In this article, we will determine Puiseux series solutions of ordinary polynomial differential e... more In this article, we will determine Puiseux series solutions of ordinary polynomial differential equations. We also study the binary complexity of computing such solutions. We will prove that this complexity bound is single exponential in the number of terms in the series. Our algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.
Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write ... more Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y=ynn≥1 such that y/z=yn/znn≥1∈E; in particular, sz0 denotes the set of all sequences y such that y/z tends to zero. Here, we consider the infinite tridiagonal matrix Br,s,t˜, obtained from the triangle Br,s,t, by deleting its first row. Then we determine the sets of all positive sequences a=ann≥1 such that EaBr,s,t˜⊂Ea, where E=ℓ∞, c0, or c. These results extend some recent results.
Journal of Nonlinear Sciences and Applications, Dec 12, 2012
In this paper, we are interested in solving infinite linear systems of differential equations of ... more In this paper, we are interested in solving infinite linear systems of differential equations of the form x (t) = T x (t) + b with x(0) = x 0 , where T is either the generalized Cesàro operator C (λ) or the weighted mean matrix N q , x 0 and b are two given infinite column matrices and λ is a sequence with non-zero entries. We use a new method based on Laplace transformations to solve these systems.
In this paper, we apply the results stated in [19] to the solvability of the sequence spaces equa... more In this paper, we apply the results stated in [19] to the solvability of the sequence spaces equations (SSE) ?+Fx = Fb, where ?,F are linear spaces of sequences and b; x are positive sequences (x is the unknown). In this way, we solve the (SSE) of the form (Ea)G(?,?) + Fx = Fb, where G (?,?) is a factorable triangle matrix defined by [G(?,?)]nk = ?n?k for k ? n and (E,F) ? {(??,c), (c0,??),(c0,c), (?p,c), (?p,??), (w0,??} with p ? 1. Then we deal with some (SSE) involving the matrices C(??), C1 and N?q. Finally, we solve the (SSE) with operator of the form (Ea)?2 + Fx = Fb.
Given any sequence z = (zn)n?1 of positive real numbers and any set E of complex sequences, we wr... more Given any sequence z = (zn)n?1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n?1 such that y/z = (yn/zn)n?1 ? E; in particular, cz = s(c) z denotes the set of all sequences y such that y/z converges. Starting with the equation Fx = Fb we deal with some perturbed equation of the form ? + Fx = Fb, where ? is a linear space of sequences. In this way we solve the previous equation where ? =(Ea)T and (E,F) ? {(l?,c), (c0,l?), (c0,c), (lp,c), (lp,l?), (w0,l?)} with p ? 1, and T is a triangle.
UDC 517.9 Given any sequence a = (a n)n≥1 of positive real numbers and any set E of complex seque... more UDC 517.9 Given any sequence a = (a n)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn) n≥1 such that y/a = (yn/an) n≥1 2 E. In particular, ca denotes the set of all sequences y such that y/a converges. We deal with sequence spaces inclusion equations (SSIE) of the form F ⇢ Ea + F 0 x with e 2 F and explicitly find the solutions of these SSIE when a = (r n) n≥1 , F is either c or s1, and E and F 0 are any sets c0, c, s1,`p, w0, and w1. Then we determine the sets of all positive sequences satisfying each SSIE c ⇢ Dr ⇤ (c0)∆ + cx and c ⇢ Dr ⇤ (s1)∆ + cx, where ∆ is the operator of the first difference defined by ∆ny = yn − yn−1 for all n ≥ 1 with y0 = 0. Then we solve the SSIE c ⇢ Dr ⇤ EC 1 + s (c) x with E 2 {c, s1} and s1 ⇢ Dr ⇤ (s1)C 1 + sx, where C1 is the Cesàro operator defined by (C1)ny = n −1 X n k=1 y k for all y. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as Dr ⇤ EC 1 + s (c) x = c with E 2 {c0, c, s1} and Dr ⇤ EC 1 + sx = s1 with E 2 {c, s1}.
Journal of Nonlinear Sciences and Applications, 2012
This paper presents a new algorithm for solving zero-dimensional parametric systems of polynomial... more This paper presents a new algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations. This algorithm is based on the computation of what we call parametric U-resultants. The parameters space, i.e., the set of values of the parameters is decomposed into a finite number of constructible sets. The solutions of the input polynomial system are given uniformly in each constructible set by Polynomial Univariate Representations. The complexity of this algorithm is single exponential in the number n of the unknowns and the number r of the parameters.
In the context of deductive program verification, supporting floatingpoint computations is tricky... more In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a first-order axiomatization of floating-point operations which allows to reduce verification to checking the validity of logic formulas, in a suitable form for a large class of provers including SMT solvers and interactive proof assistants. Experiments using the Frama-C platform for static analysis of C code are presented.
Between January 2004 and June 2007 we conducted a retrospective analysis to assess post-operative... more Between January 2004 and June 2007 we conducted a retrospective analysis to assess post-operative complications related to endoscopic pituitary surgery in a series of 150 patients. Patients were treated with an endoscopic endonasal transsphenoidal approach to the sellar region for removal of pathological sellar and suprasellar lesions. We analysed the complications in groups according to the anatomical structures of the approach and the functional systems of the pituitary gland (anterior and posterior endocrine systems), and compared them to a large historical series using the traditional microsurgical transsphenoidal approach. Overall, we observed a decreased incidence of complications with respect to the surgical trauma, the functionality of the pituitary gland and post-operative patient comfort. We believe that the reduction of the complication rate observed in this study was mainly due to the wide structural overview offered by the endoscope as well as the anatomically direct, and therefore minimally invasive, character of the procedure. Successful endoscopic pituitary surgery requires extensive training in the use of an endoscope and careful planning of the surgery. Furthermore, close cooperation between a multidisciplinary team consisting of endocrinologists, neurosurgeons, ear, nose and throat surgeons, radiologists, and radiation oncologists is of utmost importance.
Journal of clinical neuroscience : official journal of the Neurosurgical Society of Australasia, 2007
Microsurgical transsphenoidal surgery for pituitary tumors has been standard therapy for decades ... more Microsurgical transsphenoidal surgery for pituitary tumors has been standard therapy for decades and was established by Harvey Cushing in the early twentieth century. Today, endoscopy is increasingly accepted in the therapy of pituitary lesions. In this retrospective study, we analysed the surgical technique and outcome of 50 patients with pituitary lesions treated with an endoscopic endonasal trans-sphenoidal approach.
This paper presents a new algorithm for computing absolutely irreducible components of n-dimensio... more This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented
Abstract: We prove that the binary complexity of solving ordinary polynomial differential equatio... more Abstract: We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials ...
Abstract. We propose an expressive language to specify formally behavioral properties of programs... more Abstract. We propose an expressive language to specify formally behavioral properties of programs involving floating-point computations. We present a deductive verification technique, which allows to prove formally that a given program meets its specifications, ...
This paper presents a lecture on existing algorithms for solving polynomial systems with their co... more This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc.
In this article, we will determine Puiseux series solutions of ordinary polynomial differential e... more In this article, we will determine Puiseux series solutions of ordinary polynomial differential equations. We also study the binary complexity of computing such solutions. We will prove that this complexity bound is single exponential in the number of terms in the series. Our algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.
Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write ... more Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y=ynn≥1 such that y/z=yn/znn≥1∈E; in particular, sz0 denotes the set of all sequences y such that y/z tends to zero. Here, we consider the infinite tridiagonal matrix Br,s,t˜, obtained from the triangle Br,s,t, by deleting its first row. Then we determine the sets of all positive sequences a=ann≥1 such that EaBr,s,t˜⊂Ea, where E=ℓ∞, c0, or c. These results extend some recent results.
Journal of Nonlinear Sciences and Applications, Dec 12, 2012
In this paper, we are interested in solving infinite linear systems of differential equations of ... more In this paper, we are interested in solving infinite linear systems of differential equations of the form x (t) = T x (t) + b with x(0) = x 0 , where T is either the generalized Cesàro operator C (λ) or the weighted mean matrix N q , x 0 and b are two given infinite column matrices and λ is a sequence with non-zero entries. We use a new method based on Laplace transformations to solve these systems.
In this paper, we apply the results stated in [19] to the solvability of the sequence spaces equa... more In this paper, we apply the results stated in [19] to the solvability of the sequence spaces equations (SSE) ?+Fx = Fb, where ?,F are linear spaces of sequences and b; x are positive sequences (x is the unknown). In this way, we solve the (SSE) of the form (Ea)G(?,?) + Fx = Fb, where G (?,?) is a factorable triangle matrix defined by [G(?,?)]nk = ?n?k for k ? n and (E,F) ? {(??,c), (c0,??),(c0,c), (?p,c), (?p,??), (w0,??} with p ? 1. Then we deal with some (SSE) involving the matrices C(??), C1 and N?q. Finally, we solve the (SSE) with operator of the form (Ea)?2 + Fx = Fb.
Given any sequence z = (zn)n?1 of positive real numbers and any set E of complex sequences, we wr... more Given any sequence z = (zn)n?1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n?1 such that y/z = (yn/zn)n?1 ? E; in particular, cz = s(c) z denotes the set of all sequences y such that y/z converges. Starting with the equation Fx = Fb we deal with some perturbed equation of the form ? + Fx = Fb, where ? is a linear space of sequences. In this way we solve the previous equation where ? =(Ea)T and (E,F) ? {(l?,c), (c0,l?), (c0,c), (lp,c), (lp,l?), (w0,l?)} with p ? 1, and T is a triangle.
UDC 517.9 Given any sequence a = (a n)n≥1 of positive real numbers and any set E of complex seque... more UDC 517.9 Given any sequence a = (a n)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn) n≥1 such that y/a = (yn/an) n≥1 2 E. In particular, ca denotes the set of all sequences y such that y/a converges. We deal with sequence spaces inclusion equations (SSIE) of the form F ⇢ Ea + F 0 x with e 2 F and explicitly find the solutions of these SSIE when a = (r n) n≥1 , F is either c or s1, and E and F 0 are any sets c0, c, s1,`p, w0, and w1. Then we determine the sets of all positive sequences satisfying each SSIE c ⇢ Dr ⇤ (c0)∆ + cx and c ⇢ Dr ⇤ (s1)∆ + cx, where ∆ is the operator of the first difference defined by ∆ny = yn − yn−1 for all n ≥ 1 with y0 = 0. Then we solve the SSIE c ⇢ Dr ⇤ EC 1 + s (c) x with E 2 {c, s1} and s1 ⇢ Dr ⇤ (s1)C 1 + sx, where C1 is the Cesàro operator defined by (C1)ny = n −1 X n k=1 y k for all y. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as Dr ⇤ EC 1 + s (c) x = c with E 2 {c0, c, s1} and Dr ⇤ EC 1 + sx = s1 with E 2 {c, s1}.
Journal of Nonlinear Sciences and Applications, 2012
This paper presents a new algorithm for solving zero-dimensional parametric systems of polynomial... more This paper presents a new algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations. This algorithm is based on the computation of what we call parametric U-resultants. The parameters space, i.e., the set of values of the parameters is decomposed into a finite number of constructible sets. The solutions of the input polynomial system are given uniformly in each constructible set by Polynomial Univariate Representations. The complexity of this algorithm is single exponential in the number n of the unknowns and the number r of the parameters.
In the context of deductive program verification, supporting floatingpoint computations is tricky... more In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a first-order axiomatization of floating-point operations which allows to reduce verification to checking the validity of logic formulas, in a suitable form for a large class of provers including SMT solvers and interactive proof assistants. Experiments using the Frama-C platform for static analysis of C code are presented.
Between January 2004 and June 2007 we conducted a retrospective analysis to assess post-operative... more Between January 2004 and June 2007 we conducted a retrospective analysis to assess post-operative complications related to endoscopic pituitary surgery in a series of 150 patients. Patients were treated with an endoscopic endonasal transsphenoidal approach to the sellar region for removal of pathological sellar and suprasellar lesions. We analysed the complications in groups according to the anatomical structures of the approach and the functional systems of the pituitary gland (anterior and posterior endocrine systems), and compared them to a large historical series using the traditional microsurgical transsphenoidal approach. Overall, we observed a decreased incidence of complications with respect to the surgical trauma, the functionality of the pituitary gland and post-operative patient comfort. We believe that the reduction of the complication rate observed in this study was mainly due to the wide structural overview offered by the endoscope as well as the anatomically direct, and therefore minimally invasive, character of the procedure. Successful endoscopic pituitary surgery requires extensive training in the use of an endoscope and careful planning of the surgery. Furthermore, close cooperation between a multidisciplinary team consisting of endocrinologists, neurosurgeons, ear, nose and throat surgeons, radiologists, and radiation oncologists is of utmost importance.
Journal of clinical neuroscience : official journal of the Neurosurgical Society of Australasia, 2007
Microsurgical transsphenoidal surgery for pituitary tumors has been standard therapy for decades ... more Microsurgical transsphenoidal surgery for pituitary tumors has been standard therapy for decades and was established by Harvey Cushing in the early twentieth century. Today, endoscopy is increasingly accepted in the therapy of pituitary lesions. In this retrospective study, we analysed the surgical technique and outcome of 50 patients with pituitary lesions treated with an endoscopic endonasal trans-sphenoidal approach.
This paper presents a new algorithm for computing absolutely irreducible components of n-dimensio... more This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented
Abstract: We prove that the binary complexity of solving ordinary polynomial differential equatio... more Abstract: We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials ...
Abstract. We propose an expressive language to specify formally behavioral properties of programs... more Abstract. We propose an expressive language to specify formally behavioral properties of programs involving floating-point computations. We present a deductive verification technique, which allows to prove formally that a given program meets its specifications, ...
This paper presents a lecture on existing algorithms for solving polynomial systems with their co... more This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc.
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