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1. Title: Erasures and Recovery: Optimal K-Dual Frames in Redundant Systems
1. Title: Erasures and Recovery: Optimal K-Dual Frames in Redundant Systems
[Paper ID: ICPAM-DS-26-AM0001]
[Paper ID: ICPAM-DS-26-AM0001]
Presenter: Shankhadeep Mondal (Joint work with Dr. Deguang Han and Dr. R. N. Mohapatra)
Presenter: Shankhadeep Mondal (Joint work with Dr. Deguang Han and Dr. R. N. Mohapatra)
Abstract: Frames provide redundant representations that are inherently robust to data loss, making them fundamental tools in signal processing and related applications. In practical transmission systems, however, the loss of frame coefficients—known as erasures—can significantly affect reconstruction accuracy. This talk investigates optimal reconstruction strategies for K-frames in the presence of erasures, where reconstruction is constrained to the range of a bounded linear operator K. We introduce and analyze optimal K-dual frames and K-dual pairs under deterministic and probabilistic erasure models. Optimality is measured using operator-theoretic quantities, including the operator norm and the spectral radius of the associated error operators. For single and multiple erasures, we characterize when a K-dual minimizes the worst-case reconstruction error and identify conditions under which the canonical K-dual is optimal or uniquely optimal. In addition, we introduce the notions of uniform and higher-order uniform K-dual pairs, which play a key role in spectral optimality for multiple erasures. Explicit bounds for recon struction error are derived in terms of trace and spectral data of the operator K, and examples are provided to illustrate both uniqueness and non-uniqueness phenomena. These results extend classical frame-theoretic robustness principles to the broader setting of K-frames and provide a unified framework for stable reconstruction under structured data loss.
Abstract: Frames provide redundant representations that are inherently robust to data loss, making them fundamental tools in signal processing and related applications. In practical transmission systems, however, the loss of frame coefficients—known as erasures—can significantly affect reconstruction accuracy. This talk investigates optimal reconstruction strategies for K-frames in the presence of erasures, where reconstruction is constrained to the range of a bounded linear operator K. We introduce and analyze optimal K-dual frames and K-dual pairs under deterministic and probabilistic erasure models. Optimality is measured using operator-theoretic quantities, including the operator norm and the spectral radius of the associated error operators. For single and multiple erasures, we characterize when a K-dual minimizes the worst-case reconstruction error and identify conditions under which the canonical K-dual is optimal or uniquely optimal. In addition, we introduce the notions of uniform and higher-order uniform K-dual pairs, which play a key role in spectral optimality for multiple erasures. Explicit bounds for recon struction error are derived in terms of trace and spectral data of the operator K, and examples are provided to illustrate both uniqueness and non-uniqueness phenomena. These results extend classical frame-theoretic robustness principles to the broader setting of K-frames and provide a unified framework for stable reconstruction under structured data loss.
2. Title: Dynamics of a two-stage epidemiological model with post-infection mortality and transmission heterogeneity
2. Title: Dynamics of a two-stage epidemiological model with post-infection mortality and transmission heterogeneity
[Paper ID: ICPAM-DS-26-AM0002]
[Paper ID: ICPAM-DS-26-AM0002]
Presenter: B Sagar Email: b.sagar@ucf.edu Co-Author: Zhisheng Shuai
Presenter: B Sagar Email: b.sagar@ucf.edu Co-Author: Zhisheng Shuai
School of Data, Mathematical, and Statistical Sciences, University of Central Florida, Orlando, FL. 32826
School of Data, Mathematical, and Statistical Sciences, University of Central Florida, Orlando, FL. 32826
Abstract: We formulate a two-stage epidemiological model that incorporates key post-infection features, including reinfection and post-infection mortality (PIM). The model emphasizes the role of transmission heterogeneity in shaping disease dynamics, influencing both endemic levels and oscillatory behaviors. Numerical simulations show that late-stage hyper-infectivity leads to higher endemic infection levels with long-period oscillations, while early-stage hyper-infectivity results in lower infection levels with shorter-period oscillations.
Abstract: We formulate a two-stage epidemiological model that incorporates key post-infection features, including reinfection and post-infection mortality (PIM). The model emphasizes the role of transmission heterogeneity in shaping disease dynamics, influencing both endemic levels and oscillatory behaviors. Numerical simulations show that late-stage hyper-infectivity leads to higher endemic infection levels with long-period oscillations, while early-stage hyper-infectivity results in lower infection levels with shorter-period oscillations.
3. Title: Statistical and Weighted Convergence Rates for Generalized q- Bernstein-Durrmeyer Positive Linear Operators with Applications to Neural-Network-Inspired Operators
3. Title: Statistical and Weighted Convergence Rates for Generalized q- Bernstein-Durrmeyer Positive Linear Operators with Applications to Neural-Network-Inspired Operators
[Paper ID: ICPAM-DS-26-AM0008]
[Paper ID: ICPAM-DS-26-AM0008]
Department of Mathematics, Kalinga University, Naya Raipur (CG) IN-492101
Department of Mathematics, Kalinga University, Naya Raipur (CG) IN-492101
Abstract: This paper examines the statistical and weighted convergence characteristics of a generalized family of q-Bernstein-Durrmeyer positive linear operators. Employing Korovkin-type approximation theory, adequate conditions are delineated for the uniform statistical convergence of the proposed operators. Quantitative assessments of the convergence rate are obtained through the utilization of the modulus of continuity and second-order central moments. Moreover, weighted approximation results are derived, facilitating the examination of convergence on unbounded intervals. To show how general the framework is, we talk about special cases that are related to classical Bernstein-Durrmeyer operators. Due to recent advancements at the intersection of approximation theory and machine learning, an application to neural-network- inspired averaging operators is introduced, demonstrating the utility of positive linear operators in the analysis of the approximation characteristics of specific neural network layers. The findings augment and consolidate various recent advancements in the theory of q-operators and statistical approximation.
Abstract: This paper examines the statistical and weighted convergence characteristics of a generalized family of q-Bernstein-Durrmeyer positive linear operators. Employing Korovkin-type approximation theory, adequate conditions are delineated for the uniform statistical convergence of the proposed operators. Quantitative assessments of the convergence rate are obtained through the utilization of the modulus of continuity and second-order central moments. Moreover, weighted approximation results are derived, facilitating the examination of convergence on unbounded intervals. To show how general the framework is, we talk about special cases that are related to classical Bernstein-Durrmeyer operators. Due to recent advancements at the intersection of approximation theory and machine learning, an application to neural-network- inspired averaging operators is introduced, demonstrating the utility of positive linear operators in the analysis of the approximation characteristics of specific neural network layers. The findings augment and consolidate various recent advancements in the theory of q-operators and statistical approximation.
Keywords: q-Bernstein-Durrmeyer operators; Positive linear operators; Statistical convergence; Rate of convergence; Weighted approximation; Neural-network-inspired operators.
Keywords: q-Bernstein-Durrmeyer operators; Positive linear operators; Statistical convergence; Rate of convergence; Weighted approximation; Neural-network-inspired operators.
Mathematics Subject Classification (MSC 2020): 41A36
Mathematics Subject Classification (MSC 2020): 41A36
4. Title: Analytical and Dynamical Investigation of the Modified Nonlinear Landau–Ginzburg–Higgs Equation via the Generalized Exponential Rational Function Method
4. Title: Analytical and Dynamical Investigation of the Modified Nonlinear Landau–Ginzburg–Higgs Equation via the Generalized Exponential Rational Function Method
[Paper ID: ICPAM-DS-26-AM0004]
[Paper ID: ICPAM-DS-26-AM0004]
Presenter: Rajat Ranjan Priyadarshi
Presenter: Rajat Ranjan Priyadarshi
Department of Physics, VSSUT, Burla
Department of Physics, VSSUT, Burla
Abstract: In this work, we investigate the Modified Nonlinear Landau–Ginzburg–Higgs equation us- ing analytical and dynamical approaches. By employing the Generalized Exponential Rational Function Method, exact traveling wave solutions are constructed in closed form. To enrich the solution space, the analysis is further extended to obtain new Jacobi elliptic function solu- tions of sn- and cn-type, which reduce to solitary wave structures under appropriate parametric limits. Furthermore, the dynamical behavior of the governing system is examined through phase portrait analysis, revealing periodic and chaotic characteristics for suitable parameter regimes. A linear stability analysis of the obtained solutions is also carried out to discuss their physical feasibility and robustness. The results demonstrate the effectiveness of the proposed method in generating diverse nonlinear wave structures and provide deeper insight into the complex dynamics of the Modified Nonlinear Landau–Ginzburg–Higgs equation.
Abstract: In this work, we investigate the Modified Nonlinear Landau–Ginzburg–Higgs equation us- ing analytical and dynamical approaches. By employing the Generalized Exponential Rational Function Method, exact traveling wave solutions are constructed in closed form. To enrich the solution space, the analysis is further extended to obtain new Jacobi elliptic function solu- tions of sn- and cn-type, which reduce to solitary wave structures under appropriate parametric limits. Furthermore, the dynamical behavior of the governing system is examined through phase portrait analysis, revealing periodic and chaotic characteristics for suitable parameter regimes. A linear stability analysis of the obtained solutions is also carried out to discuss their physical feasibility and robustness. The results demonstrate the effectiveness of the proposed method in generating diverse nonlinear wave structures and provide deeper insight into the complex dynamics of the Modified Nonlinear Landau–Ginzburg–Higgs equation.
Keywords: Modified Landau–Ginzburg–Higgs equation; Generalized Exponential Rational Func- tion Method; Exact solutions; Jacobi elliptic functions; Chaos; Stability analysis.
Keywords: Modified Landau–Ginzburg–Higgs equation; Generalized Exponential Rational Func- tion Method; Exact solutions; Jacobi elliptic functions; Chaos; Stability analysis.
5. Title: Secure Visual Data Transmission through Neural-Based Key Synchronization and Authenticated Image Encryption [Paper ID: ICPAM-DS-26-AM0006]
5. Title: Secure Visual Data Transmission through Neural-Based Key Synchronization and Authenticated Image Encryption [Paper ID: ICPAM-DS-26-AM0006]
Department of Mathematics, National Institute of Technology Raipur, Raipur, Chhattisgarh, India.
Department of Mathematics, National Institute of Technology Raipur, Raipur, Chhattisgarh, India.
Abstract: The use of multimedia data transmission over insecure networks has also led to the challenge of facilitating digital image security which has become a major obstacle. A hybrid security frame- work is introduced in this paper that combines Neural Cryptography for secure key exchange and AES-GCM for strong image encryption. The system proposed implements a Tree Par- ity Machine (TPM) as a means of synchronizing two parties. The TPMs start with different random weights and a common input vector, and they learn from each other through the use of Hebbian, Anti-Hebbian, and Random Walk rules. The weights are then used to create a common secret key which reduces the susceptibility of the key exchange methods to the vul- nerabilities of the traditional key exchange methods when the synchronization of the parties reaches 100%. To increase the strength of the encryption, the TPM-synchronized key is then processed through HMAC-SHA256 to produce a high- entropy 256-bit key. This key is used in the Advanced Encryption Standard in Galois/Counter Mode (AES-GCM), which ensures both data confidentiality and has built-in integrity authentication. The framework is tested using a standard 8-bit grayscale image of size 256×256. The security analysis reveals that the encrypted output has an Entropy of 7.997 which is nearly ideal and signifies total randomness. The histogram analysis proves that the distribution is uniform thus successfully hiding the pixel-level statistical features. In addition, the Correlation Coefficients in horizontal, vertical, and diagonal orientations were lowered to almost zero. The resistance to differential cryptanal- ysis was verified by an NPCR of 99.58% and a UACI of 26.74%. The system guarantees the original image to be listlessly reconstructed, hence concluding that the neural-cryptographic approach proposed is very secure and computationally effective for the protection of modern visual data.
Abstract: The use of multimedia data transmission over insecure networks has also led to the challenge of facilitating digital image security which has become a major obstacle. A hybrid security frame- work is introduced in this paper that combines Neural Cryptography for secure key exchange and AES-GCM for strong image encryption. The system proposed implements a Tree Par- ity Machine (TPM) as a means of synchronizing two parties. The TPMs start with different random weights and a common input vector, and they learn from each other through the use of Hebbian, Anti-Hebbian, and Random Walk rules. The weights are then used to create a common secret key which reduces the susceptibility of the key exchange methods to the vul- nerabilities of the traditional key exchange methods when the synchronization of the parties reaches 100%. To increase the strength of the encryption, the TPM-synchronized key is then processed through HMAC-SHA256 to produce a high- entropy 256-bit key. This key is used in the Advanced Encryption Standard in Galois/Counter Mode (AES-GCM), which ensures both data confidentiality and has built-in integrity authentication. The framework is tested using a standard 8-bit grayscale image of size 256×256. The security analysis reveals that the encrypted output has an Entropy of 7.997 which is nearly ideal and signifies total randomness. The histogram analysis proves that the distribution is uniform thus successfully hiding the pixel-level statistical features. In addition, the Correlation Coefficients in horizontal, vertical, and diagonal orientations were lowered to almost zero. The resistance to differential cryptanal- ysis was verified by an NPCR of 99.58% and a UACI of 26.74%. The system guarantees the original image to be listlessly reconstructed, hence concluding that the neural-cryptographic approach proposed is very secure and computationally effective for the protection of modern visual data.
Keywords: Image Encryption, Tree Parity Machine, AES-GCM, Hebbian, Anti-Hebbian, and Random Walk rules
Keywords: Image Encryption, Tree Parity Machine, AES-GCM, Hebbian, Anti-Hebbian, and Random Walk rules
6. Title: Presenter: Abstract:
6. Title: Presenter: Abstract:
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