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Continuous water and sediment flow monitoring across river cross sections is essential for the management of flood- and sediment-related problems in watersheds. The sediment rating curve (SRC) estimates missing or uncertain sediment flow via its corresponding water discharge. Generally, a power form of relationship correlates the two quantities. The log-transformed water discharge and sediment discharge data were used to depict the SRCs developed in the present study. SRC parameter estimation via least squares regression using at-site dataset pairs can be found in the literature. However, the availability of reliable datasets at the site limits model applicability. This method does not describe the SRC on the basis of the continuity aspects of river system flow characteristics. Therefore, the current study proposes integrated SRC estimation models (Model 2 and Model 3) using modified Muskingum equations abiding by the spatial and temporal continuity of the entire river system state. These models are derived from streamflow storage balance criteria and ensure flow continuity norms. Moreover, Model 3 considers an inverse power form of the relationship depicting the water flow characteristics that govern the sediment transport phenomena through the river system. Standalone models for SRC parameter estimation (Model 1) were also developed for comparison among all three models via the root mean square error (RMSE), NRMSE (normalized root mean square error) and coefficient of determination (R2). The Mahanadi River system within Chhattisgarh state, India comprises five sections at tributaries, and the main channel was considered for the study. The improved NRMSE by Model 2 (7.53%) and Model 3 (7.14%) at Rajim and Model 3 (3.44%) at Bamnidhi in comparison to Model 1 at Rajim (9.19%) and Bamnidhi (4.80%) encouraged the application of integrated models for SRC estimation in river systems. Moreover, Model 3 outperformed Model 2 in some cases where the sediment transport process may be governed by water flow characteristics. [Display omitted] •Sediment rating curve estimate for entire river network replacing standalone model.•Muskingum model applications ensuring flow continuity, is recommended to adopt.•Water flow characteristics parameters influence sediment water relationship in river.•Both integrated models outperformed standalone model at upstream bounding section.

In this paper, we study the dynamical behavior of the solution for the stochastic reaction–diffusion equation with the nonlinearity satisfying the polynomial growth of arbitrary order p≥2 and any space dimension N. Based on the inductive principle, the higher-order integrability of the difference of the solutions near the initial data is established, and then the (norm-to-norm) continuity of solutions with respect to the initial data in H01(U) is first obtained. As an application, we show the existence of (L2(U),Lp(U)) and (L2(U),H01(U))-pullback random attractors, respectively.

Subspaces of an Orlicz space L M generated by probabilistically independent copies of a function , , are studied. In terms of dilations of f , we get a characterization of strongly embedded subspaces of this type and obtain conditions that guarantee that the unit ball of such a subspace has equi-absolutely continuous norms in L M . A class of Orlicz spaces such that, for all subspaces generated by independent identically distributed functions, these properties are equivalent and can be characterized by Matuszewska–Orlicz indices is determined.

This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in a Banach space. The results are obtained by using the resolvent operators and Krasnoselski-Schaefer type fixed point theorem. An example is given to illustrate the results.

We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant—for multiple p-norms—of a feed forward neural network composed of commonly used layer types. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that the performance of the resulting models exceeds that of models trained with other common regularisers. We also provide evidence that the hyperparameters are intuitive to tune, demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model, and show that the performance gains provided by our method are particularly noticeable when only a small amount of training data is available.

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space W^{s,p}, where s \\geq 0 and 1\\leq p\\leq \\infty . The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm \\dot{H}^{-1}, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case s=1 and 1 \\leq p \\leq \\infty (including the case of Lipschitz continuous velocities and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalars that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N -dimensional subspace of the space of continuous functions it is sufficient to use e CN sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m -term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate the application of our technique on the example of trigonometric polynomials.

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ⊆C such that the evaluation functionals φτ(f)=f(τ), τ∈Θ, are continuous on ℋ. The Berezin number of an operator X is defined by ber(X)=supτ∈Θ|X˜(τ)|=supτ∈Θ|〈Xkˆτ,kˆτ〉|, where the operator X acts on the reproducing kernel Hilbert space H=H(Θ) over some (non-empty) set Θ. In this paper, we introduce a new family involving means ∥⋅∥σt between the Berezin radius and the Berezin norm. Among other results, it is shown that if X∈L(H) and f, g are two non-negative continuous functions defined on [0,∞) such that f(t)g(t)=t,(t⩾0), then ∥X∥σ2⩽ber(14(f4(|X|)+g4(|X∗|))+12|X|2) and ∥X∥σ2⩽12ber(f4(|X|)+g2(|X|2))ber(f2(|X|2)+g4(|X∗|)), where σ is a mean dominated by the arithmetic mean ∇.

Neutrosophy consists of neutrosophic logic, probability, and sets. Actually, the neutrosophic set is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set, etc. A neutrosophic set is a mathematical notion serving issues containing inconsistent, indeterminate, and imprecise data. The notion of intuitionistic fuzzy metric space is useful in modelling some phenomena where it is necessary to study the relationship between two probability functions. In this paper, the definition of new metric space with neutrosophic numbers is given. Neutrosophic metric space uses the idea of continuous triangular norms and continuous triangular conorms in intuitionistic fuzzy metric space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. Triangular norms and triangular conorm are very significant for fuzzy operations. Neutrosophic metric space was defined with continuous triangular norms and continuous triangular conorms. Several topological and structural properties neutrosophic metric space have been investigated. The analogues of Baire Category Theorem and Uniform Convergence Theorem are given for Neutrosophic metric spaces.

Let $X, Y$ be two locally compact Hausdorff spaces and $T:C_0(X)\\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e. \\begin{equation*} \\big|\\|T(f)+T(g)\\|-\\|f+g\\|\\big|\\leq \\varepsilon,\\;{\\rm for\\;all}\\; f, g\\in C_0(X). \\end{equation*} Then there exist a homeomorphism $\\varphi:Y\\rightarrow X$ and a continuous function $\\lambda:Y\\rightarrow\\lbrace\\pm1\\rbrace$ such that \\begin{equation*} |T(f)(y)-\\lambda(y)f(\\varphi(y))|\\leq\\frac{3}{2}\\varepsilon,\\;{\\rm for\\;all}\\;y\\in Y,\\;f\\in C_0(X). \\end{equation*} The estimate ‘$\\frac{3}{2}\\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.