The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ? ^m , with m≥2 and 1≤k≤m?1, and given ?>0, we define B _? (Λ) to be the geodesic tubular neighborhood of radius ? about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$\begin{cases} \Delta u + u^p = 0 &\mbox{in } B_{\epsilon}(\varLambda) \\ u = 0 & \mbox{on } \partial B_{\epsilon}(\varLambda) , \end{cases} $$ when the parameter ? is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m?k≤2 or $p\in(1, \frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ? tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for $p>\frac{n+2}{n-2}$ , n≥3, if ? is sufficiently small.     

Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rⁿ of the quasilinear singular parabolic equations: u_t = div(|∇u|^m − 1∇u) + t^s|x|^σu^p, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that p_c ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ p_c then every non-trivial solution blows up in finite time, but for p > p_c, a small positive global solution exists.     

The semicycles of solutions of neutral difference equations

In this paper, we study the semicycles of solutions of neutral delay difference equation Δ(y_n + p_ny_n−τ) + q_ny_n−σ = 0, where {p_n} and {q_n} are sequences of nonnegative real numbers, τ and σ are positive integers. Upper bound of numbers of terms of semicycles are determined.     

Non-uniform hyperbolicity in complex dynamics

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n _c so that \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles. Let μ _max be the maximal multiplicity of the critical points.The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\) where δ _Poin (J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ _Poin (J)=HDim(J). If F is polynomially summable with the exponent α, \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.We prove that if F is summable with an exponent \(\alpha< \frac{2}{2+\mu_{\textit{max}}}\) then the Minkowski dimension of J is strictly less than 2 if \(J\neq\hat{\mathbb{C}}\) and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\) then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.     

Probabilities of dominant candidates based on first-place votes

Suppose each of an odd number n of voters has a strict preference order on the three ‘candidates’ in {1,2,3} and votes for his most preferred candidate on a plurality ballot. Assume that a voter who votes for i is equally likely to have ijk and ikj as his preference order when {i,j,k} = {1,2,3}.Fix an integer m between $\text{1}\text{2}$(n + 1) and n inclusive. Then, given that n_i of the n voters vote for i, let f_m(n₁,n₂,n₃) be the probability that one of the three candidates is preferred by m or more voters to each of the other two.This paper examines the behavior of f_m over the lattice points in L_n, the set of triples of non-negative integers that sum to n. It identifies the regions in L_n where f_m is 1 and where f_m is 0, then shows that f_m(a,b + 1, c)>f_m(a + 1,b,c) whenever a + b + c + 1 = n, a≤c≤b, a<c<m and c≤n ? m. These results are used to partially identify the points in L_n where f_m is minimized subject to f_m>0. It is shown that at least two of the n_i are equal at minimizing points.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

The knaster problem: More counterexamples

Given a continuous function\(f:\mathbb{S}^{n - 1} \to \mathbb{R}^m \) andn ?m + 1 pointsp ₁, …,p _{n?m + 1} ε\(p_1 ,...,p_{n - m + 1} \in \mathbb{S}^{n - 1} \), does there exist a rotation ? εSO(n) such thatf(?(p ₁)) = … =f(?(p _n?m+1))? We give a negative answer to this question form = 1 ifn ε {61, 63, 65} orn≥67 and form=2 ifn≥5.     

Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 2^1?n , which is attained when the polynomial is 2^1?n T _n(x), whereT _n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan²(π/2n) is quite simple, but that for |σ|> tan²(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.     

Die multivariate inkomplette Beta-Funktion

The integral $$\int_{\Delta (x)} {u_1^{p_1 ^{ - 1} } ...u_n^{p_n ^{ - 1} } (1 - u_1 - ... - u_n )^{q - 1} du_1 \wedge ... \wedge du_n } $$ extended over then-simplex $$\Delta (x) = \{ (u_1 ,...,u_n ) \in R_ + ^n :u_1 /x_1 + ... + u_n /x_n< 1\} $$ is considered as a natural generalization of the classical incomplete Beta-functionB _x (p, q). The partial differential equations of this multivariate incomplete Beta-function lead to a kind of ‘minimum principle’ for its restriction to the hypersurface with the equation:p ₁/x ₁+...+ +p _n /x _n =c=const.     

On a kind of generalized Lehmer problem

For 1 ? c ? p ? 1, let E ₁,E ₂, …,E _m be fixed numbers of the set {0, 1}, and let a ₁, a ₂, …, a _m (1 ? a _i ? p, i = 1, 2, …,m) be of opposite parity with E ₁,E ₂, …,E _m respectively such that a ₁ a ₂…a _m ≡ c (mod p). Let $$N(c,m,p) = {1 \over {{2^{m - 1}}}}\mathop {\sum\limits_{{a_1} = 1}^{p - 1} {\sum\limits_{{a_2} = 1}^{p - 1} \ldots } }\limits_{{a_1}{a_2} \ldots \equiv c{\rm{ (}}\bmod {\rm{ }}p)} \sum\limits_{{a_m} = 1}^{p - 1} {(1 - {{( - 1)}^{{a_1} + {E_1}}})(1 - {{( - 1)}^{{a_2} + {E_2}}}) \ldots } (1 - {( - 1)^{{a_m} + {E_m}}}).$$ We are interested in the mean value of the sums $$\sum\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$ where E(c, m, p) = N(c,m, p)?((p ? 1) ^m?1)/(2 ^m?1) for the odd prime p and any integers m ? 2. When m = 2, c = 1, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.     

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

LetP _n,k ^c denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[?1, 1], let $B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$ , andM _n,k ^* ?max{n(k+1),nlogn},M _n,k ?n(k+1). It is shown that $M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$ for everyx∈[?1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[?1,1] denotes the supremum norm on [?1,1]. This result should be compared with the inequalities $c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$ , for everyx∈[?1,1], wherec ₃>0 andc ₄>0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.     

Existence and nonexistence of global positive solutions for the evolution P-Laplacian equations in exterior domains

This paper deals with the existence and nonexistence of global positive solutions for two evolution P-Laplacian equations in exterior domains with inhomogeneous boundary conditions. We demonstrate that _qc=n(p−1)/(n−p)$q_c=n(p-1)/(n-p)$ is its critical exponent provided 2n/(n+1)<p<n$2n/(n+1)<p<n$. Furthermore, we prove that if max{1,p−1}<q≤_qc$max\{1,p-1\}<q\le q_c$, then every positive solution of the equations blows up in finite time; whereas for q>_qc$q>q_c$, the equations admit the global positive solutions for some boundary value f(x)$f\left(x\right)$ and some initial data _u0(x)$u_0\left(x\right)$. We also demonstrate that every positive solution of the equations blows up in finite time provided n≤p$n\le p$.     

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability p _c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p _c ~ π ²/(18?log?n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is ?(log?n)^?3/2+o(1), and moreover determining it up to a poly(log?log?n)-factor. The exponent ?3/2 corrects numerical predictions from the physics literature.     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.