Differentiation in logical form

We introduce a logical theory of differentiation for a real-valued function on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the finite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient

A pilot study to evaluate the efficacy of self-attachment to treat chronic anxiety and/or depression in Iranian women

The aim of this pilot study was to evaluate the efficacy of the new Self-Attachment Technique (SAT) in treating resistant anxiety and depression, lasting at least three years, among Iranian women from different social backgrounds. In this intervention, the participant, using their childhood photos, imaginatively creates an affectional bond with their childhood self, vows to consistently support and lovingly re-raise this child to emotional well-being. We conducted a longitudinal study with repeated measurement to evaluate the efficacy of SAT using ANOVA. Thirty-eight women (N=30) satisfying the inclusion and exclusion criteria were recruited from different parts of Tehran. To describe the SAT protocols, a total of eight one-to-one sessions were offered to the recruits, the first four were weekly while the last four were fortnightly. The participants were expected to practice the protocols for twenty minutes twice a day. Two questionnaires, GAD-7 and PHQ-9, were used to measure anxiety and depression levels before and after the intervention and in a three-month follow-up. Thirty women completed the course. The change in the anxiety level between the pre-test and the post-test was significant at p<0.001 with effect size 2.6. The change in anxiety between pre-test and follow-up test was also significant at p<0.001 with effect size 3.0 respectively. The change in anxiety between the post-test and the follow-up was significant at p<0.05 with effect size 0.6. For depression, the change between the pre-test and the post-test or the follow-up was significant at p<0.001 with effect size 2.5 for each

From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

TRPM8 is required for survival and radioresistance of glioblastoma cells

TRPM8 is a Ca$^{2+}$-permeable nonselective cation channel belonging to the melastatin sub-group of the transient receptor potential (TRP) family. TRPM8 is aberrantly overexpressed in a variety of tumor entities including glioblastoma multiforme where it reportedly contributes to tumor invasion. The present study aimed to disclose further functions of TRPM8 in glioma biology in particular upon cell injury by ionizing radiation. To this end, TCGA data base was queried to expose the TRPM8 mRNA abundance in human glioblastoma specimens and immunoblotting was performed to analyze the TRPM8 protein abundance in primary cultures of human glioblastoma. Moreover, human glioblastoma cell lines were irradiated with 6 MV photons and TRPM8 channels were targeted pharmacologically or by RNA interference. TRPM8 abundance, Ca$^{2+}$ signaling and resulting K$^{+}$ channel activity, chemotaxis, cell migration, clonogenic survival, DNA repair, apoptotic cell death, and cell cycle control were determined by qRT-PCR, fura-2 Ca$^{2+}$ imaging, patch-clamp recording, transfilter migration assay, wound healing assay, colony formation assay, immunohistology, flow cytometry, and immunoblotting. As a result, human glioblastoma upregulates TRPM8 channels to variable extent. TRPM8 inhibition or knockdown slowed down cell migration and chemotaxis, attenuated DNA repair and clonogenic survival, triggered apoptotic cell death, impaired cell cycle and radiosensitized glioblastoma cells. Mechanistically, ionizing radiation activated and upregulated TRPM8-mediated Ca$^{2+}$ signaling that interfered with cell cycle control probably via CaMKII, cdc25C and cdc2. Combined, our data suggest that TRPM8 channels contribute to spreading, survival and radioresistance of human glioblastoma and, therefore, might represent a promising target in future anti-glioblastoma therapy

Probabilistic Bisimulation: Naturally on Distributions

In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view that treats probability distributions as first-class citizens. Our definition applies in the same way to discrete systems as well as to systems with uncountable state and action spaces. Several examples demonstrate that our definition refines the understanding of behavioural equivalences of probabilistic systems. In particular, it solves a long-standing open problem concerning the representation of memoryless continuous time by memory-full continuous time. Finally, we give algorithms for computing this bisimulation not only for finite but also for classes of uncountably infinite systems

Role of synovial fibroblast subsets across synovial pathotypes in rheumatoid arthritis: a deconvolution analysis

OBJECTIVES: To integrate published single-cell RNA sequencing (scRNA-seq) data and assess the contribution of synovial fibroblast (SF) subsets to synovial pathotypes and respective clinical characteristics in treatment-naïve early arthritis. METHODS: In this in silico study, we integrated scRNA-seq data from published studies with additional unpublished in-house data. Standard Seurat, Harmony and Liger workflow was performed for integration and differential gene expression analysis. We estimated single cell type proportions in bulk RNA-seq data (deconvolution) from synovial tissue from 87 treatment-naïve early arthritis patients in the Pathobiology of Early Arthritis Cohort using MuSiC. SF proportions across synovial pathotypes (fibroid, lymphoid and myeloid) and relationship of disease activity measurements across different synovial pathotypes were assessed. RESULTS: We identified four SF clusters with respective marker genes: PRG4(+) SF (CD55, MMP3, PRG4, THY1(neg)); CXCL12(+) SF (CXCL12, CCL2, ADAMTS1, THY1(low)); POSTN(+) SF (POSTN, collagen genes, THY1); CXCL14(+) SF (CXCL14, C3, CD34, ASPN, THY1) that correspond to lining (PRG4(+) SF) and sublining (CXCL12(+) SF, POSTN(+) + and CXCL14(+) SF) SF subsets. CXCL12(+) SF and POSTN(+) + were most prominent in the fibroid while PRG4(+) SF appeared highest in the myeloid pathotype. Corresponding, lining assessed by histology (assessed by Krenn-Score) was thicker in the myeloid, but also in the lymphoid pathotype + the fibroid pathotype. PRG4(+) SF correlated positively with disease severity parameters in the fibroid, POSTN(+) SF in the lymphoid pathotype whereas CXCL14(+) SF showed negative association with disease severity in all pathotypes. CONCLUSION: This study shows a so far unexplored association between distinct synovial pathologies and SF subtypes defined by scRNA-seq. The knowledge of the diverse interplay of SF with immune cells will advance opportunities for tailored targeted treatments

Computable de Finetti measures

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor corrections. To appear in Annals of Pure and Applied Logic. Extended abstract appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23

Orbits and phase transitions in the multifractal spectrum

We consider the one dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q < 0 at some critical strength of the random field. We introduce the concept of orbits which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of random iterated function systems in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role.Comment: The article has been completely rewritten; the title has changed; a section about the typical pointwise dimension as well as several references and remarks about more general systems have been added; to appear in J. Phys. A; 25 pages, 11 figures, LaTeX2

Topological analysis of representations

International audienceComputable analysis is the theoretical study of the abilities of algorithms to process infinite objects. The algorithms abilities depend on the way these objects are presented to them. We survey recent results on the problem of identifying the properties of objects that are decidable or semidecidable, for several concrete classes of objects and representations of them. Topology is at the core of this study, as the decidable and semidecidable properties are closely related to the open sets induced by the representation

Cauchyness and convergence in fuzzy metric spaces

[EN] In this paper we survey some concepts of convergence and Cauchyness appeared separately in the context of fuzzy metric spaces in the sense of George and Veeramani. For each convergence (Cauchyness) concept we find a compatible Cauchyness (convergence) concept. We also study the relationship among them and the relationship with compactness and completeness (defined in a natural sense for each one of the Cauchy concepts). In particular, we prove that compactness implies p-completeness.Almanzor Sapena acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant TEC2013-45492-R. Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant MTM 2012-37894-C02-01.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2017). Cauchyness and convergence in fuzzy metric spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(1):25-37. https://doi.org/10.1007/s13398-015-0272-0S25371111Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 29, 1073–1078 (2006)Edalat, A., Heckmann, R.: A computational model for metric spaces. Theor. Comput. Sci. 193, 53–73 (1998)Engelking, R.: General topology. PWN-Polish Sci. Publ, Warsawa (1977)Fang, J.X.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 46(1), 107–113 (1992)George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395–399 (1994)George, A., Veeramani, P.: Some theorems in fuzzy metric spaces. J. Fuzzy Math. 3, 933–940 (1995)George, A., Veeramani, P.: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365–368 (1997)Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385–389 (1989)Gregori, V., Romaguera, S.: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485–489 (2000)Gregori, V., Romaguera, S.: On completion of fuzzy metric spaces. Fuzzy Sets Syst. 130, 399–404 (2002)Gregori, V., Romaguera, S.: Characterizing completable fuzzy metric spaces. Fuzzy Sets Syst. 144, 411–420 (2004)Gregori, V., López-Crevillén, A., Morillas, S., Sapena, A.: On convergence in fuzzy metric spaces. Topol. Appl. 156, 3002–3006 (2009)Gregori, V., Miñana, J.J.: Some concepts realted to continuity in fuzzy metric spaces. In: Proceedings of the conference in applied topology WiAT’13, pp. 85–91 (2013)Gregori, V., Miñana, J.-J., Sapena, A.: On Banach contraction principles in fuzzy metric spaces (2015, submitted)Gregori, V., Miñana, J.-J.: std-Convergence in fuzzy metric spaces. Fuzzy Sets Syst. 267, 140–143 (2015)Gregori, V., Miñana, J.-J.: Strong convergence in fuzzy metric spaces Filomat (2015, accepted)Gregori, V., Miñana, J.-J., Morillas, S.: Some questions in fuzzy metric spaces. Fuzzy Sets Syst. 204, 71–85 (2012)Gregori, V., Miñana, J.-J., Morillas, S.: A note on convergence in fuzzy metric spaces. Iran. J. Fuzzy Syst. 11(4), 75–85 (2014)Gregori, V., Morillas, S., Sapena, A.: On a class of completable fuzzy metric spaces. Fuzzy Sets Syst. 161, 2193–2205 (2010)Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metric spaces and applications. Fuzzy Sets Syst. 170, 95–111 (2011)Kramosil, I., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975)Mihet, D.: On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets Syst. 158, 915–921 (2007)Mihet, D.: Fuzzy $$\varphi$$ φ -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 159, 739–744 (2008)Mihet, D.: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431–439 (2004)Mishra, S.N., Sharma, N., Singh, S.L.: Common fixed points of maps on fuzzy metric spaces Internat. J. Math. Math. Sci. 17(2), 253–258 (1994)Morillas, S., Sapena, A.: On Cauchy sequences in fuzzy metric spaces. In: Proceedings of the conference in applied topology (WiAT’13), pp. 101–108 (2013)Ricarte, L.A., Romaguera, S.: A domain-theoretic approach to fuzzy metric spaces. Topol. Appl. 163, 149–159 (2014)Sherwood, H.: On the completion of probabilistic metric spaces. Z.Wahrschein-lichkeitstheorie verw. Geb. 6, 62–64 (1966)Sherwood, H.: Complete Probabilistic Metric Spaces. Z. Wahrschein-lichkeitstheorie verw. Geb. 20, 117–128 (1971)Tirado, P.: On compactness and G-completeness in fuzzy metric spaces. Iran. J. Fuzzy Syst. 9(4), 151–158 (2012)Tirado, P.: Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets. Fixed Point Theory 13(1), 273–283 (2012)Vasuki, R., Veeramani, P.: Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets Syst. 135(3), 415–417 (2003)Veeramani, P.: Best approximation in fuzzy metric spaces. J. Fuzzy Math. 9, 75–80 (2001