Update README.md

README.md

@@ -83,49 +83,55 @@ tokenizer_source: model:cognitivecomputations/dolphin-2.2-70b # keep exact token
 
 # 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:
+** interactively design a new derivative of transformer model architecutre **
+
+
 
-`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.
+** write a math paper **
+
+> 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.