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In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k -set agreement , n processes must decide on at most k of their input values. While n -set agreement is trivially wait-free solvable by each process deciding on its input, ( n − 1 ) -set agreement is not wait-free solvable. In M -renaming , processes must decide on distinct names in a range of size M . For any number n of processes, ( 2 n − 1 ) -renaming is wait-free solvable, but surprisingly, ( 2 n − 2 ) -renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n + 1 . In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that ( 2 p − 1 ) -adaptive renaming is wait-free solvable, while ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming is not. This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither ( n − 1 ) -set agreement nor ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming; if n is a prime power, n processes cannot wait-free solve ( 2 n − 2 ) -renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers. Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the task’s conditions. This makes the proofs easier to understand, verify, and, we hope, extend. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence of segregated vector solutions for where Ω is a bounded or unbounded domain in with , is a small parameter, is an integer, ( ) is a potential function, ( ) is constant and ( ) is coupling constant. This system describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive in optics and Bose–Einstein condensates. For , which corresponds to the synchronization case for the above system with constant potentials, we prove that the system has multiple positive vector solutions, whose components may have spikes clustering at the same point as , but the distance between them divided by ε will go to infinity. [Copyright &y& Elsevier]

Abstract: In this paper it is shown that there exists a constant such that there do not exist Cameron–Liebler line classes in with parameter x satisfying . This improves all known bounds except for small values of q. [Copyright &y& Elsevier]

Abstract: We consider nonlinear elliptic partial differential equations for quasilinear operators of the form subject to fully nonlinear boundary conditions involving boundary operators of the form, for each , The main goal of this paper is to give, under suitable assumptions on and , an explicit estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators which are, possibly, of the same order as . As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales. [Copyright &y& Elsevier]

Abstract: In this paper, we give existence, uniqueness and regularity of the solutions of problems whose prototype is where is a bounded open set of , , and . Here is a norm on , is its polar and is the anisotropic Laplacian. [Copyright &y& Elsevier]