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The classical theorem of M. Riesz about the conjugate harmonic functions is extended onto octonion-valued monogenic functions.

Let$\\mathcal{D}$be a doubly connected region limited by the infinite point and a starlike boundary component$\\Gamma $which does not reduce to a point. If$\\lambda $is a given positive number, we show there exists a unique annulus$\\omega _\\lambda \\subset \\mathcal{D}$having$\\Gamma $as one boundary component and another boundary component$\\gamma _\\lambda $such that there is a harmonic function$V$in$\\omega _\\lambda $satisfying$V \\equiv 0$on$\\Gamma $ ,$V \\equiv 1$on$\\gamma _\\lambda $and$| {{\\operatorname{grad}}V_\\lambda } | \\equiv \\lambda $on$\\gamma _\\lambda $ . We also show that$\\gamma _\\lambda $is starlike.

Let$\\mathcal{D}$be a doubly connected region in the complex plane limited by the infinite point and a convex set$\\Gamma $ . If$\\lambda > 0$ , then we study the existence, uniqueness and geometry of annuli$\\omega \\subset \\mathcal{D}$having$\\Gamma $as one boundary component and another boundary component$\\gamma $ , such that there exists a harmonic function$V$in$\\omega $satisfying : (a)$V = 0$on$\\Gamma $ , (b)$V = 1$on$\\gamma $ \\ and (c)$| \\operatorname{grad}V | = \\lambda $on$\\gamma $ .

BILL BELICHICK Head Coach UNC Chapel Hill Chapel Hill Belichick, who won six Super Bowls as the New England Patriots head coach, was given the reins of the University of North Carolina football program in December. Education: BS San Diego State University MARK BRAZIL CEO Wyndham Championship Greensboro Brazil has spent nearly 25 years shepherding the state's longest-running PGA Tour event through professional golf's rapidly changing landscape. The son of former ACC Commissioner Gene Corrigan, he is a player in the world of big-time college athletics, chairing the College Football Playoff Selection Committee in 2023-24. Education: Duke University VALERIE HILLINGS CEO North Carolina Museum of Art Raleigh Hillings manages more than 200 employees at the state's largest public art museum, which USA Today recently ranked No. 8 on its list of the country's best free museums.

Let H be the homogeneous space associated to the group PGL3(R). Let X =$\\Gamma \\backslash {\\cal H}$. where Γ = SL3(Z) and consider the first nontrivial eigenvalue λ1of the Laplacian on L2(X). Using geometric considerations, we prove the inequality λ1> 3π2/10. Since the continuous spectrum is represented by the band [1, ∞ ), our bound on λ1can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

Let H H be the homogeneous space associated to the group PGL 3 ( R ) PGL_3( R) . Let X = Γ ∖ H X = H where Γ = SL 3 ( Z ) = SL_3( Z) and consider the first nontrivial eigenvalue λ 1 _1 of the Laplacian on L 2 ( X ) L^2(X) . Using geometric considerations, we prove the inequality λ 1 > 3 π 2 / 10 _1 > 3 ^2/10 . Since the continuous spectrum is represented by the band [ 1 , ∞ ) [1,ınfty ) , our bound on λ 1 _1 can be viewed as an analogue of Selberg’s eigenvalue conjecture for quotients of the hyperbolic half space.

Occurrence of sticky and dry cerumen was determined in 483 Indians from various tribes of the United States. The elevated frequencies of the allele for dry cerumen, found in Indians of pure ancestry, support the theory of the mongoloid origin of the American Indian. Potential application of cerumen quality as a marker for genetic and anthropological studies is discussed.

Let Γ/H\\Gamma /\\mathcal {H} be a finite volume symmetric space with H\\mathcal {H} the product of half planes. Let Δi{\\Delta _i} be the Laplacian on the iith half plane, and assume that we have a cusp form ϕ\\phi, so we have Δiϕ=λiϕ{\\Delta _i}\\phi = {\\lambda _i}\\phi for i=1,2,…,ni = 1,2, \\ldots ,n. Let λ→=(λ1,…,λn)\\vec \\lambda = ({\\lambda _1}, \\ldots ,{\\lambda _n}) and let \\[ R=r12+⋯+rn2R = \\sqrt {r_1^2 + \\cdots + r_n^2} \\] with ri2+14=λir_i^2 + \\frac {1} {4} = {\\lambda _i}. Letting r→=(r1,…,rn)\\vec r = ({r_1}, \\ldots ,{r_n}), we let M(r→)M(\\vec r) denote the dimension of the space of cusp forms with eigenvalue λ→\\vec \\lambda. More generally, let M(r→,a)M(\\vec r,a) denote the number of independent eigenfunctions such that the r→\\vec r associated to an eigenfunction is inside the ball of radius aa, centered at r→\\vec r. We will define a function f(r→)f(\\vec r), which is generally equal to a linear sum of products of the ri{r_i}. We prove the following theorems. Theorem 1. \\[ M(r→)=O(f(r→)(log⁡R)n).M(\\vec r) = O\\left (\\frac {f(\\vec r)} {(\\log R)^n} \\right ). \\] Theorem 2. \\[ M(r→,A)=2nf(r→)+O(f(r→)log⁡R).M (\\vec {r}, A) = 2^n f(\\vec {r})+O\\left (\\frac {f(\\vec r)}{\\log R} \\right ). \\]

Let w=f(z)=z+∑n=2∞anznw = f(z) = z + \\sum \\nolimits _{n = 2}^\\infty {{a_n}{z^n}} be regular and univalent for |z|>1|z| > 1 and map |z|>1|z| > 1 onto a region which is starlike with respect to w=0w = 0. If r0{r_0} denotes the radius of convexity of w=f(z)w = f(z), d0=min|f(z)|d_0 = \\min |f(z)| for |z|=r0|z| = {r_0}, and d∗=inf|β|{d^ \\ast } = \\inf |\\beta | for f(z)≠βf(z) \\ne \\beta, then it has been conjectured that d0/d∗≧2/3{d_0}/{d^ \\ast } \\geqq 2/3. It is shown here that d0/d∗≧0.343…{d_0}/{d^ \\ast } \\geqq 0.343 \\ldots, which improves the old estimate d0/d∗≧0.268…{d_0}/{d^ \\ast } \\geqq 0.268 \\ldots. In addition, sharp estimates for r0{r_0} are given which depend on the value of |a2||{a_2}|.