On heat conductors with a stationary hot spot

We consider a heat conductor having initial constant temperature and zero boundary temperature at every time.The hot spot is the point at which temperature attains its maximum at each given time. For convex conductors, if the hot spot does not move in time, we prove symmetry results for planar triangular and quadrangular conductors.Then, we examine the case of a general conductor and, by an asymptotic formula, we prove that, if there is a stationary critical point, not necessarily the hot spot, then the conductor must satisfy a geometric condition. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Mathematics Subject Classification (2000) Primary 35K05, 35K20, 35J05; Secondary 35J25, 35B38, 35B40     

The conductor of one-dimensional gorenstein rings in their blowing-up

Let (A,M) be a local, one-dimensional, Cohen-Macaulay ring of multiplicity e=e(A)>1 and Hilbert function H(A). Let I=Ann_A (B/A) be the conductor of A in its blowing up B. Northcott and Matlis have proved that if the embedding dimension emdim A of A is 2 then I=M^e−1 [3; Corollary 13.8]. If emdim A>2 little is know about I. In [6] and [7] I is computed when the associated graded ring G(A) is reduced (in this case B in the integral closure of A). In this paper we compute I when A is Gorenstein. There are in general upper and lower bounds for I in terms of a power of M and we start discussing when these bounds are attained. In particular we show that in the extremal situation I=M^e−1 one has emdimA=2 (thus inverting the result of Northcott and Matlis). Then we consider the case of Gorenstein rings. We prove that if G(A) in Gorenstein then I=M^ϑ where ϑ=Min{n‖H(n)=e}. If more generally A is Gorenstein then I⊂M² or emdim A=e(A)=2. When A is the local ring of a curve at a singular point p we get, as a consequence of this last result a proof of the following conjecture of Catanese which has interesting geometric applications [1]: if the conductor J of A in its normalization is not contained in M² then p is a node.     

A two stage stochastic equilibrium model for electricity markets with two way contracts

This paper investigates generators’ strategic behaviors in contract signing in the forward market and power transaction in the electricity spot market. A stochastic equilibrium program with equilibrium constraints (SEPEC) model is proposed to characterize the interaction of generators’ competition in the two markets. The model is an extension of a similar model proposed by Gans et al. (Aust J Manage 23:83–96, 1998) for a duopoly market to an oligopoly market. The main results of the paper concern the structure of a Nash–Cournot equilibrium in the forward-spot market: first, we develop a result on the existence and uniqueness of the equilibrium in the spot market for every demand scenario. Then, we show the monotonicity and convexity of each generator’s dispatch quantity in the spot equilibrium by taking it as a function of the forward contracts. Finally, we establish some sufficient conditions for the existence of a local and global Nash equilibrium in the forward-spot markets. Numerical experiments are carried out to illustrate how the proposed SEPEC model can be used to analyze interactions of the markets.     

Pathwidth of outerplanar graphs

We are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin [3], after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a constant c such that the pathwidth of every biconnected outerplanar graph is at most c plus the pathwidth of its dual. They also conjectured that this was actually true with c being one for every biconnected planar graph. Fomin [10] proved that the second conjecture is true for all planar triangulations. First, we construct for each p ≥ 1, a biconnected outerplanar graph of pathwidth 2p + 1 whose (geometric) dual has pathwidth p + 1, thereby disproving both conjectures. Next, we also disprove two other conjectures (one of Bodlaender and Fomin [3], implied by one of Fomin [10]. Finally we prove, in an algorithmic way, that the pathwidth of every biconnected outerplanar graph is at most twice the pathwidth of its (geometric) dual minus one. A tight interval for the studied relation is therefore obtained, and we show that all cases in the interval happen. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 27–41, 2007     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation

In shock loading computations based on an implicit finite-difference scheme, the surfaces of velocity discontinuity and the discontinuity sizes are determined by computing an asymptotic (ray) expansion of the solution behind the front surfaces at every time step. The method for constructing ray expansions is based on a recurrence formulation of the geometric and kinematic consistency conditions for discontinuities of the derivatives of functions that are discontinuous on a moving surface. The algorithm is illustrated by computing a simple example of the out-of-plane motion of an incompressible elastic medium.     

Local maxima of solutions to some nonsymmetric reaction‐diffusion systems

In this study, we examine the solution profile of some reaction‐diffusion systems. The systems are derived after approximating the Arrhenius term in our system which models the sintering process and consists of two coupled equations in terms of two unknowns. The unknowns describe the temperature of the solid and the concentration of the fuel. We show the evolution over time of local temperature hot spots. The key idea is to use ‘microscopic scaling’ around the temperature hot spot at the initial time along with asymptotic analysis. We also provide some numerical results that support the efficiency of our analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Convex domains with stationary hot spots

Consider the problem of heat flow in a convex domain in ℝⁿ with Dirichlet boundary condition and constant initial temperature. We show that the solution has a fixed hot spot if the domain is invariant under the action of an essential symmetry group. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Pathwidth of Planar and Line Graphs

 We prove that for every 2-connected planar graph the pathwidth of its geometric dual is less than the pathwidth of its line graph. This implies that pathwidth(H)≤ pathwidth(H ^*)+1 for every planar triangulation H and leads us to a conjecture that pathwidth(G)≤pathwidth(G ^*)+1 for every 2-connected graph G. Received: May 8, 2001 Final version received: March 26, 2002 RID="*" ID="*" I acknowledge support by EC contract IST-1999-14186, Project ALCOM-FT (Algorithms and Complexity - Future Technologies) and support by the RFBR grant N01-01-00235. Acknowledgments. I am grateful to Petr Golovach, Roland Opfer and anonymous referee for their useful comments and suggestions.     

Ranks of twists of elliptic curves and Hilbert’s tenth problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.     

Approximate Predictions of Ice Accretion on an Overhead Transmission Line which Rotates with Constant Angular Velocity

An airflow carrying either freezing-fog droplets or wet snowimpinging on an overhead transmission line causes ice to accreteon the windward side of the conductor. In the vicinity of theconductor anchors (at the towers), axial growth, in the formof a spike of ice growing outward into the wind, is observed.Near the centre of the span this eccentric ice loading causesthe conductor to twist so that cylindrical-sleeve growth isobserved. For small-diameter conductors of low torsional stiffnessthe conductor rotation can amount to many complete revolutions.It is useful to obtain some understanding of this mode of iceaccretion prior to undertaking any investigation of the aeroelasticsystem under icing conditions and it has direct relevance toice loading on small-diameter conductors. To this end, a simple‘icing-clock’ model for sleeve growth in which theconductor rotates with constant angular velocity is proposed,enablinganalytical results on the mechanics of the ice evolution processto be obtained.     

Homogenization and concentration for a diffusion equation with large convection in a bounded domain

We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.     

Asymptotic Analysis of a Nonlinear Problem in Thermoelasticity

In this paper we employ our previously formulated constitutive relations for an elastic heat conductor to examine the nonlinear influence of the temperature on the propagation of second sound. Utilizing the method of geometric optics we ascertain that large values of second sound travel more slowly than do smaller ones. As well, numerical results for a signaling problem are displayed graphically in order to highlight the qualitative properties of the temperature and strain waves. The choice of signaling data was motivated by experimental studies.     

Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. M. T. Barlow’s research was partially supported by NSERC (Canada), the twenty-first century COE Program in Kyoto University (Japan), and by EPSRC (UK). R. F. Bass’s research was partially supported by NSF Grant DMS-0601783. T. Kumagai’s research was partially supported by the Grant-in-Aid for Scientific Research (B) 18340027 (Japan).     

Numerical simulation of the butt-fusion welding process

This study concerns an industrial application of modelling andsimulation for the butt-fusion welding process. We concentrateon investigating evolutionary heat transfer in the immediatevicinity of the weld. The model will facilitate the predictionof process parameters under certain operating conditions and,in particular, should (a) predict visible failures, such asthe joint interface not reaching the fusion temperature of thepipe material, (b) predict less visible potential failures,such as internal hot spots which should solidify before thepipe is moved, and (c) determine near-optimal operating conditions.We fmus on the construction, validation, and robustness of anenthalpy-based model with reference to simplifying assumptions,comparison with measured data, and analysis of the numericalscheme.     

Second sound and multiple shocks in superfluid helium

A temperature pulse propagating in superfluid helium is studied through the simple waves theory. Our aim is to determine the shape change of this pulse, initially represented by a gaussian profile, using a generalized non-linear Cattaneo model proposed, in the framework of Extended Thermodynamics, by Ruggeri and co-workers in the case of a rigid conductor. The theoretical basis of our arguments is given in a paper [1] where the differential system of a binary mixture of Euler’s fluids is written as a system for a single heat conducting fluid. We prove that there exist three characteristic temperatures [(q)\tilde]\tilde{\theta}, playing an essential role in the shape change of the propagating second sound wave; in particular, several families of multiple shocks (i.e. usual double shocks, double shocks only ahead or behind the wave profile, and very strange quadri-shocks) can appear, depending on the relation among the unperturbed temperature of Helium II and the characteristic temperatures and, in some cases, on the wave’s amplitude. Both the cases of a hot wave and a cold wave are discussed, proving that this last process is not symmetric with respect to the previous one. Finally, suitable choices of some parameters are suggested in order to better point out the changes of the wave profile and, in particular, the formation of multiple shocks.     

On one problem of electrophysics with discontinuous nonlinearity

We consider a conductor heating problem in the following setting: a constant voltage and a constant temperature are maintained on the conductor surface, and the electric conductivity of the material experiences jumps when passing through certain temperatures. Earlier-obtained results for problems with a spectral parameter for equations of elliptic type with discontinuous nonlinearities are applied to this problem of electrophysics. We weaken the conditions imposed on the set of points of discontinuity of the nonlinearity (the electric conductivity of the conductor) occurring in the equation.