a two-way classification model with k rows and p columns in which the residuals (ei1, ..., eip) for row i are independently distributed with a multivariate normal distribution with zero means and common covariance matrix ((ij)). They concentrate their attention particularly on the situation where the row classification corresponds to k subjects, each treated in one of two or more ways; the column classification corresponds to p repetitions, perhaps in time, from each subject, probably resulting in correlated errors. Replication of subjects within treatments enables the elements of the covariance matrix to be estimated from the data, which in turn enables maximum likelihood estimates of the row and column parameters to be computed and likelihood ratio tests to be carried out without having to make assumptions about the form of the covariance matrix. The careful analysis of data of this kind has been the subject of numerous papers, Wishart (1938) perhaps being the first to concentrate on this topic. Despite this, however, the use of erroneous methods in published material is widespread and this provided the stimulus for the publication of our earlier paper (Rowell & Walters, 1976), and also for the present communication. We note here that there is confusion in Olkin & Vaeth's row/column terminology in their worked example: the rows in their tables are referred to as columns in the text, and vice versa. In what follows, when referring to their numerical example, the rows classification will refer to the high/low factor, and the columns classification to the time factor.

A subset A A of a topological space is said to be semi-open if there exists an open set U U such that U ⊆ A ⊆ Cl ⁡ ( U ) U \subseteq A \subseteq \operatorname {Cl} (U) where Cl ⁡ ( U ) \operatorname {Cl} (U) denotes the closure of U U . The class of semi-open sets of a given topological space ( X , T ) (X,\mathcal {T}) is denoted S .O . ( X , T ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) . In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if T \mathcal {T} and T ∗ {\mathcal {T}^ \ast } are two topologies for a set X X such that S .O . ( X , T ) ⊆ S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T ⊆ T ∗ \mathcal {T} \subseteq {\mathcal {T}^ \ast } . In a corollary to this theorem, Levine states that if S .O . ( X , T ) = S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T = T ∗ \mathcal {T} = {\mathcal {T}^ \ast } . An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.

The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'

A discussion of the applicability of the concepts of T. S. Kuhn's theory of scientific revolutions, stimulated by the statement by M. J. Crowe of ten “laws” concerning change in the history of mathematics. The concepts “revolution” and “crisis” are rejected, while the concepts centering around the sociology of the scientific community are accepted for systematic use in historiography of mathematics. This is supplemented by the consideration of extra-mathematical influences. Finally the “laws” of Crowe are shown to be explainable with those concepts.

1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.

The purpose of this short paper is to present a new form of the so called early exercise premium for the American type derivatives. The decomposition we derived consists of the corresponding European derivative and a derivative with a stochastic maturity. In different particular cases we reach to the well known form for the American put option where the underlying asset is driven by a Brownian motion or a Levy process.

The paper consists of two parts. In the first part a short proof of the main theorem due to A. Aziz on the location of zeros of composite polynomials is given. In the second part some properties of a fixed length continuation of a polynomial are deduced.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.

Among other results, it is proved that if a sequence { μ n } \{ {\mu _n}\} of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all σ \sigma -compact sets or all open sets, then there exists a maximal open set U such that μ ˙ n ( U ) → 0 , { μ ˙ n } {\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\} being the associated submeasures.

In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations

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In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

The result indicated in the title of this paper (Ref. Zh. Mat. 1976, 7A225) is false. The error is corrected.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension ⩽ 1 \leqslant 1 , then finitely generated projective modules over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank > d > d over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A.