Khaleel Ahmad

We develop the mathematical foundations of a framework that attributes a disturbance agent, namely wildfire, harvest, insect defoliation, windthrow, or drought, to each disturbance event detected in standardized satellite time-series change metrics. The attribution task is posed as a statistical decision problem on a feature space built from spectral-temporal descriptors, object geometry, terrain, and climate. Within this formulation we prove the Bayes optimality of the posterior-maximizing rule and of its cost-sensitive counterpart, a finite-sample generalization bound for the empirical-risk minimizer over a finite class, the variance-reduction identity that governs ensemble estimators together with a bias-variance decomposition, an ordering between the Gini and entropy impurities, the convexity and second-order optimality conditions of gradient-boosted trees, the gradient identities of the softmax and focal objectives, the efficiency, symmetry, and null-player properties of the Shapley attribution used for explanation, the order preservation of temperature scaling, and the distribution-free lower and upper coverage bounds of the marginal and class-conditional conformal predictors used for uncertainty. We further prove an unbiasedness and variance result for the design-based accuracy estimator and a decomposition of the cross-region transfer gap, together with an estimable form of that decomposition. Many of these results are specializations of established statistical-learning theory to the attribution setting; the contribution lies in their unified formulation on a standardized metric product, in the explicit statement of the assumptions under which each guarantee holds, and in a small number of new corollaries. The paper is theoretical: all results are established by proof and illustrated with synthetic numerical examples, no empirical forest-disturbance dataset is analysed, and the empirical transferability, calibration, and accuracy of an implemented system remain to be evaluated.

Plug-and-Play methods solve imaging inverse problems by inserting a learned denoiser into a proximal splitting algorithm. Their convergence theory presently requires the denoiser, or its residual, to be nonexpansive or firmly nonexpansive, a constraint that curbs the network and, in turn, the reconstruction quality. We prove that this requirement can be weakened to enriched nonexpansivity, a strictly larger operator class in which the map is nonexpansive only after averaging. Using the displacement and transfer–return calculus for nonexpansive-type operators, we show that when the composite Plug-and-Play operator is enriched nonexpansive an inertial Mann or Ishikawa iteration converges to a fixed point of the operator, that is, to a Plug-and-Play solution, and we identify the inertial correction as the exact quantity that reduces the enriched iteration to a classical iteration of the nonexpansive generator. Around this reduction we build a certification theory. The operator is b -enriched exactly when the displacement I− T is infinitesimally cocoercive with modulus 1/(2( b +1)), so that the smallest admissible constant is one less than the supremum of a Rayleigh-type quotient of that displacement. The characterization is attained on non-scalar operators, is computable by automatic differentiation, and remains bounded in the near-identity regime where the field-of-values bound degenerates. A perturbation estimate converts spectral measurements on a finite sample into a bound over a segment or a covered set, which is what the Fejér argument needs and what a program can carry out. We add a training objective that promotes the certified inequality, a safeguarded step-size controller with a closed-form greedy rule and an amortized policy trained by unrolled differentiation, sublinear and linear rates, stability under inexact denoiser evaluation, and a splitting-agnostic extension covering Plug-and-Play ADMM. Two experiments support the theory. On a linear surrogate, for which every constant is available in closed form, the certificates are confirmed and the exact characterization widens the admissible step-size window by about 31% over the field-of-values bound. On Gaussian deblurring, motion deblurring and single-image super-resolution with trained denoisers, the certified constant is measured for each composite operator. On the deblurring task a trained composite with Lipschitz constant 2.08, admitted by no nonexpansive condition, is certified with constant 0.60, where the characterization is more than fifteen times tighter than the field-of-values bound; it restores within four hundredths of a decibel of a nonexpansive composite of identical architecture, so the larger class is reached at no measured cost in quality. The greedy relaxation halves the iteration count of the constant-step scheme. The acceleration is measured among safeguarded schemes; unrelaxed Picard, which carries no guarantee here, remains faster on the linear instance.

We develop and analyze a mesh-free deep learning method for computing normalized solutions of the quasilinear p -Laplacian Schrödinger equation with prescribed mass in the L ^p -supercritical regime, where solutions are known to exist only through non-constructive variational arguments. The method follows the variational proof: the energy is first reduced along the L ^p -preserving dilations, which removes the unstable direction of the supercritical energy and turns the mountain-pass level into the minimum of a fiber-reduced functional on the mass sphere; this reduced functional is then minimized over a neural ansatz with an augmented-Lagrangian treatment of the mass constraint, and the spectral parameter of the equation is identified with the Lagrange multiplier of that constraint. We establish existence, uniqueness, and regularity of the fiber parameter under explicit hypotheses on the potential, prove that every stationary point of the discrete problem satisfies the weak equation on the dilated tangent space of the ansatz with a multiplier given exactly by a Rayleigh quotient, and show that quasi-minimizers of the discrete problem have energies converging to the ground-state level and, under the compactness property of the existence theory, projected iterates converging strongly to a ground state; this compactness property is proved in the radial potential-free case. Regularization of the degenerate flux, whole-space truncation and quadrature are analyzed, and residual-based diagnostics are derived. For validation, we construct a family of exact manufactured normalized solutions with closed-form potential, multiplier and energy, together with radial references with and without a potential. Numerical experiments confirm the theory quantitatively: the construction is verified, the ground-state energy obeys the proved monotonicity and blow-up laws with the exact power scaling, and the solver recovers supercritical normalized solutions, their multipliers and energies, with relative errors of order 10 ⁻³ in the profile and 10 ⁻⁵ to 10 ⁻⁶ in the multiplier and the energy, for two problems, one of them with a nonzero potential. Mathematics Subject Classification (2020): 35J92; 35Q55; 65N99; 68T07; 35A15.

Sets, probability, and neutrosophic logic are all topics covered by neutrosophy. Moreover, the classical set, fuzzy set, and intuitionistic fuzzy set are generalized using the neutrosophic set. A neutrosophic set is a mathematical concept used to solve problems with inconsistent, ambiguous, and inaccurate data. In this article, we demonstrate some basic fixed-point theorems for any even number of compatible mappings in complete neutrosophic metric spaces. Our primary findings expand and generalize the findings previously established in the literature.

In this manuscript, we give sufficient conditions for a sequence to be Cauchy in the context of controlled fuzzy metric space. Furthermore, we generalize the concept of Banach’s contraction principle by utilizing several new contraction conditions and prove several fixed point results. Furthermore, we provide a number of non-trivial examples to validate the superiority of main results in the existing literature. At the end, we discuss an important application to the transformation of solar energy to electric power by utilizing differential equations.

In this article, we find the fixed point, fixed ellipse, and fixed elliptic disc theorems, we introduce the concepts of an orthogonal -contraction, an orthogonal -weak contraction, an orthogonal -weak JS-contraction, an integral-type orthogonal -weak contraction, and an integral-type orthogonal -weak JS-contraction. To further prove the validity of the postulates, we validate these using illustrated cases with geometric meanings. The reason for this work is the possibility of a geometric shape like a circle, ellipse, disc, or elliptic disc being among the set of non-unique fixed points. We provide an orthogonal -contraction application to chemical sciences.

In order to solve issues that arise in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization issues, equilibrium problems, complementarity issues, selection and matching problems, and issues proving the existence of solutions to integral and differential equations, fixed point theory provides vital tools. In this study, we discuss topological structure and several fixed-point theorems in the context of generalized neutrosophic cone metric spaces. In these spaces, the symmetric properties play an important role. We examine the existence and a uniqueness of a solution by utilizing new types of contraction mappings under some circumstances. We provide an example in which we show the existence and a uniqueness of a solution by utilizing our main result. These results are more generalized in the existing literature.

In this article, we establish the concept of intuitionistic fuzzy double-controlled metric-like spaces by “assuming that the self-distance may not be zero”; if the value of the metric is zero, then it has to be “a self-distance”. We derive numerous fixed-point results for contraction mappings. In addition, we provide several non-trivial examples with their graphical views and an application of integral equations to show the validity of the proposed results.