On q-n-gonal Klein Surfaces

We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.     

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂ ⁿ satisfying the condition R. Suppose that its Bergman kernel extends to [`(W)]×[`(W)]\overline{\Omega}\times\overline{\Omega} minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.     

A characterization of the identity function with equation f(p+q+r)=f(p)+f(q)+f(r)

Let f(n) be a multiplicative function such that there exists a prime p ₀ at which f does not vanish. In this paper, we prove that if f satisfies the equation f(p+q+r)=f(p)+f(q)+f(r) for all primes p, q and r, then f(n)=n for all integers n≥1.     

Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can be chosen so that the surface bundles M(F′,f′) and M(F,f) have the same first Betti numbers. Y. Ni is partially supported by a Centennial fellowship of the Graduate School at Princeton University. S.C. Wang is partially supported by MSTC     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

On the index of approximating sets of periodic points

Summary LetK be a compact space andf:K→K a continuous map without fixed points, i.e. Fixf=⊘. For prime numbersp, the sets Fixf ^p are freeℤ/p-spaces with theℤ/p-action induced byf. Our aim is to estimate the topological indicesi(F _p,f) of invariant subsetsF _p⊂Fixf ^p approximating a givenS⊂K. We construct an example (K,f,S) withS⊂Fixf ^q (q being some prime number) such that, for each neighborhoodU ofS, i (Fix (f|u) ^p, f) increases linearly withp. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.     

Covering homotopy 3-spheres

We prove the following theorem: for any closed orientable 3-manifoldM and any homotopy 3-sphere Σ, there exists a simple 3-fold branched coveringp:M→Σ. We also propose the conjecture that, for any primitive branched coveringp:M→N between orientable 3-manifolds,g(M)≥g(N), whereg denotes the Heegaard genus. By the above mentioned result, the genus 0 case of such conjecture is equivalent to the Poincaré conjecture.     

On the existence of entire functions of boundedl-index andl-regular growth

We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM _f(r)lnL(r),r→∞, whereM _f (r)=max {|f(z)|: |z|=r} andL(r)=∫ ₀ ^r l(t)dt. Ifl(r)=r ^p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM _f (r)≈r ^p . Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996.     

Connected components of spaces of Morse functions with fixed critical points

Let M be a smooth closed orientable surface and F = F _p,q,r be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F _f be a connected component of a function f ∈ F in F.We construct a surjection π ₀(F) → ℤ ^p+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π₀(F)| = ∞, and the component F _f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D ⁰ be the connected component of id _M in D, and D _f ⊂ D be the set of diffeomorphisms preserving F _f . Let H _f be the subgroup of D _f generated by D ⁰ and all diffeomorphisms h ∈ D preserving some function f ₁ ∈ F _f , and let H _f ^abs be its subgroup generated by D ⁰ and the Dehn twists about the components of level curves of the functions f ₁ ∈ F _f . We prove thatH _f ^abs ⊊ D _f for q ≥ 2 and construct an epimorphism D _f /H _f ^abs → ℤ₂ ^q−1 by means of the winding number. A finite polyhedral complex K = K _p,q,r associated with the space F is defined. An epimorphism μ: π ₁(K) → D _f /H _f and finite generating sets for the groups D _f /D ⁰ and D _f /H _f in terms of the 2-skeleton of the complex K are constructed.     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

A characterisation result on a particular class of non-weighted minihypers

We present a characterisation of {e₁ (q+1)+e₀,e₁ ;n,q}{\{\epsilon_1 (q+1)+\epsilon_0,\epsilon_1 ;n,q\}} -minihypers, q square, q = p ^h , p > 3 prime, h ≥ 2, q ≥ 1217, for e₀ + e₁ < \fracq^7/122-\fracq^1/42{\epsilon_0 + \epsilon_1 < \frac{q^{7/12}}{2}-\frac{q^{1/4}}{2}}. This improves a characterisation result of Ferret and Storme (Des Codes Cryptogr 25(2): 143–162, 2002), involving more Baer subgeometries contained in the minihyper.     

Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.     

Genus 2 Belyĭ surfaces with a unicellular uniform dessin

A bicoloured graph embedded in a compact oriented surface and dividing it into a union of simply connected components (faces) is known as a dessin d’enfant. It is well known that such a graph determines a complex structure on the underlying topological surface, but a given compact Riemann surface may correspond to different dessins. In this paper we deal with all unicellular (one-faced) uniform dessins of genus 2 and their underlying Riemann surfaces. A dessin is called uniform if white vertices, black vertices and faces have constant degree, say p, q and r respectively. A uniform dessin d’enfant of type (p, q, r) on a given surface S corresponds to the inclusion of the torsion-free Fuchsian group K uniformizing S inside a triangle group Δ(p, q, r). Hence the existence of different uniform dessins on S is related to the possible inclusion of K in different triangle groups. The main result of the paper states that two unicellular uniform dessins belonging to the same genus 2 surface must necessarily be isomorphic or obtained by renormalisation. The problem is approached through the study of the face-centers of the dessins. The displacement of such a point by the elements of K must belong to a prescribed discrete set of (hyperbolic) distances determined by the signature (p, q, r). Therefore looking for face-centers amounts to finding points correctly displaced by every element of K.     

New relationships between steenrod operations for certain spaces

In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then eitherS q ² n x≠0 or there exists somey withS q ² n y=S q ² n ⁺¹ x, if for somen≥3S q ² n ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective realK-Theory. This research was partially supported by NSERC.     

Plurisubharmonic functions in calibrated geometry and q-convexity

Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}Let (M, ω) be a K?hler manifold. An integrable function j{\varphi} on M is called ω ^q -plurisubharmonic if the current dd^cjùw^q-1{dd^c\varphi\wedge \omega^{q-1}} is positive. We prove that j{\varphi} is ω ^q -plurisubharmonic if and only if j{\varphi} is subharmonic on all q-dimensional complex subvarieties. We prove that a ω ^q -plurisubharmonic function is q-convex, and admits a local approximation by smooth, ω ^q -plurisubharmonic functions. For any closed subvariety Z ì M{Z\subset M} , dim_\mathbbC Z ￡ q-1{\dim_\mathbb{C} Z\leq q-1} , there exists a strictly ω ^q -plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony’s lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension ≥  p + 1.     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝ ⁿ is the restriction of a function of class 𝒞 ^p . A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a “tame topology”. We prove that, if X is a compact subanalytic set, then there exists q=q _X (p) such that the criterion of order q implies that f is 𝒞 ^p . The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces. Oblatum 21-XI-2001 & 3-VII-2002?Published online: 8 November 2002 RID="*" ID="*"Research partially supported by the following grants: E.B. – NSERC OGP0009070, P.M. – NSERC OGP0008949 and the Killam Foundation, W.P. – KBN 5 PO3A 005 21.     

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface Σ is a graph in ?⁴ if there is a unit constant 2-form ω on ?⁴ such that <e ₁∧e ₂, ω≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if $ \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} A surface Σ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that <e ₁∧e ₂, ω>≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if v ₀≥ on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M ₁×M ₂ where M ₁ and M ₂ are Riemann surfaces, if <e ₁∧e ₂, ω₁>≥v ₀>0 where ω₁ is a K?hler form on M ₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, v ₀≥ on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic. Received July 25, 2001, Accepted October 11, 2001     

On linguistic dynamical systems,families of graphs of large girth,and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f _α : K ⁿ → K ⁿ depending on “time” α ∈ {K − 0} such that f _α ⁻¹ = f _−αM, the relation f _α1 (x) = f _α2 (x) for some x ∈ Kⁿ implies α₁ = α₂, and each map f _α has no invariant points. The neighborhood {f _α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kⁿ. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α₁, υ, α₂), s ≤ d, where α_i + α_i+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ _a = f _α1 × ⋯ × f _αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems L_n(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system L_n(q). If K = F_q, the graphs L(n, F_q) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems L_n(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.     

TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Let 0 < c < s be fixed real numbers such that , and let f : E² → E ^d for d ≥ 2 be a function such that for every p, q ∈ E ² if |p − q| = c, then |f(p) − f(q)| ≤ c, and if |p − q| = s, then |f(p) − f(q)| ≥ s. Then f is a congruence. This result depends on and expands a result of Rádo et. al. [9], where a similar result holds, but for replacing . We also present a further extensions where E² is replaced by E ⁿ for n > 2 and where the range of c/s is enlarged. This revised version was published online in August 2006 with corrections to the Cover Date.