A separation logic for negative dependence

Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes.

Orbits Theory. A Complete Proof of the Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.

Kerr–Schild–Kundt metrics in generic gravity theories with modified Horndeski couplings

The Kerr–Schild–Kundt (KSK) metrics are known to be one of the universal metrics in general relativity, which means that they solve the vacuum field equations of any gravity theory constructed from the curvature tensor and its higher-order covariant derivatives. There is yet no complete proof that these metrics are universal in the presence of matter fields such as electromagnetic and/or scalar fields. In order to get some insight into what happens when we extend the “universality theorem” to the case in which the electromagnetic field is present, as a first step, we study the KSK class of metrics in the context of modified Horndeski theories with Maxwell’s field. We obtain exact solutions of these theories representing the pp-waves and AdS-plane waves in arbitrary D dimensions.

Orbits Theory. A Complete Proof of the Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.

Conformal Blocks, Generalized Theta Functions and the Verlinde Formula

In 1988, E. Verlinde gave a remarkable conjectural formula for the dimension of conformal blocks over a smooth curve in terms of representations of affine Lie algebras. Verlinde's formula arose from physical considerations, but it attracted further attention from mathematicians when it was realized that the space of conformal blocks admits an interpretation as the space of generalized theta functions. A proof followed through the work of many mathematicians in the 1990s. This book gives an authoritative treatment of all aspects of this theory. It presents a complete proof of the Verlinde formula and full details of the connection with generalized theta functions, including the construction of the relevant moduli spaces and stacks of G-bundles. Featuring numerous exercises of varying difficulty, guides to the wider literature and short appendices on essential concepts, it will be of interest to senior graduate students and researchers in geometry, representation theory and theoretical physics.

Sequent Calculi for the Propositional Logic of HYPE

In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

The Complete Proof of the Riemann Hypothesis

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

The Complete Proof of the Riemann Hypothesis

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.