Stability of volume comparison for complex convex bodies

We prove the stability of the affirmative part of the solution to the complex Busemann–Petty problem. Namely, if K and L are origin-symmetric convex bodies in \({{\mathbb C}^n}\), n = 2 or n = 3, \({\varepsilon >0 }\) and \({{\rm Vol}_{2n-2}(K\cap H) \le {\rm Vol}_{2n-2}(L \cap H) + \varepsilon}\) for any complex hyperplane H in \({{\mathbb C}^n}\) , then \({({\rm Vol}_{2n}(K))^{\frac{n-1}n}\le({\rm Vol}_{2n}(L))^{\frac{n-1}n} + \varepsilon}\) , where Vol_2n is the volume in \({{\mathbb C}^n}\) , which is identified with \({{\mathbb R}^{2n}}\) in the natural way.     

Applications of strongly convergent sequences to Fourier series by means of modulus functions

The aim of this work is to estimate sums involving P(n), the largest prime factor of an integer \({n \geqq 2}\) under digital constraints \({{f(P(n)) \equiv a}{\rm mod} b}\), for every \({a \in \mathbb{Z}}\) and an integer \({b \geqq 2}\) where f is a strongly q-additive function with integer values (i.e. \({f(aq^j + b) = f(a) + f(b)}\), with \({(a, b, j) \in \mathbb{N}^3}\), \({{0 \leqq b} < q^j}\)). We also estimate the cardinality of the set \({\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}\), where \({c \in \mathbb{Z}}\), \({k \geqq 2}\).     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Simplices whose dihedral angles are all rational multiples of <Emphasis Type="Italic">π</Emphasis>, and related topics

We prove the following: (1) For every \({n \geq 2}\), there are infinitely many, mutually non-similar n-dimensional simplices in \({\mathbb{R}^n}\) whose dihedral angles are all rational multiples of π. (2) For every \({n \geq 3}\), there are uncountably many, mutually non-similar n-simplices whose dihedral angles and π are linearly independent over the rational field. Moreover, in the set of all n-simplices in \({\mathbb{R}^n}\) the subset of such n-simplices is everywhere dense with respect to the Hausdorff distance.     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

Remarks on the Trotter–Kato Product Formula for Unitary Groups

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L ²-norm, that is, one has

$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $

for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: \({\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}\), where \({{\mathbb R}_+ := [0,\infty)}\), satisfying in addition \({{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}\) and \({{\Im{\rm m}}\,(\psi(y)) \le 0}\) for \({y \in {\mathbb R}_+}\), we prove that

$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $

holds true uniformly on \({[0,T]\ni t}\) for any T > 0.

     

On Modular Edge-Graceful Graphs

Let G be a connected graph of order \({n\ge 3}\) and size m and \({f:E(G)\to \mathbb{Z}_n}\) an edge labeling of G. Define a vertex labeling \({f': V(G)\to \mathbb{Z}_n}\) by \({f'(v)= \sum_{u\in N(v)}f(uv)}\) where the sum is computed in \({\mathbb{Z}_n}\) . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where \({n\not\equiv 2 \pmod 4}\) , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where \({n\not\equiv 2\pmod 4}\) having diameter at most 5 is modular edge-graceful.     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface

We use the variational concept of \({\Gamma}\)-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation \({u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}\), for \({(t,x)\in \mathbb{R}^+\times\Omega}\), where \({\Omega\subset\mathbb{R}^n}\), \({n\geq 1}\), supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function f are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (n ? 1)-dimensional hypersurface \({\gamma \subset \Omega}\); across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by \({f(\cdot,x)}\) having only two roots.     

The Minimal Size of Infinite Maximal Antichains in Direct Products of Partial Orders

For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø ₁ by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T ₁-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A _i ×B _i :i∈I} an infinite partition of the square X ², then at least one of the families {A _i :i∈I} and {B _i :i∈I} contains an infinite partition of X.     

Strichartz estimates via the Schrödinger maximal operator

We consider the Schrödinger operator \({e^{it\Delta}}\) acting on initial data f in \({\dot{H}^s}\). We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which \({\lim_{t\to 0}e^{it\Delta}f(x)=f(x)}\) a.e. \({x\in \mathbb {R}^n}\), would imply an affirmative answer to a question of Planchon, concerning the sharp range of q and r for which \({e^{it\Delta}}\) is bounded in \({L_x^q(\mathbb {R}^n,L^r_t(\mathbb {R}))}\). When n  =  2, we unconditionally improve the range for which the mixed norm estimates hold.     

The Representation Type of Determinantal Varieties

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves \(\mathcal {E}\) of arbitrary high rank on a general standard (resp. linear) determinantal scheme \(X\subset \mathbb {P}^{n}\) of codimension c ≥ 1, n ? c ≥ 1 and defined by the maximal minors of a t × (t + c?1) homogeneous matrix \(\mathcal {A}\). The sheaves \(\mathcal {E}\) are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type provided the degrees of the entries of the matrix \(\mathcal {A}\) satisfy some weak numerical assumptions; and (2) we determine values of t, n and n ? c for which a linear standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type.     

New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access network for 2-dimensional image transmission. There is a one-to-one correspondence between an \((m, n, w, \lambda )\)-OOSPC and a \((\lambda +1)\)-(mn, w, 1) packing design admitting an automorphism group isomorphic to \(\mathbb {Z}_m\times \mathbb {Z}_n\). In 2010, Sawa gave a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) which contains a unique element of order 2. In this paper, we study the existence of one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) having three elements of order 2. It is proved that there is a one-factor in the Köhler graph of \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\) relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where \(p\equiv 5\pmod {12}\) is a prime and \(1\le \epsilon ,\epsilon '\le 2\). Using this one-factor, we construct a strictly \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\)-invariant regular \(G^*(p,2^{\epsilon +\epsilon '},4,3)\) relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant regular G-designs, we construct more strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal \((2^{\epsilon }m,2^{\epsilon '}n,4,2)\)-OOSPC for any \(\epsilon ,\epsilon '\in \{0,1,2\}\) with \(\epsilon +\epsilon '>0\) and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set \(\{p\equiv 5\pmod {12}:p\) is a prime, \(p<\)1,500,000}.     

Nonarchimedean Green functions and dynamics on projective space

Let \({\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}\) be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function \({\hat{g}_\varphi}\) associated to \({\varphi}\) is Hölder continuous on \({\mathbb{P}^N(K)}\) and that the Fatou set \({\mathcal{F}(\varphi)}\) of \({\varphi}\) is equal to the set of points at which \({\hat{g}_\Phi}\) is locally constant. Further, \({\hat{g}_\varphi}\) vanishes precisely on the set of points P such that \({\varphi}\) has good reduction at every point in the forward orbit \({\mathcal{O}_\varphi(P)}\) of P. We also prove that the iterates of \({\varphi}\) are locally uniformly Lipschitz on \({\mathcal{F}(\varphi)}\) .     

A spectral method for nonlinear elliptic equations

Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ?Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(?, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ?Ω is a C ^∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to ?Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.