Search

This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert's tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed. [ABSTRACT FROM AUTHOR]

The analogue of Hilbert's 10th Problem for a first-order structure A with signature L asks whether there exists an algorithm that given an L -sentence of the form ∃ x → [ s = t ] decides whether ∃ x → [ s = t ] is true in A. In this paper, we consider term algebras over a finite signature with at least one constant symbol and one function symbol of arity at least two. We investigate the structure we obtain by extending the term algebra with a substitution operator. We prove undecidability of the analogue of Hilbert's 10th problem without relying on the solution to the original Hilbert's 10th Problem. [ABSTRACT FROM AUTHOR]

For a finite group G, define and o(g) denotes the order of g 2 G. In this paper, we give a criterion for p-solvability by the function 00, where p 2 f7; 11g. We prove that if G is a finite group and 00(G) > 00(PSL(2; p)), where p 2 f7; 11g, then G is a p-solvable group. [ABSTRACT FROM AUTHOR]

Hilbert's Tenth Problem asks whether there is an algorithmic procedure to solve Diophantine equations (polynomial equations with integer coefficients) for integer solutions. This famous problem was shown to be unsolvable by Yuri Matiyasevič in 1970. In other words, there is no algorithm that can decide in general whether a given Diophantine equations has integer solutions or not. This negative solution builds on a long line of work by Martin Davis, Hilary Putnam, and importantly, Julia Robinson. In this article, we briefly describe the problem, its unsolvability, and Julia Robinson's contribution. [ABSTRACT FROM AUTHOR]

In the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia‐Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field and p be an odd prime which splits in K. Let E_1 and E_2 be elliptic curves over K such that the \operatorname {Gal}(\bar {K}/K)-modules E_1[p] and E_2[p] are isomorphic. We show that under certain explicit additional conditions on E_1 and E_2, the anticyclotomic \mathbb {Z}_p-extension K_{\operatorname {anti}} of K is integrally diophantine over K. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in K_{\operatorname {anti}}. We illustrate our results by constructing an explicit example for p=3 and K=\mathbb {Q}(\sqrt {-5}). [ABSTRACT FROM AUTHOR]

This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert’s tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed.

Let K be a number field, let L be an algebraic (possibly infinite degree) extension of K , and let O K ⊂ O L be their rings of integers. Suppose A is an abelian variety defined over K such that A (K) is infinite and A (L) / A (K) is a torsion group. If at least one of the following conditions is satisfied: 1. L is a number field, 2. L is totally real, 3. L is a quadratic extension of a totally real field, then O K has a diophantine definition over O L. [ABSTRACT FROM AUTHOR]

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We establish some of the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy, an approach closely related to Smale’s program on the foundations of computational mathematics and McMullen’s results on polynomial root finding with rational maps. Infinite-dimensional problems yield an intricate infinite classification theory, determining which spectral problems can be solved and with what types of algorithms. We provide answers to many longstanding open questions on the existence of algorithms. For example, we show that spectra can be computed, with error control, from point sampling operator coefficients for large classes of partial differential operators on unbounded domains. Further results include: computing spectra of (possibly unbounded) operators on graphs and separable Hilbert spaces with error control; determining if the spectrum intersects a compact set; the computational spectral gap problem and computing spectral classifications at the bottom of the spectrum; and computing discrete spectra, multiplicities, eigenspaces and determining if the discrete spectrum is non-empty. Moreover, the positive results with error control can be used in computer-assisted proofs. In contrast, the negative results preclude computer-assisted proofs for classes of operators as a whole. Our proofs are constructive, yielding a library of new algorithms and techniques that handle problems that before were out of reach. We demonstrate these algorithms on challenging problems, giving concrete examples of the failure of traditional approaches (e.g., “spectral pollution”) compared to the introduced techniques. [ABSTRACT FROM AUTHOR]

For any subset $Z \subseteq {\mathbb {Q}}$ , consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $ -definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. [ABSTRACT FROM AUTHOR]

Given a simplicial pair (X, A), a simplicial complex Y, and a map f : A → Y , does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its n th layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n , the rank of $E(K_n)$ is equal to the rank of $E(K)$. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert's tenth problem for number rings. [ABSTRACT FROM AUTHOR]

In this paper, we consider an iterative system of Hadamard fractional boundary value problems. The sufficient conditions are derived for the existence of at least one positive solution by utilizing Guo-Krasnosel'skii fixed point theorem. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

Let M be a compact, connected, orientable and real-analytic manifold; consider closed, real-valued, real-analytic 1-forms ω 1 , ... , ω m on M and the differential complex over M × T m naturally associated to the involutive system determined by them. In the real-analytic context, we completely characterize global solvability of the operators in its first (functional setting) and last (distributional setting) levels. Analogous results are obtained simultaneously in the Gevrey framework. [ABSTRACT FROM AUTHOR]

Fix an elliptic curve E over a number field F and an integer n which is a power of 3. We study the growth of the Mordell–Weil rank of E after base change to the fields K_d = F(\!\sqrt [2n]{d}). If E admits a 3-isogeny, then we show that the average "new rank" of E over K_d, appropriately defined, is bounded as the height of d goes to infinity. When n = 3, we moreover show that for many elliptic curves E/\mathbb {Q}, there are no new points on E over \mathbb {Q}(\sqrt [6]d), for a positive proportion of integers d. This is a horizontal analogue of a well-known result of Cornut and Vatsal [ Nontriviality of Rankin-Selberg L-functions and CM points, L -functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields \mathbb {Q}(\sqrt [6]{d}). The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the "correlation trick", which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

But to what extent did Skolem have a complete and coherent view of the foundation of mathematics? Skolem 1920 One hundred years ago, the Norwegian mathematician Thoralf Albert Skolem published the first of a series of remarkable papers on logic and the foundations of mathematics [[33]]. Davies is in no doubt about what mathematics is: "Mathematics is an aspect of human culture, just as are language, law, music, and architecture" [[6], p. 101]. But we note that the "grand view" of mathematics toward which these developments aim stands in stark contrast to Skolem's skepticism and his insistence on the primacy of specific structures and doable algorithms. [Extracted from the article]

The present paper is devoted to the solvability of various two-point boundary value problems for the equation y (4) = f (t , y , y ′ , y ″ , y ‴) , where the nonlinearity f may be defined on a bounded set and is needed to be continuous on a suitable subset of its domain. The established existence results guarantee not just a solution to the considered boundary value problems but also guarantee the existence of monotone solutions with suitable signs and curvature. The obtained results rely on a basic existence theorem, which is a variant of a theorem due to A. Granas, R. Guenther and J. Lee. The a priori bounds necessary for the application of the basic theorem are provided by the barrier strip technique. The existence results are illustrated with examples. [ABSTRACT FROM AUTHOR]

The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem. [ABSTRACT FROM AUTHOR]

We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow poles at a cofinite set of places, and undecidability results when at a finite set of places. [ABSTRACT FROM AUTHOR]

We prove that Hilbert's Tenth Problem for meromorphic functions and for entire functions in several variables is unsolvable over the language of rings, together with constant symbols for two of the variables and a predicate for a place. This is the first result in the literature, which proves unsolvability of a diophantine problem for the ring of complex analytic functions in a number of variables. The proof rests upon an interplay between analytic geometry, analysis, number theory, and logic. [ABSTRACT FROM AUTHOR]

In this paper, some necessary and sufficient solvability conditions are established for the generalized Sylvester operator equations AXB-CXD = E, AXB-CYD = E and AX + YB+CZ = E on Hilbert spaces, respectively. Moreover, we give a solvability condition for the *-Sylvester operator equation AX -X*B = C, which holds for finite matrices due to Wimmer (1994). [ABSTRACT FROM AUTHOR]

The m-point non-local problem is considered for the partial differential equation of mixed-type with singular coefficients, namely fractional wave equation involving the right-hand side bi-ordinal Hilfer derivative and sub-diffusion equation with the regularized Caputo-like counterpart hyper-Bessel differential operator. The main technique of solving the problem is based on the method of separation variables. Also, the connection between the given data and the uniquely solvability of the problem is established and an explicit solution is represented by Fourier-Bessel series in the given domain. [ABSTRACT FROM AUTHOR]

Arithmetical structures on graphs were first introduced in [11]. Later in [3] they were further studied in the setting of square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given. More precisely, an arithmetical structure on a non-negative integer matrix L with zero diagonal is a pair (d , r) ∈ N + n × N + n such that (Diag (d) − L) r t = 0 t and gcd ⁡ (r 1 , ... , r n) = 1. Thus, arithmetical structures on L are solutions of the polynomial Diophantine equation f L (X) : = det ⁡ (Diag (X) − L) = 0. Therefore, it is of interest to ask for an algorithm that compute them. We present an algorithm that computes arithmetical structures on a square integer non-negative matrix L with zero diagonal. In order to do this we introduce a new class of Z-matrices, which we call quasi M -matrices. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the solvability and stability for a class of singular fractional (p; q)-difference equations. First, we obtain an existence theorem of solution for the fractional (p; q)-difference equation. Then, by using a fractional (p; q)-Gronwall inequality, some stability criteria of solution are established, which also implies the uniqueness of solution. [ABSTRACT FROM AUTHOR]

Nominal syntax is a generalisation of first-order syntax that includes names, a notion of name binding and an elegant axiomatisation of alpha-equivalence, based on nominal set theory. However, it does not take into account non-capturing atom substitution, which is not a primitive notion in nominal syntax. We consider an extension of nominal syntax with non-capturing atom substitutions and show that matching is decidable and finitary but unification is undecidable in general. The proof of undecidability of unification is obtained by reducing Hilbert's tenth problem to unification of extended nominal terms. We provide a general matching algorithm and characterise a class of problems for which matching is unitary, giving rise to expressive and efficient rewriting systems. [ABSTRACT FROM AUTHOR]

Currently, for generic MINLP for which no structure is known in advance, the best exact[2] algorithm for solving MINLP is the spatial Branch-and-Bound (sBB) algorithm. This special issue of Mathematical Programming series B collects papers authored (or co-authored) by researchers who attended the second Oberwolfach workshop on MINLP, titled "Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics", and co-organized by the guest editors of this special issue. On the one hand such I mixed-integer optimal control i (MIOC) problems are a generalization of MINLPs, because the problem formulations allow for trivial special cases with HT ht differential states that are formally equivalent to a MINLP. The sBB algorithm is a variant of the Branch-and-Bound (BB) algorithm, which was introduced in [[32]] and extended to (separable) Nonlinear Programming (NLP) in [[17]]. [Extracted from the article]

This paper considers a class of neutral impulsive stochastic integro-differential inclusion with nonlocal conditions involving the Caputo fractional derivative of order 1 < α< 2 in a Hilbert space. A new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic systems is derived utilizing solution operator, stochastic analysis, fractional calculus and fixed point theorem for a multivalued operator under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate the derived theory at the end of the paper. [ABSTRACT FROM AUTHOR]

We make several improvements to methods for finding integer solutions to x³+y³+z³=k for small values of k. We implemented these improvements on Charity Engine's global compute grid of 500,000 volunteer PCs and found new representations for several values of k, including 3 and 42. This completes the search begun by Miller and Woollett in 1954 and resolves a challenge posed by Mordell in 1953. [ABSTRACT FROM AUTHOR]

An inverse spectral problem for the Sturm-Liouville operator with singular potential of class W2-1 is solved by the method of spectral mappings. We prove a uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and non-self-adjoint cases. [ABSTRACT FROM AUTHOR]

Hilbert's Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over ℤ. We show that there is no algorithm to determine for any P(z1,...,z9) ∈ ℤ[z1,...,zg] whether the equation P(z1,...,zg) = 0 has integral solutions with z9 ⩾ 0. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,...,z11) = 0 (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over ℤ. We also show that there is no algorithm to test for any P(z1,...,z17) ∈ ℤ[z1,...,z17] whether P(z12, ..., z12, ... z217) = 0 has integral solutions, and that there is a polynomial Q(z1, ..., z20) ∈ ℤ[z1,...,z20] such that {Q(z12, ..., z202): z1, ..., z20 ∈ ℤ} ∩ {0, 1, 2, ...} coincides with the set of all primes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem. [ABSTRACT FROM AUTHOR]

MashhadBranch,IslamicAzadUniversityLet L be a Lie algebra crossed module and Actpi(L) be a pointwise inner Actor of L. In this paper, we introduce lower and upper central series of L and show that if Actpi (L/Zj(L) /InnAct (L/Zj(L) is the nilpotent of class k wherein Zj (L) denotes the nth term of the upper central series of L, then Actpi(L)/InnAct(LL) is the nilpotent of the maximum class j + k. Moreover, if dim )Li/(Li ∩ Zj(L))) ≤ (1,1), then Actnilpotentofclasspi(L) / InnAct(L) is the nilpotent of the maximum class i + j - 1. [ABSTRACT FROM AUTHOR]

The inverse problem for semilinear Eidelman type equation with unknown time dependent function in its right-hand side is considered in this paper. The initial, boundary and integral type overdetermination conditions are posed. The sufficient conditions of the existence and the uniqueness of weak solution for the problem are obtained. [ABSTRACT FROM AUTHOR]

Investigation and numerical solution of the nonlinear integro-differential equation of parabolic type is considered. Integro-differential models of this type are based on the system of Maxwell equations and appear in various diffusion problems. Unique solvability, asymptotic behavior of the solution of the initial-boundary value problem and convergence of the finite difference scheme are given. [ABSTRACT FROM AUTHOR]