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12

Jun 29, 2018
06/18

by
Taras Banakh

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Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev network if and only if $X$ is countably tight and has a countable Pykeev$^*$ network. In the paper we establish some stability properties of the class of topological spaces with the strong Pytkeev$^*$-property.

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1607.03599

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19

Jun 28, 2018
06/18

by
Taras Banakh

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For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence, generated by the subbase consisting of the sets $[K_X\times K_Y,U]=\{f\in S(X\times Y,Z):f(K_X\times K_Y)\subset U\}$, where $U$ is an open subset of $Z$ and $K_X\subset X$, $K_Y\subset Y$ are compact sets one of which is a singleton. We prove that every...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1509.05542

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6.0

Jun 29, 2018
06/18

by
Taras Banakh

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A topological space $X$ is called strongly $\sigma$-metrizable if $X=\bigcup_{n\in\omega}X_n$ for an increasing sequence $(X_n)_{n\in\omega}$ of closed metrizable subspaces such that every convergence sequence in $X$ is contained in some $X_n$. If, in addition, every compact subset of $X$ is contained in some $X_n$, $n\in\omega$, then $X$ is called super $\sigma$-metrizable. Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly...

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1611.05351

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Jun 27, 2018
06/18

by
Taras Banakh

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We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish stability properties of the classes of Maslyuchenko spaces and study the relation of these classes to known classes of generalized metric spaces (such as Piotrowski or Stegall spaces). One of our results says that for any $\aleph_0$-space $Z$ and any separately...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1506.01661

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6.0

Jun 29, 2018
06/18

by
Taras Banakh

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A topological space $X$ is called Piotrowski if every quasicontinuous map $f:Z\to X$ from a Baire space $Z$ to $X$ has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski spaces to strictly fragmentable, Stegall, and game determined spaces. Also we prove that a Piotrowski Tychonoff space $X$ contains a dense (completely) metrizable Baire subspace if and only if $X$ is Baire (Choquet).

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1609.05482

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8.0

Jun 28, 2018
06/18

by
Taras Banakh

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A topological group $G$ is called 2-swelling if for any compact subsets $A,B\subset G$ and elements $a,b,c\in G$ the inclusions $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset c(A\cap B)$ are equivalent to the equalities $aA\cup bB=A\cup B$ and $aA\cap bB=c(A\cap B)$. We prove that an (abelian) topological group $G$ is 2-swelling if each 3-generated (resp. 2-generated) subgroup of $G$ is discrete. This implies that the additive group $\mathbb Q$ of rationals is 2-swelling and each locally...

Topics: Group Theory, General Topology, Mathematics

Source: http://arxiv.org/abs/1510.05100

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7.0

Jun 29, 2018
06/18

by
Taras Banakh

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Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and for every $x\in X$ the family $(U_\alpha[x])_{\alpha\in P}$ of balls $U_\alpha[x]=\{y\in X:(x,y)\in U_\alpha\}$ is a neighborhood base at $x$. A $P$-base $(U_\alpha)_{\alpha\in P}$ for $X$ is called locally uniform if the family of entourages...

Topics: General Topology, Logic, Mathematics

Source: http://arxiv.org/abs/1607.07978

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8.0

Jun 29, 2018
06/18

by
Taras Banakh

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Answering a question of Elekes and Vidny\'anszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_\sigma$-set in $H$ that cannot be enlarged to a Haar null $G_\delta$-set.

Topics: General Topology, Group Theory, Mathematics

Source: http://arxiv.org/abs/1604.00150

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7.0

Jun 29, 2018
06/18

by
Taras Banakh

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A family of closed subsets of a topological space $X$ is called a (strict) $Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally finite in $X$. Applications of (strict) $Cld$-fans are based on a simple observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the $k$-space and Ascoli properties in spaces that naturally appear in General Topology,...

Topics: Category Theory, General Topology, Mathematics

Source: http://arxiv.org/abs/1602.04857

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19

Jun 28, 2018
06/18

by
Taras Banakh

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We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with...

Topics: Group Theory, Geometric Topology, Mathematics, General Topology

Source: http://arxiv.org/abs/1508.07297

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Jun 30, 2018
06/18

by
Taras Banakh; Arkady Leiderman

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A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1412.4268

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18

Jun 27, 2018
06/18

by
Taras Banakh; Alex Ravsky

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For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The boundedness numbers of the (Boolean operations...

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1503.04480

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14

Jun 27, 2018
06/18

by
Taras Banakh; Alex Ravsky

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We derive many upper bounds on the submetrizability number and $i$-weight of paratopological groups and topological monoids with open shifts. In particular, we prove that each first countable Hausdorff paratopological group is submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we construct an example of a zero-dimensional (and hence regular) Hausdorff paratopological abelian group $G$ with countable pseudocharacter which is not submetrizable. In fact, all results on the...

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1503.04278

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7.0

Jun 30, 2018
06/18

by
Taras Banakh; Alex Ravsky

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We define a notion of a rotund quasi-uniform space and describe a new direct construction of a (right-continuous) quasi-pseudometric on a (rotund) quasi-uniform space. This new construction allows to give alternative proofs of several classical metrizability theorems for (quasi-)uniform spaces and also obtain some new metrizability results. Applying this construction to topological monoids with open shifts, we prove that the topology of any (semiregular) topological monoid with open shifts is...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1412.2239

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8.0

Jun 30, 2018
06/18

by
Taras Banakh; Saak Gabriyelyan

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Denote by $\mathbf C_k[\mathfrak M]$ the $C_k$-stable closure of the class $\mathfrak M$ of all metrizable spaces, i.e., $\mathbf C_k[\mathfrak M]$ is the smallest class of topological spaces that contains $\mathfrak M$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_k(X,Y)$ with Lindel\"of domain. We show that the class $\mathbf C_k[\mathfrak M]$ coincides with the class of all topological spaces...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1412.2216

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4.0

Jun 30, 2018
06/18

by
Taras Banakh; Volodymyr Gavrylkiv

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A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference...

Topics: Group Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1704.02472

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7.0

Jun 30, 2018
06/18

by
Taras Banakh; Taras Radul

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For every functional functor $F:Comp\to Comp$ in the category $Comp$ of compact Hausdorff spaces we define the notions of $F$-Dugundji and $F$-Milutin spaces, generalizing the classical notions of a Dugundji and Milutin spaces. We prove that the class of $F$-Dugundji spaces coincides with the class of absolute $F$-valued retracts. Next, we show that for a monomorphic continuous functor $F:Comp\to Comp$ admitting tensor products each Dugundji compact is an absolute $F$-valued retract if and only...

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1401.2319

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5.0

Jun 30, 2018
06/18

by
Taras Banakh; Volodymyr Gavrylkiv

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A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le \Delta[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$),...

Topics: Group Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1702.02631

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Jun 30, 2018
06/18

by
Taras Banakh; Volodymyr Gavrylkiv

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A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\frac{\Delta[G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and...

Topics: Group Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1704.02471

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4.0

Jun 30, 2018
06/18

by
Taras Banakh; Dusan Repovs

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For every metric space $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces $X,Y$ are coarsely equivalent if $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that an ultrametric space $X$ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if...

Topics: Mathematics, General Topology, Metric Geometry

Source: http://arxiv.org/abs/1408.4818

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Jun 28, 2018
06/18

by
Taras Banakh; Lyubomyr Zdomskyy

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We prove that each closed locally continuum- connected subspace of a finite dimensional topological group is locally compact. This allows us to construct many 1-dimensional metrizable separable spaces that are not homeomorphic to closed subsets of finite-dimensional topological groups, which answers in negative a question of D.Shakhmatov. Another corollary is a characterization of Lie groups as finite-dimensional locally continuum-connected topological groups. For locally path connected...

Topics: Group Theory, General Topology, Mathematics

Source: http://arxiv.org/abs/1510.03604

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6.0

Jun 28, 2018
06/18

by
Taras Banakh; Igor Belegradek

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We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube....

Topics: Differential Geometry, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1510.07269

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3.0

Jun 30, 2018
06/18

by
Taras Banakh; Alex Ravsky

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We prove that a semiregular topological space $X$ is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological...

Topics: Mathematics, General Topology, Group Theory

Source: http://arxiv.org/abs/1410.1504

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6.0

Jun 30, 2018
06/18

by
Taras Banakh; Mikolaj Fraczyk

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We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)

Topics: Mathematics, Combinatorics, Group Theory

Source: http://arxiv.org/abs/1405.2151

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7.0

Jun 30, 2018
06/18

by
Taras Banakh; Filip Strobin

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Let $X$ be a universal (Urysohn) space. We prove that every topological fractal is homeomorphic (isometric) to the attractor $A_{\mathcal F}$ of a function system ${\mathcal F}$ on $X$ consisting of Rakotch contractions.

Topics: Mathematics, General Topology, Metric Geometry

Source: http://arxiv.org/abs/1407.7687

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6.0

Jun 29, 2018
06/18

by
Taras Banakh; Arkady Leiderman

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We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G$-base. A topological space $X$ has a local $\mathfrak G$-base if every point $x$ of $X$ has a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $\omega^\omega$. To construct $\mathfrak G$-bases in free topological vector spaces, we exploit a new description of the topology of a free topological...

Topics: General Topology, Functional Analysis, Logic, Mathematics

Source: http://arxiv.org/abs/1606.01967

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19

Jun 28, 2018
06/18

by
Taras Banakh; Heike Mildenberger

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We prove that for infinite cardinals $\kappa

Topics: Group Theory, Logic, Mathematics, General Topology

Source: http://arxiv.org/abs/1506.08969

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7.0

Jun 30, 2018
06/18

by
Habib Ammari; Hai Zhang

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A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment by F.Lemoult, M.Fink and G.Lerosey (Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107 (2011)). A key ingredient is the calculation of the resonances and the Green function in the half space with the presence of a system of Helmholtz resonators in the quasi-stationary regime. By using boundary integral equations and generalized...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.2513

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10.0

Jun 29, 2018
06/18

by
Habib Ammari; Faouzi Triki

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The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1609.07091

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11

Jun 29, 2018
06/18

by
Taras Banakh; Arkady Leiderman

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A topological space $X$ is defined to have an $\omega^\omega$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$ in $\omega^\omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $\omega^\omega$-bases.

Topics: General Topology, Functional Analysis, Group Theory, Mathematics

Source: http://arxiv.org/abs/1611.06438

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Hai Zhang

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A mathematical theory is developed to explain the super-resolution and super-focusing in high contrast media. The approach is based on the resonance expansion of the Green function associated with the medium. It is shown that the super-resolution is due to sub-wavelength resonant modes excited in the medium which can propagate into the far-field.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1412.2214

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6.0

Jun 29, 2018
06/18

by
Habib Ammari; Hai Zhang

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We derive an effective medium theory for acoustic wave propagation in bubbly fluid near Minnaert resonant frequency. We start with a multiple scattering formulation of the scattering problem of an incident wave by a large number of identical small bubbles in a homogeneous fluid. Under certain conditions on the configuration of the bubbles, we justify the point interaction approximation and establish an effective medium theory for the bubbly fluid as the number of bubbles tends to infinity. The...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.08409

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4.0

Jun 30, 2018
06/18

by
Igor Protasov; Taras Banakh; Ksenia Protasova

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We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta G$ of $G$, we prove, in particular, that the closure of the minimal ideal of $\beta G$ is of type $F_{\sigma\delta}$.

Topics: General Topology, Mathematics

Source: http://arxiv.org/abs/1703.00174

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5.0

Jun 30, 2018
06/18

by
Habib Ammari; Han Wang

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This paper presents premier and innovative time-domain multi-scale method for shape identification in electro-sensing using pulse-type signals. The method is based on transform-invariant shape descriptors computed from filtered polarization tensors at multi-scales. The proposed algorithm enjoys a remarkable noise robustness even with far-field measurements at very limited angle of view. It opens a door for pulsed imaging using echolocation and induction data.

Topics: Mathematics, Numerical Analysis, Computing Research Repository, Computer Vision and Pattern...

Source: http://arxiv.org/abs/1409.3714

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Jun 27, 2018
06/18

by
Taras Banakh; Magdalena Nowak; Filip Strobin

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A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal U$ of $X$ there is a number $n\in\mathbb N$ such that for any functions $f_1,\dots,f_n\in \mathcal F$, the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal U$. If, in addition, all functions $f\in\mathcal F$ have Lipschitz constant $

Topics: General Topology, Mathematics, Metric Geometry

Source: http://arxiv.org/abs/1503.06396

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5.0

Jun 28, 2018
06/18

by
Taras Banakh; Lesia Karchevska; Alex Ravsky

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We prove that any meager quasi-analytic subgroup of a topological group $G$ belongs to every $\sigma$-ideal $\mathcal I$ on $G$ possessing the closed $\pm n$-Steinhaus property for some $n\in\mathbb N$. An ideal $\mathcal I$ on a topological group $G$ is defined to have the closed $\pm n$-Steinhaus property if for any closed subsets $A_1,\dots,A_n\notin\mathcal I$ of $G$ the product $(A_1\cup A_1^{-1})\cdots (A_n\cup A_n^{-1})$ is not nowhere dense in $G$. Since the $\sigma$-ideal $\mathcal E$...

Topics: Group Theory, General Topology, Mathematics

Source: http://arxiv.org/abs/1509.09073

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Jun 26, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari

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The main focus of this work is the reconstruction of the signals $f$ and $g_{i}$, $i=1,...,N$, from the knowledge of their sums $h_{i}=f+g_{i}$, under the assumption that $f$ and the $g_{i}$'s can be sparsely represented with respect to two different dictionaries $A_{f}$ and $A_{g}$. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that $f$ and the $g_{i}$'s can be uniquely and stably reconstructed...

Topics: Mathematics, Analysis of PDEs, Numerical Analysis

Source: http://arxiv.org/abs/1502.04540

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Jun 30, 2018
06/18

by
Taras Banakh; Robert Ralowski; Szymon Zeberski

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Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu(U)

Topics: Mathematics, General Topology, Logic, Dynamical Systems

Source: http://arxiv.org/abs/1409.3922

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Jun 27, 2018
06/18

by
Taras Banakh; Igor Guran; Alex Ravsky

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We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the M\"obius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.

Topics: Group Theory, Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1503.08993

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Jun 26, 2018
06/18

by
Habib Ammari; Dehan Chen; Jun Zou

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This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on biological media. Biological tissues and cell suspensions are described as having a heteregeneous permittivity and a heteregeneous conductivity. Well-posedness of the model problem and the regularity of its solution are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.06803

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Jun 30, 2018
06/18

by
Habib Ammari; Francisco Romero; Matias Ruiz

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In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and both single and multiple particles. For close-to-touching nanoparticles, we show that the temperature field deviates significantly from the one generated by a single nanoparticle. The results of this paper open a door for solving the challenging problems of detecting plasmonic...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1703.00422

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7.0

Jun 30, 2018
06/18

by
Habib Ammari; Alden Waters; Hai Zhang

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We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.5138

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23

Jun 26, 2018
06/18

by
Habib Ammari; Simon Boulier; Pierre Millien

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We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1501.04803

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4.0

Jun 29, 2018
06/18

by
Habib Ammari; Francisco Romero; Cong Shi

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This paper aims at imaging the dynamics of metabolic activity of cells. Using dynamic optical coherence tomography, we introduce a new multi-particle dynamical model to simulate the movements of the collagen and the cell metabolic activity and develop an efficient signal separation technique for sub-cellular imaging. We perform a singular-value decomposition of the dynamic optical images to isolate the intensity of the metabolic activity. We prove that the largest eigenvalue of the associated...

Topics: Medical Physics, Numerical Analysis, Physics, Mathematics

Source: http://arxiv.org/abs/1608.04382

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10.0

Jun 30, 2018
06/18

by
Habib Ammari; Youjun Deng; Pierre Millien

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In this paper we provide a mathematical framework for localized plasmon resonance of nanoparticles. Using layer potential techniques associated with the full Maxwell equations, we derive small-volume expansions for the electromagnetic fields, which are uniformly valid with respect to the nanoparticle's bulk electron relaxation rate. Then, we discuss the scattering and absorption enhancements by plasmon resonant nanoparticles. We study both the cases of a single and multiple nanoparticles. We...

Topics: Mathematics, Analysis of PDEs, Mathematical Physics

Source: http://arxiv.org/abs/1412.3656

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Jun 30, 2018
06/18

by
Habib Ammari; Yat Tin Chow; Jun Zou

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In this paper we consider the inverse scattering problem for high-contrast targets. We mathematically analyze the experimentally-observed phenomenon of super-resolution in imaging the target shape. This is the first time that a mathematical theory of super-resolution has been established in the context of imaging high contrast inclusions. We illustrate our main findings with a variety of numerical examples. Our analysis is based on the novel concept of scattering coefficients. These findings...

Topics: Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1410.1253

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4.0

Jun 29, 2018
06/18

by
Habib Ammari; Thomas Widlak; Wenlong Zhang

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Electropermeabilization is a clinical technique in cancer treatment to locally stimulate the cell metabolism. It is based on electrical fields that change the properties of the cell membrane. With that, cancer treatment can reach the cell more easily. Electropermeabilization occurs only with accurate dosage of the electrical field. For applications, a monitoring for the amount of electropermeabilization is needed. It is a first step to image the macroscopic electrical field during the process....

Topics: Analysis of PDEs, Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1603.00764

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7.0

Jun 30, 2018
06/18

by
Taras Banakh; Artur Bartoszewicz; Malgorzata Filipczak; Emilia Szymonik

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Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q

Topics: Mathematics, General Topology

Source: http://arxiv.org/abs/1403.0098

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5.0

Jun 30, 2018
06/18

by
Habib Ammari; Faouzi Triki; Chun-Hsiang Tsou

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The multifrequency electrical impedance tomography consists in retrieving the conductivity distribution of a sample by injecting a finite number of currents with multiple frequencies. In this paper we consider the case where the conductivity distribution is piecewise constant, takes a constant value outside a single smooth anomaly, and a frequency dependent function inside the anomaly itself. Using an original spectral decomposition of the solution of the forward conductivity problem in terms...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1704.04878

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Liangdong Zhou

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This paper presents a new iterative reconstruction method to provide high-resolution images of shear modulus and viscosity via the internal measurement of displacement fields in tissues. To solve the inverse problem, we compute the Fr\'echet derivatives of the least-squares discrepancy functional with respect to the shear modulus and shear viscosity. The proposed iterative reconstruction method using this Fr\'echet derivative does not require any differentiation of the displacement data for the...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.2932

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Loc Hoang Nguyen; Laurent Seppecher

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The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acoustic-optic data absorption distributions with bounded variation. The paper extends earlier results on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.2679

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Jun 28, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Tingting Zhang

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We are aiming to identify the thin insulating inhomogeneities and small conductive inhomogeneities inside an electrically conducting medium by using multi-frequency electrical impedance tomography (mfEIT). The thin insulating inhomogeneities are considered in the form of tubular neighborhood of a curve and small conductive inhomogeneities are regarded as circular disks. Taking advantage of the frequency dependent behavior of insulating objects, we give a rigorous derivation of the potential...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1510.08494

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4.0

Jun 30, 2018
06/18

by
Taras Banakh; Igor Protasov; Dusan Repovs; Sergii Slobodianiuk

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For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$....

Topics: Mathematics, General Topology, Metric Geometry

Source: http://arxiv.org/abs/1409.3910

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4.0

Jun 30, 2018
06/18

by
Taras Banakh; Matija Cencelj; Dušan Repovš; Ihor Zarichnyi

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In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

Topics: Mathematics, Metric Geometry, Group Theory

Source: http://arxiv.org/abs/1412.4296

4
4.0

Jun 28, 2018
06/18

by
Habib Ammari; Yat Tin Chow; Jun Zou

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In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the...

Topics: Numerical Analysis, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1510.03999

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Jun 30, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Kaixi Ruan

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This paper focuses on the acousto-electromagnetic tomography, a recently introduced hybrid imaging technique. In a previous work, the reconstruction of the electric permittivity of the medium from internal data was achieved under the Born approximation assumption. In this work, we tackle the general problem by a Landweber iteration algorithm. The convergence of such scheme is guaranteed with the use of a multiple frequency approach, that ensures uniqueness and stability for the corresponding...

Topics: Mathematics, Numerical Analysis, Analysis of PDEs

Source: http://arxiv.org/abs/1410.4119

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5.0

Jun 30, 2018
06/18

by
Habib Ammari; Lingyun Qiu; Fadil Santosa; Wenlong Zhang

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In this paper we present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor by integrating magneto-acoutic tomography with data acquired from diffusion tensor imaging. Magneto-acoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid, non-invasive medical imaging technique to produce conductivity images with improved spatial resolution and accuracy. Diffusion Tensor Imaging (DTI) is also a non- invasive technique for...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1702.05187

5
5.0

Jun 28, 2018
06/18

by
Habib Ammari; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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In this paper we use the full Maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles. We mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size, shape, and arrangement of the nanoparticles, using the layer potential techniques associated with the full Maxwell equations. We present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1511.06817

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25

Jun 27, 2018
06/18

by
Habib Ammari; Pierre Millien; Matias Ruiz; Hai Zhang

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Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1506.00866

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18

Jun 26, 2018
06/18

by
Habib Ammari; Hongjie Dong; Hyeonbae Kang; Seick Kim

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We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.01804

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10.0

Jun 29, 2018
06/18

by
Habib Ammari; Hyeuknam Kwon; Seungri Lee; Jin Keun Seo

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This paper presents a static electrical impedance tomography (EIT) technique that evaluates abdominal obesity by estimating the thickness of subcutaneous fat. EIT has a fundamental drawback for absolute admittivity imaging because of its lack of reference data for handling the forward modeling errors. To reduce the effect of boundary geometry errors in imaging abdominal fat, we develop a depth-based reconstruction method that uses a specially chosen current pattern to construct reference-like...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1607.05893

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5.0

Jun 29, 2018
06/18

by
Taras Banakh; Adam Idzik; Oleg Pikhurko; Igor Protasov; Krzysztof Pszczoła

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For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $\omega$-paths of infinite diameter.

Topics: Metric Geometry, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1606.01973

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14

Jun 29, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Francisco Romero; Timothée Wintz

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This paper provides a mathematical analysis of ultrafast ultrasound imaging. This newly emerging modality for biomedical imaging uses plane waves instead of focused waves in order to achieve very high frame rates. We derive the point spread function of the system in the Born approximation for wave propagation and study its properties. We consider dynamic data for blood flow imaging, and introduce a suitable random model for blood cells. We show that a singular value decomposition method can...

Topics: Probability, Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1604.04604

6
6.0

Jun 30, 2018
06/18

by
Habib Ammari; Kyungkeun Kang; Kyounghun Lee; Jin Keun Seo

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This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({\it i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.3650

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10.0

Jun 30, 2018
06/18

by
Taras Banakh; Wieslaw Kubis; Natalia Novosad; Magdalena Nowak; Filip Strobin

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In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.

Topics: Mathematics, General Topology, Metric Geometry

Source: http://arxiv.org/abs/1405.6289

9
9.0

Jun 30, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Tingting Zhang; Liangdong Zhou

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An electrical impedance spectroscopy-based nondestructive testing (NDT) method is proposed to image both cracks and reinforcing bars in concrete structures. The method utilizes the frequency-dependent behavior of thin insulating cracks: low-frequency electrical currents are blocked by insulating cracks, whereas high-frequency currents can pass through the conducting bars without being blocked by thin cracks. Rigorous mathematical analysis relates the geometric structures of the cracks and bars...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.4582

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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This paper is concerned with the inverse problem of reconstructing a small object from far field measurements. The inverse problem is severally ill-posed because of the diffraction limit and low signal to noise ratio. We propose a novel methodology to solve this type of inverse problems based on an idea from plasmonic sensing. By using the field interaction with a known plasmonic particle, the fine detail information of the small object can be encoded into the shift of the resonant frequencies...

Topics: Spectral Theory, Mathematical Physics, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1704.04870

6
6.0

Jun 29, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; David Gontier; Hyundae Lee; Hai Zhang

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The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on the acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) is considered. It is shown that the structure behaves as an equivalent surface with Neumann...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1608.02733

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; Hyundae Lee; Sanghyeon Yu; Hai Zhang

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The aim of this paper is to show both analytically and numerically the existence of a subwavelength phononic bandgap in bubble phononic crystals. The key is an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. The main findings in this paper are illustrated with a variety of numerical experiments.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1702.05317

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55

Jun 30, 2018
06/18

by
Habib Ammari; Laure Giovangigli; Loc Hoang Nguyen; Jin-Keun Seo

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The aim of this paper is to propose an optimal control optimization algorithm for reconstructing admittivity distributions (i.e., both conductivity and permittivity) from multi-frequency micro-electrical impedance tomography. A convergent and stable optimization scheme is shown to be obtainable from multi-frequency data. The results of this paper have potential applicability in cancer imaging, cell culturing and differentiation, food sciences, and biotechnology.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1403.5708

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8.0

Jun 29, 2018
06/18

by
Habib Ammari; Matias Ruiz; Wei Wu; Sanghyeon Yu; Hai Zhang

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In this paper we derive an impedance boundary condition to approximate the optical scattering effect of an array of plasmonic nanoparticles mounted on a perfectly conducting plate. We show that at some resonant frequencies the impedance blows up, allowing for a significant reduction of the scattering from the plate. Using the spectral properties of a Neumann-Poincare type operator, we investigate the dependency of the impedance with respect to changes in the nanoparticle geometry and...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.05019

10
10.0

Jun 29, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; David Gontier; Hyundae Lee; Hai Zhang

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Through the application of layer potential techniques and Gohberg-Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1603.03982

9
9.0

Jun 29, 2018
06/18

by
Tasawar Abbas; Habib Ammari; Guanghui Hu; Abdul Wahab; Jong Chul Ye

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The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic, and electromagnetic media. In view of their promising applications in inverse problems related to mathematical imaging and elastic cloaking, the notion of elastic scattering coefficients of an inclusion is presented from the perspective of boundary layer potentials and a few properties are discussed. A reconstruction algorithm is developed and analyzed for...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1606.08070

5
5.0

Jun 29, 2018
06/18

by
Habib Ammari; Mihai Putinar; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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We prove by means of a couple of examples that plasmonic resonances can be used on one hand to classify shapes of nanoparticles with real algebraic boundaries and on the other hand to reconstruct the separation distance between two nanoparticles from measurements of their first collective plasmonic resonances. To this end, we explicitly compute the spectral decompositions of the Neumann-Poincar\'{e} operators associated with a class of quadrature domains and two nearly touching disks. Numerical...

Topics: Spectral Theory, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.05268

8
8.0

Jun 30, 2018
06/18

by
Habib Ammari; Elie Bretin; Pierre Millien; Laurent Seppecher; Jin-Keun Seo

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We provide a mathematical analysis of and a numerical framework for full-field optical coherence elastography, which has unique features including micron-scale resolution, real-time processing, and non-invasive imaging. We develop a novel algorithm for transforming volumetric optical images before and after the mechanical solicitation of a sample with sub-cellular resolution into quantitative shear modulus distributions. This has the potential to improve sensitivities and specificities in the...

Topics: Mathematics, Optimization and Control

Source: http://arxiv.org/abs/1405.4987

4
4.0

Jun 29, 2018
06/18

by
Robert Plato

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This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1607.04879

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6.0

Jun 29, 2018
06/18

by
Robert Plato

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We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1604.08703

7
7.0

Jun 30, 2018
06/18

by
Habib Ammari; Hyeuknam Kwon; Yoonseop Lee; Kyungkeun Kang; Jin Keun Seo

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Magnetic resonance electrical property tomography is a recent medical imaging modality for visualizing the electrical tissue properties of the human body using radio-frequency magnetic fields. It uses the fact that in magnetic resonance imaging systems the eddy currents induced by the radio-frequency magnetic fields reflect the conductivity ($\sigma$) and permittivity ($\epsilon$) distributions inside the tissues through Maxwell's equations. The corresponding inverse problem consists of...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.6095

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11

Jun 29, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Bangti Jin; Jin-Keun Seo; Wenlong Zhang

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This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles, and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1602.04312

7
7.0

Jun 30, 2018
06/18

by
Habib Ammari; Pol Grasland-Mongrain; Pierre Millien; Laurent Seppecher; Jin-Keun Seo

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We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1401.2337

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36

Jul 26, 2018
07/18

by
Kyriakos Zambas; Plato

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος B

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36116

48
48

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Γ

Topics: JC, Political science -- Early works to 1800, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36191

53
53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Α

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36088

53
53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Ε

Topics: JC, State, The -- Early works to 1800, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36262

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35

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Δ

Topics: State, The -- Early works to 1800, JC, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36284

114
114

Jul 25, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 3

Topics: JC, Political science -- Early works to 1800, Utopias -- Early works to 1800, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39524

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78

Jul 25, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 4

Topics: Utopias -- Early works to 1800, JC, Justice -- Early works to 1800, Political science -- Early...

Source: https://www.gutenberg.org/ebooks/39530

108
108

Jul 26, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 1

Topics: Utopias -- Early works to 1800, Political science -- Early works to 1800, JC, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39476

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Jul 26, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 2

Topics: Utopias -- Early works to 1800, Justice -- Early works to 1800, Political science -- Early works to...

Source: https://www.gutenberg.org/ebooks/39493