Time dependent neutron transport theory in multigroup-P L-approximation

Summary A method is developed for solving the time dependent neutron transport equation in multigroupP _L approximation for one-dimensional geometries. The partial differential equations in time and space are solved by means of a power series expansion in the spatial variable. The resulting ordinary differential equations are solved up to theNth order, and the last spatial coefficients are used to satisfy the boundary conditions. Integration of the purely time dependent differential equations is carried out by means of Lie series.Numerical oscillations, appearing for high ordersN, are avoided by subdividing each zone into smaller subzones. Even and odd spatial moments must be developed in opposite directions in each subzone, and the stationary solution representing the initial condition for the time dependent calculation must be developed in the same manner.Results of two calculations in spherical geometry are presented. One is the start-up of a small experimental reactor usingP ₃ theory, the other is a demonstration of neutron waves inP ₁ theory.

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Zusammenfassung Es wird eine Methode zur Lösung der zeitabhängigen Neutronentransportgleichung in der Multigruppen-P _L-Näherung für eindimensionale Geometrien entwickelt. Für die Lösung der partiellen Differentialgleichungen in Ort und Zeit wird eine Potenzreihe im Ort angesetzt. Das sich ergebende gewöhnliche Differentialgleichungssystem wird bis zu einer gewählten OrdnungN erfüllt, und die letzten Koeffizienten der Ortsentwicklungen dienen zur Befriedigung der Randbedingungen. Die Integration in der Zeit erfolgt dann mit der Lie-Reihen-Methode.Zur Vermeidung der Oszillationen, die bei hohen OrdnungenN auftreten, werden die einzelnen Zonen in kleinere Abschnitte unterteilt. Die geraden und ungeraden Momente müssen in den Abschnitten in entgegengesetzter Richtung entwickelt werden. Die stationäre Lösung, die als Anfangsbedingung für die zeitabhängige Rechnung dient, muss mit demselben Entwicklungsschema ermittelt werden.Die Methode wird angewendet zur Lösung derP ₁- undP ₃-Gleichungen in sphärischen Reaktoren. Zwei Beispiele werden damit berechnet: Das Anfahren eines kleinen Versuchsreaktors inP ₃-Näherung und die Nachweisung von Neutronenwellen inP ₁-Theorie.

     

Covering a set with homothets of a convex body

We consider two problems mentioned in the book “Research Problems in Discrete Geometry” (Brass et al. in research problems in discrete geometry, vol xii+499. Springer, New York, pp ISBN: 978-0387-23815-8; 0-387-23815-8, 2005). First, let K and L be given convex bodies in \mathbbR^d{\mathbb{R}^{d}} . We prove that if the total volume of a family of positive homothets of K is sufficiently large then they permit a translative covering of L. This problem, in the case when K = L and the dimension is two, was originally posed by L. Fejes Tóth. The previously known bound (Januszewski in proc. of the International scientific conference on mathematics, pp 29–34. Žilina, 1998) on the total volume (in the case when K = L) was of order d ^d vol(K), we prove a bound that is exponential in the dimension. The second problem is the following: Find a condition, in terms of the coefficients of homothety, that is necessary for a family of positive homothets of K to cover K. The problem was phrased by V. Soltan, who conjectured that the sum of the coefficients is at least d. We confirm an asymptotic version of this conjecture.     

Efficient Algorithms for Finding the Maximum Number of Disjoint Paths in Grids

In a rectangular grid, given two sets of nodes, (sources) and (sinks), of size each, the disjoint paths (DP) problem is to connect as many nodes in to the nodes in using a set of “disjoint” paths. (Both edge-disjoint and vertex-disjoint cases are considered in this paper.) Note that in this DP problem, a node in can be connected to any node in . Although in general the sizes of and do not have to be the same, algorithms presented in this paper can also find the maximum number of disjoint paths pairing nodes in and . We use the network flow approach to solve this DP problem. By exploiting all the properties of the network, such as planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the working grid (to be defined) from O(N²) to O(N^1.5) and solve the problem in O(N^2.5) time for both the edge-disjoint and vertex-disjoint cases, an improvement over the straightforward approach which takes O(N³) time.     

Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

We prove that arbitrary (nonpolynomial) scalar evolution equations of order    m  ≥ 7  , that are integrable in the sense of admitting the canonical conserved densities   ρ⁽¹⁾, ρ⁽²⁾  , and   ρ⁽³⁾   introduced in [ 1 ], are polynomial in the derivatives    u _{m −  i}    for  i  = 0, 1, 2. We also introduce a grading in the algebra of polynomials in     u _k     with     k  ≥  m  − 2    over the ring of functions in     x ,  t ,  u , … ,  u _{m −3}    and show that integrable equations are scale homogeneous with respect to this grading .     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

Influence of SiO2 in SiCp on the microstructure and impact strength of Al/SiCp composites fabricated by pressureless infiltration

The effect of SiO₂ in SiC_p and the following processing parameters on the microstructure and impact strength of Al/SiC_p composites fabricated by pressureless infiltration was investigated: Mg content in the aluminum alloy, SiC particle size, and holding time. Preforms of SiC_p in the form of rectangular bars (10 × 1 × 1 cm) were infiltrated at 1150°C in an argon→nitrogen atmosphere for 45 and 60 min by utilizing two aluminum alloys (Al-6 Mg-11 Si and Al-9 Mg-11 Si, wt.%). The results obtained show that the presence of SiO₂ in SiC affects the microstructure and impact strength of the composites significantly. When Al₄C₃ is formed, the impact strength decreases. However, a high proportion of SiC to SiO₂ limits the formation of the unwanted Al₄C₃ phase in the composites. Also, a higher content of Mg in the Al alloy lowers the residual porosity and, consequently, increases the composite strength. The impact strength grows with decrease in SiC particle size and increases considerably when the residual porosity is less than 1%. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 401–418, May–June, 2006.     

A note on the C2/G/1 queue and the C2/G/1 loss system

In this note, we consider the steady-state probability of delay (PW) in the C₂/G/1 queue and the steady-state probability of loss (p_loss) in the C₂/G/1 loss system, in both of which the interarrival time has a two-phase Coxian distribution. We show that, for c_X ²<1, where c_X is the coefficient of variation of the interarrival time, both p_loss and PW are increasing in β(s), the Laplace–Stieltjes transform of the general service-time distribution. This generalises earlier results for the GE₂/G/1 queue and the GE₂/G/1 loss system. The practical significance of this is that, for c_X ²<1, p_loss in the C₂/G/1 loss system and PW in the C₂/G/1 queue are both increasing in the variability of the service time. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

Feynman’s Operational Calculi: Disentangling Away from the Origin

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ⁿ . The method of disentangling follows Feynman’s heuristic rules from in (Feynman in Phys. Rev. 84:18–128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ⁿ while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.     

On one class of modules over integer group rings of locally solvable groups

We study a Z G-module A in the case where the group G is locally solvable and satisfies the condition min–naz and its cocentralizer in A is not an Artinian Z-module. We prove that the group G is solvable under the conditions indicated above. The structure of the group G is studied in detail in the case where this group is not a Chernikov group. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 44–51, January, 2009.     

Dendriform Analogues of Lie and Jordan Triple Systems

We use computer algebra to determine all the multilinear polynomial identities of degree ≤7 satisfied by the trilinear operations (a·b)·c and a·(b·c) in the free dendriform dialgebra, where a·b is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.     

The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system

The blow-up solutions of the Cauchy problem for the Davey-Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo-Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey-Stewartson system is developed in R³. Thirdly, the mass concentration is established for all the blow-up solutions of the system in R². Finally, the existence of the minimal blow-up solutions in R² is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t→T (blow-up time) is in detail investigated in terms of the ground state.     

Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their <Emphasis Type="Italic">s</Emphasis>-Adic Digits

Properties of the set T _s of “particularly nonnormal numbers” of the unit interval are studied in detail (T _s consists of real numbers x some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proved that the set T _s is residual in the topological sense (i.e., it is of the first Baire category) and is generic in the sense of fractal geometry (T _s is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. Dedicated to V. S. Korolyuk on occasion of his 80th birthday __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1163–1170, September, 2005.     

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

Features of electron density and temperature in the 500–3500 km region of the plasmasphere—(1) Dawn and dusk sectors

Dawn-dusk features of the plasmasphere are examined for intervals in February and September 1969, using electrostatic probe data ofN _e andT _e from the ISIS-I satellite. Clear plasmatrough formation is seen in the vicinity of 70° geomagnetic latitude in both dawn and dusk sectors in the 1500–3500 km region, but the plasmatrough is absent in the altitude range 500–1500 km. The plasmatrough minimum near 70°φ exhibits no asymmetry between dawn and dusk sectors in its latitudinal position. TheT _e peak associated with the plasmatrough is more pronounced in the dawn sector. DawnN _e is less than duskN _e, but dawnT _e exceeds duskT _e. The influence of processes in the magnetosphere in causing these features is examined.     

On the modelling of distributed outflow in one-dimensional models of arterial blood flow

Summary One-dimensional mathematical models of the flow of blood in arteries commonly make use of the concept of distributed outflow to simulate the loss of blood via side branches. It is shown that the usual approach leads to physically unrealistic results in certain special cases involving flow in rigid tubes, unless we introduce in the momentum equation a suitable additional term depending on the outflow. The influence of such a term on the pressure- and velocity waves calculated from an existing model of the aorta is investigated.

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Zusammenfassung Eindimensionale mathematische Modelle des Blutstromes in Arterien verwenden meist das Konzept von distribuiertem Abfluß um den Blutverlust über Verzweigungen zu simulieren. Es wird gezeigt, daß die übliche Behandlungsweise in speziellen Fällen von Strömungen in starren Röhren zu physikalisch unrealistischen Resultaten führt, es sei denn, daß in der Impulsgleichung ein passendes zusätzliches Glied eingeführt wird, welches vom Abfluß abhängt. Der Einfluß dieses Gliedes auf die Druck- und Geschwindigkeitswellen wird mit Hilfe eines bestehenden Modelles der Aorta untersucht.

     

LM-g splines

As an extension of the notion of an L-g spline, three mathematical structures called LM-g splines of types I, II, and III are introduced. Each is defined in terms of two differential operators the coefficients a_j, J = 0,…, n − 1, and b_i, I = 0,…, m, are sufficiently smooth; and b_m is bounded away from zero on [0, T]. Each of the above types of splines is the solution of an optimization problem more general than the one used in the definition of the L-g spline and hence it is recognized as an entity which is distinct from and more general mathematically than the L-g spline. The LM-g splines introduced here reduce to an L-g spline in the special case in which m = 0 and b₀ = constant ≠ 0. After the existence and uniqueness conditions, characterization, and best approximation properties for the proposed splines are obtained in an appropriate reproducing kernel Hilbert space framework, their usefulness in extending the range of applicability of spline theory to problems in estimation, optimal control, and digital signal processing are indicated. Also, as an extension of recent results in the generalized spline literature, state variable models for the LM-g splines introduced here are exhibited, based on which existing least squares algorithms can be used for the recursive calculation of these splines from the data.     

Approximation diophantienne sur les courbes elliptiques à multiplication complexe

Let $\text{E}$ be an elliptic curve with complex multiplication, defined over $\text{Q}$. We consider linear forms on $\text{Lie}\text{(}\text{E}{}^{\mathrm{n}}\text{)}$ with coefficients in the CM field of $\text{E}$. Within this framework, we present a new measure of linear independence for elliptic logarithms in (logb)(loga)ⁿ. Like recent advances in this domain (works by Ably, David, Hirata-Kohno), our result is best possible in terms of the height of the linear forms (logb) while providing a better estimate in the height of algebraic points considered (loga), removing a term in logloga. To cite this article: M. Ably, É. Gaudron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Resonant States in High-Temperature Superconductors with Impurities

A microscopic theory of resonant states for the Zn-doped CuO₂ plane in the superconducting phase is formulated in the effective t–J model. In the model derived from the original p–d model, Zn impurities are considered as vacancies for the d states at Cu sites. In the superconducting phase, in addition to the local static perturbation induced by the vacancy, a dynamical perturbation appears that results in a frequency-dependent perturbation matrix. Using the T-matrix formalism for the Green's functions in terms of the Hubbard operators, we calculate the local density of electronic states with d, p, and s symmetries.