bluuebunny/crossref_metadata_2025 huggingface dataset:
Techniques of biliary reconstruction following bile duct resection (with video)
Authors: Hirano Satoshi , Tanaka Eiichi , Tsuchikawa Takahiro , Matsumoto Joe , Shichinohe Toshiaki , Kato Kentaro
Published: November 2011

In several clinical situations, including resection of malignant or benign biliary lesions, reconstruction of the biliary system using the Roux‐en‐Y jejunum limb has been adopted as the standard procedure. The basic technique and the procedural knowledge essential for most gastroenterological surgeons are described in this article, along with a video supplement. Low complication rates involving anastomotic insufficiency or stricture can be achieved by using proper surgical techniques, even following small bile duct reconstruction. Using the ropeway method to stabilize the bile duct and jejunal limb allows precise mucosa‐to‐mucosa anastomosis with interrupted sutures of the posterior row of the anastomosis. Placement of a transanastomotic stent tube is the second step. The final step involves suturing the anterior row of the anastomosis. In contrast to the lower extrahepatic bile duct, the wall of the hilar or intrahepatic bile duct can be recognized within the fibrous connective tissue in the Glissonean pedicle. The portal side of the duct should be selected for the posterior wall during anastomosis owing to its thickness. Meticulous inspection to avoid overlooking small bile ducts could decrease the chance of postoperative intractable bile leakage. In reconstruction of small or fragile branches, a transanastomotic stent tube could work as an anchor for the anastomosis.
Ultrasound of a distended pediatric abdomen in a limited resource setting
Authors: Das Devjani , Lema Penelope Chun , Datta Anita
Published: November 2011

Background

Abdominal distension among the pediatric population is a frequently encountered diagnostic dilemma. In rural areas, the lack of resources limits the physician’s ability to make a definitive diagnosis and initiate appropriate treatment. The use of point-of-care ultrasound as a diagnostic tool has become more widespread due to its safety, portability and relatively low costs.

Case report

We present a case of an emergency medicine where physician performed an ultrasound to diagnose the etiology of a child’s distended abdomen in a remote area of Tanzania.
Grainsize dynamics of polydisperse granular segregation down inclined planes
Authors: Marks Benjy , Rognon Pierre , Einav Itai
Published: November 2011

Granular materials segregate by size when sheared, which increases the destructive power in avalanches and causes demixing in industrial flows. Here we present a concise theory to describe this phenomenon for systems that for the first time include particles of arbitrary size. The evolution of the grainsize distribution during flow is described based on mass and momentum conservation. The theory is derived in a five-dimensional space, which besides position and time, includes a grainsize coordinate. By coupling the theory with a simple constitutive law we predict the kinematics of the flow, which depends on the grainsize dynamics. We show that the underpinning mechanism controlling segregation is a stress variation with grainsize. The theory, solved by a finite difference scheme, is found to predict the dynamics of segregation consistent with results obtained from discrete element simulations of polydisperse granular flow down inclined planes. Moreover, when applied to bimixtures, the general polydisperse theory reveals the role of grainsize contrast.
Harvest Shortfalls, Grain Prices, and Famines in Preindustrial England
Authors: CAMPBELL BRUCE M.S. , GRÁDA CORMAC Ó
Published: November 2011

The frequency of bad harvests and price elasticity of demand are measured using new data on English grain yields 1268–1480 and 1750–1850 and a revised price series. The analysis shows that major harvest shortfalls were a significant component of most historical subsistence crises, as back-to-back shortfalls were of the worst famines. Although serious harvest shortfalls long remained an unavoidable fact of economic life, by c.1800 yields had become less variable and prices less harvest sensitive. By the eve of the Industrial Revolution, England had become effectively famine-free.
Warfare, Taxation, and Political Change: Evidence from the Italian Risorgimento
Authors: DINCECCO MARK , FEDERICO GIOVANNI , VINDIGNI ANDREA
Published: November 2011

We examine the relationships between warfare, taxation, and political change in the context of the political unification of the Italian peninsula. Using a comprehensive new database, we argue that external and internal threat environments had significant implications for the demand for military strength, which in turn had important ramifications for fiscal policy and the likelihood of constitutional reform and related improvements in the provision of nonmilitary public services. Our analytic narrative complements recent theoretical and econometric works about state capacity. By emphasizing public finances, we also uncover novel insights about the forces underlying state formation in Italy.

“The budget is the skeleton of the state, stripped of any misleading ideologies.”

Sociologist Rudolf Goldscheid, 19261
Depth to Bedrock and the Formation of the Manhattan Skyline, 1890–1915
Authors: BARR JASON , TASSIER TROY , TRENDAFILOV ROSSEN
Published: November 2011

New York City historiography holds that Manhattan developed two business centers—downtown and midtown—because the bedrock is close to the surface at these locations, with a bedrock “valley” in between. This article is the first effort to measure the effect of depth to bedrock on construction costs and the location of skyscrapers. We find that while depth to bedrock had a modest effect on costs (up to 7 percent), it had relatively little influence on the location of skyscrapers.

“Hour by hour the caissons reach down to the rock of the earth and hold the building to a turning planet.”

Carl Sandburg, Skyscraper
Small Machines in the Garden: Everyday technology and revolution in the Mekong Delta
Authors: BIGGS DAVID
Published: November 2011

Twentieth-century industrialization in the agricultural landscapes of the Mekong Delta in Vietnam took a very different form from other places, characterized less by a continuous spread of large-scale technology than by its destruction in mid-century and the subsequent spread of small technology which powered scooters, water pumps, and boats. The numbers of these portable motors, an everyday technology in 1960, rose from a few thousand units in 1963 to millions in the present day. The colonial and post-colonial state in Vietnam played a key role in the demise of large technology and, ultimately, of the water infrastructure. Its failures during wartime spurred farmers to adopt cheap, small engines to survive; however, the state's role was complex during this time. Several key factors, including the influence of American aid programmes and the contributions of Taiwanese agricultural advisers, especially those pushing high-yield rice, favoured the adoption of small engines. From an ecological viewpoint, the post-1960 explosion in the use of small motors, especially as water pumps, has brought people and states in Southeast Asia to an ecological impasse as unrestricted use has impacted on water tables, salinity levels, and the long-term sustainability of agriculture in many places. This paper examines the state's indirect role in shaping this silent revolution, and it considers the political and ideological factors underpinning its history.
Stiffness analysis of a compliant precision positioning stage
Authors: Jia X. , Liu J. , Tian Y. , Zhang D.
Published: November 2011

In this paper a 3-degree of freedom compliant parallel positioning stage utilizing flexure hinges is explored for nanoimprint lithography. The performance of the stage is extensively analyzed using a pseudo-rigid body model and finite element method. The position and velocity models are established. Accordingly, the stiffness at driving point of the active arm is obtained on the basis of Castigliano's first theorem. The system stiffness of the compliant stage is explored using the compliant matrix methodology and matrix transformation, and the influence of the geometry parameters on stiffness factors in three directions has been graphically evaluated as well. Finite element analysis has been conducted to verify that the established models faithfully predict device performance.
Self-organizing approach for learning the forward kinematic multiple solutions of parallel manipulators
Authors: Assal Samy F. M.
Published: November 2011

Contrary to the inverse kinematics, the forward kinematics of parallel manipulators involves solving highly non-linear equations and provides more than one feasible end-effector pose, which are called the assembly modes, for a given set of link lengths or joint angles. Out of the multiple feasible solutions, only one solution can be achieved from a certain initial configuration. Therefore, in this paper, the Kohonen's self-organizing map (SOM) is proposed to learn and classify the multiple solution branches of the forward kinematics and then provide a unique real-time solution among the assembly modes. Each solution of the multiple feasible ones is coded using IF-THEN rules based on the values of the passive joint variables. Due to not only the classification but also the associative memory learning abilities of the SOM, the passive joint variables vector, the end-effector pose vector, and this class code are associated with the active joint variables vector constituting the input vector to the SOM in the offline learning phase. In the online testing phase, only the active joint variables vector and the class code are fed to the SOM to obtain the unique end-effector pose vector. The Jacobian matrix calculated at the SOM output layer is used for further fine tuning this output to obtain an accurate end-effector pose vector. Simulations are conducted for 3-RPR and 3-RRR planar parallel manipulators to evaluate the performance of the proposed method. The results proved high accuracy of the desired unique solution in real-time.
Classes of groups related to Fa,b,c
Authors: Campbell C. M. , Robertson E. F.
Published: November 2011

Synopsis

Groups having two generators and two relations are studied. The Reidemeister-Schreier algorithm is used to determine presentations for their derived groups. This enables the orders of the groups to be found. Necessary and sufficient conditions are given for the groups to be metabelian. Certain classes closely related to the class are also discussed.
Cyclic Banach spaces and reflexive operator algebras
Authors: Gillespie T. A.
Published: November 2011

Synopsis

Let X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.
On the location of eigenvalues of second order linear differential operators
Authors: Knowles Ian
Published: November 2011

Synopsis

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.
The condition of gram matrices and related problems
Authors: Taylor J. M.
Published: November 2011

Synopsis

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.
A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology
Authors: Lodge A. S. , McLeod J. B. , Nohel J. A.
Published: November 2011

Synopsis

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.
Convergence to travelling fronts in semilinear parabolic equations
Authors: Rothe Franz
Published: November 2011

Synopsis

We study the convergence to the stationary state for the parabolic equation u, = uxx + F(u). There exist wave-type solutions u(x, t) = φ(x − ct) for a continuum of velocities c. In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data.

All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.
Acoustic tunnelling
Authors: Jones D. S.
Published: November 2011

Synopsis

General formulae are obtained for the reflection and transmission of harmonic acoustic waves by a curved interface between two media when the frequency is high. In addition to refracted rays there turn up tunnelling rays, if the surface is concave to the source, which are emitted from an evanescent region when the phenomenon of total internal reflection would be anticipated. Uniformly valid formulae dealing with the transition from refraction to tunnelling in both transmission and reflection are derived.

The theory is applied to the circular cylinder and to the top-hat circular jet. In the latter case it is suggested that radiation may tend to be more significant at inclinations of 50°-65° (downstream) and 25°-40° (upstream) to the axis of the cylinder. The augmentation due to tunnelling rays in propagation upstream is mentioned.
Weak containment and Clifford semigroups
Authors: Paterson Alan L. T.
Published: November 2011

Synopsis

Let S be a Clifford semigroup with identity. The weak containment question is posed for S, and answered affirmatively when each of the maximal groups Se in S is amenable. The amenability of S itself is characterised in terms of PL(S), the set of normalised positive definite functions on S arising from the left regular representation of S. A type of mean associated with PL(S) and satisfying a condition weaker than left invariance is introduced.
Asymptotic formulas for solutions of a singular linear ordinary differential equation
Authors: Gilbert Richard C.
Published: November 2011

Synopsis

By use of the theory of asymptotic expansions for first-order linear systems of ordinary differential equations, asymptotic formulas are obtained for the solutions of annth order linear homogeneous ordinary differential equation with complex coefficients having asymptotic expansions in a sector of the complex plane. These asymptotic formulas involve the roots of certain polynomials whose coefficients are obtained from the asymptotic expansions of the coefficients of the differential operator.
Reachable states in boundary control of the heat equation are independent of time
Authors: Fattorini H. O.
Published: November 2011

Synopsis

It is known that the class of all reachable states in boundary control of systems described by parabolic equations in one space dimension is independent of the time during which control is applied. This result is generalized here to systems governed by the heat equation in an arbitrary number of space variables.
On a class of weighted Sobolev spaces and the minimization of quadratic forms
Authors: Penning Frans , Sauer Niko
Published: November 2011

Synopsis

In this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the spaceRnor a half-space ofRn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.
On relaxation oscillations in a forced van der Pol oscillator
Authors: Habets P.
Published: November 2011

Synopsis

Relaxation oscillations of a forced van der Pol oscillator are considered. The limiting behaviour of such a solution is investigated using a priori estimates. This simplifies previous approaches of the problem.
The treatment of a Neumann boundary value problem for force-free fields by an integral equation method
Authors: Kress Rainer
Published: November 2011

Synopsis

A Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.
A coordinatisation of implicative (semi-)lattices by Baer semigroups
Authors: Blyth T. S.
Published: November 2011

Synopsis

We show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.
The essential self-adjointness of differential operators
Authors: Keller R. G.
Published: November 2011

Synopsis

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions have to be imposed only in an increasing sequence of annular regions surrounding the origin.
Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks
Authors: Boillatt Guy , Ruggeri Tommaso
Published: November 2011

Synopsis

An incident wave creates a discontinuity in the acceleration of the shock front. The amplitudes of the reflected and transmitted waves are also determined. Special attention is given to the case of the weak shocks and the characteristic shocks.
Hyponormal and essentially normal operators
Authors: Putnam C. R.
Published: November 2011

Synopsis

Let T be a hyponormal operator on a Hilbert space, so that T*T – TT*≧ 0. Let T have the Cartesian representation T = H + iJ where H has the spectral family {Et} and suppose that EtJ − JEt is compact for almost all t on a Borei set α satisfying E(α) = I. The principal result (Theorem 3) is that under these hypotheses T must be normal. In case T is hyponormal and essentially normal some sufficient conditions are given assuring that, for a fixed t, EtJ − JEt is compact.
Solution of dual integral equations of Titchmarsh type using generalised functions
Authors: McBride Adam C.
Published: November 2011

Synopsis

The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.
Some comments on a singular elliptic problem associated with a rectangle
Authors: Nye V. A.
Published: November 2011

Synopsis

The construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.
Boundary value problems for linear systems
Authors: Neuberger J. W.
Published: November 2011

Synopsis

Suppose H and K are Hilbert spaces and H′0, H′ are closed subspaces of H so that H′0 ⊂ H′. Denote by P the orthogonal projection of H onto H′0, denote by g an element of k and by C a bounded linear transformation from H to K so that CC* = I, the identity on K. Denote CPC* by M. Given w in H′ one has the problem of finding u in H′ so that

There are given conditions on M (or certain operators related to M) which imply convergence of a certain iteratively generated sequence to a solution to this problem. The equation Cu = g represents an inhomogeneous system of linear differential equations (ordinary, partial or functional) and the condition P(u − w) = u − w is an abstract representation of inhomogeneous boundary conditions for u.
On the location of the essential spectra and regularity fields of complex Sturm—Liouville operators
Authors: Race David
Published: November 2011

Synopsis

In this paper the Sturm-Liouville expression τy= −(py′)′ +qy, with complex-valued coefficients is considered, and a number of results concerning the location of the essential spectrum of associated operators are obtained. Some of these are extensions or generalizations of results due to Birman, and Glazman, whilst others are new. These lead to criteria for the non-emptiness of the regularity field of the corresponding minimal operator—a condition which is needed in the theory ofJ-selfadjoint extensions. A complete determination of the regularity field is made when the equation τy= λ0yhas two linearly independent solutions inL2[a,∞) for some complex λ0.
Nonlinear eigenvalue problems for the whirling of heavy elastic strings
Authors: Antman Stuart S.
Published: November 2011

Synopsis

This paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.
Non-trivial solutions of elliptic equations at resonance
Authors: Thews Klaus
Published: November 2011

Synopsis

In this paper we show that Dirichlet problems at resonance, being of the type −Δu(x) = λku(x) + g(u(x)), x є G, u(x) = 0 for x є ∂G, g(−u) = −g(u), admit multiple non-trivial solutions provided the non-linearity interacts in some sense with the spectrum of −Δ. In contrast to other work on this subject we deal with the case that g(u) is very small for large arguments, for instance g(u) = 0 for |u| large. On the other hand if and g satisfies a certain concavity condition at 0 the existence of infinitely many solutions is shown independent of the asymptotic behaviour of g.
Monotone continuity of the spectral resolution and the eigenvalues
Authors: Weidmann Joachim
Published: November 2011

Synopsis

Let (Tn) be a non-decreasing sequence of self-adjoint operators which are semi-bounded from below and converge to some self-adjoint operator T in the sense of strong resolvent convergence. For every λ which is eventually below the essential spectrum of Tn it is shown that ‖En(λ)−E(λ)‖→0 for n→∞, where En(·) and E(·) are the spectral resolution of Tn and T, respectively.
Normal AW*-algebras
Authors: Maitland Wright J. D.
Published: November 2011

Synopsis

In recent years it has become clear that AW*-algebras can be much more pathological and unlike von Neumann algebras than was originally expected. When AW*-algebras are monotone complete, then the work of Kadison and Pederson shows that a particularly smooth and elegant theory can be developed. A technically weaker requirement on an AW*-algebra is that it be “normal”. This condition, which says that the lattice of projections is embedded in a well-behaved way in the partially ordered set of all self-adjoint elements, can sometimes be used as a substitute for monotone completeness. In this note we prove that when an AW*-algebra is of finite type (that is x*x = 1 implies xx* = 1) then it is normal.
Existence of periodic orbits of autonomous ordinary differential equations
Authors: Smith Russell A.
Published: November 2011

Synopsis

The Poincaré-Bendixson theorem, concerning the existence of periodic orbits of plane autonomous systems, is extended to higher order systems under certain conditions. Under similar conditions, a complementary theorem on the existence of recurrent orbits is also proved. For the feedback control equation, these conditions are reduced to a form which can be easily verified in practice.
Gewöhnliche lineare Differentialgleichungen n−ter Ordnung mit Distributionskoeffizienten
Authors: Pfaff Rainer
Published: November 2011

Synopsis

We give a formula (4) for a variety of ordinary linear differential equations of order n with distributional coefficients. There appear as coefficients distributions of order k ≦ n/2, i.e. these distributions are kth distributional derivatives of locally L-integrable functions. With a suitable transformation (7) the differential equations can be transformed into first order systems (8) with integrable coefficients. From this follows the existence of a continuous solution, which can be uniquely determined by proper initial conditions.

The coefficients in the differential equations considered are chosen as general as possible but such that a transformation into a system with integrable coefficients can be performed, and that all products are defined by Leibniz' formula.
Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter
Authors: Dijksma Aalt
Published: November 2011

Synopsis

In provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.
Weighted inequalities for the Stieltjes transformation and Hubert's double series
Authors: Andersen Kenneth F.
Published: November 2011

Synopsis

The non-negative weight functions U(x), V(x) for which the generalized Stieltjes transformation Sλ is a bounded operator from LP(V) into La(U) are characterized. This extends certain classical inequalities and some recent results of Erdélyi. In particular, our results provide weighted extensions of a classical inequality known as Hubert's double series theorem.
N-series and tame near-rings
Authors: Lyons C. G. , Meldrum J.D.P.
Published: November 2011

Synopsis

Let N be a zero-symmetric near-ring with identity and let G be an N-group. We consider in this paper nilpotent ideals of N and N-series of G and we seek to link these two ideas by defining characterizing series for nilpotent ideals. These often exist and in most cases a minimal characterizing series exists. Another special N-series is a radical series, that is a shortest N-series with a maximal annihilator. These are linked to appropriate characterizing series. We apply these ideas to obtain characterizing series for the radical of a tame near-ring N, and to show that these exist if either G has both chain conditions on N-ideals or N has the descending chain condition on right ideals. In the latter case this provides a new proof of the nilpotency of the radical of a tame near-ring with DCCR, and an internal method for constructing minimal and maximal characterizing series for the radical.
Maximum modulus algebras and singularity sets
Authors: Wermer John
Published: November 2011

Synopsis

A classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.
Asymptotic methods for fourth-order differential equations
Authors: Grudniewicz C. G. M.
Published: November 2011

Synopsis

A new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equation

wherem, nare integers and 1 ≦m,n≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.
An inverse for the Gohberg-Krupnik symbol map
Authors: Costabel Martin
Published: November 2011

Synopsis

It is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.
Critical growth problems for polyharmonic operators
Authors: Gazzola Filippo
Published: November 2011

We prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower-order perturbations of the critical term. The results highlight the phenomenon of bifurcation of the critical dimensions discovered by Pucci and Serrin; moreover, we show that another bifurcation seems to appear for ‘nonresonant’ dimensions.
Eigenvalue ratios and eigenvalue gaps of Sturm–Liouville operators
Authors: Law C. K. , Huang Yu-Ling
Published: November 2011

We prove optimal lower bounds for arbitrary eigenvalue ratios (μm/μn) of the Sturm–Liouville operator with Dirichlet and Neumann boundary conditions. These imply optimal bounds for the eigenvalue gaps (μm – μn) of the corresponding problem. The method can be generalised to consider general separated endpoint boundary conditions.
On the Titchmarsh–Weyl m -function
Authors: Harris B. J.
Published: November 2011

Synopsis

The Weyl theory assets that there exist circles, possibly degenerate, in the complex plane which are associated with Sturm–Liouville equations. Closely related to these circles is the Titchmarch–Weyl m-function which plays a major role in the theory of Sturm-Liouville equations. In this paper, we investigate the asymptotic behaviour of the m-function and derive results of the form

for λ in a certain subset of the compledx plane.
Hyperjacobians, determinantal ideals and weak solutions to variational problems
Authors: Olver Peter J.
Published: November 2011

Synopsis

The problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.
Bounds for the eigenvalues of separated Dirac operators
Authors: Harris B. J.
Published: November 2011

Synopsis

We develop a systematic method for calculating real numbers Λ1, and Λ2 such that there are no non-trivial solutions of the equation

which belong to L2(a, ∞) for λ> Λ1, or λ < Λ2. We also give conditions under which this equation has no solutions in L2(a,∞) for any real number λ.
On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
Authors: Jeanjean Louis
Published: November 2011

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form

We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.
The lattice of full regular subsemigroups of a regular semigroup
Authors: Johnston Katherine G. , Jones Peter R.
Published: November 2011

Synopsis

Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.

A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.
The number of limit cycles of certain polynomial differential equations
Authors: Blows T. R. , Lloyd N. G.
Published: November 2011

Synopsis

Two-dimensional differential systems

are considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.
Some multipoint extensions of the Rayleigh–Ritz inequality
Authors: Brown Richard C. , Vrkoč Ivo
Published: November 2011

Synopsis

We obtain the best constants for multipoint extensions of the Rayleigh–Ritz inequality, and use our results to find good upper bounds for the constants involved in the standard spline approximation and interpolation error bound theory.
An extension of the Liouville–Green asymptotic formula for oscillatory second-order differential equations
Authors: Cassell J. S.
Published: November 2011

Synopsis

Conditions, generalizing the usual Liouville–Green conditions, are obtained under which the equation {r(x)y′}′ + q(x)y = 0, where q and r are positive and have derivatives of a sufficiently high order, has solutions of the form y ∼ (rq)–¼e10. 0 ∼ f (q/r)½ dx as x →∞. A related result is obtained for a system of two first-order equations y′ = A(x)y, where the eigenvalues of A are pure imaginary.
Bifurcations near a multiple eigenvalue of the rectangular plate problem
Authors: Sadovský Zoltán
Published: November 2011

Synopsis

We consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue andZ2⊓Z2symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.
On the boundary operator in electromagnetic scattering
Authors: Kress Rainer
Published: November 2011

Synopsis

For radiating solutions to the time-harmonic Maxwell equations, it is shown that the boundary operator mapping the tangential components of the electric field into the tangential components of the magnetic field is a bounded bijective operator from the space of Holder continuous tangential fields with Hölder continuous surface divergence onto itself.
An optimal control problem in exterior hydrodynamics
Authors: Sritharan S. S.
Published: November 2011

Synopsis

In this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure. An existence theorem for the speed trajectory which corresponds to the absolute minimum is provided. The results are valid for arbitrary Reynolds numbers.
Critical semilinear biharmonic equations in RN
Authors: Noussair Ezzat S. , Swanson Charles A. , Jianfu Yang
Published: November 2011

Synopsis

An existence theorem is obtained for a fourth-order semilinear elliptic problem in RNinvolving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embeddingL2N/(N–4)(RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be
On Sommerfeld radiation conditions for the diffraction problem with two unbounded media
Authors: Roach G. F. , Zhang Bo
Published: November 2011

Synopsis

A radiation condition is obtained, and is then used together with weighted Sobolev spaces and the limiting absorption method to establish the unique existence of solutions to the diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.
On an extension of Copson's inequality for infinite series
Authors: Brown B. M. , Evans W. D.
Published: November 2011

Synopsis

In 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δan = an+1 – an and Δ2an = Δ(Δan), then is convergent and the constant 4 being best possible. Equality occurs if and only if an = 0 for all n. In this paper we give a result that extends Copson's result to inequalities of the form

where Mxn =–Δ(pn_l Δxn_l)+qnxn (n = 0, 1, …). The validity of such an inequality and the best possible value of the constant K are determined in terms of the analogue of the Titchmarsh-Weyl m-function for the difference equation Mxn = λwnxn (n = 0, 1, …).
On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order
Authors: Gundersen Gary G.
Published: November 2011

Synopsis

We show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.
A remark on comparison results via symmetrization
Authors: Alvino A. , Lions P.L. , Trombetti G.
Published: November 2011

Synopsis

In this paper, we study the converse of comparison results for solutions to linear second-order elliptic equations. Namely, in the inequalities proved by G. Talenti and others, we study the case of equality and prove that “equalities are achieved only in the spherical situation”. We also present some applications of these results to semilinear elliptic equations.
On integral transforms whose kernels are solutions of singular Sturm–Liouville problems
Authors: Zayed Ahmed I.
Published: November 2011

Synopsis

In this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known that

where the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.
On the solutions of the first boundary value problem for the linear parabolic equations
Authors: Sinestrari Eugenio , von Wahl Wolf
Published: November 2011

Synopsis

The first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and Hölder-continuous only with respect to the space variables.
Parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear Sturm–Liouville boundary value problems
Authors: Cantrell Robert Stephen
Published: November 2011

Synopsis

The set of solutions to the two-parameter system

has been shown in a preceding paper of the author to exhibit a topological-functional analytic structure analogous to the structure of solution sets for nonlinear Sturm–Liouville boundary value problems. As the parameter λ and µ are varied, transitions in the solution set occur, first from trivial solutions to solutions (u, 0) with u having n nodes on (a, b) or solutions (0, v) with v having m nodes on (a, b), and then to solutions of the form (u, v), where u has n nodes on (a, b) and v has m nodes on (a, b), with n possibly different from m. Moreover, each transition is global in an appropriate bifurcation theoretic sense, with preservation of nodal structure. This paper explores these phenomena more closely, focusing on the range of parameters (λ, µ) for the existence of solutions (u, v) with u having n nodes on (a, b) and v having m nodes on (a, b) and its dependence on the assumptions placed on the coupling functions f and g. The principal tools of the analysis are the Alexander–Antman Bifurcation Theorem and a priori estimate techniques based on the maximum principle.
On polynomial interpolation at the points of a geometric progression
Authors: Schoenberg I. J.
Published: November 2011

Synopsis

This note pursues two aims: the first is historical and the second is factual.

1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.

2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.
On the L2 nature of solutions of nth order symmetric differential equations and McLeod's conjecture
Authors: Paris R. B. , Wood A. D.
Published: November 2011

Synopsis

We investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.
Some Pontrjagin rings, I
Authors: Hubbuck J. R.
Published: November 2011

Synopsis

The H-spaces considered have no homology p-torsion and are rationally equivalent as H-spaces to products of even dimensional Eilenberg–Maclane spaces. We obtain conditions which ensure that if the cohomology with coefficients in the ring of integers localized at the prime p is a polynomial algebra, then the Pontrjagin ring with these same coefficients is polynomial. A topological consequence is that BSUP has just one homotopy associative, homotopy commutative H-structure.
Asymptotic theory and deficiency indices for differential equations of odd order
Authors: Eastham M. S. P.
Published: November 2011

Synopsis

An asymptotic theory is developed for linear differential equations of odd order. The theory is applied to the evaluation of the deficiency indicesN+andN−associated with symmetric differential expressions of odd order. General conditions on the coefficients are given under which all possible values ofN+andNsubject to |N+−N| ≦ 1 are realized.
The number of positive solutions of a non-linear problem with discontinuous non-linearity
Authors: Douchet Jacques
Published: November 2011

Synopsis

For the non-linear problem

where f is a discontinuous function at 1, we show that the number of non-trivial positive solutions, for a given real number λ≧0, is related to the graph of a continuous function g. Then, by studying the function g it is possible in some special cases to give, for any λ≧0, the minimal or exact number of non-trivial positive solutions.