# ARITHMETIC METHODS IN MATHEMATICAL PHYSICS AND BIOLOGY

3-8 August 2014, Banach Center, Będlewo, Poland

## Abstracts

### Fractal analysis in digital histology

#### Helmut Ahammer

Institute of Biophysics, Medical University of Graz

Digital images are increasingly used in medicine, especially in digital pathology. Histological techniques are well established and modern digitalization systems yield high resolution digital images. Evaluation of these images is regularly accomplished by subjective inspection, but objective or numerical methods are still rare. Biological patterns and textures are mathematically hard to measure or to simulate and therefore, fractal methods and non-Euclidean geometry are very suitable to solve these tasks.

Digital images inherently have many advantages and fractal measures can be calculated by various methods, but care has to be taken in order to gain reliable and robust results. Digital images are discrete representations of specimen and can be noisy. Furthermore, an image is not always totally filled by the specimen and consequently background pixels must be considered, too. If appropriately considered, these issues do not decline the power of fractal analyses.

### Arithmetic operators and Galois symmetries. An application to mathematical physics.

#### Grzegorz Banaszak

In this lecture I will dicuss application of arithmetic methods to investigation of certain problems in mathematical physics. In particular I will explain the notion of the arithmetic operator and the Galois theory methods to investigate the spectrum of such an operator.

A classical example of an arithmetic operator is the Hamiltonian for the $N$-gone magnetic ring in the framwork of Bethe Ansatz. This approach to this Hamiltonian allows one, among other things, to consider natural structurs such as Galois qubits and Galois qudits introduced by Jan Milewski. Galois qubits and qudits admit a quantum informatic interpretation as elementary memory units of a (hypothetical) computer. I will discuss in detail the Galois structure of the Hamiltonian spectrum, related parameters and objects for the case $N =7.$

### Cellular automaton models for collective cell behaviour

#### Andreas Deutsch

Technical University, Dresden, Germany

Collective dynamics of migrating and interacting cell populations drive key processes in biological tissue formation and maintenance under normal and diseased conditions. Lattice-gas cellular automata have proven successful to model and analyze collective behavior arising from interactions of migrating cells. We introduce lattice-gas cellular automaton models of collective cell migration, clustering and invasion and demonstrate how analysis of the models allows for prediction of emerging properties at the individual cell and the cell population level. Finally, we discuss applications of the invasion models to glioma tumours.

Ref.: A. Deutsch, S. Dormann: Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis Birkhauser, Boston, 2005 (2nd ed. 2014)

### Multifractal formalism in image and time series analysis

#### Herbert Jelinek

Charles Sturt University, Albury, Australia

### P-adic Numbers from superstrings and molecular motors to cognition and psychology

#### Andrei Khrennikov

Linnaeus University, Växjö-Kalmar, Sweden

We start with a brief mathematical introduction in theory of p-adic numbers and corresponding analysis (geometrically this is analysis on homogeneous trees). Then we again briefly present applications to physics, supestring theory, quantum mechanics, and the theory of disordered systems. The next part of the talk is devoted to applications to genetics and cellular biology, in particular, we plan to present a mathematical model of functioning of molecular motors in cells. The last part is devoted to application of theory of p-adic dynamical systems to cognition and psychology with the final arrival to the p-adic dynamical model for Freud's psychoanalysis.

References:

• A. Khrennikov, Information dynamics in cognitive, psychological, social, and anomalous phenomena. Ser.: Fundamental Theories of Physics, Kluwer, Dordreht, 2004.
• V. Anashin and A. Khrennikov, Applied algebraic dynamics. De Gruyter, Berlin, 2009.
• A.Yu. Khrennikov, S.V. Kozyrev, A. Mansson, Hierarchical model of the actomyosin molecular motor based on ultrametric diffusion with drift. arXiv:1312.7528 [q-bio.BM].
• A.Yu. Khrennikov, S.V. Kozyrev, p-Adic numbers in bioinformatics: from genetic code to PAM-matrix. Journal of Theoretical Biology. 2009. V.261. P.396--406; arXiv:0903.0137 [q-bio.GN]

### New mathematics of complexity and its biomedical application

#### Andrei P. Kirilyuk

National Academy of Sciences of Ukraine, Kiev, Ukraine

We show that the unreduced, mathematically rigorous solution of the many-body problem with arbitrary interaction, avoiding any perturbative approximations and "exact" models, reveals qualitatively new mathematical properties of thus emerging real-world structures (interaction products), including dynamic multivaluedness (universal non-uniqueness of ordinary solution) giving rise to intrinsic randomness and irreversible time flow, fractally structured dynamic entanglement of interaction components expressing physical quality, and dynamic discreteness providing the physically real space origin. This unreduced interaction problem solution leads to the universal definition of dynamic complexity describing structure and properties of all real objects. The united world structure of dynamically probabilistic fractal is governed by the universal law of the symmetry (conservation and transformation) of complexity giving rise to extended versions of all particular (correct) laws and principles. We describe then the unique efficiency of this universal concept and new mathematics of complexity in application to critical problems in life sciences and related development problems, showing the urgency of complexity revolution.

### On inverted harmonic oscillators

#### Piotr Krasoń

Szczecin University, Poland

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is ${\mathbb C}$. The spectrum of the differential operator $-{\frac{d}{dx^2}}-{\omega}^{2}{x^2}$ is continuous and has physical significance only for the states which are in $L^{2}(\mathbb R)$ and correspond to real eigenvalues. To identify them we use two approaches. First we define a unitary operator between $L^{2}(\mathbb R)$ and $L^{2}$ for two copies of $\mathbb R$. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator $-i{\frac{d}{dx}}$. This shows that the (generalized) spectrum of the inverted harmonic operator is real. The second approach uses rigged Hilbert spaces.

### Bethe Ansatz, Galois symmetries, and finite quantum systems

We demonstrate some applications of arithmetic structures in Bethe Ansatz – the famous substitution which yields an exact solution of a quantum N-body problem for the linear magnetic ring of N spins ½ within the XXX model. We point out the purely arithmetic form of the eigenproblem of the associated Heisenberg Hamiltonian in the initial (calculational) basis of all magnetic configurations, together with the resulting solution, expressed in terms of a finite extension of the prime field of rationals. The Galois group of this field extension acquires the natural physical interpretation in terms of admissible permutations of rigged string configurations. The cyclotomic number field proves to be an important subfield of this extension, responsible for the translational symmetry of the magnetic ring, reflected in quasimomenta of a finite Brillouin zone. We also point out the role of the cyclotomic fields in determination of all mutually unbiased bases for a Hilbert space with the dimension N being a power of a prime. The general case of an arbitrary N is sketched in the context of decomposition of a finite quantum system into subsystems, along Weyl – Schwinger factorization scheme.

### The goodness-of-fit of the fractal regression curve is a relevant diagnostic and prognostic feature in different biological models

University of Campinas, Brasil

In biology real structures can only be approximations to an ideal fractal, since self-similarity may be more or less pronounced and vary considerably along the scaleing window. In order to estimate the degree of similarity with an ideal mathematical fractal, we introduced the variable R2, which is the the goodness-of- fit between the real measurement points and the linear regression line in the log-log plots. The biological utility of this feature was tested in different biological models.

In a set of adult patients with acute B-precursor lymphoblastic leukemia the prognostic value of the fractal features of the chromatin texture regarding overall survival was evaluated. We used pseudo-three-dimensional nuclear images of routine May-Giemsa-GrŁnwald stained cytologic images from the files. The chromatin surface fractal dimension (FD) and their respective R2 values were calculated according to four methods : Sarkat, Minkowski, Einstein and Plane-4. The prognostic relevance was evaluated with the help of Cox regressions. Increasing FD values were associated with a shorter survival using the methods of Sarkar and Plane-4, but with a longer survival applying the Einstein method, whereas FD measures according to the Minkowski algorithm had no prognostic relevance. In contrast to that, when analyzing only the R2 values, we got the same results for all four methods: higher R2 values were always associated with a longer survival of the patients. When testing simultaneously both fractal variables with clinical and laboratory features, only R2 remained in the model as a fractal variable when using the Sarkar, Minkowski or Einstein method. But when applying the Plane-4 algorithm, both FD and R2 entered the model.

In the second biologic model we tried to evaluate whether fractal features could help for the differential diagnosis between hypertrophic scars and keloids. Both entities have different prognosis and are sometimes difficult to distinguish only by histologic examination of resection material. Cases from our files were randomly selected. The final diagnosis had been established based on histological and clinical features and especially the follow up. Images from Masson-stained routine histologic slides were digitalized, gray value transformed and fractal features extracted form pseudo-tree-dimensional images. The discriminative power of the features was estimated by t-test and linear discriminant analyses. Only the fractal dimension according to Sarkar was significantly different between both entities, but not the FD according to Blanket-4, Blanket-8, Einstein, Minkowski, Plane-8 and Plane 4, whereas the R2 values of all methods used were significantly different In all cases the goodness-of-fit was higher for hypertrophic scars than for keloids. Our investigations introduce the goodness-of-fit of the FD as a new and robust variable for the characterization of fractality with some interesting diagnostic or prognostic relevance in biologic material. Supported by FAEPEX, CAPES and CNPq

### Current challenges in IT Security, with focus on Biometry.

#### Preda Mihailescu

Universitat Gottingen, Germany

The internet is inconceivable without some provisions for privacy, confidentiality and accountability. These are all provided by public key cryptography, a quite well mathematized and understood branch of applied mathematics.

For reasons which will be presented in more detail in the talk, biometrical identification developed more and more into a self-sufficient alternative to crpyptography. However the challenges and limitations of biometry are substantial. While in previous decades the main focus has been to achieve reliable recognition at all, now that this is assured to some extent, various security issues emerge and will be eventually systemized and taken into account. Emerging communities, at the border line between engineering-biometry, cryptography, theory of information and statistics will bring their contribution to the challenges.

### Hodge Structures, Weierstrass σ-function and Quantum Mechanics

#### Jan Milewski

Poznań University of Technology, Poland

Hodge structures and mixed Hodge structures are presented by means of suitable operators which are annihilated by a Weierstrass $\sigma$-function. The space of harmonic functions i two-dimensional topological quantum mechanics are considered as a deformation of Hodge structures. Some quantum numbers are connected with parameters of decomposition of Hodge structures.

### A Numerical Method to Differentiate Between "Pure" and "Impure" Fractals

#### Martin Obert

Justus-Liebig-University, Giessen, Germany

We introduce a numerical method that enables the differentiation between natural fractal image representations of objects that have only fractal properties ("pure" fractals) from those that have partial Euclidean and partial fractal properties ("impure" fractals). We evaluate our classification method on a numerically constructed fractal that serves as "pure" fractal and compare it with an "impure" digital image representation of the blood vessel system of a mouse-liver. Our classification method is based on the study of the invariance of a data set under different levels of image-data-point reductions – comparing a dissection and a random erosion method. The various image-sets were characterized by the fractal dimension D, evaluated by the mass radius method. The differentiation between "pure" and "impure" fractals is performed by their different and opposite behavior when the reduction is performed at average reduction levels: We find for the "pure" fractal that dissections lead to no D-difference between complete and reduced sets; contrary, the erosion method leads to a noticeable D-difference. These findings swap for the "impure" fractal. Here we find that dissections lead to distinct D-differences between complete and reduced sets; contrary, the erosion method leads to no change in these D-differences. We strongly assume that this differentiation can be applied successfully when automated image classifications are desired or necessary.

### Current Challenges in Designing Biometric Template Protection

#### Benjamin Tams

Georg-August-Universität, Göttingen, Germany

In a password-based authentication scheme typically cryptographic hashes of passwords are stored on the system's database to prevent them from being lost after a theft of the database content. Similarly, in biometry-based authentication systems, irrevertible transforms of biometric templates i.e. private templates, have to be generated and stored on the system's database instead of the unprotected biometric templates as a realization of biometric template protection.

In this talk, using the example of the fingerprint biometry, it is outlined how private templates can be generated using the fuzzy vault scheme. It is illustrated how limitations of the fingerprint biometry limit the security of a private template against irreversibility attacks.

Another serious attack scenario is the one of linkability attacks in which multiple private templates generated from the same individual can be used to cross-match and even revert the private templates. In the case of a fingerprint-based fuzzy vault, such a record multiplicity attack is outlined and a simple, yet effective, countermeasure is discussed which even has the positive effect that, now, significantly smaller private templates can be generated using an "improved" version of the fuzzy vault scheme. However, with the extended Euclidean algorithm the problem of linkability attacks remains but, fortunately, could be avoided as well. However, future research, that is motivated at the end of this talk, is needed to prove or disprove whether unlinkable private templates can be generated with the "improved" fuzzy vault scheme.

### On the fractal analysis of biological growth processes

#### Mihai Tanase

Politehnica Buchuresti, Romania

In the characterization of carcinomas, an important criterion is the shape of the tumor-stroma frontier and, in particular, its fractal dimension. The definition and identification of this frontier is not trivial. For the same image, the known segmentation algorithms often produce frontiers with very different fractal dimensions making this type of analysis irrelevant. We introduce a new approach to frontier analysis by redefining the tumor-stroma frontier in order to overcome this problem

### Scattering, systems and operators for quantum mechanics

#### Yoichi Uetake

In the 1960s, R. E. Kalman developed what he called a modern control theory. In this theory a dynamical system is described by a differential equation given by a collection of matrices or operators $(A,B,C,D)$. The condition of controllability and observability is important, and if this condition is satisfied then we have a spectral interpretation $p(s)=\det(sI-A)$. Here $p(s)$ is the denominator of the transfer function of the system. In the same period of 1960s, Lax and Phillips developed their scattering theory mainly for classical wave equations. The relation between control and Lax-Phillips scattering theories was discovered by J. W. Helton. He along with Adamjan and Arov also found that these two theories are related to Nagy-Foias operator theory. In this talk we describe a new relationship between control and scattering theories. To do this we relax some conditions on Lax-Phillips scattering theory. This new relationship allows one to apply control-scattering theory to quantum scattering of two-body systems. Interestingly this two-body scattering theory has a analogue in the theory of Eisenstein series, one of the origin of the Langlands program in number theory. Following this analogue we can obtain a spectral interpretation of automorphic $L$-functions including the Riemann zeta function (Hilbert-Polya's proposal). In this talk we will also mention this point briefly.

### Lower urinary tract symptoms: A challenge for science of complexity?

#### Florian Wagenlehner

Justus-Liebig-University, Giessen, Germany

### The Universal Circular Fractal Model of Carcinoma; Complexity Measures and Stratification into the Classes of Complexity

#### Przemyslaw Waliszewski

Justus-Liebig-University, Giessen, Germany

Background: Malignant epithelial cells self-organize into tissue structures of the higher order, such as glands or layers that mimic the structural organization of normal tissue. Grading was defined as a subjective measure of alterations in tumor structure to score a degree of self-affine deformations of both cells and glands. The intuitive assumption was that more altered cancer tissue represented greater dynamics of tumor growth, greater probability of metastasis formation, and, therefore, worse prognosis. None of the molecular markers characterizes patterns of tumor growth or their biological aggressiveness better than histological grading does. Therefore, grading is an important part of the clinical evaluation for malignant tumors of the epithelial origin called carcinomas.

A number of features causes that prostate cancer is an interesting oncological model. It is a prevalent clinical phenomenon. It is known for its structural, biological, and clinical variability; yet, it was possible to reduce that variability to a few patterns of growth. Monitoring of the disease is possible owing to both the identification of prostatic specific antigen (PSA), a prostate specific protein that can be measured in serum with a great accuracy, and the modern imaging technology. A choice of treatment depends on the risk of progression. The risk of prostate cancer patient is defined by both the objective and subjective criteria, i.e., PSA concentration, grading described by the Gleason score, and pTNM-stage. The subjectivity of tumor grading influences the risk assessment owing to a large inter- and intraobserver variability. This is in the range of 40%-80%, and is associated with the coefficient κ for interobserver agreement 0.15 - 0.7. In consequence, many patients get the aggressive treatment that they do not need at all. There is also a second group of patients who are not treated on time because tumors were classified as the indolent cancers, i.e. cancers without a significant influence on survival.

Pathologists propose a central prostate pathology review as a remedy for this problem; yet, the review cannot eliminate the subjectivity from the diagnostic algorithm. There is a need for the novel approaches for the quantitative evaluation of biological aggressiveness of carcinomas. Since the evaluation of tumor structure can be difficult and ambiguous in so many cancer cases, one can evaluate biological aggressiveness of carcinomas using both some measures of complexity and elements of tissue geometry associated with patterns of tumor growth. Indeed, the spatial distribution of cancer cell nuclei changes during tumor progression. This implies changes in complexity measured by the complexity measures, such as the global and local fractal dimensions, entropy, and lacunarity.

Methods: The cornerstone of the approach is a geometrical model of prostate carcinomas that is composed of the circular fractals CF (3), CF (6+0), and CF (6+1). This model is both geometrical and analytical, i.e., its structure is well-defined, the capacity fractal dimension D0 can be calculated for the infinite circular fractals, and the dimensions D0, D1, D2 as well as entropy or lacunarity can be computed for their finite counterparts representing distribution of cell nuclei. The model enables the calibration of the software or the validation of the measurements in the digitalized images of prostate, colon, or breast carcinomas. The circular fractal model also reveals some interesting universal features, such as the topological equivalence of the different structural organization underlying the emergence of a similar dynamics of growth.

Results: The spatial distribution of cancer cell nuclei changes along tumor progression. This change can be characterized quantitatively by a power law by a number of the independent methods, such as Visual Recurrence Analysis with the correlation dimension, and the fragmentation dimension. The multifractal structure in the spatial distribution of cancer cell nuclei was identified by the global capacity fractal dimension D0, the information fractal dimension D1, and the correlation fractal dimension D2. The ROC analysis of the data defined the cut-off values of the capacity fractal dimension D0 for the seven equivalence classes called classes of complexity. The subjective Gleason classification matched in part with the classification based on the D0-values. However, the mean ROC sensitivity was rather low, i.e., 81.3% and the mean ROC specificity 75.2%. Therefore, prostate carcinomas were re-stratified into seven classes of complexity according to their D0 values. 53% of prostate carcinomas was re-located from their primary structural classes according to Gleason. This increased both the mean ROC sensitivity and the mean ROC specificity to 100%. All homogeneous Gleason patterns were subordinated to the class C1, C4 or C7. D0 = 1.5820 was the cut-off D0 value between the complexity class C2 and C3. This D0 value separated the low-risk cancers with Gleason score 3+4=7a that did not require the aggressive therapy, and could be followed by the strategy called active surveillance from the intermediate-risk cancers with Gleason score 4+3=7b that should be treated actively. The similar values of D0 were found in the case of colon or breast carcinomas. As expected, the entropy of the spatial distribution was inversely proportionate to the lacunarity. Using the values of D0, D1, D2 and the Minkowski dimension, the homogeneous patterns of Gleason 3+3, Gleason 4+4, and Gleason 5+5 were classified with very high accuracy by the independent classifiers, such as Bayessian one, support vector machine or self-organized maps.

Conclusions: The above-mentioned results indicate the existence of a relationship between tumor structure and complexity in the spatial distribution of cancer cell nuclei. The coapplication of complexity measures such as the global fractal dimensions, the local fractal dimensions, the entropy or the lacunarity eliminate the subjectivity in the evaluation of tumor structure. The D0-cut-off values define the classes of equivalence. This enables the restratification of carcinomas to the well-defined complexity classes. It also defines subgroups of carcinomas with a very low malignant potential of growth or with a large risk of progression and metastasis formation. Since colon or breast carcinomas share some similarities in the structural organization, the presented model can be universal for the stratification of carcinomas according to the values of the global capacity fractal dimension. However, biology of those carcinomas is known to be different. The most important clinical information, i.e., dynamics of tumor growth cannot be predicted just on the basis of the values of the complexity measures for the spatial distribution of cancer cell nuclei.

### Sparse representation and compressed sensing

#### Przemysław Wojtaszczyk

Warsaw University, Poland

Sparse representation is the idea that seemingly complicated phenomena in a natural class can be represented in a suitably chosen representation system, using a much smaller set of parameters than generally needed i.e. in a sparse way. In practice it happens quite often. A classical example is a wavelet basis: a natural signal (voice, picture) represented by a function $f$ can be written as

(1)                          $f=\sum_{\gamma\in \Gamma} a_\gamma \Psi_\gamma$

where $\# \Gamma$ must be quite big to accommodate all possible signals. However for each individual signal we have $f\approx \sum_{\gamma \in \Gamma^\prime} a_\gamma \Psi_\gamma$ for $\Gamma^\prime\subset \Gamma$ and $\# \Gamma^\prime$ much smaller than $\#\Gamma$. We say that the representation (1) is sparse.

Compressed sensing is the elaboration of the idea that if $f$ as in (1) has few important coefficients, than the number of samples (measurements) needed to recover $f$ should be dependent on $\# \Gamma^\prime$ so hopefully be much smaller that $\#\Gamma$. The aim of this talk is to present an overview of this general ideas.