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In the present paper, we study a lifespan of solutions to the Cauchy problem for semilinear damped wave equations (DW) { ∂ t 2 u − Δ u + ∂ t u = f ( u ) , ( t , x ) ∈ [ 0 , T ( ε ) ) × R n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R n , ∂ t u ( 0 , x ) = ε u 1 ( x ) , x ∈ R n , where n ≥ 1 , f ( u ) = ± | u | p − 1 u or | u | p , p ≥ 1 , ε > 0 is a small parameter, and ( u 0 , u 1 ) is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is f ( u ) = | u | p and the nonlinear power is the Fujita critical exponent p = p F = 1 + 2 n , then the upper estimate to the lifespan is estimated by T ( ε ) ≤ exp ⁡ ( C ε − p ) for all ε ∈ ( 0 , 1 ] and suitable data ( u 0 , u 1 ) , without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35] . We also prove a sharp lower estimate of the lifespan T ( ε ) to (DW) in the critical case p = p F . [ABSTRACT FROM AUTHOR]

A fascinating problem on digraphs is the existence problem of the finite upper bound on s for all vertex-primitive s -arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper, we prove that s ≤ 2 for all vertex-primitive s -arc-transitive digraphs for almost simple groups with socles alternating groups except one case. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give new constraints on the distance distribution of doubly constant-weight (binary) codes. These constraints improve the linear programming bound on sizes of doubly constant-weight codes. Computations are done for all codes of length n ≤ 28 and all improved upper bounds are shown. We moreover show that the improved upper bounds give rise to further new upper bounds. [ABSTRACT FROM AUTHOR]

It is well known that the Liénard system x ˙ = y − F ( x ) , y ˙ = − g ( x ) with symmetry (i.e. F ( x ) and g ( x ) are odd functions) has a unique limit cycle under some hypotheses. In this paper we will show that the unique limit cycle locates in the strip region | x | < x ⁎ , where x ⁎ > 0 is uniquely and directly determined by ∫ 0 x ⁎ F ( x ) g ( x ) d x = 0 . In other words, an explicit upper bound x ⁎ is given for the amplitude (i.e. the maximal value of the x -coordinate) of the unique limit cycle. As a simple application we obtain a uniform estimate A ( μ ) < 5 ≐ 2.2361 for each μ > 0 , where A ( μ ) denotes the amplitude of the unique limit cycle in the Liénard system x ˙ = y − μ ( x 3 / 3 − x ) , y ˙ = − x for the van der Pol equation x ¨ + μ ( x 2 − 1 ) x ˙ + x = 0 . The upper bound 5 improves the existing ones. [ABSTRACT FROM AUTHOR]