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Non-invasive techniques such as seismic investigations and high-resolution multibeam sonars immensely improved our understanding of the geomorphology and sediment regimes in both the lacustrine and the marine domain. However, only few studies provide quantifications of basin wide-sediment budgets in lakes. Here, we use the combination of high-resolution bathymetric mapping and seismic reflection data to quantify the sediment budget in an alpine lake. The new bathymetric data of Lake Brienz reveal three distinct geomorphological areas: slopes with intercalated terraces, a flat basin plain, and delta areas with subaquatic channel systems. Quasi-4D seismic reflection data allow sediment budgeting of the lake with a total sediment input of 5.54 × 106 t sediment over 15 years of which three-quarter were deposited in the basin plain. Lake Brienz yields extraordinarily high sedimentation rates of 3.0 cm/yr in the basin plain, much more than in other Swiss lakes. This can be explained by (i) its role as first sedimentary sink in a high-alpine catchment, and by (ii) its morphology with subaquatic channel-complexes allowing an efficient sediment transfer from proximal to distal areas of the lake., Swiss Journal of Geosciences, 114, ISSN:1661-8734, ISSN:1661-8726

Overdeepened basins and valleys are common features in most regions that were glaciated during the Quaternary. Such structures are known from areas formerly covered by the Laurentian and Scandinavian Ice Sheet as well as for some parts of the Alps and their foreland. The phenomenon describes subsurface geomorphic features, erosionally cut into bedrock and later filled up by sediments. These structures can reach a depth of several hundred metres below the present valley floor. The sedimentary fillings of overdeepened basins and valleys are of particular interest for applied geosciences, for example in the context of groundwater resource management and subsurface constructions (i.e. tunnels). Recently, the topic of when and how overdeepening occurred in the past has gained additional attention with regard to the long-term stability of nuclear waste disposals, especially in Switzerland. In this context, it is important to note that relatively little research focussed on this issue during the past three decades. The last overview on the topic was a compilation of papers published in a special issue of the journal Eiszeitalter und Gegenwart in 1979. This special section in the present issue of the Swiss Journal of Geosciences comprises six papers related to the phenomenon of overdeepening in the Alps. The article by F. Preusser, J. Reitner and Ch. Schluchter summarises the areal distribution and geometry, reviews the present knowledge of the age of formation, and discusses possible processes responsible for overdeepened features. P. Jordan analyses the overdeepened structures in the foreland of the Swiss Alps using a Digital Elevation Model of bedrock surface based on outcrop and borehole data. It is shown that overdeepening mainly occurs within the limits of the Last Glaciation Maximum (LGM), but with a few overdeepened valleys and basin reaching beyond. M. Durst Stucki, R. Reber and F. Schlunegger interpret the buried valleys beneath the city of Bern as a network of tunnel valleys originating from subglacial high pressure melt water erosion. Valleys found in the surrounding Molasse uplands are regarded to represent proglacial melt water paths. The subsurface characteristics of three deep valleys from the Eastern Alps are investigated by E. Bruckl, J. Bruckl, W. Chwatal and Ch. Ullrich using different geophysical surveys. The authors conclude that deep erosion is not related to basal shear stress and that glacial confluence situations did in these examples not lead to important overdeepening. Rather, it is suggested that deep erosion was related to high pressured melt water at the glacier bed in settings of already tectonically weakened bedrock. The upper part of the sedimentary filling of the overdeepened Wehntal in northern Switzerland is analysed by F. Anselmetti, R. Drescher-Schneider, H. Furrer, H. R. Graf, S. Lowick, F. Preusser and M. Riedi revealing that it reaches back to at least 180,000 a. It is shown that the penultimate glaciation (pre-dating the Last Interglacial) may have reached the area M. Fiebig (&) Institute of Applied Geology, University of Natural Resources and Life sciences, Peter Jordan-Str. 70, 1190 Vienna, Austria e-mail: markus.fiebig@boku.ac.at

In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.

In this paper, via invertible radial differential operators, we characterize the closures of the Bergman–Besov spaces in the weighted Bloch spaces on the unit ball. The results of this paper generalize some previous results of Wen Xu and Ruhan Zhao. We first show on the way that the Bergman–Besov space is contained in the weighted little Bloch space. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

In this paper we study a Dirichlet double phase problem with a parametric superlinear right-hand side that has subcritical growth. Under very general assumptions on the data, we prove the existence of at least two nontrivial bounded weak solutions to such problem by using variational methods and critical point theory. In contrast to other works we do not need to suppose the Ambrosetti–Rabinowitz condition.

This paper is concerned with the existence of normalized solutions to a class of Schrödinger equations driven by a fractional operator with a parametric potential term. We obtain minimization of energy functional associated with that equations assuming basic conditions for the potential. Our work offers a partial extension of previous results to the non-local case.

For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between 0 and λ when λ → ∞, under an additional dynamical condition. (See [3, Theorem 3.5] for an early result in this direction.) In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in the 2013 paper [2]. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptotics already established, it suffices to study the resolvent as a pseudodifferential operator in order to identify and compute the second coefficient.

In this paper, we show that the position and the derivative operators, $${{\hat{q}}}$$ q ^ and $${{\hat{D}}}$$ D ^ , can be treated as ladder operators connecting various vectors of two biorthonormal families, $${{{\mathcal {F}}}}_\varphi $$ F φ and $${{{\mathcal {F}}}}_\psi $$ F ψ . In particular, the vectors in $${{{\mathcal {F}}}}_\varphi $$ F φ are essentially monomials in x, $$x^k$$ x k , while those in $${{{\mathcal {F}}}}_\psi $$ F ψ are weak derivatives of the Dirac delta distribution, $$\delta ^{(m)}(x)$$ δ ( m ) ( x ) , times some normalization factor. We also show how bi-coherent states can be constructed for these $${{\hat{q}}}$$ q ^ and $${{\hat{D}}}$$ D ^ , both as convergent series of elements of $${{{\mathcal {F}}}}_\varphi $$ F φ and $${{{\mathcal {F}}}}_\psi $$ F ψ , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.

The unlined Bedretto tunnel crosses large parts of the pre-Triassic basement of the Gotthard massif (Central Alps), giving the possibility to study late-Variscan plutonic rocks (Rotondo granite) and their Caledonian (poly-)metamorphic host rocks (Tremola and Prato series). The Rotondo granite consists mostly of an equigranular, fine-grained granite and to a lesser extent of a porphyritic granite. Commonly, the Rotondo granite is massive or only slightly foliated. Ductile deformation is localized along discrete shear zones composed of granitic or quartz-biotite-rich lithologies. This paper reviews the geology of the Bedretto tunnel with emphasis on the Rotondo granite and presents constraints based on kinematic, microstructural, and U–Pb geochronological evidence, which can be summarized as follows: (1) Both granitic and quartz-biotite-rich shear zones (QB-SZ) in the Rotondo granite generally dip moderately to steeply towards north and are related to top-to-south reverse shearing, indicating south-verging backthrusting during the exhumation of the Gotthard massif. (2) Zircons from both the equigranular and porphyritic Rotondo granite show overlapping 206Pb/238U-age ranges of 285–319 Ma and 280–335 Ma, respectively, which indicate that both are part of the same late-Variscan magmatic episode. Almost no older inherited cores are reported. (3) In zircons from a QB-SZ, 30% of the concordant age spots scatter between 339 and 589 Ma. This suggests that the parent material of the QB-SZ is unrelated to the magmatic episode that formed the Rotondo granite, but rather that the QB-SZ represent sheared xenoliths within the granite., Swiss Journal of Geosciences, 115, ISSN:1661-8734, ISSN:1661-8726

In this paper we prove the convergence of solutions to discrete models for binary waveguide arrays toward those of their formal continuum limit, for which we also show the existence of localized standing waves. This work rigorously justifies formal arguments and numerical simulations present in the Physics literature., The proof of Lemma 11 has been corrected, and the statement of Theorems 2 and 5 modified. Final version to appear on NoDEA

In this paper, convergence theorems involving convex inequalities of Copson’s type (less restrictive than monotonicity assumptions) are given for varying measures, when imposing convexity conditions on the integrable functions or on the measures. Consequently, a continuous dependence result for a wide class of differential equations with many interesting applications, namely measure differential equations (including Stieltjes differential equations, generalized differential problems, impulsive differential equations with finitely or countably many impulses and also dynamic equations on time scales) is provided.

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity mass of the order of $|\log\varepsilon|^{ -1}$. When $\varepsilon \to 0$ we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time $t$ in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper arXiv:1904.04785 [math-ph], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for $\varepsilon\to 0$ worked only for short times., Comment: 23 pages, LaTex; small changes in abstract and introduction, typos corrected, added an appendix with the proof of Lemma 4.1, and, with respect to version 2, Lemma 3.2 corrected (which implies very slight changes in other parts of the manuscript). This article is an extension of the results arXiv:1904.04785 and therefore draws heavily from arXiv:1904.04785

This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graph. More precisely, we show that the winding field generated by a random walk loop soup, when appropriately normalized, has a Gaussian limit as the loop soup intensity tends to ∞, and we give an explicit formula for the covariance kernel of the limiting field. We also derive a Spitzer-type law for windings of the Brownian loop soup, i.e., we show that the total winding around a point of all loops of diameter larger than δ, when multiplied by \(1/\log \delta \), converges in distribution to a Cauchy random variable as δ → 0. The random variables analyzed in this work have various interpretations, which we highlight throughout the paper.

The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is the construction of a weak solution enjoying some new properties. Of course, we look for properties that are global in time. The results hold assuming an initial data v0 ∈ J2(Ω).

In this paper we propose a notion of stability, which we call $$\varepsilon -N$$ ε - N -stability, for systems of particles interacting via Newton’s gravitational potential, and orbiting a much bigger object. For these systems the usual thermodynamical stability condition, ensuring the possibility to perform the thermodynamical limit, fails, but one can use as relevant parameter the maximum number of particles N that guarantees the $$\varepsilon -N$$ ε - N -stability. With some judicious but not particularly optimized estimates, borrowed from the classical theory of equilibrium statistical mechanics, we show that our model has a good fit with the data observed in the Solar System, and it gives a reasonable interpretation of some of its global properties.

In heritage conservation, a gap is often observed between the theory of conservation as a durable process that aims at the preservation of a historic building and the practice of restoration as a single intervention that aims at a fast and convincing result. This paper describes the proposed approach for the conservation of the main façade of the Shoemakers Chapel (in Dutch: Schoenmakerskapel) in Antwerp (Belgium), a listed monument since 1976. It serves as an example of how to develop a durable and realistic approach for the conservation of a sixteenth century façade. The basis for the conservation approach is the understanding that each intervention should take the stratified significance of the historic façade into account. In this paper, it will be shown how to combine a study of the façade from a technical point of view with an analysis of the façade as a carrier of cultural significance. ispartof: International Journal for Restoration of Buildings and Monuments vol:21 issue:4 pages:137-148 status: published

The aim of this paper is to prove the existence of at least one nontrivial weak solution for equations involving the (p(· ) , q(· ) ) -Laplace operator. The approach is variational and based on the critical point theory.

The purpose of this expository paper is to highlight the starring role of techniques from time–frequency analysis in some recent contributions concerning the mathematical theory of Feynman path integrals. We hope to draw the interest of mathematicians working in time–frequency analysis on this topic, as well as to illustrate the benefits of this fruitful interplay for people working on path integrals

In this paper, we study existence, nonexistence, and properties of solutions for some Kirchhoff–Schrodinger–Maxwell systems as (1.3). The solutions can be seen as saddle points of functionals which are unbounded both from above and from below.

In this paper we study local regularity properties of weak solutions to a class of nonlinear noncoercive elliptic Dirichlet problems with $$L^1$$ datum. The model example is $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p(w)+b(x)|Dw|^{p-1}=f(x)&{}\text {in }\Omega ,\\ w=0&{}\text {on }\partial \Omega . \end{array}\right. } \end{aligned}$$Here $$\Omega \subset {\mathbb {R}}^N$$ is a bounded open subset, $$N>1$$, $$-\Delta _p$$ is the well known p-Laplace operator, $$1

In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardys inequalities. Second, we present Hörmander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and Lorentz spaces. Both restricted weak- and strong-type results are obtained. In the case of regular weights necessary and sufficient conditions are given. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

In this paper we compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner with results obtained in ordinary thermal field theory, by means of real time and Matsubara formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. In order to cope with this smearing, the Matsubara propagator needs to be modified. We obtain an expansion of the correlation functions of the interacting equilibrium state as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real time formalism and the Matsubara formalism as particular cases. Finally, we show that a particular factorisation which is used to derive the ordinary real time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions and we notice that corrections to the self-energy in a $\lambda\phi^4$ at finite temperature theory are expected., Comment: 40 pages

The aim of this article is to explain the design and construction processes employed in the ribbed vault prolonged by translation which Diego de Riaño (†1534) built in the antesacristy of the church of Santa María in Carmona. The vault represents an outstanding synthesis between the Late Gothic diagonal ribs and severies and the emerging systems that adopt the form of orthogonal grids and tend towards a unitary volumetric composition. The work reflects the coexistence of two different architectural languages in Andalucía in the first half of the sixteenth century, a transitional situation which, in our view, Riaño deliberately embraces in his design. Based upon rigorous photogrammetric mapping, this paper analyses the formal and geometric characteristics of this exceptional work. The analysis conducted reveals that this was not simply an experiment with form, because beneath their singular appearance the geometric designs point to the investigation of new solutions.

Spatial ability is considered a key ability for STEM disciplines and has been correspondingly studied extensively, not least by the development of a vast number of psychometric tests. Based on the assumption that spatial ability is also central for architecture, the present paper presents an ongoing research on the development of a new test, specifically made to capture the spatial ability of architects. The test is based on aspects of the architectural process of design that have been broken down into single test items. The output of the research project will be both a finalized version of the test and a book for training purposes, based on this test. As spatial ability is malleable, it can be improved by exercise. This appears particularly relevant in relation to the gender gap emerging from this and former studies on spatial ability. The project is intended as a contribution to a better understanding of the nature of spatial ability in architecture in relation to the better-known visual ability as developed by art historians, among others.

We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry., Comment: 16 pages; replaces the part of arXiv:1009.0736 dealing with dynamical systems; results related to L-series have been moved to a separate preprint by the authors and Bart de Smit, of which the preliminary section (fixing notations) overlaps with that of the current paper

In this paper, we prove the existence of Fujita-type critical exponents for x-dependent fully nonlinear uniformly parabolic equations of the type $$(*)\quad \partial_{t}u=F(D^{2}u,x)+u^{p}\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ These exponents, which we denote by p(F), determine two intervals for the p values: in ]1,p(F)[, the positive solutions have finite-time blow-up, and in ]p(F), +∞[, global solutions exist. The exponent p(F) = 1 + 1/α(F) is characterized by the long-time behavior of the solutions of the equation without reaction terms $$\partial_{t}u=F(D^{2}u,x)\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ When F is a x-independent operator and p is the critical exponent, that is, p = p(F). We prove as main result of this paper that any non-negative solution to (*) has finite-time blow-up. With this more delicate critical situation together with the results of Meneses and Quaas (J Math Anal Appl 376:514–527, 2011), we completely extend the classical result for the semi-linear problem.

In this paper we consider the problem of determining all non-trivial combinations of integers a and b such that the corresponding power mean Pk, defined by Pk = a+b 2 1/k , if k 6= 0, √ ab, if k = 0, becomes integer valued. By using a variant of Fermat’s Last Theorem, proved in 1997, we show that there are no such integers for integer values of k satisfying |k| ≥ 3. All combinations of such integers are found for the cases k = −2, k = −1, k = 0, k = 1 and k = 2. Several related aspects are also discussed. Throughout this paper a and b will denote positive real numbers. The k-th power mean Pk = Pk(a, b) (with equal weights) of a and b is defined by Pk = ⎨⎩ 3 a+b 2 1/k , if k 6= 0, √ ab, if k = 0. The most well known power means are the arithmetic, the geometric and the harmonic mean given by the formulae A = a+ b 2 , G = √ ba, H = 2ba b+ a , respectively. For more detailed information on power means, see e.g. the books [3] and [7]. In this paper we consider the problem of determining integers a and b such that the corresponding power mean Pk becomes integer valued. For integer values of k satisfying |k| ≥ 3 the answer follows easily by the following variant of Fermat’s Last Theorem. We recall that a solution (x, y, z) is called trivial if xyz = 0 or ±1. Theorem 1 The equation x+y = 2z has no non-trivial solution for integer values of n ≥ 3. ∗Narvik University College, P.O.B. 385 N-8505 Narvik, Norway †Narvik University College, P.O.B. 385 N-8505 Narvik, Norway, and Norut Narvik, P.O. Box 250, N-8504 Narvik, Norway

application/pdf, Matroid theory was invented in the middle of the 1930s by two mathematicians independently, namely, Hassler Whitney in the USA and Takeo Nakasawa in Japan. Whitney became famous, but Nakasawa remained anonymous until two decades ago. He left only four papers to the mathematical community, all of them written in the middle of the 1930s. It was a bad time to have lived in a country that had become as eccentric as possible. Just as Nazism became more and more flamboyant in Europe in the 1930s, Japan became more and more esoteric and fanatical in the same time period. This book explains the little that is known about Nakasawa’s personal life in a Japan that had, among other failures, lost control over its military. We do not know what forces caused him to be discharged from the Tokyo University of Arts and Sciences. His work was considered brilliant, his papers superb, if somewhat unconventional and mysterious in notation. We do know that, in the latter half of the 1930s, forced to give up his mathematical career, he chose to live as a bureaucrat in Manchuria, at that time a puppet state of Japan. He died in 1946 at Khavarovsk, at the age of 33, after one year of forced labor in Siberian and other USSR camps, without sufficient food or shelter to protect his health. This book contains his four papers in German and their English translations as well as some extended commentary on the history of Japan during those years. The book also contains 14 photos of him or his family. Although the veil of mystery surrounding Nakasawa’s life has only been partially lifted, the work presented in this book speaks eloquently of a tragic loss to the mathematical community., As to Chapter2-5, only the first page is free to open, 9783764385729

In this paper a two-step reduction method for spectral problems on a star graph with n+1 edges e_{0}, e_{1}, \ldots , e_{n} and a self-adjoint matching condition at the central vertex v is established. The first step is a reduction to the problem on the single edge e_0 but with an energy depending boundary condition at v. In the second step, by means of an abstract inverse result for Q-functions, a reduction to a problem on a path graph with two edges e_0, \widetilde{e}_1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?

Recently, Sun (Two q-analogues of Eulers formula (2) = /6. arXiv:1802.01473, 2018) obtained q-analogues of Eulers formula for (2) and (4). Suns formulas were based on identities satised by triangular numbers and properties of Eulers q-Gamma function. In this paper, we obtain a q-analogue of (6) = /945. Our main results are stated in Theorems 2.1 and 2.2 below. Refereed/Peer-reviewed (VLID)4914700 Version of record

This paper describes a system for analysis of aerial images of urban areas using multiple images from different viewpoints and its evaluation. The proposed approach combines bottom-up and top-down processing. In this paper the emphasis is on the discussion of the experimental evaluation. To evaluate statistically the performance of the system, a set of 100 realisations of 5 images from different viewpoints was used, which was generated by combining real and ray-traced images. The experiments show a significant improvement of reliability and accuracy if multi-view imagery is used instead of single-view.

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for C^1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions., 13 pages, an example has been added at the end of Section 4. To appear in "Complex Analysis and Operator Theory"

In this paper, we study the homogenization of a Smoluchowski system of periodic discrete diffusion-coagulation equations, when the diffusion coefficients depend on all variables, in particular on the microscopic variable. This system modelizes the aggregation and diffusion of the ˇ-amyloid peptide Abeta 42 in the cerebral tissue, a process associated with the development of Alzheimer’s disease. Our homogenization result, based on Allaire-Nguetseng two-scale convergence, is meant to pass from a microscopic model to a macroscopic one.

This paper is concerned with nonlinear diffusion equations driven by the p(·)-Laplacian with variable exponents in space. The well-posedness is first checked for measurable exponents by setting up a subdifferential approach. The main purposes are to investigate the large-time behavior of solutions as well as to reveal the limiting behavior of solutions as p(·) diverges to the infinity in the whole or in a subset of the domain. To this end, the recent developments in the studies of variable exponent Lebesgue and Sobolev spaces are exploited, and moreover, the spatial inhomogeneity of variable exponents p(·) is appropriately controlled to obtain each result.

Compact Riemannian 7- and 8-manifolds with holonomy G(2) arid Spin(7) were first constructed by the author in 1994-5, by resolving orbifolds T-7/Gamma and T-8/Gamma. This paper describes a new construction of compact 8-manifolds with holonomy Spin(7). We start with a Calabi-Yau 4-orbifold Y with isolated singularities of a special kind. We divide by an antiholomorphic involution a of Y to get a real 8-orbifold Z = Y/. Then we resolve tire singularities of Z to get a compact 8-manifold M, which has metrics with holonomy Spin(7). Manifolds constructed in this way typically have large fourth Betti number b(4)(M).

This paper will present a selective overview of some uses of mathematics in fiction, expanding and updating Mann (2010), before considering the rather unusual case of Catherine Shaw’s detective novels, set in the late Victorian mathematical community.

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: $${\partial_V \psi^t (u{^\prime}(t)) + \partial_V \varphi(u(t)) + B(t, u(t)) \ni f(t)}$$ in V*, 0 < t < T, u(0) = u 0, where $${\partial_V \psi^t, \partial_V \varphi : V \to 2^{V^*}}$$ denote the subdifferential operators of proper, lower semicontinuous and convex functions $${\psi^t, \varphi : V \to (-\infty, +\infty]}$$ , respectively, for each $${t \in [0,T]}$$ , and f : (0, T) → V* and $${u_0 \in V}$$ are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T) × V into V*. We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., $${B \equiv 0}$$ ) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.

In this paper we use a calculus of differential forms which is defined using an axiomatic approach. We then define integration of differential forms over chains in a new way and we present a short proof of Stokes’ formula using distributional techniques. We also consider differential forms in Clifford analysis, vector differentials and their powers. This framework enables an easy proof for a Cauchy formula on a k-surface. Finally, we discuss how to compute winding numbers in terms of the monogenic Cauchy kernel and the vector differentials with a new approach which does not involve cohomology of differential forms.

In this paper we describe the foundation of a new kind of discrete geometry and calculus called Script Geometry. It allows to work with more general meshes than classic simplicial complexes. We provide the basic definitions as well as several examples, like the Klein bottle and the projective plane. Furthermore, we also introduce the corresponding Dirac and Laplace operators which should lay the groundwork for the development of the corresponding discrete function theory.

This book contains a selection of papers presented at the session "Quaternionic and Clifford Analysis" at the 10th ISAAC Congress held in Macau in August 2015. The covered topics represent the state-of-the-art as well as new trends in hypercomplex analysis and its applications.

In this paper we work in the 'split' discrete Clifford analysis setting, i.e., the m-dimensional function theory concerning null-functions of the discrete Dirac operator partial derivative, defined on the grid Z(m), involving both forward and backward differences. This Dirac operator factorizes the (discrete) Star-Laplacian (Delta* = partial derivative(2)). We show how the space H-k of discrete k-homogeneous spherical harmonics, which is a reducible so(m, C)-representation, may explicitly be decomposed into 2(2m) isomorphic copies of irreducible so( m, C)representations with highest weight ( k, 0,..., 0). The key element is the introduction of 2(2m) idempotents, dividing the discrete Clifford algebra in 2(2m) subalgebras of dimension ((k) (k+m-1)) - ((k) (k+m-3)).

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-It\^o chaos expansion for the nodal length shows that it is dominated by its $4$th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component., Comment: Accepted for publication on Geometric and Functional Analysis

During the course of centuries mathematics has interacted in many ways with culture and human activities, and among these a place of privilege has been reserved for art and architecture. In this paper I will show several examples of the existence of three levels of interaction between mathematics and art: the presence of a mathematical substrate in various archaeological and artistic relics from antiquity; the conscious or unconscious application by artists of mathematics principles whose theories had not yet been fully developed; and finally, the relationship established by some mathematicians with artists and art theorists that permitted an awareness and acquisition of mathematical knowledge and rules that were then applied to artistic creations.

The present paper intends to summarize the main results of Harbrecht and Tausch (Inverse Probl 27:065013, 2011; SIAM J Sci Comput 35:A104–A121, 2013) on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization problem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equation are expressed as parabolic boundary integral equations and solved using a Nystrom discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. To demonstrate the similarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.

In this paper, we prove the maximal regularity property of an abstract fractional differential equation with finite delay on periodic Besov and Triebel–Lizorkin spaces and use these results to guarantee the existence and uniqueness of periodic solution of a neutral fractional differential equation with finite delay. The main tool used to achieve our goal is an operator-valued version of Miklhin’s Fourier multiplier theorem and fixed-point argument.

This paper proves that every zero of any nth, n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation \({f(x) + f(2x) + \cdots + f(nx) = 0, x \in \mathbb{R}}\). The continuity of the solutions depends on the sign of the real part of each zero.

This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as a direct sum of [n,n] and a K-invariant irreducible subspace., Second version has an improved introduction

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called \(r\)-rectangles (rectangles with two semicircles of small radius \(r\)) in the critical strip of each function \(L_{n}(z)\!:=\!\)\(1-\sum _{k=2}^{n}k^{z}\), \(n\!\ge \!2\), containing exactly \(\left[ \dfrac{T\log n}{2\pi }\right] +1\) zeros of \(L_{n}(z)\), where \(T\) is the height of the \(r\)-rectangle and \(\left[\cdot \right]\) represents the integer part.

Preprint version This paper presents a signature-based approach to quantifying perceptual nearness of images. A signature is defined as a set of descriptors, where each descriptor consists of a real-valued feature vector associated with a digital image region (set of pixels) combined with a region-based weight. Tolerance near sets provide a formal framework for our application of near sets to image retrieval. The tolerance nearness measure tNM was created to demonstrate application of near set theory to the problem of image correspondence. A new form of tNM has been introduced in this work, which takes into account the region size. Our method is compared to two other well-known image similarity measures: earth movers distance (EMD) and integrated region matching (IRM). https://link.springer.com/article/10.1007/s11786-013-0145-x