On higher order effects in marginally separated flows

If the angle of attack α of a slender airfoil reaches a critical value α_s flow separation is known to occur at the upper s surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field – a phenomenon known as marginal separation – but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integro-differential equation. This so-called interaction equation contains a single controlling parameter which relates the angle of attack to the Reynolds number, with a value Γ_s corresponding to α_s. Some recent results concerning higher order s s corrections to this theory and their effect on the stability of steady solutions will be presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One-dimensional Singular Integral Equations

A class of singular integral equations having solution in closed form is considered. This class contains the characteristic equation and other equations associated with it. Equations are solved by their reduction to some Riemann boundary value problems in generalized Hölder spaces H_φ (Γ), where Γ is a closed rectifiable Jordan curve for which the Plemelj–Privalov theorem holds.     

Die Fundamental Gruppen Siegelscher Modul Varietäten

Let Γ be a subgroup of finite index of Siegels modular group Γ_g = Sp(g,Z) and Y(Γ) a proper modification of the Satake compactification (S_g/Γ)^* of Siegels modular space S_g/Γ of degree g≧2. It will be shown that Y(Γ) is simply connected for all principal congruence subgroups Γ = Γ_g(λ). Moreover the fundamental group of Y is always finite. These results are based on J. Mennickes work showing that Γ_g(λ) is the normal hull in Γ_g of a single matrix μ_g(λ). Secondly it will be proved that for g=2,3 and λ≧3 this fact is even equivalent to the simply connectedness of any desingularisation of (Sg/Γ_g(λ)).     

A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X‐symmetric graph admitting an X‐invariant partition ?? on V(Γ) such that Γ_?? is connected and (X, 2)‐arc transitive. A characterization of (Γ, X, ??) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where |B|>|Γ(C)∩B|=2 for an arc (B, C) of Γ_??.We con‐sider in this article the case where |B|>|Γ(C)∩B|=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K ₂, C ₆ or K _{3, 3}. As a byproduct, we prove that each connected tetravalent (X, 2)‐transitive graph is either the complete graph K ₅ or a near n‐gonal graph for some n?4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set ?? of m‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in ??. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ₂ of valency r and girth 2m. We will show that this construction can be extended to one that yields a polygonal graph Γ₃ of valency r and girth 3m, but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ_k) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011     

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.     

Über Gruppen Von Iterativen Wurzeln Der Formalen Potenzreihe F(x) = x

Let ?¦x¦ be the ring of formal power series in one indeterminate x over ?, denote by Γ the group of invertible series in ?¦x¦, and by E_Γ the set of all iterative roots of x in Γ. Then we will show that E_Γ is neither a subgroup of Γ nor a family of commuting series. We describe all subgroups of Γ lying in E_Γ, they are abelian and isomorphic to subgroups of the group E of complex roots of unity. Furthermore we determine the maximal subgroups of Γ in E{Γ} and use them to investigate how the subgroups in E _I are related.     

Upper and lower bounds for the Voronoi-Minkowski constant

We study the behavior of the constant C(s) as s → ∞ arising in the estimate of the number of relative minima of s-dimensional integer lattices Γ via the logarithm log₂ N raised to the s ? 1th power, where N = D(Γ) is the determinant of Γ.     

Numerical analysis of a mixed elliptic problem with flux and convective boundary conditions to obtain a discrete solution of nonconstant sign

We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝⁿ, with regular boundary Γ = Γ₁ ∪ Γ₂ (with Γ ∩ Γ = ∅︁) with meas (Γ₁) = |Γ₁| > 0 and |Γ₂| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady‐state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ₂ and the heat transfer coefficient on Γ₁, respectively. In a previous article (Tabacman‐ Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ₁, Γ₂ to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady‐state, two‐phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ₁, and Γ₂ to obtain a change of phase (steady‐state, two‐phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999     

Regarding Some Problems of the Kutta — Joukovskii Condition in Lifting Surface Theory

We are interested in finding the velocity distribution at the wings of an aeroplane. Within the scope of a three — dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation (Δ +k²)Φ = 0 in ?³\s where Φ denotes the perturbation velocity potential, induced by the presence of the wings and s :=L UW with the projection L of the wings onto the (x,y)- plane and the wake W. Not all Cauchy data are given explicitly on L, respectively W. These missing Cauchy data depend on the wing circulation Γ· Γ has to be fixed by the Kutta–Joukovskii condition: Λ Φ should be finite near the trailing edge x_t of L. To fulfil this condition in a way that all appearing terms can be defined mathematically exactly and belong to spaces which are physically meaningful, we propose to fix Γ by the condition of vanishing stress intensity factors of Φ near x_t up to a certain order such that ΛΦ|_xt ?W₂^?(x_t)? L₂(x_t),?>0. In the two–dimensional case, and if L is the left half–plane in ?², we have an explicit formula to calculate Γ and we can control the regularity of Γ and Φ.     

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

Concentration on curves for nonlinear Schrödinger Equations

We consider the problem where p > 1, ε > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫_Γ V^σ, where σ = (p + 1)/(p ? 1) ? 1/2. We prove the existence of a solution u_? concentrating along the whole of Γ, exponentially small in ε at any positive distance from it, provided that ε is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two‐dimensional case. © 2006 Wiley Periodicals, Inc.     

Remarkable valuation of the dichromatic polynomial of planar multigraphs

Let Γ be a finite multigraph; we denote by χ(Γ, x, y) the dichromatic polynominal of Γ, as defined by W. T. Tutte in 1953. We prove that, for any planar multigraph Γ with m edges, χ(Γ, ?1, ?1) = (?1)^m · (?2)^k, where $\text{0 ≤ k ≤}\text{m}\text{2}$. Furthermore, if Γ is connected, s = k ? 1 turns out to be a pertinent invariant of the medial of Γ.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

Sonar echo analysis

Pulse mode sonar operation is analyzed under the assumption that the scattering object Γ lies in the far fields of both the transmitter and the receiver. It is shown that, in this approximation, the sonar signal is a plane wave s(x · θ₀ – t) near Γ, where θ₀ is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e(x · θ – t) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, if Γ is stationary with respect to the sonar system then it is shown that where ?(ω) is the Fourier transform of S(τ) and T+(ωθ,ωθ₀) is the scattering amplitude in the direction θ due to the scattering by Γ of a time-harmonic plane wave with frequence ω and propagation direction θ₀. A generalization of this relation is derived for moving scatterers.     

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady‐state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a ε‐neighborhood of a portion Γ of the boundary. We assume that this ε‐neighborhood shrinks to Γ as the small parameter ε goes to zero. Also, we suppose the upper boundary of this ε‐strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Abstract

Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver Γ_A of A in which each Γ_A-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τ_A-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α?γ and multiple coray-ray insertions of type α?γ.     

Approximation on limit compact sets of kleinian groups

Let Γ be a geometrically finite or a quasi-Fuchsian Kleiman group such that ∞ ? $\mathop \Omega \limits^o \left( v \right)$ . We establish the relation $X = clos_X L\left( {\frac{1}{{1 - a}},a \in \Xi } \right)$ for some countable sets Ξ?ω(Γ) connected with actions of elements of Γ, and for the space X=C(Γ) or for the Hölder classes X=L^α(Λ), 0<α<1, where Λ=Λ(Γ)=?\Ω is the limit set of Γ. Bibliography: 6 titles.