Ehrenfest's Theorem

A one dimensional presentation of Ehrenfest's theorem is presented.

The Kramer sampling theorem revisited

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space H through an H -valued kernel K defined on an appropriate domain.

From the Farkas Lemma to the Hahn–Banach Theorem

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn–Banach–Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn–Banach theorem and the Mazur–Orlicz theorem for extended sublinear functions.

An approximate Hahn–Banach theorem for positively homogeneous functions

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one. Given p : X → R∪{+∞} such a function defined on the real vector space X, and a linear function defined on a subspace V of X and dominated by p (i.e. (x) ≤ p(x) for all x ∈ V), we say that can approximately be p-extended to X, if is the pointwise limit of a net of linear functions on V, every one of which can be extended to a linear function defined on X and dominated by p. The main result of this note proves that can approximately be p-extended to X if and only if is dominated by p∗∗, the pointwise supremum over the family of all the linear functions on X which are dominated by p.

A Quantile-Based Probabilistic Mean Value Theorem

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

al- Bustān fī dhikr al-awliyāʼ wa-al-ʻulamāʼ bi-Tilimsān

taʼlīf Abī ʻAbd Allāh Muḥammad ibn Muḥammad ibn Aḥmad, al-mulaqqab bi-Ibn Maryam al-Sharīf al-Malītī al-Madyūnī al-Tilimsānī ; wāfaqa ʻalá ṭabʻihi wa-iʻtaná bi-murājaʻati aṣlihi Muḥammad ibn Abī Shanab.

Hādhā kitāb Qiṣaṣ al-ʻulamāʼ

min taʼlīfāt al-ʻālim al-fāḍil al-kāmil nukhbat al-ʻulamāʼ wa-al-mujtahidīn al-Mīrzā Muḥammad ibn Sulaymān al-Tunukābunī.

Kitāb al-Arbaʻīn ʻan arbaʻīn min arbaʻīn li-arbaʻīn fī arbaʻīn / taʼlīf al-Imām al-Ḥāfiẓ Muḥaddith al-Shām Nāṣir al-Sunnah afḍal al-Mutaʼakhkhirīn awḥad al-ʻUlamāʼ ʻAlī ibn al-Ḥasan ibn Hibat Allāh ibn ʻAbd Allāh ibn al-Ḥusayn ibn ʻAsākir al-Dimashqī al-Shāfiʻī : manuscript, 1287

Written in one column, from 17 to 23 lines per page, in black.

Hādhihi Risālah fī taḥqīq ṣalāt al-wusṭá / min taʼlīf shaykhinā al-ʻālim al-muḥaqqiq wa-al-fāḍil al-mudaqqiq Imām al-ʻulamāʼ Qāḍī al-Mulk Mawlawī Ṣibghat Allāh : manuscript, 1862

Written in one column, 13 lines per page, in black.

Hādhā Risālah fī ṣadāq Sayyidatinā Fāṭimah al-Zahrāʼ bint Sayyid al-Mursalīn / lil-Fāḍil al-ʻAllāmah Imām al-ʻulamāʼ Mawlawī Ṣibghat Allāh Qāḍī al-Mulk : manuscript, 1863 or 4

Title from f. 1r.

Ḥāshiyat malik al-ʻulamāʼ Mawlānā ʻAbd al-ʻAlī ʻalá ḍābiṭat Tahdhīb al-manṭiq : manuscript, 1854 or 5

Written in one column, 9 lines per page, in black and red.

Risālah dar bayān-i shurūṭ-i iqtidā / min taʼlīf janāb Imām al-ʻulamā Qāḍī al-Mulk Bahādur : manuscript, 1862

Title from f. 1r.

Risālat ṭabaqāt al-ʻulamāʼ / lil-marḥūm Aḥmad ibn Kamāl Pāshā : manuscript, undated

Cream laid paper. 20 x 13 cm. (16 x 8.5 cm.).

Hādhā Risālah fī bayān ḥadīth ikhtilāf al-ummah / min al-Fāḍil al-ʻAllāmah Imām al-ʻulamāʼ Mawlawī Ṣibghat Allāh Qāḍī al-Mulk : manuscript, 1863 or 4

Title from f. 13r.

Les 'ulamã' à la veille de la bataille d'al-Kasr al-Kabir: fatwa contre l'alliance de mawlay al-Mutawakkil avec D. Sebastião

Osʿulamā’ exercem uma infuência sobre a comunidade, através do controlo do espaço religioso, a fim de dar a conhecer as normas do islão e garantir a sua boa aplicação. Do quadro social e cultural, onde se inserem estes sabios religiosos, abordamos o seu papel jurídico e religioso, mediante un aviso jurídico, « fatwā», na legitimação do poder. Num contexto marcado pela crise política e pela ameaça ibérica e otomana, osʿulamā’ unificam as linhas em torno do poder para melhor combater o perigo português.