Endomorphism rings of some Young modules

Let \({\Sigma_r}\) be the symmetric group acting on \({r}\) letters, \({K}\) be a field of characteristic 2, and \({\lambda}\) and \({\mu}\) be partitions of \({r}\) in at most two parts. Denote the permutation module corresponding to the Young subgroup \({\Sigma_\lambda}\), in \({\Sigma_r}\), by \({M^\lambda}\), and the indecomposable Young module by \({Y^\mu}\). We give an explicit presentation of the endomorphism algebra \({{\rm End}_{k[\Sigma_r]}(Y^\mu)}\) using the idempotents found by Doty et al. (J Algebra 307(1):377–396, 2007).     

Nonparametric signal detection with small type I and type II error probabilities

In the problem of signal detection in the heteroscedastic Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L ₂- norms of kernel estimators. The sets of alternatives are defined by the sets of all signals such that L ₂- norms of signals smoothed by the kernel exceed some constants \({\rho_\epsilon}\) . The constants \({\rho_\epsilon}\) depend on the power \({\epsilon}\) of noise and \({\rho_\epsilon \to 0}\) as \({\epsilon \to 0}\) . The setup is considered in the zone of moderate deviation probabilities. We suppose that type I or type II error probabilities of tests tend to zero as \({\epsilon \to 0}\).     

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Let \({\alpha}\) be a bounded linear operator in a Banach space \({\mathbb{X}}\), and let A be a closed operator in this space. Suppose that for \({\Phi_1, \Phi_2}\) mapping D(A) to another Banach space \({\mathbb{Y}}\), \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) are generators of strongly continuous semigroups in \({\mathbb{X}}\). Assume finally that \({A_{|{\rm ker}\, \Phi_\text{a}}}\), where \({\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}\) and \({\beta = I_\mathbb{X} - \alpha}\), is a generator also. In the case where \({\mathbb{X}}\) is an L ¹-type space, and \({\alpha}\) is an operator of multiplication by a function \({0 \le \alpha \le 1}\), it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of \({A_{|{\rm ker}\, \Phi_1}}\) with probability \({\alpha (x)}\) and is subject to the rules of \({A_{|{\rm ker}\, \Phi_2}}\) with probability \({\beta (x)= 1 - \alpha (x)}\). We provide an approximation (a singular perturbation) of the semigroup generated by \({A_{|{\rm ker}\, \Phi_\text{a}}}\) by semigroups built from those generated by \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.     

An additive ($${\alpha, \beta}$$)-functional equation and linear mappings in Banach spaces

In this paper, we investigate the additive (\({\alpha, \beta}\))-functional equation \({f(x+y) + \bar{\alpha}f({\alpha}z) = \beta^{-1}f(\beta(x+y+z))}\) for all complex numbers \({\alpha}\) with \({|\alpha| = 1}\) and for a fixed nonzero complex number \({\beta}\). Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of this additive (\({\alpha, \beta}\))-functional equation in complex Banach spaces.     

The imaginary part of the characteristic function

We consider the conditions under which a continuous function \({\varphi \colon {\mathbb{R}}^n \to \mathbb {R}}\) is the imaginary part \({\Im f}\) of the characteristic function f of a probability measure on \({{\mathbb{R}}^n}\). A similar problem about such an \({\varphi}\) that it is the argument of the characteristic function was solved by Ilinskii [Theory Probab. Appl. 20 (1975), 410–415]. In this paper, a characterization of what \({\varphi}\) might serve as the imaginary part of the characteristic function f is given. As a consequence, we provide an answer to the following question posed by N. G. Ushakov [7]: Is it true that f is never determined by its imaginary part \({\Im f}\) ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that \({\Im f\equiv \Im g}\) but \({ f\not\equiv g}\) ? We prove that the answer to this question is negative. In addition, several examples of characteristic functions which are uniquely determined by their imaginary parts are given.     

Remarks on the nonexistence of biharmonic maps

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that \({\phi : (M, g) \to (N, h)}\) is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that \({\phi}\) is a harmonic map if one of the following conditions holds: (i) \({|d\phi|}\) is bounded in L^q(M) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 \leq q \leq \infty}\), \({1 < p < \infty}\); or (ii) \({Vol(M) = \infty}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). In addition, if N has strictly negative sectional curvature, we assume that \({rank\phi(q) \geq 2}\) for some \({q \in M}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).     

{\mathbb{Z}_2}-bordism and the Borsuk–Ulam Theorem

The purpose of this work is to classify, for given integers \({m,\, n\geq 1}\), the bordism class of a closed smooth \({m}\)-manifold \({X^m}\) with a free smooth involution \({\tau}\) with respect to the validity of the Borsuk–Ulam property that for every continuous map \({\phi : X^m \to \mathbb{R}^n}\) there exists a point \({x\in X^m}\) such that \({\phi (x)=\phi (\tau (x))}\). We will classify a given free \({\mathbb{Z}_2}\)-bordism class \({\alpha}\) according to the three possible cases that (a) all representatives \({(X^m, \tau)}\) of \({\alpha}\) satisfy the Borsuk–Ulam property; (b) there are representatives \({({X_{1}^{m}}, \tau_1)}\) and \({({X_{2}^{m}}, \tau_2)}\) of \({\alpha}\) such that \({({X_{1}^{m}}, \tau_1)}\) satisfies the Borsuk–Ulam property but \({({X_{2}^{m}}, \tau_2)}\) does not; (c) no representative \({(X^m, \tau)}\) of \({\alpha}\) satisfies the Borsuk–Ulam property.     

Convergence in Variation for Bernstein-Type Operators

In this paper, we deal with Bernstein-type operators defined by Cárdenas-Morales et al. as \({B_{n}(f \circ \tau^{-1}) \circ \tau}\), where \({B_{n}}\) is the nth Bernstein polynomial (Comput Math Appl 62(1):158–163, 2011). Assuming that \({\tau}\) and f are absolutely continuous functions on \({[0, 1]}\) and inf \({\tau ^{\prime} (x) \geq m > 0}\) as well as \({\tau (0) = 0}\) and \({\tau (1) = 1,}\) we study the convergence of Bernstein-type operators to f in variation seminorm. Moreover, we give a Voronovskaja-type formula and a Jackson-type estimate in the sense of Bardaro et al. (Analysis 23:299–340, 2003).     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

Fully developed laminar slip and no-slip flow in rough microtubes

The effect of surface roughness on developed laminar flow in microtubes is investigated. The tube boundary is defined by \({r=R\left[{1+\varepsilon\, {\rm sin}\left( {\lambda \theta }\right)}\right]}\), with R representing the reference radius and \({\varepsilon}\) and λ the roughness parameters. The momentum equation is solved using Fourier–Galerkin–Tau method with slip at the boundary. A novel semi-analytical method is developed to predict friction factor and pressure drop in corrugated rough microtubes for continuum flow and slip flow that are not restricted to small values of \({\varepsilon \lambda }\) . The analytical solution collapses onto the perturbation solution ofDuan and Muzychka (J. Fluids Eng., 130:031102, 2008) for small enough values of \({\varepsilon \lambda }\) .     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Σ = {0} × S ¹ of operators of the form \({L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}\), \({b\not\equiv0}\) and a(0) ≠ 0, defined on \({\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}\), \({\epsilon >0 }\), where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon)}\) and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of \({C^\infty(\Omega_\epsilon)}\) there is a solution \({u\in L^\infty}\) of the equation Lu = f in a neighborhood of Σ; moreover, the L ^∞ regularity is sharp.     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

Cantor’s intersection theorem for <Emphasis Type="Italic">K</Emphasis>-metric spaces with a solid cone and a contraction principle

We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenovi? and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone. Next we show that our Cantor’s theorem yields a special case of a generalization of Banach’s contraction principle given very recently by Cvetkovi? and Rako?evi? (2014): we assume that a mapping \({T}\) satisfies the condition “\({d(Tx, Ty) \preceq \Lambda (d(x, y))}\)” for \({x, y \in X}\), where \({\preceq}\) is a partial order induced by \({K}\), and \({\Lambda : E \rightarrow E}\) is a linear positive operator with the spectral radius less than one. We also obtain new characterizations of convergence in the sense of Huang and Zhang in a \({K}\)-metric space.     

Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).     

Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators

We consider a finite region of a d-dimensional lattice, \({d \in \mathbb{N}}\), of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size \({\varepsilon}\). Each oscillator weakly interacts by force of order \({\varepsilon}\) with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as \({\varepsilon \rightarrow 0}\) behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order \({\varepsilon^{-1}}\) and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next, we assume that the interaction potential is of size \({\varepsilon \lambda}\), where \({\lambda}\) is another small parameter, independent from \({\varepsilon}\). Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit \({\varepsilon \rightarrow 0}\), the main order in \({\lambda}\) of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space–time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

Short-Time Existence of the Möbius-Invariant Willmore Flow

In this article the author proves existence and uniqueness of a smooth short-time solution of the “Möbius-invariant Willmore flow” Eq. (9) starting in a \(C^{\infty }\)-smooth immersion \(F_0\) of a fixed smooth compact torus \({\varSigma }\) into \(\mathbb {R}^n\) without umbilic points. Hence, for some sufficiently small \(T^*>0\) there exists a unique smooth family \(\{f_t\}\) of smooth immersions of the torus \({\varSigma }\) into \(\mathbb {R}^n\), with \(f_0=F_0\), which solve the evolution Eq. (9) for \(t \in [0,T^*]\) and whose tracefree parts \(A^{0}_{f_t}(x)\) of their second fundamental forms do not vanish in any \((x,t) \in {\varSigma }\times [0,T^*]\). The right-hand side of Eq. (9) has the specific property that any family \(\{f_t\}\) of umbilic-free \(C^4\)-immersions \(f_t:{\varSigma }\longrightarrow \mathbb {R}^n\) solves Eq. (9) if and only if its composition \({\varPhi }(f_t)\) with any applicable Möbius-transformation \({\varPhi }\) of \(\mathbb {R}^n\) solves Eq. (9) as well.     

On the Rationality of the Spectrum

Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].