Differential Operators on Graphs and Photonic Crystals

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: –u=u, where (x) (the dielectric constant) is a periodic function that assumes a large value near a periodic graph in R ² and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.     

Sharp Large Deviation Estimates for the Stochastic Heat Equation

We consider the family {X , 0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is _t X = X +b({X })+({X }) . Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({X }A) as 0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Inverse Problems in the Theory of Singular Perturbations

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation ² y'=(1+² (x, ))y with a small parameter , where is analytic and even with respect to . It is well known that it has two formal solutions of the form y^±(x,)=e^±x/h^±(x,), where h^±(x,) is a formal series in powers of whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations ^r y' = A(x,) y in the neighborhood of a nonturning point and for second-order (analytic) equations y' - 2xy'-g(x,) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution such that the coefficients have no poles at x=0.     

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[ω]C

Suppose thatk, rz₊, W _o ^r H[]_C= {ff is a 2-periodic function,f C^r [–, ], (f^(r), ) ()}, T_k is the space of trigonometric polynomials of order k, p_k(f)T_k is the polynomial of best uniform approximation to f, and E_k(f) is the error of the best approximation. It is shown that for an arbitrary > 0 we have,where for 0<&#x2A7D;(1),k > 0.R () is the root of the equation , and for k = 0 or > (1) we have R()=.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 85–101, July, 1977.The author thanks S. B. Stechkin for posing the problem and for his attention to this work.     

Linear foliations ofT n

We present a definition of diophantine matrix and use this concept to distinguish two classes of minimal linear foliations ofT ⁿ, the diophantine and the Liouville one. Let _p , 1pn–1, denote a minimal (all leaves are dense) linearp-dimensional foliation ofT ⁿ, andH ^om(T ⁿ, _p ), 1mp, the cohomology group of type (0,m) of the foliated manifold (T ⁿ, _p ). Our main result is the computation of these groups.H ^om(T ⁿ, _p ) is isomorphic to if _p is diophantine and is an infinite dimensional non-Hausdorff vector space if _p is Liouville. Some of these groups were computed before, see [4], [6] and [9].     

Approximation Schemes for Scheduling Jobs with Common Due Date on Parallel Machines to Minimize Total Tardiness

The problem of scheduling n nonpreemptive jobs having a common due date d on m, m 2, parallel identical machines to minimize total tardiness is studied. Approximability issues are discussed and two families of algorithms {A } and {B } are presented such that (T ⁰ – T*)/(T* + d) holds for any problem instance and any given > 0, where T* is the optimal solution value and T ⁰ is the value of the solution delivered by A or B . Algorithms A and B run in O(n ^2m / ^m–1) and O(n ^m+1/ ^m ) time, respectively, if m is a constant. For m = 2, algorithm A can be improved to run in O(n ³/) time.     

Singular perturbations of variational problems arising from a two-phase transition model

Given that, are two Lipschitz continuous functions of to ₊ and thatf(x, u, p) is a continuous function of × × ^N to [0, + [ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two points(x) and(x), we prove the convergence ofF (u) = (1/) f (x, u, Du) dx when the perturbation parameter goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.     

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ² u _xx –u=f(x), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for 1 because the homogeneous solutions are exp(±x/), which have boundary layers of thickness O(1/). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([x–1]/).) Two strategies for small are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f(x) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when is very, very tiny.     

Amalgams of Weyl algebras and theA (V, δ, Γ) conjecture

Summary We prove a long standing conjecture ([5, Conjecture 5.6]) concerning the algebrasA (V, , ). Namely, two such algebrasA (V, , ),A (W, , ) are isomorphic if and only if there is an isomorphism between the triples (V, , ), (W, , ) from which they are constructed. As a consequence, to each primitive ideal in the enveloping algebra of a solvable Lie algebra there is associated a unique (V, , ).     

Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

For a sectorial operator A with spectrum (A) that acts in a complex Banach space B, we prove that the condition (A) i R = Ø is sufficient for the differential equation where is a small positive parameter, to have a unique bounded solution x for an arbitrary bounded function f: R B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as 0+ to the unique bounded solution of the differential equation x(t) = Ax(t) + f(t).     

Asymptotic behavior of constrained stochastic approximations via the theory of large deviations

Summary Let G be a bounded convex set, and _G the projection onto G, and a bounded random process. Projected algorithms of the types , where 0<a _n0, a _n =) occur frequently in applications (among other places) in control and communications theory. The asymptotic convergence properties of {X _n } as 0, n, have been well analyzed in the literature. Here, we use large deviations methods to get a more thorough understanding of the global behavior. Let be a stable point of the algorithm in the sense that X _n in distribution as 0, n. For the unconstrained case, rate of convergence results involve showing asymptotic normality of , and use linearizations about . In the constrained case is often on G, and such methods are inapplicable. But the large deviations method yields an alternative which is often more useful in the applications. The action functionals are derived and their properties (lower semicontinuity, etc.) are obtained. The statistics (mean value, etc.) of the escape times from a neighborhood of are obtained, and the global behavior on the infinite interval is described.Research has been supported in part by the US Army Research Office under Contract #DAAG 29-84-K-0082, and in part by the Office of Naval Research under Contract #N00014-83-K0542Research has been supported in part by the National Science Foundation Grant #ECS 82-11476, and the Air Force Office of Scientific Research under Contract #AF-AFOSR 81-0116     

On differentiability of SRB states for partially hyperbolic systems

Consider a one parameter family of diffeomorphisms f such that f ₀ is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let be any u-Gibbs state for f . We prove (Theorem 1) that for any C function A the map (A) is differentiable at =0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to . We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

On the formulation of mechanical balance laws for structured continua

Force and moment balance laws are derived for structured continua. The approach consists in the use of two hypotheses: the availability of a microscope, mathematized by the introduction of a scale parameter; the existence of total and fine power expenses and the invariance of the corresponding power functionals in the limit 0+.

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Sommario Vengono dedotte leggi di bilancio delle forze e dei momenti per continui con struttura. L'approccio consiste nell'uso di due ipotesi: la disponibilità di un microscopio, resa matematicamente tramite l'introduzione di un parametro di scala; l'esistenza di forme della potenza spesa totale e fine, e l'invarianza di queste forme per 0+.

     

The generalized inverse of a perturbed singular matrix

It is shown that, whenA is singular, (A+A ₁)^–1 can be expanded into a Laurent's series in. The coefficients of the expansion are given in an explicit form. The case where A+A ₁ vanishes identically in is also studied and a generalized inverse ofA+A ₁ is given.

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Zusammenfassung Es wird gezeigt, daß wennA singulär ist (A+A ₁)^–1 auf eine Laurent-Serie in erweitert werden kann. Die Koeffizienten der Erweiterung werden in expliziter Form gegeben. Der Fall wo A+A ₁ sich identisch in auflöst wird ebenfalls untersucht und eine verallgemeinerte Umkehrung vonA+A ₁ wird angegeben.

     

Über Abstand 1 erhaltende Abbildungen in Minkowski-Ebenen

By using a computer the following theorem is proved: Consider K=GF(q), q {32,64,81,128}, :K² K² bijective such that P,Q K². Then is a semi-isometry. The assumption bijective can be dropped if q {32,128}.     

Poisson kernel is the unique approximate identity,asymptotically multiplicative on H∞

Let for anyf H(R), where (x): = ^–1(x^–1). Then (x) P (x + h) for some h R and > 0; P denotes the Poisson kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 82–89, 1989.