PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

Positive Derivations on Archimedean Almost f-Rings

It is shown by P. Colville, G. Davis and K. Keimel that if R is an Archimedean f-ring then a positive group endomorphism D on R is a derivation if and only if the range of D is contained in N(R) and the kernel of D contains R ², where N(R) is the set of all nilpotent elements in R and R ² is the set of all products uv in R. The main objective of this paper is to establish the result corresponding to the Colville–Davis–Keimel theorem in the almost f-ring case. The result obtained in this regard is that if D is a positive derivation in an Archimedean almost f-ring, then the range of D is contained in N(R) and the kernel of D contains R ³, where R ³ is the set of all products uvw in R. Examples are produced showing that, contrary to the f-ring case, the converse is in general false and the third power is the best possible.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D _N ≡D∩ (ℤ ^d /N) (N∈ℤ⁺), D a suitable subset of ℝ ^d , i.e. the continuous spin Gibbs measure ℙ _N on ℝ ^ℤd/N with Hamiltonian given by H(ϕ) = ∑ _{x,y:|x−y|=1} V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D _N ^C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D _N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ _N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ _N } and of the induced family of gradient fields ∇ ^N ξ _N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ _N } and show that the corresponding rate function is given by ∫ _D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ _N under the volume constraint, i.e. the constraint that (1/N ^d ) ∑ _x∈DN ξ _N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ _N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ _N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ ^N ξ _N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? _p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sj?strand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields. Received: 3 March 1999 / Revised version: 9 August 1999 / Published online: 30 March 2000     

Valuation-like maps and the congruence subgroup property

Let D be a finite dimensional division algebra and N a subgroup of finite index in D ^×. A valuation-like map on N is a homomorphism ϕ:N?Γ from N to a (not necessarily abelian) linearly ordered group Γ satisfying N _<-α+1⊆N _<-α for some nonnegative α∈Γ such that N _<-α≠=?, where N _<-α={x∈N|ϕ(x)<-α}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (v-adically) open. We then show that if N is normal in D ^× and the diameter of the commuting graph of D ^×/N is ≥4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D ^×. The notion of a valuation-like map is inspired by [27], and in fact is closely related to part (U3) of the U-Hypothesis in [27]. Oblatum 14-VII-2000 & 22-XI-2000?Published online: 5 March 2001     

PSPACE complexity of modal logic <Emphasis Type="Italic">K D</Emphasis>45<Subscript>n</Subscript>

We study sequent calculus for multi-modal logic K D45_n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □_i^•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus. We concentrate on the complexity analysis of the introduced sequent calculus for multi-modal logic K D45_n. We prove that the space complexity of the given calculus is polynomial (O(l ³)). We show the maximum height of the constructed derivation tree that leads to the reduction of the time and space complexity. We present a decision algorithm for multi-modal logic K D45_n and some nontrivial examples to improve the introduced loop-check free sequent calculus.     

Wellposedness of abstract Cauchy problems for second order differential equations

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A ^k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A ^k)] toX, associated with an exponentially wellposed ACP onD(A ^k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H _k) onX. Research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (No. 63540139), Ministry of Education, Science and Culture.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

Large bipartite graphs with given degree and diameter

We give constructions of bipartite graphs with maximum Δ, diameter D on B vertices, such that for every D ≥ 2 the lim inf_Δ→∞B. Δ^1-D = b_D > 0. We also improve similar results on ordinary graphs, for example, we prove that lim_Δ→∞N · Δ^?D = 1 if D is 3 or 5. This is a partial answer to a problem of Bollobás.     

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

Inverse of the distance matrix of a block graph

A connected graph G, whose 2-connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its distance matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D)?≠?0. We give a formula for both the determinant and the inverse, D ^?1 of D.     

Free groups in a normal subgroup of the field of fractions of a skew polynomial ring

Let k(t) be the field of rational functions over the field k, let σ be a k-automorphism of K = k(t), let D = K(X;σ) be the ring of fractions of the skew polynomial ring K[X;σ], and let D^? be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^?, then N contains a free subgroup. We also prove that when k is algebraically closed and σ has infinite order, there exists a specialization from D to a quaternion algebra. This allows us to explicitly present free subgroups in D^?.     

A note on the optimality of the N- and D-policies for the M/G/1 queue

This note considers the N- and D-policies for the M/G/1 queue. We concentrate on the true relationship between the optimal N- and D-policies when the cost function is based on the expected number of customers in the system.     

A mean value formula for the M/G/1 queues controlled by workload

We present a mean value formula for the M/G/1 queues controlled by workload (such as the D-policy queues). We first prove the formula and then demonstrate its application. This formula also works for the conventional vacation systems which are controlled by number of customers (such as the N-policy queues).     

Standardness of *-representations of the algebra of smooth finite functions

We study strongly continuous *-representations by unbounded operators of the algebra of smooth finite functions on R^N. We establish that the closure of a strongly continuous *-representation of the algebraD(R^N), satisfying a certain condition, is self-adjoint and standard. We also study the correspondence between the strongly continuous *-representations ofD(R^N) and the continuous representations of the group R^N with respect to addition by symmetric operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 713–716, May, 1990.     

A fractional porous medium equation

We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion, with m>m_?=(N−1)/N, N?1 and f∈L¹(RN). An L¹-contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all x∈RN, t>0.     

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.     

Applications of Operator Space Theory to Nest Algebra Bimodules

Recently, Blecher and Kashyap have generalized the notion of W ^*-modules over von Neumann algebras to the setting where the operator algebras are σ closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras. We prove that Y is a right weak* rigged module over a nest algebra Alg(M){\rm{Alg}(\mathcal M)} if and only if there exists a completely isometric normal representation F{\Phi } of Y and a nest algebra Alg(N){\rm{Alg}(\mathcal N)} such that Alg(N) F(Y)Alg(M) ì F(Y){\rm{Alg}(\mathcal N) \Phi (Y)\rm{Alg}(\mathcal M)\subset \Phi (Y)} while F(Y){\Phi (Y)} is implemented by a continuous nest homomorphism from M{\mathcal M} onto N{\mathcal N} . We describe some properties which are preserved by continuous CSL homomorphisms.     

On the Hasse principle for Shimura curves

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X ₀ ^D (N)_/ℚ or X ₁ ^D (N)_/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.     

Investigating the scale effects in strength of fractured rock mass

Based on the method of fractal geometry, the relationship between quantity and size of fracture in a rock mass can be studied. It was found that this relationship is power: N_δ=N₀δ^-D. Based on this result and theoretical analysis, the nonlinear relationship between rock mass strength and fracture quantity is found: Rc=Rc_m+ΔRcExp[-BN_δ].the formula includes the residual strength of fractured rock mass Rc_m, the difference in value (ΔRc) between the intact rock strength Rc₀ and the residual strength, and fracture quantity. Then, combining the relationship between fracture quantity and the size of rock mass, the formula of rock mass strength is found: . The formula includes fracture quantity, the fractal dimension of fracture quantity distribution and rock mass size. The results show that the attenuation index of rock mass strength is related to the fractal dimension of fracture quantity distribution: α=2-D.