A note on slant submanifolds of nearly trans-Sasakian manifolds

The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), 125–138] and [GUPTA, R. S.—KHURSHEED HAIDER, S. M.—SHARFUDIN, A.: Slant submanifolds of a trans-Sasakian manifold, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (2004), 45–57] are generalized.     

Almost contact Einstein–Weyl structures

 Almost contact Weyl manifolds are introduced: in dimension at least 5 they naturally lead to locally conformal cosymplectic spaces. We analyze them from the point of view of Weyl geometry considering in particular the case of compact Einstein–Weyl manifolds. Received: 6 July 2001/Revised version: 5 March 2002     

Antisymmetric tensor fields and the Weyl gauge theory

The aim of the present paper is to find a spinor current—a source—in the Weyl non-Abelian gauge theory whose distinguishing feature is that it involves no abstract gauge space. It is shown that the desired spinor representation of the Weyl gauge group can be constructed in the space of antisymmetric tensor fields in the form of a 16-component quantity for which a gauge-invariant Lagrangian is established. The relationship between the Weyl non-Abelian gauge potential and the Cartan torsion field, and the question of where the interactions in question could manifest are discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 112–123, October, 1997.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

Nonholonomic Riemann and Weyl tensors for flag manifolds

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007.     

Some argument inequalities for certain analytic functions

For analytic functions f(z) in the open unit disk U and convex functions g(z) in U, Nunokawa et al. [NUNOKAWA, M.—OWA, S.—NISHIWAKI, J.—KUROKI, K.—HAYAMI, T: Differential subordination and argumental property, Comput. Math. Appl. 56 (2008), 2733–2736] have proved one theorem which is a generalization of the result [POMMERENKE, CH.: On close-toconvex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176–186]. The object of the present paper is to generalize the theorem due to Nunokawa et al..     

Asymptotic Geometry and Conformal Types of Carnot-Carathéodory Spaces

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot—Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further applications are discussed, such as the relation between local and global invertibility properties of quasiconformal immersions (the global homeomorphism theorem). Submitted: November 1997, revised: November 1998.     

A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends

It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147–1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.     

Conformal Riemannian Maps between Riemannian Manifolds,Their Harmonicity and Decomposition Theorems

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.     

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.     

Generalized difference sequence s pace defined by |$$ \bar N $$, pk| summability and an Orlicz function in seminormed s pace

The object of this paper is to introduce a new difference sequence space which arise from the notions of |$ \bar N $ \bar N , p _k | summability and an Orlicz function in seminormed complex linear space. Various algebraic and topological properties and certain inclusion relations involving this space have been discussed. This study generalizes results: [ALTIN, Y.—ET, M.—TRIPATHY, B. C.: The sequence space |$ \bar N_p $ \bar N_p |(M, r, q, s) on seminormed spaces, Appl. Math. Comput. 154 (2004), 423–430], [BHARDWAJ, V. K.—SINGH, N.: Some sequence spaces defined by |$ \bar N $ \bar N , p _n | summability, Demonstratio Math. 32 (1999), 539–546] and [BHARDWAJ, V. K.—SINGH, N.: Some sequence spaces defined by |$ \bar N $ \bar N , p _n | summability and an Orlicz function, Indian J. Pure Appl. Math. 31 (2000), 319–325].     

Partition function of periodic isoradial dimer models

Isoradial dimer models were introduced in Kenyon (Invent Math 150(2):409–439, 2002)—they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of (Kenyon in Invent Math 150(2):409–439, 2002), namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case (Kenyon et al. in Ann Math, 2006). Supported by Swiss National Fund under grant 47102009.     

Four-dimensional manifolds with degenerate self-dual Weyl curvature operator

It is shown that any four-dimensional Walker metric of nowhere zero scalar curvature has a natural almost para-Hermitian structure. In contrast to the Goldberg–Sachs theorem, if this structure is self-dual and *-Einstein, it is symplectic but not necessarily integrable. This is due to the non-diagonalizability of the self-dual Weyl conformal curvature tensor.      

Conformal quaternionic contact curvature and the local sphere theorem

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern–Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-Sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes.     

Prediction with Expert Advice: A PDE Perspective

This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite horizon and random stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor’s and the opponent’s minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin et al. (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, SODA ’16, (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2016). Our techniques are similar to those of Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

On very weak solutions of the stationary Navier–Stokes equations

We study the stationary Navier–Stokes equations in a bounded domain Ω of R ³ with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) to obtain solvability results for the Navier–Stokes equations with very irregular data. In this article, we prove a complete solvability result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono (Preprint).     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.