Free subgroups, the elements of which have ample cycles in their cycle decompositions, are enlarged to free subgroups of an infinite symmetric group by the specific construction of one more …
Free subgroups, the elements of which have ample cycles in their cycle decompositions, are enlarged to free subgroups of an infinite symmetric group by the specific construction of one more free generator.
Free subgroups, the elements of which have ample cycles in their cycle decompositions, are enlarged to free subgroups of an infinite symmetric group by the specific construction of one more …
Free subgroups, the elements of which have ample cycles in their cycle decompositions, are enlarged to free subgroups of an infinite symmetric group by the specific construction of one more free generator.
(1979). Comments and Complements. The American Mathematical Monthly: Vol. 86, No. 10, pp. 836-841.
(1979). Comments and Complements. The American Mathematical Monthly: Vol. 86, No. 10, pp. 836-841.
Let $\operatorname {Sym} M$ be the symmetric group of an infinite set $M$. What is the smallest subgroup of $\operatorname {Sym} M$ containing a given element if the subgroup is …
Let $\operatorname {Sym} M$ be the symmetric group of an infinite set $M$. What is the smallest subgroup of $\operatorname {Sym} M$ containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on $M$? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime $p$ and show that such components must be bounded abelian groups.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S y m upper M"> <mml:semantics> <mml:mrow> <mml:mi>Sym</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sym} M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the symmetric group …
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S y m upper M"> <mml:semantics> <mml:mrow> <mml:mi>Sym</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sym} M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the symmetric group of an infinite set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. What is the smallest subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S y m upper M"> <mml:semantics> <mml:mrow> <mml:mi>Sym</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sym} M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show that such components must be bounded abelian groups.
FRANKLIN I[_AIMO 0. Introduction In this pper, 11 rings re to be ssocitive, nd 11 mppings re to be written (t least in spirit) to the let.We shll consider certain …
FRANKLIN I[_AIMO 0. Introduction In this pper, 11 rings re to be ssocitive, nd 11 mppings re to be written (t least in spirit) to the let.We shll consider certain portions of Q(U), the group of quasi-regular (q.r.) elements of ring U, portions which, in prticulr, extend the Jcobson rdicl J(U) of U. The portion under con- sidertion will depend upon the wy in which U is regarded s n lgebr.Our principal device will be the introduction of different multiplication on U, n introduction brought bout by employing q.r.operator on U. We shll show (Theorem 1 that if suitable change of multiplication is introduced into the bimultipliction ring M(U), [5], on U, then this modified M(U) cn be injected into the bimultipliction ring of modified version of U which is ob- tined from U by making related change of multiplication theorem.By era- employing properly chosen commutative subring S(U) of M(U), it is pos- sible to turn U into n S( U)-lgebrm If U is n lgebr over commutative ring T, nd if U hs trivial bicenter [5], [3], then the mp a which effects the ction of T on U cn be fctored through S(U) (Theorem 2).Let ( U, a, T) be the set of 11 q.r.eleInents of U which re lso q.r. with respect to M1 the changes of multiplication on U which re induced by the members of Q(T).One finds that : >_ J(U).If Q(U) is central in U, nd if U is treated s n S( U)-lgebr, the resulting ., here clled (U), is (Theorem 3) subring of U, n lgebr over certain of S(U).If U is without divisors of zero, is commutative, is not rdicl ring, nd hs its underlying belin group U + irreducible s U-module, then 9(U) vnishes Theorem 4).If a ring extension is not too formidable, it is possible to obtain information about its Jacobson rdical.We select an uncomplicated extension V( U, a, T)of U by T (going bck to Dorroh [1]) which happens to be splitting.If T is an integral domMn, and if ( U, a, T) has been turned into an appropriate algebra, then the obvious K of V is a related extension of ( U, a, T) by certain of T (Theorem 5).If T is commutative and if the members of J(T) operate on U in such way that ru ru for all r e J(T) and all u e U, then J(V) is a related extension (Theorem 6) of J(U) by J(T).Finally, if U + is an irreducible [/'-module T commutative nd U a T-algebra), and if the members of J(T) do not ct as utomorphisms on U+, then (Theorem 7) J(V) reduces to the algebra direct sum of J(T) nd U.
If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A …
If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A , and if β ∈ End B , then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒ β be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.
Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let …
Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let H 1 be the set of all elements of G which have only a finite number of conjugates in G, let H 2 be the set of those elements of G, the conjugates of each of which lie in only a finite number of cosets of H 1 in G; and in this fashion define H 3 , H 4 , …. We shall show that the H i are strictly characteristic subgroups of G.
In this paper, we shall show that if is a nilpotent [ 5 ] group and if M, a positive integer, is a uniform bound on the number of conjugates …
In this paper, we shall show that if is a nilpotent [ 5 ] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → x n is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [ 1 ], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.
1. The content. The success of Stone [5] in representing Boolean algebras by fields of sets leads one to search for other representations. (See also [2, p. 159], [4].) Using …
1. The content. The success of Stone [5] in representing Boolean algebras by fields of sets leads one to search for other representations. (See also [2, p. 159], [4].) Using splitting endomorphisms on a type of Abelian group with an order relation, we are led to a faithful representation of a given Boolean algebra B. The group employed will be called a Boolean group. A special case of a Boolean group is a Boolean ring, and we shall show that 8 can be represented by a set of endomorphisms on its corresponding Boolean ring. If ? is complete the group on which it operates will be lower complete and upper conditionally complete. The Boolean groups are themselves special cases of a wider class of groups with an order relation, the vector ordered groups. Vector ordered groups generalize direct sums of Abelian groups, and their splitting endomorphisms form Boolean algebras. A simple example illustrates virtually the entire theory: Let ZJ be the ring of integers modulo 2, and let (M be the Cartesian product of some infinite collection of copies of 2*. 5 can be made into the strong direct sum of the 32 by introducing component-wise addition and multiplication. Then @ becomes a Boolean ring with a unity. It is also a Boolean algebra [2, p. 154] if one writes x ? y whenever each component of y ? (5 equals the corresponding component of x ? (M or is zero. Let 'P (() be the set of all endomorphisms V of the group structure of (M for which 12 _ v and V (x) ? x for every x P (M. If an order relation is defined in 'P in the obvious way, 'P becomes a Boolean algebra isomorphic to the Boolean algebra 5, so that the algebra 05 is repre
By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈 A ; F 〉 where A and F are nonvoid sets and every element of F …
By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈 A ; F 〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [ 1 ] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [ 6 ], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [ 8 ] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.
E. C. Posner ( 5 ) has shown that a ring R is primitive if and only if the corresponding matrix ring M n (R) is primitive. From this result …
E. C. Posner ( 5 ) has shown that a ring R is primitive if and only if the corresponding matrix ring M n (R) is primitive. From this result he is able to deduce that the primitive ideals in M n (R) are precisely those ideals of the form M n (P) , where P is a primitive ideal in R . This affords an alternative proof that the Jacobson radical of M n (R) is M n (J) , where J is the Jacobson radical of R . But Patterson ( 3, 4 ) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R . Finally the Jacobson radical of this ring is determined.
If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of …
If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of the location of S in G. Thus, for example, famed theorems of Frobenius and Burnside give criteria for the existence of a normal subgroup N of G such that G = NS and 1 = N ⋂ S, and hence in particular for the non-simplicity of G. To aid in locating S in G, and to facilitate exploitation of the transfer, we single out a descending chain of normal subgroups of S. Namely, we introduce the focal series of S in G by means of the recursive formulae
The object of this paper is to determine all cases in which two or more finitely generated abelian groups have the same holomorph (').Let G and G' be finitely generated …
The object of this paper is to determine all cases in which two or more finitely generated abelian groups have the same holomorph (').Let G and G' be finitely generated abelian groups and let H be the holomorph of G. Then it will be shown that H is the holomorph of G' if and only if G' is an invariant maximal-abelian subgroup of H isomorphic to G.All such subgroups of H are determined.There are at most four.If G does not contain any elements of order 2, or if G has at least three independent generators of infinite order, then G itself is the only such subgroup(2).1. Definitions.Let G be a group.If a and r are two automorphisms of G, then CTT is defined to be the automorphism such that (or)g =o,(rg) for all gGG.Under this composition the automorphisms of G form a group A. Consider the set H of all pairs (g, a), g(E.G, aG-<4.We define a composition in H by (a, o-)(b, t) = (aab, <jt).Under this composition the set H forms a group.If e is the identity of G and / is the identity of A, then (e, I) is the identity of II.Furthermore the inverse of (a, a) is (o-_1a_1, a-1).The group H is called the holomorph(3) of G.The mapping g->(g, I) gives an imbedding of G in the group H.We identify the element g in G with the element (g, I) in H. Then G is an invariant subgroup of H.If G is abelian, then it is a maximal-abelian subgroup of H, that is, an abelian subgroup not properly contained in any abelian subgroup of H.
In this paper, we shall show that if is a nilpotent [ 5 ] group and if M, a positive integer, is a uniform bound on the number of conjugates …
In this paper, we shall show that if is a nilpotent [ 5 ] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → x n is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [ 1 ], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.
SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie …
SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie algebras rather than topological groups, topological algebras, and so on.The method of study is also purely algebraic but is the replica of an algebraic process which has been widely used in topology, thus the words "toplogical methods" could be replaced by "algebraic methods suggested by algebraic topology."These purely algebraic theories do, however, have several applications in topology.The algebraic process borrowed from topology is the following.Consider a sequence of abelian groups {C q ) and homomorphisms 8,such that ôô = 0.In each group C q two subgroups are distinguished Z q = kernel of 8:C q -* 0 +1 , B q = image of ô:C*~l ~» C q
It is not the object of this address to introduce you to new theories or to tell of great discoveries.Quite on the contrary; I intend to speak of unsolved problems …
It is not the object of this address to introduce you to new theories or to tell of great discoveries.Quite on the contrary; I intend to speak of unsolved problems and of conjectures.In order to describe these, certain concepts will have to be discussed ; and for obtaining a proper perspective it will be necessary to mention a number of theorems, some of them new.The proofs of the latter will be relegated to appendices so that the hurried reader may skip them easily.The bibliography is in no sense supposed to be complete.We just selected convenient references for facts mentioned and beyond that just enough to be a basis for further reading.
Introduction.Let { E } denote the class of generalized euclidean spaces E (that is, E (I { E \ provided all finite dimensional subspaces of E are euclidean spaces).The problem …
Introduction.Let { E } denote the class of generalized euclidean spaces E (that is, E (I { E \ provided all finite dimensional subspaces of E are euclidean spaces).The problem of characterizing metrically the class { E \ with respect to the class { B } of all Banach spaces has been solved in many different ways.x Fre'chet's characteristic conditions [δ] was immediately weakened by Jordan and von Neumann [6] to ( * * ) llp t + p 2 II 2 + I I P I -P 2 H 2 = 2 C I | P l || 2 + ll P a | | 2 ) ( P l , p 2 G B ) .This relation has now become a kind of standard to which others repair by showing that it is implied by newly postulated conditions [3,4, 10], and it has been, apparently, the motivation of work in which it does not enter directly [7,9], Perhaps the best possible result in this direction, however, is due to Aronszajn [ 1 ] who assumed merely that ||(χ+y)/2|| = -0(||*||, ||y||, ||*-y|| (*,y GB),with φ unrestricted except for being nonnegative and φ(r,0,r)=r, r >_ 0.These conditions, and others like them, are all equivalent in a Banach space, for each is necessary and sufficient to insure the euclidean character of all subspaces.In a more general environment, however, this is not the case, and so the desirability of making a comparative study of such conditions in more general spaces is suggested.In this note the larger environment is furnished by the class { M } of complete, metrically convex and externally convex, This note deals exclusively with norrned linear spaces over the field of reals.
The setting of the problem The cohomology theory of rings, in the form recently introduced [15], will here be shown appropriate to the systematic treatment of the general extension problem …
The setting of the problem The cohomology theory of rings, in the form recently introduced [15], will here be shown appropriate to the systematic treatment of the general extension problem for rings.The treatment is parallel to the known theory [5] of the extensions of groups.If G is a normal subgroup of a group E, the assignment to each e e E of the operation of conjugation by e in G induces a homomorphism 0 of the quotient group Q E/G into the group of automorphism classes of G.The converse problem of group extensions therefore starts with the data: groups Q and G plus a homomorphism 0. These data are called a "Q-kernel" by Eilenberg-Mac Lane [5].On the center C of G the homomorphism 0 assigns to each element of Q a well defined automorphism of C; thus C may be regarded as a module over the integral group ring of Q, and the cohomology groups H'(Q; C) are then available.To 0 one assigns an element of H3(Q, C) as "obstruction"; there exists an extension E of G by Q which realizes 0 if and only if this obstruction is zero.When the obstruction is zero, the usual description of extensions by factor sets yields a one-one correspondence be- tween H2(Q, C) and the set of those equivalence classes of extensions of G by Q which realize 0. These results [5] yield an algebraic interpretation of the two-and three-dimensional cohomology groups and provide a refinement of the usual extension theory (normally attributed to Schreier [17], but actually initiated by HSlder [14]) in which the map 0 and the factor sets are all treated together, in a somewhat indigestible lump.There are subsequent and parallel studies for the extensions of associative algebras (Hochschild [11]) and of Lie algebras (Hochschild [12, 13]).In both cases, the algebras are taken over a field and hence have the additive structure of a vector space over that field.Consequently the extension problem for the additive structure involved is trivial, and only the multiplicative structure is substantially involved in the cohomology theory.The new cohomology for rings to be used here has as its object precisely the simultaneous treatment of additive and multiplicative structures.For example, Everett [10] has de- veloped the analogue of the Schreier extension theory for the case of rings,
Here R is derived from any representation, G = F/R, with F a free group, and Hom(R, K) is the group of all homomorphisms of R into K. The group …
Here R is derived from any representation, G = F/R, with F a free group, and Hom(R, K) is the group of all homomorphisms of R into K. The group Hn+2(G, K) is defined in terms of an arbitrarily given system of right-operators G on K; the definition of Hh(G, Hom(R, K)) then requires specification of a system of operators G on Hom(R, K). For this, let it first be noted that the given right-operators G on K induce right-operators G on Hom(R, K):
Background and summary.The determination of "optimum" solutions of systems of linear inequalities is assuming increasing importance as a tool for mathematical analysis of certain problems in economics, logistics, and the …
Background and summary.The determination of "optimum" solutions of systems of linear inequalities is assuming increasing importance as a tool for mathematical analysis of certain problems in economics, logistics, and the theory of games [l;5] The solution of large systems is becoming more feasible with the advent of high-speed digital computers; however, as in the related problem of inversion of large matrices, there are difficulties which remain to be resolved connected with rank.This paper develops a theory for avoiding assumptions regarding rank of underlying matrices which has import in applications where little or nothing is known about the rank of the linear inequality system under consideration.The simplex procedure is a finite iterative method which deals with problems involving linear inequalities in a manner closely analogous to the solution of linear equations or matrix inversion by Gaussian elimination.Like the latter it is useful in proving fundamental theorems on linear algebraic systems.For example, one form of the fundamental duality theorem associated with linear inequalities is easily shown as a direct consequence of solving the main problem.Other forms can be obtained by trivial manipulations (for a fuller discussion of these interrelations, see [13]); in particular, the duality theorem [8; 10; 11; 12] leads directly to the Minmax theorem for zero-sum two-person games [id] and to a computational method (pointed out informally by Herman Rubin and demonstrated by Robert Dorfman [la]) which simultaneously yields optimal strategies for both players and also the value of the game.The term "simplex" evolved from an early geometrical version in which (like in game theory) the variables were nonnegative and summed to unity.In that formulation a class of "solutions" was considered which lay in a simplex.The generalized method given here was outlined earlier by the first of the authors (Dantzig) in a short footnote [lb] and then discussed somewhat more fully at the Symposium of Linear Inequalities in 1951.Its purpose, as we have
The object of this paper is to determine all cases in which two or more finitely generated abelian groups have the same holomorph (').Let G and G' be finitely generated …
The object of this paper is to determine all cases in which two or more finitely generated abelian groups have the same holomorph (').Let G and G' be finitely generated abelian groups and let H be the holomorph of G. Then it will be shown that H is the holomorph of G' if and only if G' is an invariant maximal-abelian subgroup of H isomorphic to G.All such subgroups of H are determined.There are at most four.If G does not contain any elements of order 2, or if G has at least three independent generators of infinite order, then G itself is the only such subgroup(2).1. Definitions.Let G be a group.If a and r are two automorphisms of G, then CTT is defined to be the automorphism such that (or)g =o,(rg) for all gGG.Under this composition the automorphisms of G form a group A. Consider the set H of all pairs (g, a), g(E.G, aG-<4.We define a composition in H by (a, o-)(b, t) = (aab, <jt).Under this composition the set H forms a group.If e is the identity of G and / is the identity of A, then (e, I) is the identity of II.Furthermore the inverse of (a, a) is (o-_1a_1, a-1).The group H is called the holomorph(3) of G.The mapping g->(g, I) gives an imbedding of G in the group H.We identify the element g in G with the element (g, I) in H. Then G is an invariant subgroup of H.If G is abelian, then it is a maximal-abelian subgroup of H, that is, an abelian subgroup not properly contained in any abelian subgroup of H.
ON GROUP EXTENSIONS WITH OPERATORS Get access J. H. C. WHITEHEAD J. H. C. WHITEHEAD Oxford Search for other works by this author on: Oxford Academic Google Scholar The Quarterly …
ON GROUP EXTENSIONS WITH OPERATORS Get access J. H. C. WHITEHEAD J. H. C. WHITEHEAD Oxford Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 1, Issue 1, 1950, Pages 219–228, https://doi.org/10.1093/qmath/1.1.219 Published: 01 January 1950 Article history Received: 20 July 1949 Published: 01 January 1950
1. Introduction.In a previous paper [l], the notion of subharmonic functions was generalized in a manner corresponding to Beckenbach's [2] generalization of convex functions^ This generalization was accomplished by replacing …
1. Introduction.In a previous paper [l], the notion of subharmonic functions was generalized in a manner corresponding to Beckenbach's [2] generalization of convex functions^ This generalization was accomplished by replacing the dominating family of harmonic functions by a more general family of functions.In [l] the discussion was restricted to continuous subfunctions.In the present paper we shall give some further properties of the dominating functions and extend the definition of subfunctions to permit upper semi continuous subfunctions.We shall then show that results of J. W. Green [3] on approximately subharmonic functions extend to our subfunctions. {F [-functions and sub-{ F} functions.Let D be a given plane domain and let { γ \ be a given family of contours bounding subdomains Γ of D such that Γ = y + Γ C D where Γ indicates the closure of Γ.We assume that { γ \ contains all circles of radii less than a fixed number which lie, together with their interiors, in D. We shall use the Greek letter K to represent a circle of ί γ \ and K its interior.We shall use single small Roman letters to represent points in the plane.Let there be given a family of functions \Fix)} which we shall call { F }-functions satisfying the following postulates.POSTULATE 1.For any γ£ {γ} and any continuous boundary value function hix) on γ, there is a uniquePOSTULATE 2. If hι(x) and h 2 ix) are continuous on y and if hι(x)h 2 (x) < M on y, M > 0, then F{x;h ι ;γ)-F(x;h 2 ;γ) <M
Σ, \n M l I _oo un exist. Similar results are valid for the circumstance where does not exist. Then we deal only with the case where an ~ 0 …
Σ, \n M l I _oo un exist. Similar results are valid for the circumstance where does not exist. Then we deal only with the case where an ~ 0 for n . t >_ 0 a strongly continuous semi-group of operators acting either on the space of bounded functions (M) or integrable functions ( L ) with | | Γ ( ί ) . | | < 1. Let A denote the infinitesimal generator of Tit) and let df it) define a nonlatt ice distribution with finite first moment on [0, col. If u belongs to iU) we consider u ( t ) (°° T ( t ) d f ( t ) ] u = v L J o J where v belongs to ( L ). The linear operator ί°° Tit)dfit) Jo is well defined either over iM) or ( L ) into itself. Put rit) = 1 f it), then r G L and the Fourier transform of r never vanishes . Since r is monotonic decreasing and in L it can be easi ly shown that