Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebra...
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Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation. Computing Grobner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Grobner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugere and Mou, is used to find a Grobner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is O(mD(2)), where D is the degree of the ideal generated by the input and m is the number of non-trivial columns of a certain D x D matrix. While asymptotic estimates are known for m in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems. By assuming Frob erg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Socias by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity m by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems. We consider the ideal inside the polynomial ring K[x(1), ... , xn], where K is some infinite field, generated by p generic polynomials of degree d and the maximal minors of a p x (n - 1) polynomial matrix with generic entries of degree d - 1. Then, in this setting, for the case d = 2 and for n >>
Background: There is a growing body of literature describing the properties of marketed drugs, the concept of drug-likeness and the vastness of chemical space. In that context, enumerative combinatorics with simple at...
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Background: There is a growing body of literature describing the properties of marketed drugs, the concept of drug-likeness and the vastness of chemical space. In that context, enumerative combinatorics with simple atomic components may be useful in the conception and design of structurally novel compounds for expanding and enhancing high-throughput screening (HTS) libraries. Results: A random combination of mono- and diatomic carbon, hydrogen, nitrogen, and oxygen containing components in the absence of molecular weight constraints but with the ability to form rings affords virtual compounds that fall in bulk physicochemical space typically associated with drugs, but whose ring assemblies fall in new or under-represented areas of chemical shape space. When compared against compounds in the ChEMBL_14, MDDR, Drug Bank and Dictionary of Natural Products, the percentage of virtual compounds with a Tanimoto index of 1.0 (ECFP_4) was found to be as high as 0.21. Depending on therapeutic target, this value may be in range of what might be expected from an experimental HTS campaign in terms of a true hit rate. Conclusion: Virtual compounds derived through enumerative combinatorics of simple atomic components have drug-like properties with ring assemblies that fall in new or under-represented areas of shape space. Structures derived in this manner could provide the starting point or inspiration for the design of structurally novel scaffolds in an unbiased fashion.
The complexity of priority proofs in recursion theory has been growing since the first priority proofs in [1] and [11]. Refined versions of classic priority proofs can be found in [18]. To this date, this part of recu...
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The complexity of priority proofs in recursion theory has been growing since the first priority proofs in [1] and [11]. Refined versions of classic priority proofs can be found in [18]. To this date, this part of recursion theory is at about the same stage of development as real analysis was in the early days, when the notions of topology, continuity, compactness, vector space, inner product space, etc., were not invented. There were no general theorems involving these concepts to prove results about the real numbers and the proofs were repetitive and *** priority method contains an unprecedent wealth of combinatorics which is used to answer questions in recursion theory and is bound to have applications in many other fields as well. Unfortunately, very little progress has been made in finding theorems to formulate the combinatorial part of the priority method so as to answer questions without having to reprove the combinatorics in each *** and Lerman in [10] provide an overview of the subject. The entire edifice of definitions and theorems which formulate the combinatorics of the priority method has acquired the name Priority Theory. From a different vein, Groszek and Slaman in [2] have initiated a program to classify priority constructions in terms of how much induction or collection is needed to carry them out. This program studies the complexity of priority proofs and can be called Complexity Theory of Priority Proofs or simply Complexity.
We elaborate on a recently conjectured relation of Painleve transcendents and 2D conformal field theory. General solutions of Painleve VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular...
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We elaborate on a recently conjectured relation of Painleve transcendents and 2D conformal field theory. General solutions of Painleve VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in N = 2 supersymmetric gauge theories with N-f = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painleve functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.
This paper proposes the use of several classes of simple combinatorial problems that share the same solution for teaching problem equivalence and recursion. Our focus is on counting problems that involve Fibonacci num...
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ISBN:
(纸本)9781605580784
This paper proposes the use of several classes of simple combinatorial problems that share the same solution for teaching problem equivalence and recursion. Our focus is on counting problems that involve Fibonacci numbers. While these problems have simple recursive solutions, we propose that for teaching purposes - they can also be solved by finding other isomorphic problems for which the solution is known.
We solve the normal ordering problem for (A(+)A)(n) where A and A(+) are one mode deformed ([A, A(+)] = [N + 1] - [N]) bosonic ladder operators. The solution generalizes results known for canonical bosons. It involves...
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We solve the normal ordering problem for (A(+)A)(n) where A and A(+) are one mode deformed ([A, A(+)] = [N + 1] - [N]) bosonic ladder operators. The solution generalizes results known for canonical bosons. It involves combinatorial polynomials in the number operator N for which the generating function and explicit expressions are found. Simple deformations provide examples of the method.
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