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.     

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.     

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.

     

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.     

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 second order accuracy for a full discretized time-dependent Navier–Stokes equations by a two-grid scheme

We study a second-order two-grid scheme fully discrete in time and space for solving the Navier–Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step Δt and, in the second step, in discretizing the linearized problem around the velocity u _H computed in the first step, in space on a fine grid with mesh-size h and the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm. This strategy is motivated by the fact that under suitable assumptions, the contribution of u _H to the error in the non-linear term, is measured in the L ² norm in space and time, and thus has a higher-order than if it were measured in the H ¹ norm in space. We present the following results: if h ² = H ³ = (Δt)², then the global error of the two-grid algorithm is of the order of h ², the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.     

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 .     

Dynamics in dumbbell domains II. The limiting problem

In this work we continue the analysis of the asymptotic dynamics of reaction-diffusion problems in a dumbbell domain started in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2) (2006) 551-597]. Here we study the limiting problem, that is, an evolution problem in a “domain” which consists of an open, bounded and smooth set Ω⊂RN with a curve R₀ attached to it. The evolution in both parts of the domain is governed by a parabolic equation. In Ω the evolution is independent of the evolution in R₀ whereas in R₀ the evolution depends on the evolution in Ω through the continuity condition of the solution at the junction points. We analyze in detail the linear elliptic and parabolic problem, the generation of linear and nonlinear semigroups, the existence and structure of attractors.     

On Levels in Arrangements of Lines, Segments, Planes, and Triangles%

We consider the problem of bounding the complexity of the k th level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk ^5/3 ) , on the complexity of the k th level in an arrangement of n planes in R ³ , or on the number of k -sets in a set of n points in three dimensions, and we show that the complexity of the k th level in an arrangement of n line segments in the plane is , and that the complexity of the k th level in an arrangement of n triangles in 3-space is O(n ² k ^5/6 α(n/k)) . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p315.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received February 7, 1997, and in revised form May 15, 1997, and August 30, 1997.     

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.     

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.     

Linear Stochastic Systems: A White Noise Approach

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems.We further study ℓ ₁-ℓ ₂ stability in the discrete time case, and L ₂-L _∞ stability in the continuous time case.     

Maximum principle for fully nonlinear equations via the iterated comparison function method

We present various versions of generalized Aleksandrov–Bakelman–Pucci (ABP) maximum principle for L ^p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the “iterated” comparison function method, which was introduced in our previous paper (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5–6), 967–983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293–1326, 1998; Comm. Partial Diff. Eq. 25, 1997–2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27–76, 1992) and (Crandall and Święch in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case. Dedicated to Hitoshi Ishii on the occasion of his 60th birthday.     

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.     

Embeddings of submanifolds and normal bundles

This paper studies the embeddings of a complex submanifold S inside a complex manifold M; in particular, we are interested in comparing the embedding of S in M with the embedding of S as the zero section in the total space of the normal bundle NS of S in M. We explicitly describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho, Movasati and Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results hold for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho, Movasati and Sad.     

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.     

Ionization of Atoms by Intense Laser Pulses

The process of ionization of a hydrogen atom by a short infrared laser pulse is studied in the regime of very large pulse intensity, in the dipole approximation. Let A denote the integral of the electric field of the pulse over time at the location of the atomic nucleus. It is shown that, in the limit where |A| → ∞, the ionization probability approaches unity and the electron is ejected into a cone opening in the direction of −A and of arbitrarily small opening angle. Asymptotics of various physical quantities in |A|⁻¹ is studied carefully. Our results are in qualitative agreement with experimental data reported in Eckle et al. (Science 322, 1525–1529; 2008, Nature (physics) 4, 565–570 2008).     

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.