Qualitative analysis of second-order models of tumor–immune system competition

This paper deals with the qualitative analysis, existence of equilibria and asymptotic behavior of some second-order models of the competition between tumor and immune cells. The background model belongs to d’Onofrio [A. d’Onofrio, A general framework for modeling tumor–immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D 208 (2005) 220–235; A. d’Onofrio, Tumor–immune system interaction: Modelig the tumor-stimulated proliferation of effectors and immunotherapy, Math. Models Methods Appl. Sci. 16 (2006) 1375–1401]. Various developments proposed in this paper are focussed on the hiding–learning dynamics, followed by the qualitative analysis.     

Radial Index and Euler Obstruction of a 1-Form on a Singular Variety

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres.Mathematics Subject Classifications (2000). 14B05, 32S60, 58A10*Partially supported by the DFG-programme “Global methods in complex geometry” (Eb 102/4–2), grants RFBR–04–01–00762, NSh–1972.2003.1     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

A Clifford–Finslerian physical unification and fractal dynamics

A Clifford–Finslerian physical unification is proposed based on Clifford–Finslerian mathematical structures and three physical principles. In the Clifford–Finslerian mathematical structure, spontaneous symmetry breaking is automatically embedded in fractal branches. With the action principle, connection principle and computation principle, physics can be unified, in which the Riemman–Einstein system and Euclid–Newton system are naturally included when quaternion are reduced to complex and real phases.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

On the Rate of Almost Everywhere Convergence of Certain Classical Integral Means, II

In this sequel to previous work of A. Stokolos and W. Trebels (1999, J. Approx. Theory98, 203–222) we indicate at the example of the Gauss–Weierstrass and the Abel–Poisson means the sharpness of some results obtained there. This is achieved by modifying methods of K. I. Oskolkov (1977, Math. USSR-Sb.32, 489–514) and A. A. Soljanik (1986, Ph.D. Thesis, Odessa) developed for the periodic case.     

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.     

Corrigendum to “A note on the Loewenstein–Prelec theory of intertemporal choice” [Math. Social Sci. 52 (1) (2006) 99–108]

In a critique of the Loewenstein and Prelec [Loewenstein G., Prelec D., 1992. Anomalies in intertemporal choice: Evidence and an interpretation. The Quarterly Journal of Economics 107, 573–597] theory of intertemporal choice, [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] point out four errors. One of the alleged errors was that the elasticity of the value function in prospect theory is decreasing. But it is in fact increasing. We provide a correction and a formal proof. As a corollary, we show that the elasticity of the value function is bounded between zero and one. Nevertheless, all the remaining points in [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] remain valid     

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.     

Experimental study of young's modulus of glass-plastics

The results of study of Young's modulus of glass-plastics in the 10^–1–10⁵-Hz frequency range are presented. A stepwise variation in the modulus, which governs the relaxation process corresponding to a low-temperature internal-friction peak in the filler, is obtained.Rybinsk Institute of Aircraft Technology. Translated from Mekhanika Polimerov, No. 6, pp. 1089–1091, November–December, 1975.     

Infrared spectroscopic study of the prefracture state of rigid polymers

A correspondence has been established between the effects of stress and temperature on the half-width and the position of the maximum of the 972-cm^–1 absorption band for polyethylene terephthalate. It is shown that the prefracture state of polyethylene terephthalate is a softened state.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 520–523, May–June, 1976.     

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar–Marcus–Tardos method (see [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A 107 (2004) 153–160]) to prove that, if a hereditary property of partitions has super-exponential speed, then, for every k-permutation π, contains the partition of [2k] with parts {{i,π(i)+k}:i[k]}. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs.Our results generalize the Stanley–Wilf conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar [M. Klazar, The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture, in: D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 250–255] and Marcus and Tardos [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture, J. Combin. Theory Ser. A 107 (2004) 153–160]. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.     

Procedure for measuring the thermophysical properties of polymers at high hydrostatic pressures

A procedure is suggested for determining the thermophysical properties of solids under hydrostatic pressure conditions by the instantaneous cylindrical heat source method without allowance for heat exchange at the sample-liquid boundary. A calculation performed showed that the temperature field of a cylinder of finite radius is identical to the temperature field of an infinite body up to Fo=0.05–0.07 with an accuracy far better than 0.5%. It is shown that the sample dimensions can be small thereupon at measurement times sufficient for performing the measurements on a standard apparatus of the order of 40–70 sec.Moscow Chemical Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 568–570, May–June, 1973.     

Integrable Chains and Hierarchies of Differential Evolution Equations

A class of integrable differential–difference systems is constructed based on auxiliary linear equations defined on sequences of Zakharov–Shabat formal dressing operators. We show that the auxiliary equations are compatible with the evolution equations for the Kadomtsev–Petviashvili (KP) hierarchy. The investigation results are used to elaborate a modified version of Krichever rational reductions for KP hierarchies.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.     

Real Quadratic Fields and the Ankeny–Artin–Chowla Conjecture

A survey of results related to the Ankeny–Artin–Chowla conjecture. Bibliography: 43 titles.     

A bound for orders in differential Nullstellensatz

We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed in [A. Seidenberg, An elimination theory for differential algebra, Univ. of California Publ. in Math. III (2) (1956) 31–66] but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz [G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1) (1926) 736–788; E.W. Mayr, A.W. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (3) (1982) 305–329; W.D. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. 126 (3) (1987) 577–591; J. Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (4) (1988) 963–975; L. Caniglia, A. Galligo, J. Heintz, Some new effectivity bounds in computational geometry, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Rome, 1988, in: Lecture Notes in Comput. Sci., vol. 357, Springer, Berlin, 1989, pp. 131–151; N. Fitchas, A. Galligo, Nullstellensatz effectif et conjecture de Serre (théorème de Quillen–Suslin) pour le calcul formel, Math. Nachr. 149 (1990) 231–253; T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (3) (2001) 521–598; Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (1) (2005) 1–17; T. Dubé, A combinatorial proof of the effective Nullstellensatz, J. Symbolic Comput. 15 (3) (1993) 277–296].This paper is dedicated to the memory of Eugeny Pankratiev, who was the advisor of the first three authors at Moscow State University.     

Discrete decomposition of type III von Neumann algebras

A. R. Beruni Tashkent Polytechnical Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 60–62, April–June, 1991.     

K-Homology Relative to Semisplit Ideals

Given a C^*-algebra A, and an ideal J of A, we define a relative group K^⁰(A,J) in terms of a relative universal C^*-algebra for the pair (A,J). We show that the natural restriction map K^⁰(A,J) K⁰(J) is an isomorphism, and that, if J is a semisplit ideal of A, the Baum–Douglas–Taylor relative K-homology is recovered. This provides a generalization of one of the main results of the Baum–Douglas–Taylor theory.