DR. ZULFIQAR ALI KHAN

Residual Optimality and Classical Recovery under Zulfia Invexity

Classical optimization theory is formulated on feasibility, where admissible sets aredefined through constraint satisfaction and treated as given. Optimality conditionssuch as the Karush–Kuhn–Tucker (KKT) system are then derived on this predefineddomain.In this paper, we introduc...

Invariant Statistical Inference: A Framework Beyond Classical Statistical Inference

Classical statistical inference is traditionally formulated through probabilitymodels, likelihood functions, and estimation procedures. While these methodshave proven extremely successful, many statistical procedures implicitly rely oninvariance properties under transformations o...

Zulfia Invexity: Structural Origin and Emergence of Invexity

Invexity generalizes convexity by replacing linear displacement with an admissible direction map-ping, ensuring global optimality of stationary points beyond convex settings. However, the admissibledirection is externally specified, and its structural origin remains unexplained.I...

Zulfia Invexity: Structural Origin and Emergence of Invexity

Invexity generalizes convexity by replacing linear displacement with an admissible direction map-ping, ensuring global optimality of stationary points beyond convex settings. However, the admissibledirection is externally specified, and its structural origin remains unexplained.I...

A Class of Generalized Zulfia Invex Functions with In-Built Structural Emergence

Convexity was extended to invexity by replacing the fixed direction with a mapping,and Zulfia invexity further structured admissible mappings. However, a single admissiblemapping is insufficient to capture regime-dependent behavior.We introduce generalized Zulfia invexity by cons...

A Class of Generalized Zulfia Invex Functions with In-Built Structural Emergence

Convexity was extended to invexity by replacing the fixed direction with a mapping,and Zulfia invexity further structured admissible mappings. However, a single admissiblemapping is insufficient to capture regime-dependent behavior.We introduce generalized Zulfia invexity by cons...

A Class of Invariant Programming Problems: A Convexity-Free and KKT-Independent Framework for Optimality

In this paper we introduce a class of invariant programming problems in math-ematical programming. The proposed framework is based on an invariant identityrelation connecting the objective structure with a reference expression through a non-negative remainder termF (x) − Φ(x, u)...

A Class of Zulfia Programming via KKT Conditions Without Assumptions

In classical optimization, convexity and invexity are widely used to ensure thatstationary points are globally optimal. In particular, invexity guarantees the suf-ficiency of Karush–Kuhn–Tucker (KKT) conditions through an inequality-basedframework. However, such conditions are ty...

A Class of Zulfia Programming via KKT Conditions Without Assumptions

In classical optimization, convexity and invexity are widely used to ensure thatstationary points are globally optimal. In particular, invexity guarantees the suf-ficiency of Karush–Kuhn–Tucker (KKT) conditions through an inequality-basedframework. However, such conditions are ty...

Invariant Statistical Inference: A Framework Beyond Classical Statistical Inference

Classical statistical inference is traditionally formulated through probabilitymodels, likelihood functions, and estimation procedures. While these methodshave proven extremely successful, many statistical procedures implicitly rely oninvariance properties under transformations o...

A Certification-Based Formalism for the Event Horizon Boundary

The event horizon is traditionally defined as a geometric surface beyond which causal signalscannot escape. In this note we propose an alternative, purely structural vocabulary for describingthe same object. We introduce a certification framework, in which a carrier domain C is e...

A Certification-Based Formalism for the Event Horizon Boundary

The event horizon is traditionally defined as a geometric surface beyond which causal signalscannot escape. In this note we propose an alternative, purely structural vocabulary for describingthe same object. We introduce a certification framework, in which a carrier domain C is e...

The Ziwave – Pre-Ziwave Duality: End of 100 Years Wave-Particle Duality Photoelectric Effect as a Special Case of Avatar Effect

We prove wave-particle duality is Ziwave – Pre-Ziwave duality. Based on Zulfia Bi-Face Probability Theory (ZBPT) [2] built on Zulfia Decomposition Mapping (ZDM) [1]Z(t) = MZ [I] + ∆(t), the same identity Z(t) sustains two irreversible regimes dividedby certified face ΩC = ∆−1(0):...

The Zulfia Quantum Ziwave Equation: Schr¨odinger as a Special Case

For more than a century since Schr¨odinger, the quantum wave equation has been as-sumed to hold on a fully accessible domain Ω without certified origin. No extension in theavailable literature has revisited its foundational probability assumption.We replace this assumption with a...

The Pre-Ziwave Equation: Before Collision on a Certified Domain d'Alembert as a Special Case

For more than a century since d’Alembert, the classical wave equation has been assumedto hold on a fully accessible domain Ω. No extension in the available literature has revisitedits foundational domain assumption.We replace this assumption with a certified origin. Using Zulfia...

The Pre-Ziwave Gravity Equation: Certified Domain Filtering of Mass-Information – Newton and Einstein as Special Cases

We derive the Pre-Ziwave Gravity Equation on a certified domain ΩC underZulfia Bi-Face Probability Theory (ZBPT). Classical gravity assumes∇2Φ = 4πGρ and □hμν = − 16πGc4 Tμνon full possibility space Ω, i.e. certification C(ω) ≡ 1 and filtering trace ∆ ≡ 0. Weprove that on Z-Domai...

Generalized Zulfia Ziwave Equation Beyond Failure Domain

In Zulfia’s Certified Probability Theory (ZBPT Framework [1]), we define theFailure Domain as Df = Ω \ Dc, where Ω is the total probability space and Dc is thecertified domain with C(Dc) = 1. The classical wave equation [3] fails in Df becauseit assumes Ω = Dc. We show that failu...

Fundamentals of Reflection Generated Ziwave Beyond the Rumour Concept of Quantum Wave

For a century, Quantum Mechanics has been predicated on an unverified assump-tion – the existence of the quantum wave. We demonstrate that this concept is amisidentification. While classical waves such as water waves and acoustic wavesexist as physical reflections, a particle-gen...

Zulfia Bi-Face Probability Theory (ZBPT): Probability Generated on Certified Domains, Not Assumed on Full Sample Spaces

Classical probability theory assigns uncertainty directly on a full sample space Ω, implicitlyassuming every ω ∈ Ω is accessible for measurement. This assumption is never verified — it isimposed. This paper introduces Zulfia Bi-Face Probability Theory (ZBPT), a completereformulat...

Generalized Zulfia Ziwave Equation Beyond Failure Domain

In Zulfia’s Certified Probability Theory (ZBPT Framework [1]), we define theFailure Domain as Df = Ω \ Dc, where Ω is the total probability space and Dc is thecertified domain with C(Dc) = 1. The classical wave equation [3] fails in Df becauseit assumes Ω = Dc. We show that failu...