Analytic Semigroups on H 1 (Ω) Generated by Degenerate Elliptic Operators

One proves that Ay=aΔ y+b⋅ \nabla y , D(A)={y∈ H ¹ (Ω ); aΔ y+b⋅ \nabla y∈ H ₀ ¹ (Ω ), \sqrt a Δ y∈ L ² (Ω )} generates, under suitable conditions on a and b , a C ₀ -analytic semigroup on H ¹ (Ω) . September 5, 1999     

On the Hausdorff dimensions of certain attractors

For any a,b∈R let ϕ_a,b(x)=ax+b(x∈R). Suppose 0<a<1. Let C_a,b be the generalized a-Cantor set with generating iterated function systme {ϕ_a,0, ϕ_a,b; ϕ_a,l}. Then we prove the Hausdorff dimension of C_a,c² C_{a,c^2 } is \fracln(3 - ?5 - ln2lna\frac{{ln(3 - \sqrt 5 - ln2}}{{lna}} when 0<a≤2 cos 80°.     

On a problem of Cilleruelo and Nathanson

Let ℤ denote the set of all integers and ℕ the set of all positive integers. Let A be a set of integers. For every integer u, we denote by d _A (u) and s _A (u) the number of solutions of u=a − a′ with a,a′ ∈ A and u=a+a′ with a,a′ ∈ A and a≤a′, respectively.     

Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

Bricks and pseudo MV-algebras are equivalent

We will show that the bricks (of Bosbach) and the pseudo MV-algebras are each term equivalent to the class of semigroups with a pair of unary operations ^ and ˘ satisfying the equations: (aa)^b = b = b(aă)˘ and a( a)˘ = (bă)^b and also show that a brick is an interval [0, u] of the positive cone of a unital lattice ordered group. We further extend the notion of implications to a pseudo MV-algebra and study the algebra of such implications.      

DENSEST CIRCLE PACKING ON THE FLAT TORUS

Melissen [1] considered packings of k congruent circles in the symmetric flat torus T ² = [0, 1)² and determined the largest possible radius for k ≤ 4. In the present paper the analogous problem is studied for an arbitrary asymmetric torus T′² = [0,b) × [0,a), 0 < a ≤ b and the maximal radius is found again for k ≤ 4. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new polynomial-time algorithm for linear programming

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n ^3.5 L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n ^2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time. This is a substantially revised version of the paper presented at the Symposium on Theory of Computing, Washington D. C., April 1984.     

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C _t = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C ¹-family of circles in the plane such that lim _t→0+ C _t = {a}, lim _t→1− C _t = {b}, a ≠ b, and |c′(t)|² + |r′(t)|² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation ̅c′(t)w ² + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.     

Exceptional 0-Alia algebras

Analgebra (A, ∘) with the identity [a, b]∘c + [b, c]∘a + [c, a]∘b = 0, where [a, b] = a∘b−b∘a, is called 0-Alia. We prove that the algebra (ℂ[x], ∘) with multiplication a ∘ b = ∂2(2a∂(b)+∂(a)b) is a simple, exceptional 0-Alia algebra. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

On the solutions of a boundary value problem arising in free convection with prescribed heat flux

For given , c < 0, we are concerned with the solution f _b of the differential equation f ′′′ + ff ′′ + g(f ′) = 0 satisfying the initial conditions f(0) = a, f ′ (0) = b, f ′′ (0) = c, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists b _* > 0 such that f _b exists on [0, + ∞) and is such that as t → + ∞, if and only if b ≥ b _*. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.      

Some results on the majorization theorem of connected graphs

Let π = (d ₁, d ₂, ..., d _n ) and π′ = (d′ ₁, d′ ₂, ..., d′ _n ) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ _i=1 ⁿ d _i = Σ _i=1 ⁿ d′ _i , and Σ _i=1 ^j d _i ≤ Σ _i=1 ^j d′ _i for all j = 1, 2, ..., n. Weuse C _π to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C _π and C′ _π , respectively, then ρ(G) < ρ(G′).     

Absolute upper semicontinuity

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ _t′ ^t″ g (s) ds. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Equivalent proportionally modular Diophantine inequalities

We study Diophantine inequalities of the form ax mod b ≤ cx. In particular, we prove that there exists a positive integer such that for every integer n ≥ N there exist a′, c′ (positive integers dependent of n) such that a′ x mod n ≤ c′ x has the same solutions as the above inequality. Received: 23 March 2007     

Zur Theorie der Stetigen Teilbarkeitshalbgruppen

O. CallS:=(S,·,∩) a d-semigroup ifS satisfies the axioms (A1) (S,·) is a semigroup, (A2) (S,∩) is a semilattice (A3), (S,·,∩) is a semiring, (A4) a ≤b⇒bε aS ∩ Sa. Call tεS positive if ÅaεS: ta ≥a≤at. Let S⁺ denote the set {t‖t positive}. Every d-semigroup is closed under sup and (s,·,∪) is a semiring, (S, ∩, ∪) is a distributive lattice. Denote by D□X the implication s=Xa_i⇒x□s□y=X(x□a_i□y) where □ε{·,∩,∪} and Xε{∪,∩}. CallS continuous ifS satisfies all D□X. The theory of d-semigroups (divisibility-semigroups) was established in [3], [4], [5], and is continued here by some contributions to the theory of continuous d-semigroups the main results of which are the two propositions: (1) LetS be a d-semigroup with 1. ThenS satisfies D□X iffS ⁺ satisfies this axiom. (2) LetS be continuous. Then (S,·) is commutative. Obviously Proposition (2) is an improvement of Iwasawa's theorem concerning conditionally complete lattice ordered groups.

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Zur Theorie der Stetigen Teilbarkeitshalbgruppen

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Klaus Wagner zum 70. Geburtstag gewidmet     

Invariant sets for a class of perturbed differential equations of retarded type

LetX be a Banach space and leta, b, q be real numbers such thata<b,q>0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b ₁],X),a<b ₁<b, to the differential equationdu(t)/dt=Au(t)+f(t, u _t), such thatu:[a,b ₁]→D, u _a=ϕ is given. The linear operatorA is the generator of aC ₀ semigroupT(t), t≧0, withT(t) compact fort>0,f: [a, b)×C([−q,0],D _λ)→X is continuous and ϕ∈C([−q,0],D _λ) with ϕ(0)∈D. D _λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.     

Characterization of separating couples

Let A, B be two archimedean ℓ-algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V) is separating with respect to A and B if |a||b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f-algebra with unit elment e, if B is an ℓ-algebra and if (U,V) is a separating couple with respect to A and B then (U ^∼∼,V ^∼∼), where U ^∼∼ (resp V ^∼∼) is the bi-adjoint of U (resp of V), is again a separating couple with respect to the order continuous order biduals (A′)′ _n and (B′)′ _n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B′ separates the points of B, we give a characterization of any separating couple with respect to A and B.      

On the Hausdorff dimensions of certain attractors

For any a,b∈R let ϕ_a,b(x)=ax+b(x∈R). Suppose 0<a<1. Let C_a,b be the generalized a-Cantor set with generating iterated function systme {ϕ_a,0, ϕ_a,b; ϕ_a,l}. Then we prove the Hausdorff dimension of is when 0<a≤2 cos 80°.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).