H. Sato, Takuya Ikeda, Koichi Nishiwaki · arXiv (Cornell University) · 2023
In recent years, a deep learning framework has been widely used for object pose estimation. While quaternion is a common choice for rotation representation, it cannot represent the ambiguity of the observation. In order to handle the ambiguity, the Bingham distribution is one promising solution. However, it requires complicated calculation when yielding the negative log-likelihood (NLL) loss. An alternative easy-to-implement loss function has been proposed to avoid complex computations but has difficulty expressing symmetric distribution. In this paper, we introduce a fast-computable and easy-to-implement NLL loss function for Bingham distribution. We also create the inference network and show that our loss function can capture the symmetric property of target objects from their point clouds.
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Apoloniusz Tyszka · 2025
For n∈N, f(n) denotes the smallest b∈N such that if a system of equations S⊆{1=x_k, x_i+x_j=x_k, x_i·x_j=x_k: i,j,k∈{0,...,n}} has a solution in N^{n+1}, then S has a solution in {0,...,b}^{n+1}. The author proved earlier that the function f:N→N is computable in the limit and eventually dominates every computable function g:N→N. We present a simple code in MuPAD which for n∈N prints the sequence {f_i(n)}_{i=0}^∞ of non-negative integers converging to f(n).
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Chew, Paul, Machtey, Michale · Purdue e-Pubs (Purdue University System) · 1978
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Walter Massing · Die Naturwissenschaften · 1997
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Walter Massing · Die Naturwissenschaften · 1997
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Margarita Korovina, Oleg Kudinov · Electronic Notes in Theoretical Computer Science · 2008
In this paper we study different approaches to computability over effectively enumerable topological spaces. We introduce and investigate the notions of computable function, strongly-computable function and weakly-computable function. Under natural assumptions on effectively enumerable topological spaces the notions of computability and weakly-computability coincide.
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Amir M. Ben-Amram · ACM SIGACT News · 2005
"A function is mechanically computable (that is: computable by means of a machine) if and only if it is Turing-computable."
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Dag Normann · Logical Methods in Computer Science · 2011
We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.
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George Boolos, John P. Burgess, Richard C. Jeffrey · Cambridge University Press eBooks · 2002
A function is effectively computable if there are definite, explicit rules by following which one could in principle compute its value for any given arguments. This notion will be further explained below, but even after further explanation it remains an intuitive notion. In this chapter we pursue the analysis of computability by introducing a rigorously defined notion of a Turing-computable function. It will be obvious from the definition that Turing-computable functions are effectively computable. The hypothesis that, conversely, every effectively computable function is Turing computable is known as Turing's thesis. This thesis is not obvious, nor can it be rigorously proved (since the notion of effective computability is an intuitive and not a rigorously defined one), but an enormous amount of evidence has been accumulated for it. A small part of that evidence will be presented in this chapter, with more in chapters to come. We first introduce the notion of Turing machine, give examples, and then present the official definition of what it is for a function to be computable by a Turing machine, or Turing computable.
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James D. Lear, James E. Sturm · Mathematics of Computation · 1967
erfc2and z2 erfc z correct to 6 significant figures (7 s.f. for z > 1) using single precision on a computer with word length of 8 decimal places, for all z for which e~z* can be calculated correctly.To obtain greater accuracy, it is necessary either to use double precision or to use more than two different expansions for each function.From Gautschi's formula [2] for the number of terms required for calculation by backward recurrence, we see that that method will be better (for 7 s.f.accuracy) if all the z's of interest are greater than about 2.5.The advantage accruing from the use of Chebyshev approximations would be still greater for multiple-precision calculations of very high accuracy.
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Apoloniusz Tyszka · Preprints.org · 2025
For n∈N, f(n) denotes the smallest b∈N such that if a system of equations S⊆{1=x_k, x_i+x_j=x_k, x_i·x_j=x_k: i,j,k∈{0,...,n}} has a solution in N^{n+1}, then S has a solution in {0,...,b}^{n+1}. The author proved earlier that the function f:N→N is computable in the limit and eventually dominates every computable function g:N→N. We present a short program in MuPAD which for n∈N prints the sequence {f_i(n)}_{i=0}^∞ of non-negative integers converging to f(n). The previously known computer programs by other authors do not compute in the limit non-computable functions from N to N. We present a short program in MuPAD that computes in the limit a function β:N→N of unknown computability which eventually dominates every function δ:N→N with a single-fold Diophantine representation.
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Apoloniusz Tyszka · 2025
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Tang, Alvin, Pattinson, Dirk · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2025
We investigate a representation of computable functions as total functions over 2^∞, the set of finite and infinite sequences over {0,1}. In this model, infinite sequences are interpreted as non-terminating computations whilst finite sequences represent the sum of their digits. We introduce a new definition principle, function space corecursion, that simultaneously generalises minimisation and primitive recursion. This defines the class of computable corecursive functions that is closed under composition and function space corecursion. We prove computable corecursive functions represent all partial recursive functions, and show that all computable corecursive functions are indeed computable by translation into the untyped λ-calculus.
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Cameron Freer, Bjørn Kjos-hanssen, Andre Nies et al. · 2015
Abstract. We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Beyond these principal results, we show that a real z is Schnorr random if and only if every Lipschitz function with L1-computable derivative is differentiable at z. 1.
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Jeremy Avigad · Figshare · 2012
V’yugin has shown that there are a computable shift-invariant measure on 2N and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages Anf. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given ε.
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Alec Matheson, Timothy H. McNicholl · Proceedings of the American Mathematical Society · 2007
We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.
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Kenshi Miyabe · 2016
Brattka, Miller and Nies [1] showed that some randomness notions are char-acterized by dierentiability of some classes of functions. They also proposed to study which class corresponds to which randomness notion. Pathak, Rojas and Simpson [3] and independently Rute [4] showed that a real in the unit interval is Schnorr random if and only if the Lebesgue dierentiation theorem for the point holds for all eective version of L1-computable functions. Then its other randomness versions are of our interest. The author [2] gave several character-izations of the class of the eective version of L1-computable functions. Then we also study its other randomness versions. Let (X; d; ) be a computable metric space and be a computable measure on it. The following denition and result are by [2]. A integral test for Schnorr randomness is a nonnegative lower semicomputable function f: X! R whose integral is computable. A function f is L1-computable with an eective code if there exists a computable sequence fsng of nite rational step functions such that f = limn sn and jjsn+1 snjj1 2 n for all n. De nition 1 ([2]). Let f: X! R be a function whose domain is the set of Schnorr random points. Then f is L1-computable with an eective code i f is the dierence between two integral tests for Schnorr randomness. The following is the Martin-lof randomness verions of this result. Recall that an integral test is a nonnegative lower semicomputable function t: X! R withR td < 1. De nition 2. A function f: X! R is weakly L1-computable if there exists a computable sequence fsng of nite rational step functions such that f(x) = limn sn(x) and
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Avigad, Jeremy · KiltHub Repository · 2012
V’yugin has shown that there are a computable shift-invariant measure on 2N and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages Anf. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given ε.
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Jeremy Avigad · Notre Dame Journal of Formal Logic · 2012
V’yugin has shown that there are a computable shift-invariant measure on 2N and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages Anf. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given ε.
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Tyszka, Apoloniusz · PhilPapers (PhilPapers Foundation)
The theorem of Royer and Case states that there exists a limit-computable function β_1:N→N which eventually dominates every computable function δ_1:N→N. We present an alternative proof of this theorem. K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent and publicly available. Any theorem of any mathematician from past or present forever belongs to K. We prove: (1) there exists a limit-computable function f:N→N of unknown computability which eventually dominates every function δ:N→N with a single-fold Diophantine representation. We present both constructive and non-constructive proof of (1). Statement (1) claims that there exists a function f:N→N such that (f is computable in the limit)∧(¬K(f is computable))∧(¬K(f is uncomputable))∧(f eventually dominates every function δ:N→N with a single-fold Diophantine representation). Since Martin Davis' conjecture on single-fold Diophantine representations disproves Statement (1), Statement (1) justifies the title of the article.
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