Global Solutions to a Nonlocal Fisher-KPP Type Problem

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\), \(\beta\). More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\), \(\alpha<1+2/n\) is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem.     

A Tight Upper Bound on Acquaintance Time of Graphs

In this note we confirm a conjecture raised by Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) on the acquaintance time of graphs, proving that for all graphs G with n vertices it holds that \(\mathcal {AC}(G) = O(n^{3/2})\). This is done by proving that for all graphs G with n vertices and maximum degree \(\varDelta \) it holds that \(\mathcal {AC}(G) \le 20 \varDelta n\). Combining this with the bound \(\mathcal {AC}(G) \le O(n^2/\varDelta )\) from Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) gives the uniform upper bound of \(O(n^{3/2})\) for all n-vertex graphs. This bound is tight up to a multiplicative constant. We also prove that for the n-vertex path \(P_n\) it holds that \(\mathcal {AC}(P_n)=n-2\). In addition we show that the barbell graph \(B_n\) consisting of two cliques of sizes \({\lceil n/2\rceil }\) and \({\lfloor n/2\rfloor }\) connected by a single edge also has \(\mathcal {AC}(B_n) = n-2\). This shows that it is possible to add \(\varOmega (n^2\)) edges a graph without changing its \(\mathcal {AC}\) value.     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

Some Ricci-flat ($$\alpha ,\beta $$)-metrics

In this paper, we study a special class of Finsler metrics, \((\alpha ,\beta )\)-metrics, defined by \(F=\alpha \phi (\beta /\alpha )\), where \(\alpha \) is a Riemannian metric and \(\beta \) is a 1-form. We find an equation that characterizes Ricci-flat \((\alpha ,\beta )\)-metrics under the condition that the length of \(\beta \) with respect to \(\alpha \) is constant.     

Estimates for large solutions of a quasilinear elliptic equation

Let \(\Omega \) be a bounded smooth domain of \(R^{n}\). We study the asymptotic behaviour of the solutions to the equation \(\triangle u-|Du|^{q}=f(u)\) in \(\Omega , 1<q<2,\) which satisfy the boundary condition \(u(x)\rightarrow \infty \) as \(x\rightarrow \partial \Omega \). These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio \(\psi (u)/\delta \), where \(\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt\), \(F'=f\), \(\delta =dist(x,\partial \Omega )\) or for the ratio \(u/\delta ^{(2-q)/(1-q)}\). When in particular the ratio \(f/F^{q/2}\)is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).     

Hankel Determinants of Random Moment Sequences

For \(t \in [0,1]\) let \(\underline{H}_{2\lfloor nt \rfloor } = (m_{i+j})_{i,j=0}^{\lfloor nt \rfloor }\) denote the Hankel matrix of order \(2\lfloor nt \rfloor \) of a random vector \((m_1,\ldots ,m_{2n})\) on the moment space \(\mathcal {M}_{2n}(I)\) of all moments (up to the order 2n) of probability measures on the interval \(I \subset \mathbb {R}\). In this paper we study the asymptotic properties of the stochastic process \(\{ \log \det \underline{H}_{2\lfloor nt \rfloor } \}_{t\in [0,1]}\) as \(n \rightarrow \infty \). In particular weak convergence and corresponding large deviation principles are derived after appropriate standardization.     

Phase transition in loop percolation

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters \(\alpha \) and \(\kappa \). The non-negative parameter \(\alpha \) measures the amount of loops, and \(\kappa \) plays the role of killing on vertices penalizing (\(\kappa >0\)) or favoring (\(\kappa <0\)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed \(\kappa \) and large enough \(\alpha \), there exists an infinite cluster in the loop percolation on \({\mathbb {Z}}^d\). In the present article, we show a non-trivial phase transition on the integer lattice \({\mathbb {Z}}^d\) (\(d\ge 3\)) for \(\kappa =0\). More precisely, we show that there is no loop percolation for \(\kappa =0\) and \(\alpha \) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of \(\alpha \), namely, for \(\kappa =0\) and any sub-critical value of \(\alpha \), the probability of one-arm event decays at most polynomially. For \(d\ge 5\), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For \(\alpha \) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For \(d=3\) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity \(\alpha \).     

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any \(d\ge 2\) and any prime \(p>(d^2-3d+4)^2\) there is no complete mapping polynomial in \(\mathbb {F}_p[x]\) of degree d. For arbitrary finite fields \(\mathbb {F}_q\), we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if \(n<\lfloor q/2\rfloor \), then there is no complete mapping in \(\mathbb {F}_q[x]\) of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank \(n<\lfloor q/2\rfloor \) are from being complete, by studying value sets of \(f+x.\) We provide examples of complete mappings if \(n=\lfloor q/2\rfloor \), which shows that the above bound cannot be improved in general.     

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.     

Beatty sequences and prime numbers with restrictions on strongly <Emphasis Type="Italic">q</Emphasis>-additive functions

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes \(p=\lfloor \alpha n +\beta \rfloor \) and \(f(p)\equiv a (\mathrm{mod\,}b)\), with \(n\ge N \), where \(\alpha \), \(\beta \) are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer \(k\ge 3 \), all sufficiently large \(N\equiv k (\mathrm{mod\,}2) \) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.     

Bilinear Fractional Integral Along Homogeneous Curves

The boundedness of the bilinear fractional integrals along homogeneous curves \(\gamma (t)=(t^{\alpha _1},t^{\alpha _2})\) with \(\alpha _2>\alpha _1\ge 1\) is obtained. The authors extend the results of the bilinear fractional integrals of Kenig and Stein (Math Res Lett 6:1–15, 1999) and Grafakos and Kalton (Math Ann 319(1):151–180, 2001) to integrals along the curves.     

Stark points and the Hida–Rankin <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-function

This article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a p-adic avatar of the leading term at \(s=1\) of the Hasse–Weil–Artin L-series \(L(E,\varrho _1\otimes \varrho _2,s)\) of an elliptic curve \(E/\mathbb {Q}\) twisted by the tensor product \(\varrho _1\otimes \varrho _2\) of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a \(2\times 2\) p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where \(\varrho _1\) and \(\varrho _2\) are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.     

On quasi-inversions

Given a bounded domain \(D \subset {\mathbb R}^n\) strictly starlike with respect to \(0 \in D\,,\) we define a quasi-inversion w.r.t. the boundary \(\partial D \,.\) We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every “tangent line” of \(\partial D\) is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1,  when \(\partial D\) approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the \(\alpha \)-tangent condition due to Gehring and Väisälä (Acta Math 114:1–70,1965). This condition is shown to be equivalent to the bi-Lipschitz and quasiconformal extension property of what we call the polar parametrization of \(\partial D\). In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto \(\partial D\), is bi-Lipschitz if and only if D satisfies the \(\alpha \)-tangent condition.     

Improved Moser–Trudinger inequality of Tintarev type in dimension <Emphasis Type="Italic">n</Emphasis> and the existence of its extremal functions

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$

for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)

     

Dirichlet series with functional equations and arithmetical identities

Exploiting the functional equation of Hecke-type associated with a function satisfying a modular relation with a residual function as developed in Bochner (J Indian Math Soc 16:99–102, 1952), Chandrasekharan and Narasimhan (Ann Math 74:1–23, 1961) derived the equivalence of the functional equation to two arithmetical identities. Hawkins and Knopp (Contemp Math 143:451–475, 1993) showed the equivalence of the functional equation to modular integrals with rational period functions of weight 2k, \(k \in \mathbb {Z}^+\) on the theta group \(\Gamma _\vartheta \). The aim of the current work is to show that results analogous to those of Chandrasekharan and Narasimhan can be developed in the Hawkins and Knopp context, but with respect to the full modular group \(\Gamma (1)\), rather than the theta group \(\Gamma _\vartheta \).     

Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

An algorithmic upper bound on the domination number \(\gamma \) of graphs in terms of the order n and the minimum degree \(\delta \) is proved. It is demonstrated that the bound improves best previous bounds for any \(5\le \delta \le 50\). In particular, for \(\delta =5\), Xing et al. (Graphs Comb. 22:127–143, 2006) proved that \(\gamma \le 5n/14 < 0.3572 n\). This bound is improved to 0.3440 n. For \(\delta =6\), Clark et al. (Congr. Numer. 132:99–123, 1998) established \(\gamma <0.3377 n\), while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to \(\gamma <0.3340 n\). Here the bound is further improved to \(\gamma < 0.3159n\). For \(\delta =7\), the best earlier bound 0.3088n is improved to \(\gamma < 0.2927n\).     

Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Few associative triples,isotopisms and groups

Let Q be a quasigroup. For \(\alpha ,\beta \in S_Q\) let \(Q_{\alpha ,\beta }\) be the principal isotope \(x*y = \alpha (x)\beta (y)\). Put \(\mathbf a(Q)= |\{(x,y,z)\in Q^3;\) \(x(yz)) = (xy)z\}|\) and assume that \(|Q|=n\). Then \(\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})\), and for every \(\alpha \in S_Q\) there is \(\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2\), where \(f_x=|\{y\in Q;\) \( y = \alpha (y)x\}|\). If G is a group and \(\alpha \) is an orthomorphism, then \(\mathbf a(G_{\alpha ,\beta })=n^2\) for every \(\beta \in S_Q\). A detailed case study of \(\mathbf a(G_{\alpha ,\beta })\) is made for the situation when \(G = \mathbb Z_{2d}\), and both \(\alpha \) and \(\beta \) are “natural” near-orthomorphisms. Asymptotically, \(\mathbf a(G_{\alpha ,\beta })>3n\) if G is an abelian group of order n. Computational results: \(\mathbf a(7) = 17\) and \(\mathbf a(8) \le 21\), where \(\mathbf a(n) = \min \{\mathbf a(Q);\) \( |Q|=n\}\). There are also determined minimum values for \(\mathbf a(G_{\alpha ,\beta })\), G a group of order \(\le 8\).     

Weakly horospherically convex hypersurfaces in hyperbolic space

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\). Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\). In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\), we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\).     

Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal

We consider an integro-PDE model from evolutionary biology. The solution \(u_\epsilon (x,\alpha )\) is structured by two variables \(x\in D \subset {\mathbb {R}}^k\) and \(\alpha \in (\underline{\alpha },{\overline{\alpha }})\subset \subset {\mathbb {R}}_+\). The diffusion coefficient in the x direction depends on \(\alpha \) and the diffusion coefficient in the \(\alpha \) direction is a constant \(\epsilon ^2\). A special feature of this model is the appearance of the integral \({\hat{u}}_\epsilon (x)\) of the solution in the \(\alpha \) variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by \(\alpha \). As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.