A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ _j ?E[ξ _j ], 0<α?Var[ξ _j ] andE[|ξ _j ?μ _j |^2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ₀ are two random variables such thatE[ξ ₀ ² ]<∞ andE[|U|ξ ₀ ² ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ _j :j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereS _n ?ξ₁+ξ₂+?+ξ _n ,μ _j ?E[ξ _j ],s _n ² ?Var[S _n ], andc _n =O(s _n ).     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

Stability of semilinear ill-posed problems with a prescribed energy bound

In the present paper we discuss the stability of semilinear problems of the form A^αu + G^α(u) = ? under assumption of an a priori bound for an energy functional E^α(u) ? E, where α is a parameter in a metric space M. Following [11] the problem A^αu + G^α(u) = ?, E^α(u) ? E is called stable in a Hilbert space H at a point α ? M if for any ??H, E, ? > 0 there exists δ > 0 such that for any functions u^α1, u^α2 satisfying A^αju^αj + G^αj(u^αj) = ?^αj, E^αj(u^αj) ? E, j = 1,2 we have ‖u^α1 ? u^α2_H ? ? provided ρ_M(α_j, α) ? δ, ‖?^αj ? ?‖_H ? δ, j = 1,2. In the present paper we obtain stability conditions for the problem A^αu + G^α(u) = ?, E^α(u) ? E.     

The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk

We study the asymptotic tail behavior of the maximum M = max{0,S _n ,n ≥ = 1} of partial sums S _n = ξ₁ + ? + ξ _n of independent identically distributed random variables ξ₁,ξ₂,... with negative mean. We consider the so-called Cramer case when there exists a β > 0 such that E e ^βξ1 = 1. The celebrated Cramer-Lundberg approximation states the exponential decay of the large deviation probabilities of M provided that Eξ₁ e ^βξ1 is finite. In the present article we basically study the critical case Eξ₁ e ^βξ1 = ∞.     

Generalized Hilbert Functions

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciuperc?. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients 𝔧₀,…, 𝔧 _d?2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of 𝔧 _d?1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the ‘expected’ shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.     

Convex mappings on some Reinhardt domains

In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define
DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}
Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.     

A perturbative method for the spectral analysis of an acoustic multistratified strip

We consider the acoustic propagator A=−∇·c²∇ in the strip Ω={(x, z)∈ℝ²∣0<z<H} with finite width H>0. The celerity c depends for large ∣x∣ only on the variable z and describes the stratification of Ω: it is assumed to be in L^∞(Ω), bounded from below by c_min>0, such that there exists M>0 with c(x, z)=c₁(z) if x< −M and c(x, z)=c₂(z) if x>M. We look at the propagator A as a ‘perturbation’ of the free propagators A_j in Ω associated to the velocities c_j, j=1, 2, and implement a ‘perturbative’ method, adapting ideas of Majda and Vainberg. The spectrum of A is defined in section 2, a limiting absorption principle is proved in section 3 outside of a countable set Γ(A). The points of Γ(A) can only accumulate at the left of the thresholds of the free propagators. The needed material about A_j, j=1, 2, and some technical estimates for A are given in Appendix. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ₀, ξ₁, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ _j=0 ^∞ a(j)ξ _j was studied under different assumptions on the rate of decrease of the probability ?(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ?(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ?(ξ < x) ～ bx ^α as x → 0, b > 0, a > 0, the asymptotics ?(S < x) is obtained in an explicit form up to the factor x ^o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].     

Monotone subsequences in (0, 1)-matrices

A (0, 1)-matrix contains anS ₀(k) if it has 0-cells (i, j ₁), (i + 1,j ₂),..., (i + k – 1,j _k) for somei andj ₁ < ... < j_k, and it contains anS ₁(k) if it has 1-cells (i ₁,j), (i ₂,j + 1),...,(i _k ,j + k – 1) for somej andi ₁ < ... <i _k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 m n whose largestk for anS ₀(k) isk ₀ m, thenM must have anS ₁(k) withk m/(k ₀ + 1). Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS ₀(k) (in the cells on or below the main diagonal) isk ₀ m, thenM has anS ₁(k) withk m/(k ₀ + 1). Moreover, these results are best-possible.     

Existence of quantum diffusions

Summary A quantum diffusion (A, A, j) comprises of unital *-algebras A and A and a family of identity preserving *-homomorphisms j=(j _t : t0) from A into A. Also j satisfies a system of quantum stochastic differential equations dj _t (x ₀=j _t( _j ⁱ (x ₀))dM _i ⁱ , j ₀(x ₀)=x ₀I for all x ₀A where _j ⁱ , 1i, jN are maps from A to itself and are known as the structure maps. In this paper an existence proof is given for a class of quantum diffusions, for which the structure maps are bounded in the operator norm sense. A solution to the system of quantum stochastic differential equations is first produced using a variation of the Picard iteration method. Another application of this method shows that the solution is a quantum diffusion.     

On Complex Oscillation Theory

Suppose that A is a transcendental entire function with $\rho(A)<{1\over 2}$ . Suppose that k ≥ 2 and y^(k) + Ay = 0 has a solution ? with λ(?) < ρ(A), and suppose that A₁ = A + h where h ? 0 is an entire function with ρ(h) < ρ(A). Then y^(k) + A₁y = 0 does not have a solution g with λ(g) < ρ(A).     

On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?

     

Robust interpolation of random fields homogeneous in time and isotropic on a sphere,which are observed with noise

We study the problem of optimal linear estimation of the functional $$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(k, x),k?Z,x?S _n homogeneous in time and isotropic on a sphereS _n, by observations of the field ξ(k,x)+η(k,x) with k? Z{0, 1, ...,N},x?S_n (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere S_n). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalA _Nξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalA _Nξ.     

Über gerade unimodulare Gitter der Dimension 32, III

Let A_j, B_j be complex B-spaces, j = 0, 1, A_θ and B_θ–the complex-interpolation spaces generated by the couples (A₀, A₁) and (B₀, B₁), resp., by CALDERON'S/LIONS'S method. Let T: A₀ ∧ A₁ → B₀ → B₁ be an operator satisfying some conditions such as continuity, estimates etc. in terms of the norms of A_j, B_j (j = 0, 1). We consider the question which one of these properties is inherited to T when A₀ → A₁ and B₀ → B₁ are equipped with the norm of A_θ and B_θ.     

Simultaneous approximation to algebraic numbers by elements of a number field

A special case of the main result is as follows. Given a number fieldK a number ?>0 and real or complex algebraic numbers ξ₁,...,ξ _n with 1, ξ₁,...,ξ _n linearly independent overK, there are only finitely many α=(α₁,...,α _n ) with components inK and with |ξ₁,...,α₁| whereH(α) is a suitably defined height.     

Betti numbers of lex ideals over some Macaulay-Lex rings

Let A=K[x ₁,…,x _n ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R=A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl–Füredi–Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

ON THE STRUCTURE OF THEATTRACTORS OF HYPERBOLICITERATED FUNCTION SYSTEM

1.IntroductionThefractalsgeneratedbytheattractorsofiteratedfunctionsystems(i.f.s.)havebeenresearchedbymanyauthorsfl--3'6'7'9].ByaniteratedfunctionsystemwemeanacompactmetricspaceXtogetherwithacollectionofcontinuousmapsTI,T2,'tTNonit,denotedby(X,TI,',TN).IfalltheTi'sarecontractionswecall(X;TI,',TN)ahyperboliciteratedfunctionsystem(h.i.f.s.).NForanh.i.f.s.thereexistsacompactsubsetAofX,suchthatA=.UTi(A).Aiscalledtheattractoroftheh.i.f.s.DenoteZ=(1,2,',N)N,anddefineametricdonZby…