Orbit Spaces of Free Involutions on the Product of Two Projective Spaces

Let X be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on X and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration ${X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}}$ . We also give an application of our result to show that if X has the mod 2 cohomology algebra of the product of two real projective spaces (respectively, complex projective spaces), then there does not exist any ${\mathbb{Z}_2}$ -equivariant map from ${\mathbb{S}^k \to X}$ for k ≥ 2 (respectively, k ≥ 3), where ${\mathbb{S}^k}$ is equipped with the antipodal involution.     

Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle ξ over a finite connected CW-complex X is by definition the largest integer ${k, 0 \leq k \leq \mathrm{dim}(X)}$ , such that every cohomology class ${x \in H^{j}(X;\mathbb{Z}_2), 0 \leq j \leq k}$ , is a polynomial in the Stiefel–Whitney classes w _i (ξ). In this note we compute the characteristic rank of vector bundles over the Stiefel manifold ${V_k(\mathbb{F}^n), \mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}}$ .     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Conformal blocks,Berenstein–Zelevinsky triangles,and group-based models

Work of Buczyńska, Wi?niewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group \(\mathbb {Z}/2\mathbb {Z}\) with the Wess–Zumino–Witten (WZW) model of conformal field theory associated to \(\mathrm {SL}_2(\mathbb {C})\) . In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group \(\mathbb {Z}/m\mathbb {Z}\) and the WZW model for the special linear group \(\mathrm {SL}_m(\mathbb {C}).\) We use this relationship to also show how a combinatorial device from representation theory, the Berenstein–Zelevinsky triangle, corresponds to elements in the affine semigroup algebra of the \(\mathbb {Z}/3\mathbb {Z}\) phylogenetic statistical model.     

Unbounded positive solutions for m-point time-scale boundary value problems on infinite intervals

In this work, we consider the following second-order m-point boundary value problem on time scales $$\left\{\begin{array}{@{}l}(\phi_{p}(u^{\triangle}(t)))^{\nabla}+h(t)f(t,u(t),u^{\triangle }(t))=0,\quad t\in(0,+\infty)_{\mathbb{T}},\\[4pt]\displaystyle u(0)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i}),\qquad u^{\triangle}(+\infty)=\sum_{i=1}^{m-2}\beta_{i}u^{\triangle}(\eta_{i}).\end{array}\right.$$ We establish new criteria for the existence of at least three unbounded positive solutions. Our results are new even for the corresponding differential $({\mathbb{T}}={\mathbb{R}})$ , difference equation $({\mathbb{T}}={\mathbb{Z}})$ and for the general time-scale setting. An example is given to illustrate our results.     

A Barrier Principle for Surfaces with Prescribed Mean Curvature and Arbitrary Codimension

We consider two dimensional surfaces ${X : \Omega\to\mathbb R^{n+2}, \Omega\subset \mathbb C, w=u+iv\mapsto X(w)}$ with arbitrary codimension n and prove a barrier principle for strong (possibly branched) subsolutions ${X\in C^1(\Omega, \mathbb {R}^{n+2})\cap H_{2,{\rm loc}}^2(\Omega,\mathbb R^{n+2})}$ of the integral inequality $$\int_{\Omega} \Big\lbrace \langle \nabla X, \nabla \varphi\rangle +2W \sum_{k=1}^n H_k \langle N_k,\varphi \rangle \Big\rbrace \; dudv\ge 0$$ with mean curvature functions (H _k ) _k=1,...,n which lie locally on one side of a supporting hypersurface S. We show under suitable assumption on the 2-mean curvature of the supporting surface S that X is locally contained in S. This generalizes a corresponding result for surfaces in ${\mathbb R^3}$ , cf. (Dierkes et al., Regularity of Minimal Surfaces, §4.4, 2010).     

Algebraic Relations between the Hypergeometric E-Function and Its Derivatives

In this paper, we consider the generalized hypergeometric function $$\sum\limits_{n = 0}^\infty {\frac{1}{{\left( {{\lambda }_{1} + 1} \right)_n ...\left( {{\lambda }_t + 1} \right)_n }}} \left( {\frac{z}{t}} \right)^{tn} ,{ \lambda }_{1} ,...,{\lambda }_{t} \in \mathbb{Q}\backslash \left\{ { - 1, - 2,...} \right\},$$ where t is an even number, and its derivatives up to the order t- 1 inclusive. In the case of algebraic dependence between these functions over $\mathbb{C}$ (z), a complete structure of algebraic relations between them is given.     

Frames for Weighted Shift-invariant Spaces

In this paper we prove the equivalence of the frame property and the closedness for a weighted shift-invariant space $$ V^p_\mu(\Phi) = \left\{\sum \limits^{r}_{i=1} \sum \limits_{j \in \mathbb{Z}^d} c_{i}(j)\phi_{i}(\cdot-j)\left \vert {\{c_{i}(j)\}}_{j \in \mathbb{Z}^{d}} \in {\ell_{\mu}^{p}}\right.\right\}, \quad p \in [1, \infty], $$ which corresponds to ${{\Phi = \Phi^r = (\phi_1, \phi_2, . . . , \phi_r)^T \in (W^{1}_\omega)^r}}$ . We, also, construct a sequence Φ^2k+1 and the sequence of spaces ${{V^{p}_{\mu} (\Phi^{2k+1})}}$ , ${k \in {\mathbb N}}$ , on ${\mathbb R}$ , with the useful properties in sampling, approximations and stability.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

Stable cohomology of alternating groups

We determine the stable cohomology groups ( $H_S^i \left( {{{\mathfrak{A}_n ,\mathbb{Z}} \mathord{\left/ {\vphantom {{\mathfrak{A}_n ,\mathbb{Z}} {p\mathbb{Z}}}} \right. \kern-0em} {p\mathbb{Z}}}} \right)$ of the alternating groups $\mathfrak{A}_n$ for all integers n and i, and all odd primes p.     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$\begin{gathered} E (H_\Phi ^\omega ; P (\Delta _n )) = sup || f - l_{p(\Delta _n )} (f)||, \hfill \\ f \in H_\Phi ^\omega \hfill \\ \end{gathered} $$ and also give an asymptotic estimate of the quantities $$\begin{gathered} E_n (H_\Phi ^\omega ) = infinf E (H_\Phi ^\omega ; P (\Delta _n )). \hfill \\ \Delta _n P(\Delta _n ) \hfill \\ \end{gathered} $$      

Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

Zero-temperature Glauber dynamics on {\mathbb{Z}^d}

We study zero-temperature Glauber dynamics on ${\mathbb{Z}^d}$ , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define ${p_c(\mathbb{Z}^d)}$ to be the infimum over p such that the system fixates at ???+??? with probability 1. It is a folklore conjecture that ${p_c(\mathbb{Z}^d) = 1/2}$ for every ${2 \le d \in \mathbb{N}}$ . We prove that ${p_c(\mathbb{Z}^d) \to 1/2}$ as d ?? ??.     

Homotopy and nonlinear boundary value problems involving singular {\phi}-Laplacians

Homotopy methods are used to find sufficient conditions for the solvability of nonlinear boundary value problems of the form $$(\phi(u^\prime))^\prime = f(t, u, u^\prime), \quad g(u(\alpha), \phi(u^\prime(\beta))) = 0,$$ where (α, β) = (0, 1), (1, 0), (0, 0) or (1, 1), ${\phi}$ is a homeomorphism from the open ball ${B(a) \subset \mathbb{R}^n}$ onto ${\mathbb{R}^n}$ , f is a Carathéodory function, ${g : \mathbb{R}^n \times \, \mathbb{R}^n \rightarrow \mathbb{R}^m}$ is continuous and m ≤ 2n.     

An explicit growth condition for middle-degree cuspidal cohomology of arithmetically defined quaternionic hyperbolic n-manifolds

Let ${G/\mathbb Q}$ be the simple algebraic group Sp(n, 1) and ${\Gamma=\Gamma(N)}$ a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group ${G(\mathbb R)}$ . Then a double quotient ${\Gamma\backslash G(\mathbb R)/K}$ is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology ${H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)}$ in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkhäuser, Boston), pp. 1–48, 1984).