This paper studies algebraic residual intersections in rings with Serre's condition . It demonstrates that a wide class of residual intersections is set theoretically perfect. This fact leads to determining a uniform upper bound for the multiplicity of residual intersections. In positive characteristic, it follows that residual intersections are cohomologically complete intersection, and hence, their variety is connected in codimension 1.
In this paper, we consider the following Chern-Simons-Schrödinger system
where (u in H^{1}(mathbb{R}^{2})), (p > 4), (A_{alpha }: mathbb{R}^{2} rightarrow mathbb{R}) are the components of the gauge potential, (N: mathbb{R}^{2} rightarrow mathbb{R}) is a neutral scalar field, (V(x)) is a periodic potential function, the parameters (kappa , q>0) represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and (e>0) is the coupling constant. We prove that system ((P)) has a nontrivial solution by using a new infinite-dimensional linking theorem.
Let (ell ^m) be a power with (ell ) a prime greater than 3 and (m) a positive integer such that 3 is a primitive root modulo (2ell ^m). Let (mathbb {F}_3) be the finite field of order 3, and let (mathbb {F}) be the (ell ^{m-1}(ell -1))-th extension field of (mathbb {F}_3). Denote by (text {Tr}) the absolute trace map from (mathbb {F}) to (mathbb {F}_3). For any (alpha in mathbb {F}_3) and (beta in mathbb {F}), let (D) be the set of nonzero solutions in (mathbb {F}) to the equation (text {Tr}(x^{frac{q-1}{2ell ^m}} + beta x) = alpha ). In this paper, we investigate a ternary code (mathcal {C}) of length (n), defined by (mathcal {C}:= {(text {Tr}(d_1x), text {Tr}(d_2x), dots , text {Tr}(d_nx)): x in mathbb {F}}) when we rewrite (D = {d_1, d_2, dots , d_n}). Using recent results on explicit evaluations of exponential sums, the Weil bound, and combinatorial techniques, we determine the Hamming weight distribution of the code (mathcal {C}). Furthermore, we show that when (alpha = beta = 0), the dual code of (mathcal {C}) is optimal with respect to the Hamming bound.
In this paper, we provide new multiplicity results for a class of impulsive fractional Schrödinger-Kirchhoff-type equations involving p-Laplacian and Riemann-Liouville derivatives. By using the variational method and critical point theory, we obtain that the impulsive fractional problem has infinitely many solutions under appropriate hypotheses when the parameter (lambda ) lies in different intervals.
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