Order one invariants of immersions of surfaces into 3-space

We classify all order one invariants of immersions of a closed orientable surface F into ³, with values in an arbitrary Abelian group . We show that for any F and and any regular homotopy class of immersions of F into ³, the group of all order one invariants on is isomorphic to is the group of all functions from a set of cardinality . Our work includes foundations for the study of finite order invariants of immersions of a closed orientable surface into ³, analogous to chord diagrams and the 1-term and 4-term relations of knot theory.Partially supported by the Minerva FoundationMathamatics Subject Classification (2000):57M, 57R42     

Power series whose coefficients form homogeneous random processes

Summary Power series f(z) = a _izⁱ are considered, where the sequence {a _i} forms a homogeneous random process. If the sequence is exchangeable and the variance of the marginal distributions exists, it is proved that r, the random radius of convergence of f(z), takes the values 0 and 1. If the sequence is a second order stationary time series then r=1 with probability 1. If {a _i} is a regular denumerable Markov chain, it can be proved that r=c=1 with probability 1, but both c=0 and c=1 can arise. A number of criteria are given for deciding the value of c in this situation.     

Functional equations in the theory of semigroups

It is established that the collection of regular solutions of the equationf(t+s)=G[f(t),f(s)], where G(, ) is a symmetric polynomial, coincides with the collection of semigroups.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 635–640, May, 1970.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable _boxclose/2{{\mathbb Z}/2} -equivariant map h(F) such that the stable \mathbb Z/2{{\mathbb Z}/2} -equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map F of an immersion f : M^m \looparrowright Nⁿ{f : M^m \looparrowright N^n} in terms of the double point set of f. We interpret the Smale–Hirsch–Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m <  2n – 1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.     

Expansion of derivatives in one-dimensional dynamics

We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ _f attracts a finite setCrit of non-flat critical points. Assuming that for eachcεCrit, either |D f ⁿ(f(c))|→∞ (iff is real) orb _n·|Df ⁿ(f(c))|→∞ for some summable sequence {b _n} (iff is complex; this is equivalent to summability of |D f ⁿ(f(c))|⁻¹), we show that for everyxεJ _f\U _i f ⁻ⁱ(Crit), there existℓ(x)≤max _c ℓ(c) andK′(x)>0

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.     

Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions , P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j.For immersion and diffeomorphism such that i and i○h are regularly homotopic we show:

P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h_∗−Id)+ε(deth∗∗))mod2     

Travelling front solution for a class of competition-diffusion system with high-order singular point

In this paper, the existence of travelling front solution for a class of competition-diffusion system with high-order singular point

Singular series of genus 2 for sums of squares

Let f₁, ..., f_h be representatives of classes in the genus of a positive quadratic form f, r_i(k) the number of representations of k by the form f_i, (k) the value of the singular Hardy-Littlewood series, E_i the number of integer automorphisms of f_i. We derive an expression for in terms of the values of singular series of genus 1 and 2. This expression is evaluated for the simplest model f(x)=x ₁ ² +...+x _m ² , m1 (mod 8), k is a prime in-integer, k=1(mod 8).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 26–39, 1986.     

Point sets with distinct distances

For positive integersd andn letf _d (n) denote the maximum cardinality of a subset of then ^d -gird {1,2,...,n} ^d with distinct mutual euclidean distances. Improving earlier results of Erds and Guy, it will be shown thatf ₂ (n)c·n ^2/3 and, ford3, thatf _d (n)c _d ·n ^2/3 ·(lnn)^1/3, wherec, c _d >0 are constants. Also improvements of lower bounds of Erds and Alon on the size of Sidon-sets in {1²,22²,...,n ²} are given.Furthermore, it will be proven that any set ofn points in the plane contains a subset with distinct mutual distances of sizec ₁·n ^1/4, and for point sets in genral position, i.e. no three points on a line, of sizec ₂·n ^1/3 with constantsc ₁,c ₂>0. To do so, it will be shown that forn points in ² with distinct distancesd ₁,d ₂,...,d _t , whered _i has multiplicitym _i , one has _i=1 ^t m _i ² c·n _3.25 for a positive constantc. If then points are in general position, then we prove _i=1 ^t m _i ² c·n ₃ for a positive constantc and this bound is tight.Moreover, we give an efficient sequential algorithm for finding a subset of a given set with the desired properties, for example with distinct distances, of size as guaranteed by the probabilistic method under a more general setting.     

Efficiency in integral facility design problems

An example of design might be a warehouse floor (represented by a setS) of areaA, with unspecified shape. Givenm warehouse users, we suppose that useri has a known disutility functionf _isuch thatH _i(S), the integral off _iover the setS (for example, total travel distance), defines the disutility of the designS to useri. For the vectorH(S) with entriesH _i(S), we study the vector minimization problem over the set {H(S) :S a design} and call a design efficient if and only if it solves this problem. Assuming a mild regularity condition, we give necessary and sufficient conditions for a design to be efficient, as well as verifiable conditions for the regularity condition to hold. For the case wheref _iis thel _p-distance from warehouse docki, with 1<p<, a design is efficient if and only if it is essentially the same as a contour set of some Steiner-Weber functionf =₁ f ₁++ _m f _m ,when the _i are nonnegative constants, not all zero.This research was supported in part by the Interuniversity College for PhD Studies in Management Sciences (CIM), Brussels, Belgium; by the Army Research Office, Triangle Park, North Carolina; by a National Academy of Sciences-National Research Council Postdoctorate Associateship; and by the Operations Research Division, National Bureau of Standards, Washington, D.C. The authors would like to thank R. E. Wendell for calling Ref. 16 to their attention.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints

It is shown that a Lagrange multiplier rule involving the Michel-Penot subdifferentials is valid for the problem: minimizef ₀(x) subject tof _i (x) 0,i = 1, ,m;f _i (x) = 0,i = m + 1,,n;x Q where all functionsf are Lipschitz continuous andQ is a closed convex set. The proof is based on the theory of fans.     

The minimal number of singularities of stable maps between surfaces

For a C^∞ stable map of a closed and connected surface M into the sphere, let c(φ), i(φ) and n(φ) denote the numbers of cusps, fold curves components and nodes respectively. In this paper, in a given homotopy class, we will determine the minimal pair (i,c+n) with respect to the lexicographic order.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

Line search by curve fitting in minimization algorithms

Summary Consider an unconstrained minimization of an uniformly convex functionf(z). Let be an algorithm that, for solving it constructs a sequence {z _i} withz _i+1 =z _i + ⁽ⁱ⁾ h _i ,h _i R ⁿ , ⁽ⁱ⁾ R ⁼ and –h _i ^T f(z _i ) > 0. For any algorithm that converges linearly and that uses parabolic or cubic interpolations for the line search, upper bounds on the number of function evaluations needed to approximate the minimum off(z) within a given accuracy, are calculated. The obtained results allow to compare the line search procedure under investigation.     

Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.