Not Identically Equal to U(X, -Y) in R, and Downloaded by Guest on September 29, 2021 268 MA THEMA TICS: J

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* John Simon Guggenheim Memorial Fellow. l Birkhoff, G., "Lattice Theory," Am. Math. Soc. Colloq. Pub., 25, 48 (1940). 2 For the case of abelian groups with-operators from a group Q, this property is used in Eilenberg, S., and MacLane, S., "Homology Theory of Spaces with Operators II," forthcoming in Trans. Am. Math. Soc. 3Eilenberg, S., and Steenrod, N., PRoC. NAT. AcAD. Sci., 31, 117-120 (1945). (The writer has also profited by reading further unpublished work of these authors on this subject.) 4 Pontrjagin, L., Topological Groups, Princeton, 1939. Weil, A., L'Integration dans les groupes topologiques et ses applications, Paris, 1938. aVeblen, O., and Young, J. W., Projective Geometry, Boston, 1910. * Eilenberg, S., and MacLane, S., PRoc. NAT. ACAD. SCm., 28, 537-543 (1942); Trans. Am. Math. Soc., 58, 231-294 (1945). 7 The formulation with bicategories does not yet indicate the duality between center and factor commutator groups, and similar dual concepts of verbal and marginal sub- groups; Hall, P., J. f. d. reine und angew. Math., 182, 156-157 (1940). 8 A careful treatment, emphasizing the equality approach, appears in the unjustly neglected book by Haupt, O., Einfahrung in die Algebra, Leipzig, 1929.

METHODS OF SYMMETRY AND CRITICAL POINTS OF HARMONIC FUNCTIONS By J. L. WALSH DEPARTMENT oF MATHEMATICS, HARvARW UNIVERSITY Communicated March 19, 1948 The most powerful method known for the study of the location of the critical points of harmonic functions is the expression of the gradient of a given harmonic function as the force in a field due to a suitable distribution of matter.' Nevertheless simpler methods involving less machinery, based on topological considerations involving symmetry, yield some sur- prisingly deep results, as we wish to indicate in the present note. Our principal result is THEOREM 1. Denote by I1, and II2 the open upper and lower half-planes respectively. Let u(x, y) be harmonic in a region R cut by the axis of reals, and let the relktion u(x, y) > u(x, -y) for (x, y) in II, (1) hold whenever both (x, y) and (x, -y) lie in R. Then u(x, y) has no critical* point in R on the axis of reals. Alternate sufficient conditions that u(x, y) have no critical point in R on the axis of reals are that R be bounded by a Jordan configuration B, that u(x, y) be harmonic and bounded in R, continuous in R + B except perhaps for afinite number ofpoints, u(x, y) not identically equal to u(x, -y) in R, and Downloaded by guest on September 29, 2021 268 MA THEMA TICS: J. L. WALSH PROC. N. A. S.

1. R symmetric in the axis of reals, with u(x, y) > 0 on B HI and u(x, y) = O on B*112. 2. R symmetric in the axis of reals, with u(x, y) > 0 on B .l1 and u(x, y) < 0 on B*112. 3. R symmetric in the axis of reals, with u(x, y) _ u(x, -y) (2) at every point (x, y) of B*11. 4. (x, y) lies in R whenever (x, -y) lies in R1H2; u(x, y) _ 0 on B 111 and u(x,y) = OonB*II2. 5. R' is a region whose boundary. B' is symmetric in the axis of reals, and R is a subregion of R' whose boundary in lk is denoted by B' 11I + B,(k = 1, 2), where Bk is disjoint from B'; we have (2) at every point of B'- i1; we denote by Uk(X, y) thefunction hatmonic and bounded in R'"1, defined by thc boundary values u(x, y) on the axis of reals and on B'- Ilk; we suppose u(x, y) > u1(x, y) on BI, u(x, y) . u2(x, y) on B2. 6. R is cut by the axis of reals and u(x, y) has boundary values unity on B *1],, zero on B*1112. 7. R is cut by the axis of reals; the boundary values of u(x, y) on B - 1, are not less than I u b [u(x, 0) in R]; the boundary values on B .H2 are not greater than g I b [u(x, 0) in R]. By a Jordan configuration we mean a set composed of a finite number of Jordan arcs. As a matter of convention, segments of the axis of reals belonging to B (or Bk) are considered to belong to both B. i1 and B 112 (or Bk* ilk), and boundary values may of course be different on B 1-il and B*112. To prove the main part of Theorem 1 we set U(x, y) = u(x, y) -u(x, -y), whence in III we have U(x, y) > 0 and in 112 we have U(x, y) <0; on the axis of reals U(x, 0) = 0, 1U(x, 0)/6x = 0. In the neighborhood of a critical point (xo, yo) of U(x, y) of order k the locus U(x, y) = U(xo, yo) consists of k + 1 analytic Jordan arcs intersecting at (xo, yo) at successive angles of 7r/(k + 1); no such arc of the locus U(x, y) = 0 can lie in R Hil or R- 12, so no critical point of U(x, y) lies on the axis of reals in R. On the axis we have aU/ax = 0, whence aU/ly s 0, and aiU/ly > 0 follows from the behavior of U(x, y) in II, and 112. We also have I.U(x, 0)/dy = 2 bu(x, O)/by, so 1u(x, O)/by > 0 (3) follows in R. Both parts 1 and 2 are contained in part 3, so we proceed to establish part 3. Again we set U(x, y) = u(x, y) -. u(x, -y), so at every point of B* II1 we have U(x, y) _ 0; at every point of B H2 we have U(x, y) = -U(x, -y) . 0. The function U(x, y) does not vanish identically in R Downloaded by guest on September 29, 2021 VoL. 34, 1948 MATHEMATICS: J. L. WALSH 269

but vanishes on the axis of reals, so we have U(x, y) > 0 in R H1, U(x, y) < 0 in R- I12. From the main part of Theorem 1 we now have at any point (x, 0) of R: 6U(x, 0)/by = 2bu(x, 0)/by > 0, so (3) followsinR. In part 4, let B1 denote the reflection in the axis of reals of B *112. On both B1 and B* 1 we have u(x, y) 2 0, and on B* I2 we have u(x, y) = 0. Each point of R on the axis of reals lies in a subregion of R symmetric in the axis bounded wholly by points of B1 and B-. 12, so the conclusion follows from part 1. In part 4 examples show that the condition u(x, y) = 0 on B - 112 cannot be replaced by the weaker condition u(x, y) . 0. To prove part 5 we note that on the boundary of R' II we have u(x, y) > u(x, -y), both on B'.H1 and on the axis of reals, whence ul(x, y) 2 u2(x, -y) in R' -1H. From the relations u(x, y) > ul(x, y) on B1 and u(x, y) < u2(x, y) on B2 we deduce u(x, y) 2 ul(x, y) in R-HI and u(x, y) < U2(X, y) in R112. In sum we have for (x, y) in R.lI: u(x, y) 2 u1(x, y) > u2(x, -y) _ u(x, -y), provided (x, -y) lies in R1II2. Throughout a suitable neighborhood of any point of R on the axis of reals we have (2) satisfied in II1, so the conclusion follows from part 3. Part 6 is contained in part 7, so we proceed to prove the latter' We set bi = I u b[u(x, 0) in R], b2 = g I b[u(x, 0) in R], so we have u(x, y) _ bi on B * H1, u(x, y) _ b2 in R-1]I, u(x, y) _ b2 on B*II2, u(x, y) < b1 in R*112- We set U(x, y) = u(x, y) - u(x, -y). If U(x, y) is defined, at a point of B * H, we have U(x, y) _ 0 and at a point of the reflection of B .1u* we have U(x, y) _ 0; at a point of B * 12 we have U(x, y) < 0, and at a point of the reflection of B 112 we have U(x, y) _ 0. Every point (x, 0) in R lies in some subregion R' of R symmetric in the axis of reals bounded wholly by points of B I1, BI112, and their reflections; on the boundary of R' in II, we have U(x, y) 2 0 and on the boundary of R' in II2 we have U(x, y) < 0. It is to be noted that the relation U(x, y) = 0 in one region R' would imply that relation in every region R', and throughout R. The conclusion now follows as in the proof of part 3. In each case under Theorem 1 we have established (3) throughout R; consequently if each function Uk(x, y) satisfies the conditions of either the main part of Theorem 1 or one of the supplementary parts, with ref- erence to a region Rk, it follows that the function XkUk(x,k Y), Xk >0, has no critical point on the axis of reals in R1 - R2 . .. R.. Indeed we need

not require Uk(x, y) - Uk(x, -y) if we have U(x, y) - U(x, -y). As an illustration of the power of Theorem 1, we prove in detail THEOREM 2. Let R be the interior of the Jordan curve C, and let R be provided with a NE (non-euclidean) geometry by means of a conformal map onto the interior of a circle. Let the function u(x, y) be harmonic but not identically zero in R, continuous on R + C, non-negative on an arc Co of C and zero on C-CO. Then all critical points of u(x, y) in R lie in the subregion of R bounded by Co and the NE linejoining the end-points of Co. Downloaded by guest on September 29, 2021 270 MATHEMATICS: J. L. WALSH PROC. N. A. S.

Let (xo, yo) be an arbitrary point of R not in the subregion of R mentioned. Map R onto the interior of the unit circle IwI = 1 so that (xo, yo) is trans- formed into the origin, and rotate the plane so that the image of Co lies in the closed upper half-plane. The conclusion now follows from part 1. Theorem 2 is not new,2 and is presented here to show both the range of Theorem 1 and the kind of result that can be proved by its use. For an arbitrary Jordan region R, an arbitrary NE line behaves conformally like an axis of symmetry, for it is the image of such an axis under a conformal map of the interior of a circle onto R. Thus each part of Theorem 1 -is of significance in connection with such a map, and a number of the results obtained by mapping are new. To show the connection of Theorem 1 with fields of force we give an independent proof of THEOREM 3. Let the region R be the interior of a Jordan curve C, and let B be a Jordan configuration in R which together with C bounds a region R'. Let the function u(x, y) be harmonic in R', superharmonic in R, continuous on R' + C and zero on C. Then all critical points of u(x, y) in R' lie in the smallest NE convex region of R containing B. Choose C as the unit circle, and choose an arbitrary NE line wholly in R' as the axis of reals, with an arbitrary preassigned point of the NE line at the origin and with B in the upper half-plane. It follows from a classical result due to F. Riesz (1930) on superharmonic functions that u(x, y) can be written in R as u(x, y) = fRG(z, t, R)dyu + h(x, y), d >-O0 (4) where G(z, t, R) is Green's function for R with pole in t and h(x, y) is harmonic in R; since u(x, y) is harmonic in R', the function d,u vanishes in R', and the integral can be taken over R - R' and represents a function harmonic in R'. Of course Green's function can be extended harmonically across C by reflection; at corresponding points the values are the negatives of each other. Consequently the integral converges not merely in the neighborhood of C in R but also in the neighborhood of C exterior to R; the function represented by the integral is continuous and vanishes on C. It follows from (4) that h(x, y) is continuous on C and vanishes there, hence vanishes identically. Study of the field of force' defined as the conjugate of the function f'(z), where f(z) is an analytic function whose real part is u(x, y), now completes the proof. Theorem 3 is contained in part 5 of Theorem 1; the latter requires (notation of Theorem 1) u(x, y) _ ul(x, y) on B,, which is closely related to the requirement of Theorem 3 that u(x, y) be superharmonic. Riesz's results are of still further significance in connection with part 5 of Theorem 1, where we use both positive and negative mass ,. We postpone for another occasion the discussion and formulation in Downloaded by guest on September 29, 2021 VOL. 34, 1948 PA THOLOGY: E. R. LONG 271

detail of further applications of Theorem 1, but mention that symmetry in a point may be used in a manner similar to our present use of symmetry in an axis. Moreover, still other methods are available for the study of critical points. For instance, topological methods, similar to our proof of the main part of Theorem 1 and involving the consideration of level curves, yield the following two results: THioRiz 4. Let the function u(x, y) be harmonic but not identically zero in the region R whose boundary is the Jordan curve J, continuous on R + J. Let u(x, y) be respectivdy non-negative and non-positive on ttwo complmentary arcs of J. Then no critical point of u(x, y) lies on the locus u(x, y) = 0 in R. THEOREM 5. Let thefunction u(x, y) be harmonic but not identically zero in the annuwar region R bounded by the disjoint Jordan curves J1 and 12, continuous on R + J1 + J2. Let u(x, y) be non-negative on J1, non-positive on J2. Then u(x, y) has no critical point on the locus u(x, y) = 0 in R. Theorems 4 and 5 extend to the case where there are admitted other boundary components, disjoint from the Jordan curves already mentioned, on which u(x, y) is assumed to take the boundary value zero. IWah, J. L., these PROCEBEDINGS, 34, 111-119, 1948. sWalsh, J. L., BuU. Am. Math. Soc., 54, 196-205,1948.

TUBERCULOSIS IN GERMANY BY ESMOND R. LONG THE HENRY PHIPPS INSTITUTE OF T¶B UNIVERSITY OF PENNSYLVANLA, PHILADELPHIA PENNSYLVANIA Read before the Academy, April 26, 1948 A rise in tuberculosis is a usual concomitant of war and prolonged disas- ter. Germany experienced such an increase in both world wars, as did most countries of Europe. A study of the rise in tuberculosis mortality in Ger- many, where records of unusual accuracy were maintained in spite of ad- ministrative upheaval, is of considerable interest for the general under- standing of the epidemiology of the disease. The analysis here made is based on German documents printed from 1920-1940 and on observations made in a series of visits to Germany since April, 1945, and particularly on studies performed by a commission* ap- pointed by the Secretary of the Army in February, 1948, to "investigate the incidence of and recommend control measures for tuberculosis in the German civilian population." Like all countries with a continuing rise in the standard of living and simultaneous development of a public health program, Germany experi- Downloaded by guest on September 29, 2021