Relaxation results for functions depending on polynomials changing sign on rank-one matrices

In this paper, we are interested in computing the different convex envelopes of functions depending on polynomials, especially those having it is main part change sign on rank-one matrices. Our main result applies to functions of the type W(F)=φ(P(F)), W(F)=φ(P(F))+f(detF) or W(F)=φ(P(F))+g(adjnF) defined on the space of matrices, where φ, f:R→R and g:R³→R are three continuous functions, and P=P₀+P₁+?+Pd is a polynomial such that Pd has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.     

Fractional weak discrepancy and split semiorders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wd_F for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.     

Topological equivalence of complex polynomials

The following numerical control over the topological equivalence is proved: two complex polynomials in n≠3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions f_s:Cⁿ→C with isolated singularities such that the degree, the number of vanishing cycles and the number of atypical values are constant in the family.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Lane-Emden-Fowler equations with convection and singular potential

We are concerned with singular elliptic problems of the form −Δu±p(d(x))g(u)=λf(x,u)+μa|∇u| in Ω, where Ω is a smooth bounded domain in RN, d(x)=dist(x,∂Ω), λ>0, μ∈R, 0<a?2, and f is a nondecreasing function. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the origin. Taking into account the competition between the anisotropic potential p(d(x)), the convection term a|∇u|, and the singular nonlinearity g, we establish various existence and nonexistence results.     

Nonradial Large Solutions of Sublinear Elliptic Equations

We show that the equation Δu = p(x)f(u) has a positive solution on R ^N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup _{s ≥ 1} f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial.     

Representation sets for integral binary quadratic forms over the rationals

Let f be an integral binary form of discriminant d which represents n integrally. Two rational representations (r, s) and (r′, s′), with denominators prime to n, of n by f are called semiequivalent with respect to f if there is a rational automorph of f with determinant 1 and denominator m which takes (r, s) into (r′, s′) where (m, n) = 1 and m contains no factors p of d such that $\text{d}\text{p}{}^2$ is a discriminant. The number of such equivalence classes for a given f and n is sometimes finite. This number is obtained for forms with negative discriminants which have one class in each primitive genus.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

Fractional weak discrepancy and interval orders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in [A. Shuchat, R. Shull, A. Trenk, The fractional weak discrepancy of a partially ordered set, Discrete Applied Mathematics 155 (2007) 2227-2235] as the minimum nonnegative k for which there exists a function satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in [A. Shuchat, R. Shull, A. Trenk, Range of the fractional weak discrepancy function, ORDER 23 (2006) 51-63; A. Shuchat, R. Shull, A. Trenk, Fractional weak discrepancy of posets and certain forbidden configurations, in: S.J. Brams, W.V. Gehrlein, F.S. Roberts (Eds.), The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer, New York, 2009, pp. 291-302] on the range of the wd_F function for semiorders (interval orders with no induced ) to interval orders with no , where n≥3. In particular, we prove that the range for such posets P is the set of rationals that can be written as r/s, where 0≤s−1≤r<(n−2)s. If wd_F(P)=r/s and P has an optimal forcing cycle C with and , then r≤(n−2)(s−1). Moreover when s≥2, for each r satisfying s−1≤r≤(n−2)(s−1) there is an interval order having such an optimal forcing cycle and containing no.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

Intersection theory of algebraic obstructions

Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤r≤d, we define obstruction groups E^r(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E^r(A,A)×E^s(A,A)→E^r+s(A,A). For a projective A-module Q of rank n≤d, with an orientation , we define a Chern class like homomorphism

w(Q,χ):E^d−n(A,L^′)→E^d(A,LL^′),     

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f   can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0)$f:(X,0)\to (\mathbb{C},0)$ and g:(X,0)→(C,0)$g:(X,0)\to (\mathbb{C},0)$. We give applications when g is a generic linear form and when f and g have isolated singularities.     

On gaps and unoccupied urns in sequences of geometrically distributed random variables

This paper continues the study of gaps in sequences of n geometrically distributed random variables, as started by Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239], who concentrated on sequences which were gap-free. Now we allow gaps, and count some related parameters.Our terminology of gaps just means empty “urns” (within the range of occupied urns), if we think about an urn model. This might be called weak gaps, as opposed to maximal gaps, as in Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239]. If one considers only “gap-free” sequences, both notions coincide asymptotically, as n→∞.First, the probability p_n(r) that a sequence of length n has a fixed number r of empty urns is studied; this probability is asymptotically given by a constant p^*(r) (depending on r) plus some small oscillations. When , everything simplifies drastically; there are no oscillations.Then, the random variable ‘number of empty urns’ is studied; all moments are evaluated asymptotically. Furthermore, samples that have r empty urns, in particular the random variable ‘largest non-empty urn’ are studied. All moments of this distribution are evaluated asymptotically.The behavior of the quantities obtained in our asymptotic formulæ is also studied for p→0 resp. p→1, through a variety of analytic techniques.The last section discusses the concept called ‘super-gap-free.’ A sample is super-gap-free, if r=0 and each non-empty urn contains at least 2 items (and d-super-gap-free, if they contain ?d items). For the instance , we sketch how the asymptotic probability (apart from small oscillations) that a sample is d-super-gap-free can be computed.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth in fluid-like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation of tumor cells. The proliferation rate μ and the cell-to-cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius r=R. For a sequence μ/γ=Mn(R) there exist symmetry-breaking bifurcation branches of solutions with free boundary r=R+εYn,0(θ)+O(ε²) (n even ?2) for small |ε|, where Yn,0 is the spherical harmonic of mode (n,0). Furthermore, the smallest Mn(R), say Mn_∗(R), is such that n_∗=n_∗(R)→∞ as R→∞. In this paper we prove that the radially symmetric stationary solution with R=RS is linearly stable if μ/γ<N_∗(RS,γ) and linearly unstable if μ/γ>N_∗(RS,γ), where N_∗(RS,γ)?Mn_∗(RS), and we prove that strict inequality holds if γ is small or if γ is large. The biological implications of these results are discussed at the end of the paper.