| Name | dividebyzero JSON |
| Version |
0.1.2
JSON |
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| home_page | None |
| Summary | A NumPy extension implementing division by zero as dimensional reduction |
| upload_time | 2024-12-29 09:03:30 |
| maintainer | None |
| docs_url | None |
| author | None |
| requires_python | >=3.8 |
| license | None |
| keywords |
mathematics
numpy
quantum
tensor
|
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# DivideByZero: Dimensional Reduction Through Mathematical Singularities
## Foundational Framework for Computational Singularity Analysis
DivideByZero (`dividebyzero`) implements a novel mathematical framework that reconceptualizes division by zero as dimensional reduction operations. This paradigm shift transforms traditionally undefined mathematical operations into well-defined dimensional transformations, enabling new approaches to numerical analysis and quantum computation.
## Core Mathematical Principles
### Dimensional Division Operator
The framework defines division by zero through the dimensional reduction operator $\oslash$:
For tensor $T \in \mathcal{D}_n$:
```
T ∅ 0 = π(T) + ε(T)
```
Where:
- $\pi(T)$: Projection to lower dimension
- $\epsilon(T)$: Quantized error preservation
- $\mathcal{D}_n$: n-dimensional tensor space
### Installation
```bash
pip install dividebyzero
```
## Fundamental Usage Patterns
### Basic Operations
```python
import dividebyzero as dbz
# Create dimensional array
x = dbz.array([[1, 2, 3],
[4, 5, 6]])
# Divide by zero - reduces dimension
result = x / 0
# Reconstruct original dimensions
reconstructed = result.elevate()
```
### Key Features
#### 1. Transparent NumPy Integration
- Drop-in replacement for numpy operations
- Preserves standard numerical behavior
- Extends functionality to handle singularities
#### 2. Information Preservation
- Maintains core data characteristics through reduction
- Tracks quantum error information
- Enables dimensional reconstruction
#### 3. Advanced Mathematical Operations
```python
# Quantum tensor operations
from dividebyzero.quantum import QuantumTensor
# Create quantum-aware tensor
q_tensor = QuantumTensor(data, physical_dims=(2, 2, 2))
# Perform gauge-invariant reduction
reduced = q_tensor.reduce_dimension(
target_dims=2,
preserve_entanglement=True
)
```
## Theoretical Framework
### Mathematical Foundations
The framework builds on several key mathematical concepts:
1. **Dimensional Reduction**
- Singular Value Decomposition (SVD)
- Information-preserving projections
- Error quantization mechanisms
2. **Quantum Extensions**
- Tensor network operations
- Gauge field computations
- Holonomy calculations
3. **Error Tracking**
- Holographic error encoding
- Dimensional reconstruction algorithms
- Quantum state preservation
## Advanced Applications
### 1. Quantum Computing
```python
# Quantum state manipulation
state = dbz.quantum.QuantumTensor([
[1, 0],
[0, 1]
])
# Preserve quantum information during reduction
reduced_state = state / 0
```
### 2. Numerical Analysis
```python
# Handle singularities in numerical computations
def stable_computation(x):
return dbz.array(x) / 0 # Returns dimensional reduction instead of error
```
### 3. Data Processing
```python
# Dimensionality reduction with information preservation
reduced_data = dbz.array(high_dim_data) / 0
reconstructed = reduced_data.elevate()
```
## Technical Requirements
- Python ≥ 3.8
- NumPy ≥ 1.20.0
- SciPy ≥ 1.7.0
### Optional Dependencies
- networkx ≥ 2.6.0 (for quantum features)
- pytest ≥ 6.0 (for testing)
## Development and Extension
### Contributing
1. Fork the repository
2. Create feature branch
3. Implement changes with tests
4. Submit pull request
### Testing
```bash
pytest tests/
```
## Mathematical Documentation
Detailed mathematical foundations are available in the [Technical Documentation](docs/theory.md), including:
- Formal proofs of dimensional preservation
- Quantum mechanical extensions
- Gauge field implementations
- Error quantization theorems
## Citation
If you use this framework in your research, please cite:
```bibtex
@software{dividebyzero2024,
title={DivideByZero: Dimensional Reduction Through Mathematical Singularities},
author={Michael C. Jenkins},
year={2024},
url={https://github.com/jenkinsm13/dividebyzero}
}
```
## License
MIT License - see [LICENSE](LICENSE) for details.
---
**Note:** This framework reimagines fundamental mathematical operations. While it provides practical solutions for handling mathematical singularities, users should understand the underlying theoretical principles for appropriate application.
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"description": "# DivideByZero: Dimensional Reduction Through Mathematical Singularities\n\n## Foundational Framework for Computational Singularity Analysis\n\nDivideByZero (`dividebyzero`) implements a novel mathematical framework that reconceptualizes division by zero as dimensional reduction operations. This paradigm shift transforms traditionally undefined mathematical operations into well-defined dimensional transformations, enabling new approaches to numerical analysis and quantum computation.\n\n## Core Mathematical Principles\n\n### Dimensional Division Operator\nThe framework defines division by zero through the dimensional reduction operator $\\oslash$:\n\nFor tensor $T \\in \\mathcal{D}_n$:\n```\nT \u2205 0 = \u03c0(T) + \u03b5(T)\n```\nWhere:\n- $\\pi(T)$: Projection to lower dimension\n- $\\epsilon(T)$: Quantized error preservation\n- $\\mathcal{D}_n$: n-dimensional tensor space\n\n### Installation\n\n```bash\npip install dividebyzero\n```\n\n## Fundamental Usage Patterns\n\n### Basic Operations\n```python\nimport dividebyzero as dbz\n\n# Create dimensional array\nx = dbz.array([[1, 2, 3],\n [4, 5, 6]])\n\n# Divide by zero - reduces dimension\nresult = x / 0\n\n# Reconstruct original dimensions\nreconstructed = result.elevate()\n```\n\n### Key Features\n\n#### 1. Transparent NumPy Integration\n- Drop-in replacement for numpy operations\n- Preserves standard numerical behavior\n- Extends functionality to handle singularities\n\n#### 2. Information Preservation\n- Maintains core data characteristics through reduction\n- Tracks quantum error information\n- Enables dimensional reconstruction\n\n#### 3. Advanced Mathematical Operations\n```python\n# Quantum tensor operations\nfrom dividebyzero.quantum import QuantumTensor\n\n# Create quantum-aware tensor\nq_tensor = QuantumTensor(data, physical_dims=(2, 2, 2))\n\n# Perform gauge-invariant reduction\nreduced = q_tensor.reduce_dimension(\n target_dims=2,\n preserve_entanglement=True\n)\n```\n\n## Theoretical Framework\n\n### Mathematical Foundations\n\nThe framework builds on several key mathematical concepts:\n\n1. **Dimensional Reduction**\n - Singular Value Decomposition (SVD)\n - Information-preserving projections\n - Error quantization mechanisms\n\n2. **Quantum Extensions**\n - Tensor network operations\n - Gauge field computations\n - Holonomy calculations\n\n3. **Error Tracking**\n - Holographic error encoding\n - Dimensional reconstruction algorithms\n - Quantum state preservation\n\n## Advanced Applications\n\n### 1. Quantum Computing\n```python\n# Quantum state manipulation\nstate = dbz.quantum.QuantumTensor([\n [1, 0],\n [0, 1]\n])\n\n# Preserve quantum information during reduction\nreduced_state = state / 0\n```\n\n### 2. Numerical Analysis\n```python\n# Handle singularities in numerical computations\ndef stable_computation(x):\n return dbz.array(x) / 0 # Returns dimensional reduction instead of error\n```\n\n### 3. Data Processing\n```python\n# Dimensionality reduction with information preservation\nreduced_data = dbz.array(high_dim_data) / 0\nreconstructed = reduced_data.elevate()\n```\n\n## Technical Requirements\n\n- Python \u2265 3.8\n- NumPy \u2265 1.20.0\n- SciPy \u2265 1.7.0\n\n### Optional Dependencies\n- networkx \u2265 2.6.0 (for quantum features)\n- pytest \u2265 6.0 (for testing)\n\n## Development and Extension\n\n### Contributing\n1. Fork the repository\n2. Create feature branch\n3. Implement changes with tests\n4. Submit pull request\n\n### Testing\n```bash\npytest tests/\n```\n\n## Mathematical Documentation\n\nDetailed mathematical foundations are available in the [Technical Documentation](docs/theory.md), including:\n\n- Formal proofs of dimensional preservation\n- Quantum mechanical extensions\n- Gauge field implementations\n- Error quantization theorems\n\n## Citation\n\nIf you use this framework in your research, please cite:\n\n```bibtex\n@software{dividebyzero2024,\n title={DivideByZero: Dimensional Reduction Through Mathematical Singularities},\n author={Michael C. Jenkins},\n year={2024},\n url={https://github.com/jenkinsm13/dividebyzero}\n}\n```\n\n## License\n\nMIT License - see [LICENSE](LICENSE) for details.\n\n---\n\n**Note:** This framework reimagines fundamental mathematical operations. While it provides practical solutions for handling mathematical singularities, users should understand the underlying theoretical principles for appropriate application.",
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