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In this paper we study a class of sub-spaces of loop spaces which have appeared in the calculus of variations. Generalizing a result of Smale, we show that the space of loops tangent to a distribution satisfying Hδrmander's condition is weakly homotopic to the space of all loops. If the distribution is fat, we resolve the end point map from the space of horizontal paths. This resolution has two applications: (1) the proof that the cut-locus on an analytic fat Carnot-Caratheodory manifold is sub-analytic; (2) a study of the singularity of the horizontal loop space. At the end we study the geometry of left-invarian t Carnot-Caratheodory metrics on fact nilpotent groups.

This paper considers the delayed age replacement policy, in which the lifetimes of all units are assumed to be uncertain variables, and the lifetime of the first unit has an uncertainty distribution which is different from the others. A delayed age replacement model which is concerned with finding the optimal replacement time to minimize the expected cost is developed. In the policy, the optimal replacement time is irrelevant to the uncertain distribution of lifetime of the first unit over the infinite time span.

In a former papert the author communicated a number of properties of every real function f (x) ; these were stated in terms of the successive saltus functions associated with a given function. The present paper makes no use of the saltus functions, and the new properties are direct qualifications of f (x). Since f (x) is entirely unrestricted, except, of course, that it is defined t and therefore finite for every real x-these qualifications are consequences of nothing else than that f (x) is a function. A new light is thus thrown upon the nature of a function. The new properties are of two types, descriptive and metric; the former are concerned with density, and the latter with measure (Lebesgue). For the sake of greater concreteness of exposition, we shall discuss, for the most part, planar sets and real functions of two real variables.

Two point sets are mutually separated if they are mutually exclusive and neither of them contains a limit point of the other. A point set M is said to be not connected or connected according as M is or is not the sum of two non-vacuous mutually separated point sets. A subset X of a connected point set M is said to be a cutting of M, or is said to cut M, provided the set of points M—X is not connected; X is called an irreducible cutting of M provided X cuts M but no proper subset of X cuts M. If A and B are subsets of a connected point set Af, the subset X of M is said to separate A and B in My or to cut M between A and B, provided that M~X is the sum of two mutually separated sets Ma(X) and Mb(X) containing A and B respectively. In this paper the ordinary notation of point set theory will be employed, for example, X = X + X ' , whereX is the set of all limit points of the set X, K c H means that the set K is a subset of the set H, K H means the set of points common to K and Hy F(R) denotes the boundary of the set i?, etc. In addition, if X is a cutting of a connected point set M then, unless otherwise stated, the equation M—X — Ma(X) + Mb(X) is to be interpreted as meaning that M—X is the sum of two mutually separated sets Ma(X) and Mb(X) which contain the sets A and B respectively. 1. Cuttings of Connected Sets in General. The point sets considered in this section are assumed to lie in a separable metric space which will be denoted by 5. It is apparent from the proofs of some of the theorems, however, that they hold in even more general space.

In this paper is presented a partial list of formulas giving the orders of the ruled hypersurfaces and ruled surfaces whose rulings are incident with given curves and surfaces in 4-space. If the number of incidences is equivalent to six simple conditions, the number of lines having these incidences is finite. The formulas herein presented are analogous to those giving the orders of the surfaces whose rulings are lines satisfying three conditions and the number of lines satisfying four conditions in 3-space.* Some of these formulas are obvious, others require proof. Those that are not herein included are left out for future consideration. We shall let h be the number of apparent double points, p the deficiency, of a given curve C; H the number of apparent double points, P the deficiency, of a 3-space section c of a given surface F; and t the number of apparent triple points of F or the number of lines through a given point meeting F three times, f First consider the hypersurface F" whose lines satisfy four conditions. For a line to meet a given curve once is equivalent to two conditions. Hence the °o lines cutting across two given curves Cs C* form a hypersurface whose order is obviously

In his fundamental paper on a posteriori probability,* Bayes considered a certain event M having an unknown probability p of its occurring in a single trial. In deriving his a posteriori formula he assumed that all values of p are equally likely, and he recommended this assumption for similar problems in which nothing is known concerning p. In the corollary to proposition 8 he derives the value r/n\ 1

Let z=f(x, y) be an arbitrary surface S, in the sense that f(x, y) is an arbitrary one-valued real function of the real variables x and y. By the grade of a segment joining two points A and B, we understand the absolute value of the tangent of the angle which AB makes with the :ry-plane. The point A = (£, rj, f ) of the surface S is said to be of bounded grade—or S is said to be of bounded grade at the point (£, rj)—if the grade of AB is bounded for all B = (x, y, z) of 5 at a sufficiently small "horizontal" distance [{x — %) + (y — rj)] from A. HA does not satisfy this condition, S is said to be of unbounded grade at (£, rj). It is the object of the present paper to prove the following theorem, which identifies the aggregate—for the totality of arbitrary surfaces—of sets of points of unbounded grade with the aggregate of sets of type Go.f

on the basis of the general analytic properties of the kernel K (x, ~) such as im tegrability, continuity, differentiability, analyticity and the like? The l i tera ture where this and analogous questions are t rea ted is v e r y considerable [HELLI~G~n-To~eLITZ, I]. ~ A relatively small par t of this l i terature, however, has points of contact with the present paper, the discussion of the majori ty of papers published on the subject being based on various special properties of the kernels. I t is assumed frequent ly tha t the kernel belongs to some special class of functions, or tha t it coincides with the Green 's funct ion of a different ial or integro-differential boundary value problem. Problems of this sort will be excluded f rom the scope of our paper a l though they are in teres t ing f rom a theoret ical point of view and impor tan t for the applications.

Presented to the Society April 14, 1939, under the title A characterization of plane isothermic maps; received by the editors March 4, 1940. (1) Some of the results of this paper have been summarized in the authors' note of the same title in the Proceedings of National Academy of Sciences, vol. 25 (1939), pp. 92-97. (2) It is to be understood throughout this paper that the domains under consideration are finite. 354

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

The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently.

In the development of lattice theory considerable work has been devoted to the study of the arithmetical properties of modular and distributive lattices. Indeed most of the decomposition theorems of abstract algebra have been extended to these more general domains. Nevertheless, there are lattices with very simple arithmetical properties which come under neither of these classifications. For example, the lattices with unique irreducible decompositions, which were studied by the author in a previous paper [3]1 satisfy the Birkhoff condition2 which is even less restrictive than the modular axiom. Furthermore, there are important algebraic systems which give rise to non-modular, Birkhoff lattices. Thus, since every exchange lattice (Mac Lane [4]) is a Birkhoff lattice, the systems which satisfy Mac Lane’s exchange axiom form lattices of the type in question. In this paper we shall study the arithmetical structure of general Birkhoff lattices and in particular determine necessary and sufficient conditions that certain important arithmetical properties hold.

Several integral formulas referring to convex plane curves, notable for their great generality, were obtained by W. Crofton in 1868 and successive years from the theory of geometrical probability [6], [7], [8], [9], and [10]. A direct and rigorous exposition of Crofton's principal results, adding some (new formulas, was made in 1912 by H. Lebesgue [12]. Another systematic exposition of Crofton's most interesting formulas, together with the generalization of many of them to space, is found in the two volumes on integral geometry by Blaschke [2]. The purpose of the present paper is to give a generalization of Crofton's formulas to the surface of the sphere. This is what we do in part I. We find further integral formulas on the sphere (for instance, (16), (17), (20), (21)) which have no equivalent in the plane. Other formulas, if we consider the plane as the limit of a sphere whose radius increases indefinitely, give integral formulas referring to plane convex curves (e. g., (34), (35)) which we think are new. In part II, with simple methods of integral geometry [2], we obtain three inequalities referring to spherical curves. Inequality (38) is the generalization to the sphere of an inequality that Hornich [11] obtained for plane curves. (52) and (58) contain the classical isoperimetric inequality on the sphere. Finally, inequality (61) gives a superior limitation for the "isoperimetric deficit" of convex curves on the sphere https://doi.org/10.1215/S0012-7094-42-00949-9

In an earlier paper we considered homogeneous polynomials ƒ and g whose degrees are relatively prime, and solved the Diophantine equation f(x)=g(y). These results are generalized in the present paper. The solutions are given in terms of arbitrary parameters and if the parameters are integral the solutions are also. We begin our discussion with the hypothesis that the functions J\X) J\Xll, * * j X\m, , Xnif , Xnm), g\X) = g\Xn, *, X\m, , Xni, • • • i Xnm) are polynomials with integral coefficients, homogeneous in each set of variables Xk\Xk2 * Xkm ; ƒ being of degree ak e 0, g being of degree /3k^0 in the sets XkiXk2 • • * xkm and dk = ak—Pk. We suppose further that integers Xt, jUt^O exist such that 2

We use the expression Integral geometry in the sense given it by Blaschke in “Vorlesungen über Integrtüçeomelrie”. In a previous paper (“Integral formulas in Croflon style on the sphere and some inequalities referring to spherical curve” (1942), Duke Mathematical Journal, vol. 9, pp. 707) we generalized to the sphere many formulas of plane integral geometry and at the same time applied these to the demonstration of certain inequalities referring to spherical curves. The present paper considers analogous qüestions for surfaces of constant negative curvature and consequently for hyperbolic geometry https://doi.org/10.1215/S0012-7094-43-01064-6

This paper consists of three parts. In the first part we complete one of the deepest theorems due to Littlewood and Paley [3, 4].1 Our theorem was already proved by Marcinkiewicz and Zygmund [4]. But the proof given here is very simple and direct. In the second part a strong summability theorem is proved. It is the completion of our former theorem [5]. We use the method due to Zygmund [8] for the proof. Finally we prove a strong summability theorem concerning lacunary sequences of partial sums (Zalcwasser [6]). 1. Letf(6) be an integrable function with period 2ir and its Fourier series be f(I)'f= (an cos nO+b,, sin nO), assuming ao=O for the sake of simplicity. If we put z=pei0 and

This paper is primarily concerned with fibre mappings into an absolute neighborhood retract. Theorem 3 is a converse of the covering homotopy theorem; it characterizes fibre mappings (into a compact ANR) as mappings for which the covering homotopy theorem holds. Theorem 4 is Borsuk's fibre theorem; the proof which I present here is new. It seems to me that this theorem is a promising tool in function-space theory. Also I think that it furnishes conclusive justification for the generality of the Hurewicz-Steenrod definition of a fibre space. In fact, a fibre space of the type constructed by Borsuk's theorem almost never has a compact base space and almost never has its fibres of the same topological type. The common denominator of the proofs of Theorems 3 and 4 is a property which I call local equiconnectivity. Local equiconnectivity is a strengthened form of local contractibility and a weakened form of the absolute neighborhood retract property (Theorems 1 and 2). Definitions and notations are those of FS. I. Let A be the diagonal subset ]C&e#(^ &) °f 5 X 5 . I shall call the space B locally equiconnected (or, to be specific, (U, F)-equiconnected) if there are neighborhoods U and F of A and a homotopy X in B between the two projections of U which does not move the points of A and which is uniform with respect to V. Precisely: (1) X*(eo, fa) is defined for all (&0, fa) £ Uy (2) Xo(6o, &i)=*o

1. S. Ramanujan, Congruence properties of partitions. Math. Zeit. vol. 9 (1921) pp. 147-153. Reprinted in Collected papers of Srinivasa Ramanujan, Cambridge, 1927, pp. 232-238. 2. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. vol. 22 (1916) pp. 159-184. Reprinted in Collected papers of Srinivasa Ramanujan, pp. 136-162. 3. H. S. Zuckerman, Identities analogous to Ramanujan's identities involving the partition f unction, Duke Math. J. vol. 5 (1939) p. 89, equation (1.15).

Introduction. Comparatively few of the plane sextic curves are known. None of the papers written about them gives a complete treatment or a classification of these curves. The sextics appear more or less isolated in the different treatises. Apparently the methods employed produce only a very limited number of sextics. The projective method of deriving higher plane curves described by Dr. H. P. Pett i t in his Projective description of some higher plane curves allows a more systematical study of sextics and especially a direct construction point by point. I t is the purpose of this paper to give a projective description of some of the sextics that can be derived by this method, if conies are the base curves. The number of types of these curves which could be generated by this method is here limited to 354 by giving the triangle of reference special positions with respect to the base conies.

Here sn x is an abbreviation for sin (=rx/2). This paper gives other conditions for the validity of these identities. The previous conditions permitted qk to have various types of discontinuity. The present paper is concerned with smooth functions; however, the growth at 0 and oo is permitted to be greater than before. Theorems 1, 2, and 3 of the previous paper together with Theorems 2 and 3 of this paper form a fairly complete elementary theory of these identities. The proofs given here do not depend on the previous paper. The results of this paper hinge on the possibility of defining the Fourier sine transform for functions which do not vanish at infinity. Theorem 1 below shows that this is possible merely by employing summability. It is to be noted that Theorem 1 is not true, as it stands, for the cosine transform. For example, the cosine transform of any constant evaluated by such a definition would vanish. Hence the inversion formula could not apply. A theory of "generalized Fourier

Eilenberg and MacLane's paper 'General theory of natural equivalences' defines CATEGORY, FUNCTOR, and NATURAL TRANSFORMATION, thereby giving birth to category theory ortheory of naturality.

The purpose of the present paper is to give a new geometrical construction for the following affine diferencial invariants of a space curve: geometrical elements associated to an ordinary point of a space curve; geometrical characterization of the affine fundamental trihedral of a space curve at an ordinary point; geometrical interpretation of the affine curvature and geometrical interpretation of the affine torsion

In this paper, we study a fractional order COVID-19 model using different techniques and analysis. The sumudu transform is applied with the environment as a route of infection in society to the proposed fractional-order model. It plays a significant part in issues of medical and engineering as well as its analysis in community. Initially, we present the model formation and its sensitivity analysis. Further, the uniqueness and stability analysis has been made for COVID-19 also used the iterative scheme with fixed point theorem. After using the Adams-Moulton rule to support our results, we examine some results using the fractal fractional operator. Demonstrate the numerical simulations to prove the efficiency of the given techniques. We illustrate the visual depiction of sensitive parameters that reveal the decrease and triumph over the virus within the network. We can reduce the virus by the appropriate recognition of the individuals in community of Saudi Arabia.

In the classical Lie theory it is shown how to construct a differential equation invariant under a given group, and how to solve an equation when a group leaving the equation invariant is known. However, little is said about the problem of determining the group for a given differential equation, which is by far the most interesting problem. In the present paper, necessary and sufficient conditions for the existence of an infinitesimal contact transformation leaving a given equation invariant are determined along with the general form of the characteristic function of the group. It will also be shown how to reduce, by a proper change of variables, the infinitesimal contact transformation to a point transformation. This enables one to solve the transformed differential equation by Lie's methods. Passing back to the original variables, a new differential equation is obtained which combined with the original equation gives its solution in parametric form. Let

Such equations have an interesting history. In art. 358 of the Disquisitiones [1, a], Gauss determines the Gaussian sums (the so-called cyclotomic “periods”) of order 3, for a prime of the form p = 3n + 1, and at the same time obtains the number of solutions for all congruences ax− by ≡ 1 (mod p). He draws attention himself to the elegance of his method, as well as to its wide scope; it is only much later, however, viz. in his memoir on biquadratic residues [1, b], that he gave in print another application of the same method; there he treats the next higher case, finds the number of solutions of any congruence ax − by ≡ 1 (mod p), for a prime of the form p = 4n + 1, and derives from this the biquadratic character of 2 mod p, this being the ostensible purpose of the whole highly ingenious and intricate investigation. As an incidental consequence (“coronodis loco,” p. 89), he also gives in substance the number of solutions of any congruence y ≡ ax − b (mod p); this result includes as a special case the theorem stated as a conjecture (“observatio per inductionem facta gravissima”) in the last entry of his Tagebuch [1, c]; and it implies the truth of what has lately become known as the Riemann hypothesis, for the function–field defined by that equation over the prime field of p elements. Gauss’ procedure is wholly elementary, and makes no use of the Gaussian sums, since it is rather his purpose to apply it to the determination of such sums. If one tries to apply it to more general cases, however, calculations soon become unwieldy, and one realizes the necessity of inverting it by taking Gaussian sums as a starting point. The means for doing so were supplied, as early as 1827, by Jacobi, in a letter to Gauss [2, a] (cf. [2, b]). But Lebesgue, who in 1837 devoted two papers [3, a,b] to the case n0 = · · · = nr of equation (1), did not succeed in bringing out any striking result. The whole problem seems then to have been forgotten until Hardy and Littlewood found it necessary to obtain formulas for the number of solutions of the congruence ∑ i x n i ≡ b (mod p) in their work on the singular series for Waring’s problem [4]; they did so by means of Gaussian sums. More recently, Davenport and Hasse [5] have applied the same method to the case r = 2, b = 0 of equation (1) as well as to other similar equations; however, as they were chiefly concerned with other aspects of the problem, and in particular with its relation to the Riemann

On the euclidean plane one-parametric motion is in general a roulett motion, exceptions being a translation at each instant and a rotation with a fixed center; here we mean by a roulett motion a motion in which a certain curve rolls on another fixed curve without slipping. In this paper we extend this fact to the case of Klein spaces and investigate in detail especially the cases of the euclidean space and the projective space.