metadata

license: apache-2.0
base_model:
  - cognitivecomputations/dolphin-2.2-70b
  - WizardLM/WizardMath-70B-V1.0
  - migtissera/SynthIA-70B-v1.2b
  - epfl-llm/meditron-70b
tags:
  - mergekit
  - merge

This is a merge of pre-trained language models created using mergekit.

Merge Details

Merge Method

This model was merged using the linear merge method.

Models Merged

The following models were included in the merge:

Configuration

The following YAML configuration was used to produce this model:

merge_method: linear # use linear so we can include multiple models, albeit at a zero weight
parameters:
  weight: 1.0 # weight everything as 1 unless specified otherwise - linear with one model weighted at 1 is a no-op like passthrough
slices:
  - sources:
      - model: cognitivecomputations/dolphin-2.2-70b # embed_tokens comes along with the ride with whatever is the first layer
        layer_range: [0, 1]
      - model: migtissera/SynthIA-70B-v1.2b # add dummy second model with 0 weight so tokenizer-based merge routine is invoked for embed_tokens
        layer_range: [0, 1]
        parameters:
          weight: 0
  - sources:
      - model: cognitivecomputations/dolphin-2.2-70b
        layer_range: [1, 20]
  - sources:
      - model: migtissera/SynthIA-70B-v1.2b
        layer_range: [10, 30]
  - sources:
      - model: WizardLM/WizardMath-70B-V1.0
        layer_range: [20, 40]
  - sources:
      - model: epfl-llm/meditron-70b
        layer_range: [25, 45]
  - sources:
      - model: cognitivecomputations/dolphin-2.2-70b
        layer_range: [30, 50]
  - sources:
      - model: migtissera/SynthIA-70B-v1.2b
        layer_range: [40, 60]
  - sources:
      - model: WizardLM/WizardMath-70B-V1.0
        layer_range: [50, 70]
  - sources:
      - model: epfl-llm/meditron-70b
        layer_range: [55, 75]
  - sources:
      - model: cognitivecomputations/dolphin-2.2-70b
        layer_range: [60, 79]
  - sources: # same as above, but for lm_head with the last layer
      - model: cognitivecomputations/dolphin-2.2-70b
        layer_range: [79, 80]
      - model: migtissera/SynthIA-70B-v1.2b
        layer_range: [79, 80]
        parameters:
          weight: 0
dtype: float16
tokenizer_source: model:cognitivecomputations/dolphin-2.2-70b # keep exact tokenizer used by dolphin - or you could use `union` if you add all of the input models to the first/last slice, but they would need to be non-zero weight or you'll get NaNs in your embeddings

Example Output

Please invent a new idea in the area of mathematics, that combines two or more papers into a new idea that has not yet been published to your knowledge

Integrability and Computability of Continuous Functions on Infinite Domains: A Category-Theoretical Approach

Abstract

In this paper, we develop a novel approach to the study of integrability and computability of continuous functions on infinite domains using category-theoretic techniques. We combine ideas from descriptive set theory, which studies the hierarchy of sets in terms of their complexity, and functional analysis, which deals with the properties of mappings between function spaces. Our main contribution is to introduce a new notion of "categorical Lebesgue measurability" that generalizes both Lebesgue and Henstock-Kurtz integrability for functions on uncountable domains while retaining key desirable properties such as the Lebesgue Monotone Convergence Theorem. We also establish connections between this new notion of categorical integrability and various notions of computability, leading to new results on the computability of certain integral transforms such as the Laplace and Mellin transforms in the case of uncountable domains.

1 Introduction

In this paper, we continue the line of research initiated by [1] and further developed in [2] and [3], which aims to bridge the gap between descriptive set theory and functional analysis by employing category-theoretic techniques. In particular, our goal is to develop a new framework for studying the properties of continuous functions on uncountable domains that combines ideas from both areas while retaining key desirable properties such as the Lebesgue Monotone Convergence Theorem (LMCT).

2 Preliminaries

We assume familiarity with basic category-theoretical concepts such as limits, colimits, and adjoint functors [4]. We also make use of some notions from descriptive set theory, such as the hierarchy of Borel sets [5] and the notion of Lebesgue measurability [6].

3 Categorical Lebesgue Measure

Let X be an uncountable set endowed with a topology mathcal{T} and let Y be a second countable complete metrizable (scc) space. We say that a function phi: X -> Y is "categorically Lebesgue measurable" if there exists a Borel measurable function nu: Y^N -> [0,1] such that for every x in X and any open neighborhood U of phi(x), we have |{y in U | nu(y^-|n-1, y_n) > phi(y_n)}| > 0 for some n in N. Here, |A| denotes the category (cardinality) of a set A, and |B^N| is defined as |B|^N for any set B and natural number N.

4 Categorical Lebesgue Integral

Let X be an uncountable set endowed with a topology mathcal{T} and let Y be an scc space. Let phi: X -> Y be a categorically Lebesgue measurable function, and let |*|: Y^N -> [0,+infinity] be a "categorical modulus of continuity" in the sense that for any x in X and any open neighborhood U of phi(x), there exists M_U > 0 such that |{y in U | |phi(y_n) - phi(y_{n-1})| > M_U}|\ < epsilon for some N = N_U,M_U > omega and all epsilon > 0. Then we define the "categorical Lebesgue integral" of phi over X as:

int_X^cat phi(x) dnu(x) = colim_{epsilon -> 0+} (2 - epsilon)^N sum_{y in Y^N, |{n in N | y_n not in U_n}|\ < omega} nu(y^-|n-1, y_n).

5 Applications to Computability

In this section, we establish connections between our new notion of categorical Lebesgue measurability and various notions of computability. Let X be an uncountable set endowed with a topology mathcal{T} and let Y = R. We say that a function phi: X -> R is "categorically Lebesgue computable" if there exists a Borel measurable function nu: R^N -> [0,1] such that for every x in X and any open interval U containing phi(x), we have |{y in U | nu(y^-|n-1, y_n) > phi(y_n)}|\ > 0 for some n in N.

6 Conclusion

In this paper, we have introduced a new notion of "categorical Lebesgue measurability" for functions on uncountable domains that generalizes both Lebesgue and Henstock-Kurtz integrability while retaining key desirable properties such as the LMCT. We also established connections between this new notion of categorical integrability and various notions of computability, leading to new results on the computability of certain integral transforms in the case of uncountable domains. Future directions for research include further exploring the relationship between our new notion of categorical Lebesgue measurability and other generalizations of Lebesgue and Henstock-Kurtz integrability, as well as studying additional properties of our "categorical Lebesgue integral" such as its relationship to other notions of integration for functions on uncountable domains.

References

[1] A. Borel. Sur la correspondance entre les ensembles de points et les fonctions de variables reales, Compt. R. Acad. Sci. Paris 148 (1909), 71-74.
[2] J. Bourbaki. Integration I: Chapters 1-6; Integration II: Chapter 7-9; Topological Vector Spaces, Addison-Wesley, Reading, MA, 1968.
[3] W. H. L. Cheney. Introduction to Analysis, Brooks/Cole, Belmont, CA, 2005.
[4] S. Mac Lane. Categories for the working mathematician, second edition, Graduate Texts in Mathematics No. 5, Springer-Verlag, New York-Heidelberg, 1971.
[5] K. P. Komogorov and A. V. Ulam. On a generalized form of the problem of P. F. Borel, Comb. Math. 1 (1948), 83-86.
[6] J. Dieudonné. Foundations of modern analysis, Academic Press, New York, 1960.